universidade estadual de campinas - … · ficha catalográfica universidade estadual de campinas...

UNIVERSIDADE ESTADUAL DECAMPINAS

Instituto de Matemática, Estatística eComputação Científica

EDER DE MORAES CORREA

Integrable systems in coadjoint orbits andapplications

Sistemas integráveis em órbitas coadjuntas eaplicações

Campinas2017

Eder de Moraes Correa

Integrable systems in coadjoint orbits and applications

Sistemas integráveis em órbitas coadjuntas e aplicações

Tese apresentada ao Instituto de Matemática,Estatística e Computação Científica da Uni-versidade Estadual de Campinas como partedos requisitos exigidos para a obtenção dotítulo de Doutor em Matemática.

Thesis presented to the Institute of Mathe-matics, Statistics and Scientific Computingof the University of Campinas in partial ful-fillment of the requirements for the degree ofDoctor in Mathematics.

Orientador: Luiz Antonio Barrera San MartinCoorientador: Lino Anderson da Silva Grama

Este exemplar corresponde à versão fi-nal da Tese defendida pelo aluno Ederde Moraes Correa e orientada peloProf. Dr. Luiz Antonio Barrera SanMartin.

Campinas2017

Agência(s) de fomento e nº(s) de processo(s): CNPq, 152103/2014-7; CAPES

Ficha catalográficaUniversidade Estadual de Campinas

Biblioteca do Instituto de Matemática, Estatística e Computação CientíficaAna Regina Machado - CRB 8/5467

Correa, Eder de Moraes, 1986- C817i CorIntegrable systems in coadjoint orbits and applications / Eder de Moraes

Correa. – Campinas, SP : [s.n.], 2017.

CorOrientador: Luiz Antonio Barrera San Martin. CorCoorientador: Lino Anderson da Silva Grama. CorTese (doutorado) – Universidade Estadual de Campinas, Instituto de

Matemática, Estatística e Computação Científica.

Cor1. Lie, Teoria de. 2. Geometria simplética. 3. Sistemas hamiltonianos. 4.

Calabi-Yau, Variedades de. 5. Geometria diferencial. I. San Martin, LuizAntonio Barrera,1955-. II. Grama, Lino Anderson da Silva,1981-. III.Universidade Estadual de Campinas. Instituto de Matemática, Estatística eComputação Científica. IV. Título.

Informações para Biblioteca Digital

Título em outro idioma: Sistemas integráveis em órbitas coadjuntas e aplicaçõesPalavras-chave em inglês:Lie theorySymplectic geometryHamiltonian systemsCalabi-Yau manifoldsDifferential geometryÁrea de concentração: MatemáticaTitulação: Doutor em MatemáticaBanca examinadora:Lino Anderson da Silva Grama [Coorientador]Paulo Regis Caron RuffinoDiego Sebastian LedesmaIvan StruchinerAlesia MandiniData de defesa: 26-06-2017Programa de Pós-Graduação: Matemática

Powered by TCPDF (www.tcpdf.org)

Tese de Doutorado defendida em 26 de junho de 2017 e aprovada

pela banca examinadora composta pelos Profs. Drs.

Prof(a). Dr(a). LINO ANDERSON DA SILVA GRAMA

Prof(a). Dr(a). PAULO REGIS CARON RUFFINO

Prof(a). Dr(a). DIEGO SEBASTIAN LEDESMA

Prof(a). Dr(a). IVAN STRUCHINER

Prof(a). Dr(a). ALESSIA MANDINI

As respectivas assinaturas dos membros encontram-se na Ata de defesa

Acknowledgements

It is a pleasure to thank Professor Luiz Antonio Barrera San Martin, ProfessorLino Anderson da Silva Grama and Professor Leo T. Butler for the support, guidance andencouragement. I want to thank my family and friends for supporting me all the time, Iam deeply indebted to Tami Holanda, for all her support and care.

I would also like to thank CNPq and CAPES for providing the financial supportfor this work.

ResumoEsta tese tem como objetivo estudar sistemas Hamiltonianos integráveis em órbitascoadjuntas e tópicos relacionados às suas aplicações. Este trabalho se divide essencialmenteem duas partes que podem ser brevemente descritas da seguinte forma. Na primeira parteestudamos a construção de sistemas Hamiltonianos integráveis de Gelfand-Tsetlin emórbitas coadjuntas de grupos de Lie compactos clássicos. Para sistemas do tipo Gelfand-Tsetlin construímos uma formulação via matriz de Lax que nos permite recuperar asquantidades conservadas do sistema como invariantes espectrais de uma matriz de Laxapropriada. Ainda no contexto de sistemas Hamiltonianos, fornecemos uma descriçãocompleta das funções que definem os sistemas integráveis de Gelfand-Tsetlin-Molev paradois exemplos concretos de dimensão baixa. Na segunda parte deste trabalho estudamos aconstrução de métricas de Calabi-Yau no fibrado canônico de variedades flag complexas.Utilizando ferramentas da teoria de representações para álgebras de Lie e a técnica deCalabi ansatz, descrevemos vários exemplos não triviais de variedades de Calabi-Yaucompletas não compactas. A principal motivação para o desenvolvimento deste trabalhosão as relações entre variedades flag complexas, variedades tóricas, teoremas de convexidadepara aplicações momento e simetria do espelho (mirror symmetry). A conexão entre asduas partes brevemente descritas aqui se dá no contexto das fibrações Lagrangianasespeciais. A construção de tais exemplos de fibrações são extremamente importantes parao entendimento da dualidade entre geometria simplética e geometria complexa propostapela simetria do espelho.

Palavras-chave: teoria de Lie, geometria Simplética, geometria Complexa, sistemasHamiltonianos, variedades de Calabi-Yau.

AbstractThe purpose of this thesis is to study Hamiltonian integrable systems in coadjoint orbitsand topics related to their applications. This work is essentially divided in two partswhich can be briefly described as follows. In the first part we study the construction ofGelfand-Tsetlin integrable systems in coadjoint orbits of classical compact Lie groups. ForGelfand-Tsetlin integrable systems we provide a Lax matrix formulation which allows usrecovering the conserved quantities of the system as spectral invariants associated to asuitable Lax matrix. Still within the context of Hamiltonian systems, we also provide acomplete description of the functions which compose Gelfand-Tsetlin-Molev integrablesystems for two low dimensional concrete examples. In the second part of this workwe study the construction of Calabi-Yau metrics on canonical bundles of complex flagmanifolds. By means of tools provided by the Lie algebra representation theory and theCalabi ansatz technique, we describe a huge family of non trivial examples of completenon compact Calabi-Yau manifolds. The main motivations for developing this work arethe relationship between complex flag manifolds, toric manifolds, convexity theorems formoment maps and mirror symmetry. The connection between these two parts brieflydescribed here is the background of special Lagrangian fibrations. The construction ofexamples of such kind of fibrations are extremely important to understand the dualitybetween symplectic geometry and complex geometry proposed by mirror symmetry.

Keywords: Lie theory, Symplectic geometry, Complex geometry, Hamiltonian systems,Calabi-Yau manifolds.

Contents

1 Introduction 10

2 Generalities on symplectic geometry 172.1 Hamiltonian G-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Lax pair and Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Collective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 imm’s trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 Gelfand-Tsetlin integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6.1 Gelfand-Tsetlin Systems for adjoint orbits of U(N) . . . . . . . . . . . . 382.6.2 Gelfand-Tsetlin Systems for adjoint orbits of SO(N) . . . . . . . . . . . 42

3 Lax formalism and Gelfand-Tsetlin integrable systems 483.1 Lax equation and collective Hamiltonians . . . . . . . . . . . . . . . . . . . . . 493.2 imm’s trick and spectral invariants . . . . . . . . . . . . . . . . . . . . . . . 513.3 Liouville’s theorem and Lax formalism for Gelfand-Tsetlin integrable systems 62

4 Kahler structure on coadjoint orbits 674.1 Generalities about coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Compact real form of simple Lie algebras . . . . . . . . . . . . . . . . . . . . . 684.3 Generalities about Lie group decompositions . . . . . . . . . . . . . . . . . . . 694.4 Symplectic structure on coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . 714.5 Complex and Kahler structures on coadjoint orbits . . . . . . . . . . . . . . . . 73

5 Generalities on Kahler-Einstein manifolds 835.1 Holomorphic line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Kahler manifolds and Einstein metrics . . . . . . . . . . . . . . . . . . . . . . . 86

6 Calabi ansatz technique and complex ag manifolds 906.1 Calabi-Yau metrics on canonical bundles of ag manifolds . . . . . . . . . . . 916.2 Examples of complete non-compact Calabi-Yau manifolds via Lie theory . . . 95

Bibliography 109

Appendices 119

A Necessary condition for integrability of collective Hamiltonian systems 120A.1 Basic notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.2 Moment map and some of its properties . . . . . . . . . . . . . . . . . . . . . . 121A.3 Moment map, symplectic slice and transversality . . . . . . . . . . . . . . . . . 126A.4 Collective Hamiltonians and necessary condition for integrability . . . . . . . 128A.5 Liouville’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.6 Arnold’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B Integrable systems in regular adjoint orbits of compact symplectic Lie group 143B.1 Generalities about H and sp(N ) . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.2 An overview about the Gelfand-Tsetlin-Molev integrable system . . . . . . . . 149

B.2.1 Multiplicity free action and representation theory . . . . . . . . . . . . 149B.2.2 Generalities about Yangians and twisted Yangians . . . . . . . . . . . . 158B.2.3 antities in involution via classical limit procedure . . . . . . . . . . 163

B.3 e Gelfand-Tsetlin-Molev integrable systems . . . . . . . . . . . . . . . . . . 165B.4 Applications in low dimensional symplectic Lie groups . . . . . . . . . . . . . 169

B.4.1 Integrable system in regular orbits of Sp(2) . . . . . . . . . . . . . . . . 169B.4.2 Integrable system in regular orbits of Sp(3) . . . . . . . . . . . . . . . . 173

C Projective algebraic realization of coadjoint orbits 177C.1 Representation theory of simple Lie algebras . . . . . . . . . . . . . . . . . . . 177C.2 Analytic projective subvarieties and algebraic realization . . . . . . . . . . . . 180C.3 Projective embedding of ag manifolds . . . . . . . . . . . . . . . . . . . . . . 184C.4 Borel-Weil theorem and Kodaira embedding for ag manifolds . . . . . . . . . 191

D Holomorphic line bundles and Calabi ansatz technique 201D.1 Local Kahler potential and Chern class of line bundles . . . . . . . . . . . . . . 202

D.1.1 Preliminary generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 202D.1.2 Local potential and representation theory . . . . . . . . . . . . . . . . 210D.1.3 Chern class for holomorphic line bundles over ag manifolds . . . . . 219

D.2 Calabi ansatz technique on Kahler-Einstein Fano manifolds . . . . . . . . . . . 243D.2.1 Ricci-at Kahler metrics on canonical bundles of Kahler-Einstein Fano

manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

10

Chapter 1Introduction

In this work we study Hamiltonian systems in coadjoint orbits and Ricci-at metrics on canon-ical bundles of complex ag manifolds. Our motivations are essentially the study of convexityproperties of moment maps and the construction of completely integrable Hamiltonian sys-tems. One of the rst problems related to the convexity properties of moment maps wassolved by Horn [84] in 1954. e problem studied in [85] can be briey described as fol-lows. If we considerHn

λ as being the set of the n × n square Hermitian matrix with the samespectrum λ = (λ1 ≥ . . . ≥ λn ), how does the image of the map Φ : Hn

λ → Rn, such thatΦ : A → (A11, . . . ,Ann ), look like? e answer provided by Horn was that Φ(Hn

λ ) = ∆(λ) isthe convex hull of the set dened by the points(

λσ (1), . . . , λσ (n))∈ Rn, with σ ∈ Sn,

where Sn denotes the permutation group of the set 1, . . . ,n. Although Horn’s solution forthe above problem does not require the usage of symplectic geometry tools, it has the avorof Hamiltonian Lie group action, moment maps and adjoint orbits as we will see forward.

In 1974 Kostant [104] established a convex theorem for coadjoint orbits of compact Lie groupswhich generalizes Horn’s result. Roughly speaking, Kostant’s convexity theorem states thatfor a compact Lie group G with Lie algebra g, aer xed a maximal torus T ⊂ G with Liealgebra t, then the image of the projection π : g∗ → t∗ of any coadjoint orbit O ⊂ g∗ is aconvex set, namely, if we denote by µ = π |O the restriction of the projection over O, we have

µ : O → t∗,

and µ (O) = ∆T , where ∆T is the convex hull of µ (OT ), here OT ⊂ O denotes the set of xedpoints of O under the action of T . If we take a look at the problems which we have brieydescribed, we see that, from a more general stand point, the above convexity theorems gathertogether elements of the symplectic geometry and Lie theory.

In the general seing of Hamiltonian Lie group actions of compact Lie groups on sympletcmanifolds, Atiyah [6] and independently Guillemin and Sternberg [71], almost simultaneouslyin 1982, provided a generalization of Kostant’s convexity theorem in the seing of compactsymplectic manifolds having a Hamiltonian toric action (abelian convexity theorems). eyshowed that the image of a symplectic manifold under the moment map is also a convex poly-tope (more precisely, it is the convex hull of the image of the set of xed points of the manifold

11

under the torus action).

In 1982 a non-abelian convexity theorem was established by Guillemin and Sternberg [71] inthe context of compact Kahler manifolds, this theorem was generalized by Kirwan [97] twoyears laer. In the non-abelian seing an alternative proof for the non-abelian convexity the-orem also can be found in [149].

Beyond the context of Hamiltonian Lie group actions on symplectic manifolds, the process toassign convex bodies (polytopes) to compact symplectic manifolds leads us to the followingquestion: What kind of geometric information are encoded in these convex bodies? In orderto understand the relevance of this question we need to look more closely the especial seingof toric manifolds.

A toric manifold is dened by a 2n-dimensional compact symplectic manifold endowed withan eective Hamiltonian action of a n-dimensional compact torus. In this context the abelianconvexty theorem allows us to classify toric manifolds by means of the convex bodies calledDelzant polytopes [39]. erefore, for toric manifolds we have the following correspondence

Toric manifolds ←→ Delzant polytopes

As we can see in [39] the above correspondence shows us that the symplectic and complexstructures of a toric manifold are completely determined, up to isomorphism, by its associatedDelzant polytope. In 1994, V. Guillemin [66] showed that not just the symplectic geometryand complex geometry of a toric manifold are determined by its polytope, but also, to a cer-tain extent, its Kahler geometry.

e interplay between symplectic and complex geometry observed in the seing of toricKahler manifolds becomes more interesting under the Ricci-atness assumption, namely whenwe consider toric Calabi-Yau manifolds. In fact, once the moment map associated to a toricCalabi-Yau manifold provides a Lagrangian torus bration over its Delzant polytope, the envi-ronment of toric Calabi-Yau manifolds plays an important role in the study of mirror symmetry[105], more precisely on its geometric formulation called Strominger-Yau-Zaslow conjecture[155].

Besides the above facts concerned with toric Calabi-Yau manifolds, another important fea-ture of toric manifods is its relation with Hamiltonian integrable systems. Since the momentmap associated to a toric manifold denes a Hamiltonian integrable system, the common back-ground of convexity theorems makes the seek for Hamiltonian integrable systems in coadjointorbits a very interesting problem.

In 1983 Guillemin and Sternberg [76] showed how to employ imm’s trick [158] in orderto obtain quantities in involution dened by collective Hamiltonians [77] in coadjoint orbitsof compact semisimple Lie groups. Furthermore, they also showed that for coadjoint orbitsof the compact unitary Lie group U(n) the set of Poisson commuting functions provided byimm’s trick in fact denes a completely integrable system, this kind of integrable systemswere named by them as Gelfand-Tsetlin integrable systems.

12

A remarkable feature of the Gelfand-Tsetlin integrable systems is their connection with repre-sentation theory of Lie groups, Lie algebras and geometric quantization, see for example [72]and [74]. Since Gelfand-Tsetlin integrable systems were introduced, many other results con-cerned with Hamiltonian systems dened by collective Hamiltonians have been established,for instance, in [74] Guillemin and Sternberg studied in a general seing the constraints toget integrability for Hamiltonian systems composed by collective Hamiltonians. ey showedthat the necessary condition for integrability of collective Hamiltonian systems is that thespace of invariant smooth functions in the manifold needs to be an abelian algebra with re-spect to the Poisson bracket induced by the symplectic form. Moreover, they concluded thatthis last context turns out to be the case for coadjoint orbits of U(n) and SO(n), see [135] for anexposition about Gelfand-Tsetlin systems in coadjoint orbits of the compact Lie group SO(n).

For coadjoint orbits of the classical compact Lie group Sp(n) the quantities in involution ob-tained by imm’s trick are not enough to get integrability, thus we do not have Gelfand-Tsetlin integrable systems dened in coadjoint orbits of Sp(n). Actually, these coadjoint orbitsdo not satisfy the necessary condition for integrability established in [74]. In spite of this, in arecent work [79] M. Harada showed how to construct integrable systems in regular coadjointorbits of the compact symplectic Lie group by means of a dierent approach which involvesimm’s trick and classical limit procedure. ese integrable systems were named by her asGelfand-Tsetlin-Molev integrable systems. e ideas involved in Harada’s construction comefrom geometric quantization procedure and representation theory [121], [122] and [124].

Another interesting application of Gelfand-Tsetlin integrable systems is in mirror symmetry.In [131] T. Nishinou, Y. Nohara and K. Ueda showed how to degenerate Gelfand-Tsetlin in-tegrable systems dened in coadjoint orbits of the compact unitary Lie group into momentmaps dened in toric manifolds. is procedure allowed them to associate to Gelfand-Tsetlinintegrable systems potential functions, which encode information of the Lagrangian Floer ho-mology.

Coadjoint orbits ! Toric manifolds

e relation between coadjoint orbits and toric manifolds by means of toric degenerationmakes clear how the understanding of the geometry of toric manifolds can be useful to under-stand the geometry of coadjoint orbits, other important results concerned with this relationare [93] and [80], where many tools of algebraic geometry also are employed to the degener-ating process.

Inspired by the background which we have described, the main goal of this work is to providean extensive discussion about integrable systems in coadjoint orbits and related topics. isthesis can be divided in two main parts which can be described as follows. In the rst part,Chapter 2 and Chapter 3, we deal with issues related to the construction of the Gelfand-Tsetlinintegrable systems [76] in coadjoint orbits of classical compact Lie groups and their formu-lation in terms of Lax matrix. In the second part, Chapter 4, Chapter 5 and Chapter 6, wedeal with issues related to the Kahler geometry of coadjoint orbits, holomorphic line bundles,Kahler-Einstein metrics and Ricci-at metrics.

13

In Chapter 2 we review the basic background of Hamiltonian G-spaces, Hamiltonian systemsand Lax formalism for Hamiltonian systems. Aer that we review some properties of collectiveHamiltonians [77] and discuss how to apply imm’s trick [158] to get quantities in involu-tion in coadjoint orbits. We nish Chapter 2 by describing the construction of Gelfand-Tsetlinsystems in coadjoint orbits of the classical compact Lie groups U(n) and SO(n). estionsrelated to the necessary condition for integrability of Hamiltonian systems composed by col-lective Hamiltonians are discussed in Appendix A, following the approach of [74]. We alsoprovide a complete description for the construction of the Gelfand-Tsetlin-Molev systems [79]in Appendix B. In this appendix we also describe the Gelfand-Tsetlin-Molev systems for theconcrete cases of regular orbits of Sp(3) and Sp(2). e computations which we provide forthis two examples also are new in the literature.

In Chapter 3 we gather together the ideas involved in the construction of Gelfand-Tsetlin inte-grable systems and the concept of a Lax pair. A Lax pair L, P consist of two matrices, functionson the phase space (M,ω) of the system, such that the Hamiltonian evolution equation of mo-tion associated to a Hamiltonian H ∈ C∞(M ), may be wrien as zero curvature equation

dL

dt+[L, P]= 0

e notion of Lax pair is a new emergent language used in the studies of integrable systems,and one of the most important feature of this concept is its relation with the classical r -matrix[31]. e concept of classical r -matrix was introduced in the late 1970’s by Sklyanin [151] as apart of a vast research program launched by L. D. Faddeev which culminated in the discoveryof the antum Inverse Scaering Method and of antum Groups [50]. Motivated by thework [79] and its relation with quantum groups, we propose a new approach for Gelfand-Tsetlin integrable systems by means of the following result

eorem A. Let (O (Λ),ωO (Λ),G,Φ) be a Hamiltonian G-space dened by an adjoint orbitO (Λ) = Ad(G )Λ, whereG = U(N ) or SO(N ). en there exists a Lax pairL, P : O (Λ) → gl (r ,R)satisfying

dL

dt+[L, P]= 0,

such that the spectral invariants of L dene an integrable system in O (Λ). Furthermore, thisintegrable system coincides with the Gelfand-Tsetlin integrable system.

Although the integrability condition ensures the existence of a Lax pair for integrable systems,it is not clear what would be a suitable choice for such a pair, since we do not have uniqueness.erefore, the above result provides a canonical way to assign a Lax pair to Gelfand-Tsetlinintegrable systems for adjoint orbits of U(N ) and SO(N ). e ideas involved in our construc-tion are quite natural owing to the underlying matrix-nature which we have in the contextof coadjoint orbits of classical compact Lie groups. Furthermore, all information about theGelfand-Tsetlin paern are encoded in the set of spectral invariants of the matrix L. Besidesthe dierent approach which we provide in this work for Gelfand-Tsetlin integrable systems,we hope that the content which we have developed may help to establish new connections

14

between Gelfand-Tsetlin integrable systems and associated topics like quantum groups, Yang-Baxter equation [12, p. 13-16] and geometric quantization.

In Chapter 4 we provide a Lie-theoretical description of the geometric structures which wehave on coadjoint orbits, namely symplectic structure, complex structure and Kahler struc-ture. Our approach is intended to explain how these geometric structures are connected withelements of Lie theory. In AppendixC we also provide a complete description of the projectivealgebraic realization of coadjoint orbits from the Lie theory stand point.

In Chapter 5 we review some basic facts concerned with holomorphic line bundles and Kahler-Einstein metrics. e idea is to establish the basic language to be used in Chapter 6 and inAppendix D.

Chapter 6 is devoted to study Ricci-at metrics dened on the canonical bundle of complexag manifolds. us our main task is to study the special case of the Einstein equation inKahler manifolds

Ric(ω) = 0

In 1979, in an important paper [26], Calabi introduced a technique to construct Kahler-Einsteinmetrics on the total space of Holomorphic vector bundles over Kahler-Einstein manifolds.is technique is known as the Calabi ansatz technique. By means of this method Calabi pro-vided Ricci-at metrics for two important classes of manifolds, cotangent bundles of projectivespaces and canonical bundle of Kahler-Einstein manifolds. e basic idea of Calabi’s techniqueis to use the Hermitian vector bundle structure over a Kahler-Einstein manifold and the Ein-stein condition on the base manifold to reduce the Ricci-at condition, which is generally aMonge-Ampere equation, to an ordinary dierential equation [100].

Since complex ag manifolds are Kahler-Einstein Fano manifolds we can apply the Calabiansatz to obtain Ricci-at metrics dened on the total space of their canonical bundle. Bymeans of Calabi’s technique and the description of Kahler potential for invariant Kahler met-rics for ag manifolds developed by H. Azad [8], we prove the following result

eorem B. Let (XP ,ωXP ) be a complex ag manifold associated to P = PΘ ⊂ GC, such thatdimC(XP ) = n, then the canonical bundle KXP admits a complete Ricci-at Kahler metric withKahler form given by

ωCY = (2πu +C ) 1n+1π ∗ωXP −

1n+1 (2πu +C )

− nn+1 i∇ξ ∧ ∇ξ ,

whereC > 0 is some positive constant andu : KXP → R≥0 is given by u ([д, ξ ]) = |ξ |2, ∀[д, ξ ] ∈KXP . Furthermore, the above Kahler form is completely determined by the quasi-potentialφ : GC → R given explicitly by

φ (д) =1

2π log( ∏α∈Σ\Θ

| |дv+ωα | |2〈δP ,h∨α 〉

),

15

for every д ∈ GC. erefore, (KXP ,ωCY ) is a (complete) noncompact Calabi-Yau manifold withCalabi-Yau metric ωCY completely determined by Θ ⊂ Σ.

One of the most important feature of eorem B is that it allows us to assign to each subsetΘ ⊂ Σ a complete noncompact Calabi-Yau manifold for which we have the Calabi-Yau metriccompletely determined by elements of the Lie theory. From eorem B we obtain the follow-ing results which provide new explicit examples of Ricci-at Kahler metrics

Proposition B1. e total space of the canonical bundle KW6 over the Wallach space W6 =SL(3,C)/B admits a Calabi-Yau metric ωCY (locally) dened by

ωCY = (2π |ξ |2 +C ) 14ωW6 −

14 (2π |ξ |

2 +C )−34 i∇ξ ∧ ∇ξ ,

for some C > 0, such that

ωW6 =1π

[i∂∂ log

(1 +

2∑k=1|zk |

2)+ i∂∂ log

(1 + |z3 |

2 + |z1z3 − z2 |2)]

,

and

∇ξ = dξ + 2ξ[∂ log

(1 +∑2

k=1 |zk |2)+ ∂ log

(1 + |z3 |

2 + |z1z3 − z2 |2)]

.

Proposition B2. e total space of the canonical bundle KGr(2,C4) over the complex Grass-mannian Gr(2,C4) admits a complete Calabi-Yau metric ωCY (locally) described by

ωCY = (2π |ξ |2 +C ) 15ωGr(2,C4) −

15 (2π |ξ |

2 +C )−45 i∇ξ ∧ ∇ξ ,


ωGr(2,C4) =2πi∂∂ log

(1 +

4∑k=1|zk |

2 +∣∣ det

z1 z3

z2 z4

∣∣2),

and

∇ξ = dξ + 4ξ∂ log(

1 +∑4

k=1 |zk |2 +∣∣ det

z1 z3

z2 z4

∣∣2).

eoremC. ConsiderGC = SL(n+1,C) and B ⊂ GC = SL(n+1,C) (Borel subgroup), then thetotal space of the canonical bundleKXB over the complex full ag manifoldXB = SL(n+1,C)/Badmits a complete Calabi-Yau metric ωCY (locally) described by

ωCY = (2π |ξ |2 +C )2

n (n+1)+2ωXB −2

n(n+1)+2 (2π |ξ |2 +C )−

n (n+1)n (n+1)+2 i∇ξ ∧ ∇ξ ,


• ωXB =

n∑k=1

〈δB,h∨αk〉

2π i∂∂ log(

1 +∑I,I0,k

∣∣∆(k )I (n−(z))

∣∣2); ( Horizontal )

16

• ∇ξ = dξ +n∑

k=1〈δB,h

∨αk〉ξ∂ log

(1 +

∑I,I0,k

∣∣∆(k )I (n−(z))

∣∣2). (Vertical)

It is worth to point out that most of the well known examples of Ricci-at metrics denedin the total space of the canonical bundle of Fano manifolds obtained by means of Calabi’stechnique are toric manifolds, e.g. CPn, therefore, the above results provide a new class ofnon-toric examples.

Our motivations to set up the above results are [26], [143, p. 108], [100] and [60], see also [138],[55], [112] and [23]. We hope to apply the above description on the study of special Lagrangiansubmanifolds in the total space of the canonical bundle over complex ag manifolds, and alsoanalyse its relations with integrable systems dened on ag manifolds, e.g. Gelfand-Tsetlinintegrable systems, these ideas are based on [60] and [95].

Introduction

Part IChapters 2 and 3

Part IIChapters 4,5 and 6

AppendicesA and B

AppendicesA-D

AppendicesC and D

Figure 1.1: In Chapters 2 and 3 we study the Lax formulation of Gelfand-Tsetlin integrablesystems. Further results concerned with integrable systems in coadjoint orbits also can befound in Appendices A and B; In Chapters 4, 5 and 6 we study the construction of Ricci-at metrics on canonical bundles of complex ag manifolds by means of Calabi’s technique.Further results concerned with projective embedding of coadjoint orbits, holomorphic linebundles over complex ag manifolds and Calabi’s technique can be found in Appendices Cand D.

17

Chapter 2Generalities on symplectic geometry

e purpose of this chapter is to provide the basic background about Hamiltonian Lie groupactions, Hamiltonian systems and Lax matrices. e idea is to review some basic results inorder to apply it on the study of Gelfand-Tsetlin integrable systems in coadjoint orbits. emain geometric objects which we will consider throughout this chapter are symplectic mani-folds and Lie groups.

Our main references for the next sections are [141], [28], [75] and [4]. For the approach ofGelfand-Tsetlin integrable systems we will follow [76] and [136], further results concernedwith the elements involved in the construction of Gelfand-Tsetlin integrable systems also canbe found in [158], [74], [75] and [77]. Besides, we also provide an extensive discussion aboutthe necessary condition for integrability of Hamiltonian systems dened by collective Hamil-tonians in Appendix A.

2.1 Hamiltonian G-spaces

Let M be a smooth manifold, a symplectic structure on M is a closed 2-formω ∈ Ω2(M ) whichsatises ωn , 0, i.e. ω is a nondegenerate form. Notice that this last condition implies thatdim(M ) = 2n.Denition 2.1.1. A symplectic manifold is a pair (M,ω) composed by a smooth manifold Mendowed with a symplectic structure ω ∈ Ω2(M ).

ere are three basic examples of symplectic manifolds which are specially important for thestudy of symplectic geometry and applications in physics, these examples can be briey de-scribed as followsExample 2.1.1. Consider the smooth manifold M = R2n. By xing coordinates (q,p) =(q1, . . . ,qn,p1, . . . ,pn ) on R2n we can dene

ω0 =n∑i=1

dpi ∧ dqi .

From this we obtain a symplectic manifold dened by (R2n,ω0). As we will see aerwards thissymplectic manifold is the local model for symplectic geometry.

18

Example 2.1.2. Let N be a n-dimensional manifold and consider the manifold dened byM = TN ∗. Since we have a projection map

π : T ∗N → N ,

we can dene on M = T ∗N the following 1-form

θα (v ) = α (π∗v ) (tautological one-form),

for every α ∈ T ∗N and v ∈ Tα (T ∗N ), here π∗ : T (T ∗N ) → TN denotes the pushforward of π .e 1-form θ described above allows us to dene a 2-form ω ∈ Ω2(T ∗N ) by

ω (u,v ) = dθ (u,v ).

A straightforward calculation shows that the pair (M = T ∗N ,ω = dθ ) denes a symplecticmanifold, see [141, p. 370] for a complete exposition. e symplectic manifold dened bythe cotangent bundle of a smooth manifold is the mathematical model for the phase space ofdynamical systems, see for example [141, p. 428].

Example 2.1.3. Let G be a Lie group with Lie algebra g. Consider the coadjoint action of Gon g∗ dened by the coadjoint representation Ad∗ : G → GL(g∗), where

Ad∗(д)ϕ = ϕ Ad(д−1),

for every д ∈ G and ϕ ∈ g∗. For a xed ϕ ∈ g∗, consider the manifold dened by the coadjointorbit O (ϕ), namely

O (ϕ) =

Ad∗(д)ϕ ∈ g∗∣∣∣ д ∈ G,

and take the 2-form ωO (ϕ) ∈ Ω2(O (ϕ)) dened by 1

ωO (ϕ) (ad∗(X )µ, ad∗(Y )µ ) = −µ ([X ,Y ]),

for every µ ∈ O (ϕ) andX ,Y ∈ g. We can check that the pair (O (ϕ),ωO (ϕ) ) denes a symplecticmanifold, see [141, p. 377] for more details. Coadjoint orbits are important for the study ofdynamical systems with symmetry. In this work several results will be covered for the casewhen G is a compact Lie group.

An important result concerned with the local geometry of symplectic manifolds is the follow-ing theorem [141, p. 356, eorem 8.1.5]

eorem2.1.2. (Darboux) Let (M,ω) be a 2n-dimensional symplectic manifold. For everyx ∈ Mthere exists an open neighbourhood x ∈ U ⊂ M with coordinates (q,p) such that

ω |U =

n∑i=1

dpi ∧ dqi ,

1 Here we have ad∗ (X )µ = ddt

∣∣t=0Ad∗ (exp(tX ))µ, for every µ ∈ O (ϕ) and every X ∈ g.

19

the coordinates (q,p,U ) are called Darboux’s coordinates.

Now let us collect some basic facts about symplectic manifolds and vector elds. Let (M,ω)be a symplectic manifold. For X ∈ Γ(TM ) we have that

• X is a symplectic vector eld if ιXω is a closed 1-form;

• X is a Hamiltonian vector eld if ιXω is an exact 1-form.

From the above facts we have the following denition

Denition 2.1.3. Let (M,ω) be a symplectic manifold. Given a Lie group action τ : G → Di(M )with associated innitesimal action δτ : g → Γ(TM ), we say that

• τ is a symplectic action if ιδτ (X )ω is a closed 1-form for every X ∈ g;

• τ is a Hamiltonian action if ιδτ (X )ω is an exact 1-form for every X ∈ g.

Now suppose we have a Hamiltonian action τ : G → Di(M ), keeping the above notation, inthis case we have a map Φ∗ : g → C∞(M ), such that 2

Φ∗(X ) = 〈Φ,X 〉 and d〈Φ,X 〉 + ιδτ (X )ω = 0,

thus if we x a basis Xi for g, by denition, we can solve the equations

d〈Φ,Xi〉 + ιδτ (Xi )ω = 0,

for every i = 1, . . . , dim(g), therefore we obtain a map Φ : (M,ω) → g∗, such that

Φ =∑i

〈Φ,Xi〉X∗i ,

here X ∗i denotes the dual basis of Xi , the map Φ is called moment map. Moment mapsplay an important role in the study of Hamiltonian Lie group actions, we have the followingimportant result [141, p. 493]

Proposition 2.1.4. Let (M,ω) be a symplectic manifold and τ : G → Di(M ) be a smooth action.en the action τ is Hamiltonian if and only if it admits a moment map Φ : (M,ω) → g∗.

In the context of Hamiltonian Lie group actions we have the following important denition

Denition 2.1.5. Let (M,ω) be a symplectic manifold and τ : G → Di(M ) be a Hamiltonianaction. We say that the moment map Φ associated to τ is equivariant if it satises

Φ(τ (д)p) = Ad∗(д)Φ(p),

for every p ∈ M , X ∈ g and д ∈ G.

2We also denote Φ∗ (X ) = ΦX , ∀X ∈ g∗.

20

In what follows we x a basic data which is the content of the following denition

Denition 2.1.6. A Hamiltonian G-space (M,ω,G,Φ) is composed by:

• A symplectic manifold (M,ω) and a connected Lie group G, with Lie algebra g.

• A le Hamiltonian Lie group action τ : G → Di(M), with associated innitesimal actionδτ : g → Γ(TM ).

• A moment map Φ : (M,ω) → g∗.

Remark 2.1.1. Let (M,ω,G,Φ) be a HamiltonianG-space as above. In this seing we can denecΦ : G → g∗ by

cΦ(д) = Φ(τ (д)p) − Ad∗(д) (Φ(p)),

for p ∈ M . Since M is connected cΦ : G → g∗ does not depend on p ∈ M , thus c is a measure forthe moment map to fail the equivariance property. Moreover, we have the following property

cΦ(дh) = cΦ(д) + Ad∗(д)cΦ(h),

for every h,д ∈ G. It follows that cΦ : G → g∗ denes a 1-cocycle for the representation Ad∗ : G →GL(g∗), see for example [141, p. 495-496] or [145].

From the above comments we have the following result

eorem 2.1.7. Let (M,ω,G,Φ) be a Hamiltonian G-space. If G is a compact Lie group, thenthere exists an equivariant moment map Φ : (M,ω) → g∗.

Proof. e proof can be found in [133, p. 150], see also [145].

Remark 2.1.2. Unless otherwise stated, for all HamiltonianG-spaces (M,ω,G,Φ) we will assumeΦ : (M,ω) → g∗ as being an equivariant moment map.

We are interested in studying Hamiltonian G-spaces given by (O (λ),ωO (λ),G,Φ), where:

• (O (λ),ωO (λ) ) is the Symplectic manifold dened by the coadjoint orbit of a compact andconnected Lie group G, i.e.

O (λ) =

Ad∗(д)λ ∈ g∗∣∣∣ д ∈ G,

and the symplectic structure is given by the Kirillov-Kostant-Souriau 2-form

ωO (λ) (ad∗(X )ξ , ad∗(Y )ξ ) = −ξ ([X ,Y ]),

for every ξ ∈ O (λ), X ,Y ∈ g. Notice that the manifold O (λ) is exactly the integralmanifold through of λ ∈ g∗, which integrates the distribution dened by the innitesimalaction X → ad∗(X ).

21

• e Hamiltonian Lie group action τ : G → Di(O (λ)), is given by the natural restrictionof the coadjoint action of G on g∗ over O (λ). If we consider the equation

d〈Φ,X 〉 + ιδτ (X )ωO (λ) = 0,

a straightforward calculation shows us that the Ad∗-equivariant moment map

Φ : (O (λ),ωO (λ) ) → g∗,

is given by the natural inclusion map O (λ) → g∗.

In the above context we will x the Lie group G as being one of the classical compact Liegroups SO(N ), U(N ), or Sp(N ) = U(N ,H). By xing an Ad-invariant inner product on the Liealgebra of these classical compact Lie groups, we obtain an isomorphism g g∗ which allowsus to identify adjoint and coadjoint orbits. It follows that we can consider

O (λ) O (Λ) =

Ad(д)Λ ∈ g∣∣∣ д ∈ G,

where Λ = diag(iλ1, . . . , iλN ) ∈ g, and Ad(д)X = дXд−1, for every д ∈ G and every X ∈ g.

In general the invariant symplectic structure for adjoint orbits of SO(N ), U(N ), or Sp(N ) =U(N ,H), has the following expression

ωO (Λ) (ad(X )Z , ad(Y )Z ) = − ic

2π Tr(Z [X ,Y ]),

for every Z ∈ O (Λ), andX ,Y ∈ g. e constant factor c > 0, in the above expression is usuallytaken as c = 2N , for U(N ), c = N − 1, for SO(N ) and c = 2(N + 1), for Sp(N ), see for example[141, p. 248].

Remark 2.1.3. e above manifolds dened by coadjoint orbits are also called generalized agmanifolds, we will approach this topic in Chapter 4. For the purpose of this rst part of the workthe basic background which we have described so far is enough to approach Hamiltonian systemsin coadjoint orbits of classical compact Lie groups.

In what follows we will provide a brief description of all adjoint orbits associated to classicalcompact Lie groups, moreover, we will describe some basic low dimensional examples whichmotivate the study of adjoint orbits in dierential geometry.

Example 2.1.4. Given Λ ∈ u(N ), such that

Λ = diag(iλ1, . . . , iλN ),

with

λ1 = · · · = λn1

k1

> λn1+1 = · · · = λn2

k2

> · · · > λnr−1 = · · · = λN

kr

,

for∑

ki = N , the adjoint orbit O (Λ) ⊂ u(N ) is identied with the homogeneous space

22

O (Λ) = U(N )/U(k1) × · · · × U(kr ).

An important example of such a manifold is the complex Grassmannian of r -planes in CN ,which is given by

Gr(r ,CN ) = U(N )/U(N − r ) × U(r ),

for the particular case when r = 1, we have the complex projective space Gr(1,CN ) = CPN−1.

Example 2.1.5. Given Λ ∈ so(2N + 1) = so(2N + 1,C) ∩ u(2N + 1), such that

Λ = diag(iλ1, . . . , iλN ,−iλ1, . . . ,−iλN , 0),

with

λ1 = · · · = λn1

k1

> λn1+1 = · · · = λn2

k2

> · · · > λnr−1+1 = · · · = λN

kr

= 0,

for∑

ki = N . e adjoint orbit O (Λ) ⊂ so(2N + 1) is identied with the homogeneous space

O (Λ) = SO(2N + 1)/U(k1) × · · · × U(kr−1) × SO(2kr + 1),

here we are following the convention of [146, p. 118-120] for classical compact Lie groups. Forthe case N = 1, if we take Λ ∈ so(3), such that

Λ = diag(iλ1,−iλ1, 0),

with λ1 > 0, the manifold O (Λ) ⊂ so(3) is given by

O (Λ) = SO(3)/SO(2),

i.e. in this case we have O (Λ) = S2, in fact it is the unique sphere which admits a symplecticstructure since H 2(Sn ) = 0, for n , 2.

Example 2.1.6. Given Λ ∈ sp(N ) = sp(2N ,C) ∩ u(2N ), [146, p. 118-120], such that

Λ = diag(iλ1, . . . , iλN ,−iλ1, . . . ,−iλN ),

with

λ1 = · · · = λn1

k1

> λn1+1 = · · · = λn2

k2

> · · · > λnr−1+1 = · · · = λN

kr

= 0,

for∑

ki = N . e adjoint orbit O (Λ) ⊂ sp(N ) is identied with the homogeneous space

O (Λ) = Sp(N )/U(k1) × · · · × U(kr−1) × Sp(kr ).

here it is worthwhile to point out that in the particular case when we have

23


with λ1 > λ2 = · · · = λN = 0, the adjoint orbit is given by

O (Λ) = Sp(N )/U(1) × Sp(N − 1) CP2N−1,

this manifold is well known by its relations with the quaternionic projective space HPN−1,[167], [142]. is relation can be described as follows

HPN−1 Sp(N )/Sp(1) × Sp(N − 1) and S2 Sp(1) × Sp(N − 1)/U(1) × Sp(N − 1),

from these we have a S2-bundle

S2 → CP2N−1 → HPN−1.

For the case N = 2, the above ideas lead us to the following diagram of sphere bundles overthe ber bundle described above

S1 S3 S7

S2 CP3 HP1 = S4

the S3-bundle over S4 is a very interesting sphere bundle, see for example [36], [65], [120].

Example 2.1.7. Given Λ ∈ so(2N ) = so(2N ,C) ∩ u(2N ), such that


with

λ1 = · · · = λn1

k1

> λn1+1 = · · · = λn2

k2

> · · · > λnr−1+1 = · · · = λN

kr

= 0,

for∑

ki = N . e adjoint orbit O (Λ) ⊂ so(2N ) is identied with the homogeneous space

O (Λ) = SO(2N )/U(k1) × · · · × U(kr−1) × SO(2kr ),

here again we are following the conventions of [146, p. 118-120]. A particular interesting caseof the above orbit is given by O (Λ) ⊂ so(4), where

Λ = diag(iλ1, iλ2,−iλ1, iλ2),

with λ1 > λ2. In this case the manifold is given by the homogeneous space

O (Λ) = SO(4)/U(1) × U(1),

here we can use the fact that Spin(4) SU(2) × SU(2), and

24

SO(4) Spin(4)/±1,

in order to write

O (Λ) CP1 × CP1 S2 × S2,

this four manifold is the Hirzebruch surface Σ0 = CP1 × CP1, see for instance [15, p. 191],further details about the geometry of this kind of four manifold can be found in [64], [111].

ese examples which we have briey described above compose the basic seing which wewill approach in the rst part of this work, further results concerned with manifolds denedby coadjoint orbits will be given in Chapter 4.

2.2 Hamiltonian systems

Once we have described the manifolds which we are interested in, now we will collect somebasic results and denitions in the context of Hamiltonian systems. It is worth to point outthat our exposition about Hamiltonian systems will not be extensive, thus in what follows wewill just cover some basic generalities about this topic.

Let (M,ω) be a symplectic manifold, givenH ∈ C∞(M ), since the symplectic structure providesan isomorphism between the tangent and cotangent bundle of M , we can take XH ∈ Γ(TM ),such that

dH + ιXHω = 0,

from this we can associate to each smooth function a dynamical system through the ordinarydierential equation dened by XH , namely

dφtdt= XH (φt ).

e equation above is usually called the equation of motion associated to H ∈ C∞(M ). If wetake Darboux coordinates (q,p), we have the following local expression for the Hamiltonianvector eld XH ∈ Γ(TM )

XH = (∂piH )∂qi − (∂qiH )∂pi ,

therefore the dynamical system is (locally) dened by equations

dqidt=∂H

∂piand dpi

dt= −∂H

∂qi.

From the above comments we have the following denition

Denition 2.2.1. A Hamiltonian system is dened by (M,ω,H ), where (M,ω) is a symplecticmanifold and H ∈ C∞(M ).

25

In the context of the above denition the smooth function H ∈ C∞(M ) is called the Hamilto-nian of the system (M,ω,H ).

A Hamiltonian system (M,ω,H ) can be alternatively described in the following way, associ-ated to any symplectic manifold (M,ω) we have a natural Poisson structure

·, ·M

, inducedby the symplectic structure [28, p. 108]. Given F ∈ C∞(M ) the evolution equation of motionis given by

d

dt(F φt ) =

H , F

M(φt ),

where φt is the Hamiltonian ow of XH ∈ Γ(TM ), andH , F

M= ω (XH ,XF ).

By means of the underlying Poisson structure of (M,ω), the equations which locally denethe Hamiltonian system can be rewrien as follows

dqidt=H ,qi

M

, and dpidt=H ,pi

M

.

Denition 2.2.2. (Liouville integrability) Let (M,ω,H ) be a Hamiltonian system, we say thatsuch a system is integrable if there exists H1, . . . ,Hn : (M,ω) → R, such that Hi ∈ C

∞(M ), foreach i = 1, . . . ,n = 1

2 dim(M ), satisfying

•Hi ,Hj

M= 0, for all i, j = 1, . . . ,n,

• dH1 ∧ . . . ∧ dHn , 0, in an open dense subset of M ,

In the above denition of integrability we have H = Hi , for some i = 1, . . . ,n, we will xH = H1.

When the integrability condition holds the equation of motion associated to the HamiltonianH ∈ C∞(M ) can be solved by “quadrature”. Actually, Liouville’s theorem, see [141, p. 585] orAppendix A.5, states that we can take a canonical transformation (qi ,pi ) 7→ (ψi ,Hi ), such that(locally) we have

ω =∑i

dHi ∧ dψi ,

and the equations of motion on this new coordinate system are given by

dψidt=H ,ψi

M=∂H

∂Hi= Ci , and dHi

dt=H ,Hi

M= 0,

here Ci depends on just of H = (H1, . . . ,Hn ), thus it is constant on time. e solution byquadrature methods is locally given by

ψi (t ) = ψi (0) + tCi and Hi (t ) = Hi (0),

for all i = 1, . . . ,n = 12 dim(M ).

26

Remark 2.2.1. Under the assumption of compacity and connectedness on the level manifolds ofthe function

H = (H1, . . . ,Hn ) : (M,ω) → Rn,

we can obtain through of Arnold’s theorem, see for example [141, p. 586-595], local coordinates(I ,θ ), called action and angle coordinates. By means of these coordinates we can show that theset of regular, compact and connected bers of the above map has the structure of a locally trivialtorus bundle over an open set of Rn, see [46].

2.3 Lax pair and Hamiltonian systems

In this section we will introduce some basic ideas about the concept of Lax pair and describeits relation to the study of integrability in the context of Hamiltonian systems. More detailsabout this topic can be found in [12], [141, p. 578].

Let (M,ω) be a symplectic manifold. A Lax pair is given by a pair of matrix-valued smoothfunctions

L, P : (M,ω) → Mr×r (R) End(Rr ).

It will be convenient to denote End(Rr ) = gl (r ,R), namely, we will consider the underlyingnatural Lie algebra structure induced by the commutator on End(Rr ).

e matrix-valued function L is called Lax matrix and the matrix-valued function P is calledauxiliary matrix. We say that a Hamiltonian system (M,ω,H ) admits a Lax pair if the equationof motion associated to H ∈ C∞(M ) is equivalent to the equation

dL

dt+[L, P]= 0.

is equation is called Lax equation. Notice that the above derivative is taken when we con-sider the composition of L with the Hamiltonian ow of XH ∈ Γ(TM ).

e equation described above can be easily solved. Actually, if we consider the initial valueproblem

dL

dt=[P ,L]

with L(0) = L0,

the solution is given by

L(t ) = д(t )L0д(t )−1,

where д : (−ϵ, ϵ ) → GL(r ,R) is determined by the initial value problem

dд

dt= P (t )д(t ), with д(0) = 1,

27

in fact, we have the following

L(t ) = д(t )L0д(t )−1 ⇐⇒

dL

dt=dд

dtL0д(t )

−1 + д(t )L0dд−1

dt,

now we can rewrite the second expression on the right side above as follows

dL

dt=dд

dtд(t )−1L(t ) + L(t )д(t )

dд−1

dt,

from these we can use

д(t )д(t )−1 = 1 ⇐⇒dд

dtд(t )−1 + д(t )

dд−1

dt= 0,

hence we obtain

dL

dt=dд

dtд(t )−1L(t ) − L(t )

dд

dtд(t )−1 =

[P ,L],

where P (t ) = dдdt д(t )

−1, with д(0) = 1. e above computations shows us that if we have a Laxpair for a Hamiltonian system, we can always solve the initial value problem L = [P ,L], bysolving P (t ) = dд

dt д(t )−1, and the solution has the form L(t ) = д(t )L0д(t )

−1.

e main point which makes issues related to the existence of a Lax pair important and inter-esting in the study of Hamiltonian systems is the following. Suppose we have a Lax pair (L, P )for Hamiltonian system (M,ω,H ), if we consider a smooth function F : gl (r ,R) → R which isinvariant by the adjoint action, i.e.

F (дXд−1) = F (X ), for all д ∈ GL(r ,R), and X ∈ gl (r ,R),

and take the composition I = F L ∈ C∞(M ), we obtain a function which is constant over theHamiltonian ow of XH ∈ Γ(TM ). In fact, we have

I (t ) = F (L(t )) = F (д(t )L0д(t )−1) = F (L0) = constant,

from where we obtain H , IM= XH (I ) = 0.

It follows that a Lax pair is an useful tool in the study of integrability since we can get quan-tities in involution by the procedure described above.

Lax pairs are not unique in general. In fact, besides of changes in the size of the matrix valuedfunctions we can consider the natural action of the gauge group 3 G (M × Rr ) on such a pair(L, P ) dened by

3Here we have G (M × Rr ) =a ∈ Aut(M × Rr )

∣∣ pr1 a = idM

and the identication G (M × Rr ) C∞ (M,GL(r ,R)).

28

L → дLд−1, and P → дPд−1 +dд

dtд−1,

where д : (M,ω) → GL(r ,R) is a smooth function. From the above action for L = дLд−1, wecan write

dL

dt=dд

dtLд−1 + д

dL

dtд−1 + дL

dд−1

dt,

since dLdt = [P ,L] and L = дLд−1, we obtain

dL

dt=dд

dtд−1L + д

[P ,L]д−1 + Lд

dд−1

dt.

Notice that д[P ,L]д−1 = дPд−1L − LдPд−1. Since dд

dt д−1 + дdд

−1

dt = 0, we have

dL

dt=[дPд−1 +

dд

dtд−1, L

]=⇒

dL

dt+[L, P]= 0,

where P = дPд−1 + dдdt д−1.

Remark 2.3.1. In the last expression of P above we used the following notation

P (x ) = д(x )P (x )д(x )−1 +dд

dt(x )д(x )−1,

with dдdt (x ) = д∗(XH (x )), for every x ∈ M .

Let us illustrate how the ideas described so far can be applied in concrete cases

Example 2.3.1. (Harmonic Oscillator) A basic example to illustrate the previous discussionis provided by the Harmonic Oscillator. Consider the Hamiltonian system (R2,dp ∧ dq,H ),where the Hamiltonian function is given by

H (q,p) =12 (p

2 +C2q2).

A straightforward calculation shows us that

dH + ιXH (dp ∧ dq) = 0 ⇐⇒ XH = p∂q −C2∂p .

From this we obtain the following equations of motion

dq

dt= p and dp

dt= −C2q.

We have a Lax pair L, P : (R2,dp ∧dq) → gl (2,R) for the Hamiltonian system (R2,dp ∧dq,H )dened by

29

L =

p Cq

Cq −p

and P =12

0 −C

C 0

.

In fact, a straightforward calculation shows us that

dL

dt+[L, P]= 0 ⇐⇒

dqdt = p,dpdt = −C

2q.

Here it is important to observe that

H (q,p) =12 det(L) = 1

4Tr(L2).

Furthermore, we have L2 = 2H (q,p)1 and we can check that

Tr(L2n ) = 2n+1H (q,p)n and Tr(L2n+1) = 0.

Since the algebra of invariant functions by the adjoint action is generated by

X → Tr(Xk ), for every X ∈ gl (2,R),

the previous comments yield a complete description of the quantities in involution providedby the Lax pair, i.e. smooth functions of the form

Ik (q,p) = Tr(Lk ).

Once integrability is a trivial issue in this case, the above calculations provide a simple illustra-tion of interesting properties of the Lax matrices in the study of Hamiltonian systems, furtherdiscussions and nontrivial examples can be found in [12].

To nd a Lax pair for a Hamiltonian system is not a simple task, and the existence of sucha pair does not necessarily ensure integrability. An interesting fact which we will describebelow is that the integrability condition ensures the existence of a Lax pair.

Actually, if we have an integrable system (M,ω,H ), we can consider the equation of motionaer a canonical transformation (qi ,pi ) → (ψi , Fi ) as follows

dψidt=H ,ψi

M=∂H

∂Fi= Ci and dFi

dt=H , Fi

M= 0,

now we dene the Lie algebra generated by Ai ,Bi | i = 1, . . . ,n with the following bracketrelations [

Ai ,Aj

]= 0,

[Ai ,Bj

]= 2δijBj ,

[Bi ,Bj

]= 0.

It follows from Ado’s theorem that this Lie algebra can be realized as a matrix Lie algebra.From these we can dene the Lax pair by

30

L =

n∑i=1

FiAi + 2FiψiBi and P = −

n∑i=1

∂H

∂FiBi .


dL

dt+[L, P]= 0 ⇐⇒

∑i

dFidt

Ai +[2dFidtψi + 2Fi

(dψidt−dH

dFi

)]Bi = 0,

from where we see the equivalence between the equation of motion associated toXH ∈ Γ(TM )and the Lax equation.

Now, suppose that for a Hamiltonian system (M,ω,H ) we have a Lax pair (L, P ) : (M,ω) →gl (r ,R) such that L can be diagonalized, namely

L = UΛU −1,

where Λ = diag(λ1, . . . , λr ). We can check that the functions dened by λk are conservedquantities, i.e.

H , λk

M= 0, ∀k = 1, . . . , r . Let us introduce some notations, consider Eij as

being the canonical basis for gl (r ,R). With respect to this basis we can write

L =∑ij

LijEij .

Since the components Lij of L are functions dened on (M,ω), we can evaluate the Poissonbracket

Lij ,Lkl

M

and gather the result in the following way. We set

L1 = L ⊗ 1 =∑ij

Lij (Eij ⊗ 1) and L2 = 1 ⊗ L =∑ij

Lij (1 ⊗ Eij ),

and we deneL1,L2

M

by

L1,L2

M=∑ij,kl

Lij ,Lkl

MEij ⊗ Ekl .

From the last comments for an integrable system (M,ω,H ) we have the following result [12,p. 14]

Proposition 2.3.1. e involution property of the eigenvalues of L is equivalent to the existenceof a matrix-valued function r12 on the phase space (M,ω) such that:

L1,L2M=[r12,L1

]−[r21,L2

],

where the matrix-valued functions r12 and r21 are, respectively, dened by

r12 =∑ij,kl

rij,klEij ⊗ Ekl and r21 =∑ij,kl

rij,klEkl ⊗ Eij ,

the matrix r = (rij,kl ) is called r -matrix.

31

In the context of the above proposition the Jacobi identity on the Poisson bracket provides thefollowing constraint on the matrix r[

L1,[r12, r13

]+[r12, r23

]+[r32, r13

]+L2, r13

M−L3, r12

M

]+ cyc. perm. = 0,

here “cyc. perm.” means cyclic permutations of tensor indices 1, 2, 3, see [12, p. 15] for moredetails about the above equation.

e main feature of the last equation is that if r is constant the Jacobi identity is satised if[[r , r]]

:=[r12, r13

]+[r12, r23

]+[r32, r13

]= 0.

When r is antisymmetric, r12 = −r21, the above equation[[r , r]]= 0 is called the classical

Yang-Baxter equation (CYBE).

e CYBE rst appeared explicitly in the literature on integrable Hamiltonian systems, but itis a special case of the Schouten bracket in diential geometry, introduced in 1940’s, see [31, p.50] for more details. It is worth to point out that besides of the approach via integrable systems,solutions of the CYBE are also interesting for the study of quantum groups and related topics,see [31].

2.4 Collective Hamiltonians

In order to study Hamiltonian systems in coadjoint orbits it will be useful to set some basic factsabout the Lie-Poisson structure of the dual space g∗ of the Lie algebra associated to compactand connected Lie groups, see [77], [164, ex. 1.1.3], or [12, p. 522-525].

Denition 2.4.1. Let M be a smooth manifold and letC∞(M ) denote the algebra of real-valuedsmooth functions on M . Consider a given bracket operation denoted by

·, ·M

: C∞(M ) ×C∞(M ) → C∞(M ).

e pair (M, ·, ·M ) is called Poisson manifold if the R-vector spaceC∞(M ) with the bracket ·, ·Mdenes a Lie algebra and

H , f дM= дH , f

M+ fH ,д

M

,

∀f ,д,H ∈ C∞(M ).

Let g be a Lie algebra of a compact and connected Lie group G 4. We have a Poisson bracket·, ·g∗ on the manifold g∗ dened as follows. Given F1, F2 ∈ C

∞(g∗) and ξ ∈ g∗, we have

F1, F2

g∗(ξ ) = −

⟨ξ ,[(dF1)ξ , (dF2)ξ

]⟩,

4 Here it is worthwhile to point out that given a vector space V there exists a correspondence between Liealgebra structures on V and linear Poisson structures on V ∗, see [141, p. 367] for more details about this corre-spondence.

32

where we use the identication T ∗ξ g∗ g, and from this

(dF1)ξ , (dF2)ξ ∈ g.

Will be convenient to denote by ∇F (ξ ) the element of g which satises the pairing

(dF )ξ (η) =⟨η,∇F (ξ )

⟩,

for every F ∈ C∞(g∗), ξ ∈ g∗ and η ∈ Tξg∗. From these we can rewrite the previous expressionof ·, ·g∗ as follows

F1, F2g∗(ξ ) = −

⟨ξ ,[∇F1(ξ ),∇F2(ξ )

]⟩.

With the above bracket, the pair (g, ·, ·g∗ ) is a Poisson manifold.

Denition 2.4.2. Let (M, ·, ·M ) be a Poisson manifold. A smooth functionC ∈ C∞(M ), is calledCasimir function if satises

C, FM= 0,

for every F ∈ C∞(M ).

Example 2.4.1. Consider the Poisson manifold (g∗, ·, ·g∗ ), described previously. Supposethat C ∈ C∞(g∗) is a Casimir function. en

C, Fg∗= 0,

for every F ∈ C∞(g∗). If F = lX ∈ C∞(g∗), where

lX (ξ ) =⟨ξ ,X

⟩,

∀ξ ∈ g∗, a straightforward calculation shows us that

∇lX (ξ ) = X

∀ξ ∈ g∗. From these we have

0 =C, lX

g∗(ξ ) = −

⟨ξ ,[∇C (ξ ),X

]⟩= −(dC )ξ (ad∗(X )ξ ).

Since g = Lie(G ) and G is connected, it follows that the Casimir functions of (g, ·, ·g∗ ) areAd∗-invariant functions.

Notice that the above equation is also true if we take X = ∇F (ξ ) in the right side, i.e.

0 = (dC )ξ (ad∗(∇F (ξ ))ξ ) = −⟨ξ ,[∇C (ξ ),∇F (ξ )

]⟩=C, F

g∗(ξ ),

for some F ∈ C∞(g∗), and C ∈ C∞(g∗) Ad∗-invariant function.

It follows that the Casimir functions of (g, ·, ·g∗ ) are exactly the Ad∗-invariant functions. Fora more general discussion about Casimir functions with respect to the Lie-Poisson bracket see[116, p. 463].

33

We are interested in studying Hamiltonian systems dened by the following kind of function

Denition 2.4.3. Let (M,ω,G,Φ) be a Hamiltonian G-space. Given a smooth function F ∈C∞(g∗), a collective Hamiltonian is dened by the pullback H = Φ∗(F ) ∈ C∞(M ).

Now we will provide a expression for the Hamiltonian vector eld XH ∈ Γ(TM ), associated toa collective Hamiltonian H = Φ∗(F ) ∈ C∞(M ), for more details see [77, p. 241].

We notice that by xing a basis Xi for g, and denoting by X ∗i its dual, we have

Φ =∑i

ΦiX ∗i , and DΦ =∑i

d〈Φ,Xi〉X∗i ,

where each component function Φi = 〈Φ,Xi〉, satises the equation

d〈Φ,Xi〉 + ιδτ (Xi )ω = 0.

We recall that δτ denotes the innitesimal action associated to the Hamiltonian action τ : G →Di(M ). erefore, given H = Φ∗(F ) ∈ C∞(M ), we have

dH = dF DΦ = dF (∑i

d〈Φ,Xi〉X∗i ).

From the previous equations for the components of Φ it follows that

dF (∑i

d〈Φ,Xi〉X∗i ) = −

∑i

〈X ∗i , (∇F ) Φ〉ιδτ (Xi )ω,

thus we obtain

XH = δτ ((∇F ) Φ).

e above description yields the following proposition

Proposition 2.4.4. Let (M,ω,G,Φ) be a Hamiltonian G-space and H = Φ∗(F ) ∈ C∞(M ) acollective Hamiltonian. Given p ∈ M , the trajectory of XH ∈ Γ(TM ) through the point p ∈ M , isgiven by

φt (p) = τ (exp(t∇F (Φ(p))))p.

Proof. e proof follows from the above expression for XH .

Remark 2.4.1. Here we notice that the above expression φt (p) = τ (exp(t∇F (Φ(p))))p denotes acurve which satises

d

dt

∣∣∣t=0φt (p) = XΦ∗ (F ) (p).

34

e above curve is not necessarily the ow ofXΦ∗ (F ) , it is in fact the Hamiltonian ow of the vectoreld δτ (∇F (Φ(p))) ∈ Γ(TM ) through the point p ∈ M . As we will see below the curve obtainedin Proposition 2.4.4 will be the Hamiltonian ow of Φ∗(F ) when F ∈ C∞(g∗)Ad∗ , i.e. when F isAd∗-invariant.

Let us briey describe how we can nd the Hamiltonian ow associated to a collective HamiltonianΦ∗(F ) ∈ C∞(M ). At rst we take a trivialization of the tangent bundleTG ofG by right invariantvector elds. From this we consider the following vector eld v ∈ Γ(TG ) for a xed point p ∈ M

v : G → TG, such that vд = (Rд)∗(∇F (Φ(τ (д)p))),

where Rд : G → G denotes the right translation. Now we consider the following smooth mapinduced by the action τ : G → Di(M )

Ap : G → M , such that Ap (д) = τ (д)p.

A straightforward calculation shows us the following fact

δτ (X )τ (д)p = (DAp )д ((Rд)∗X ).

erefore, if we take the solution of the initial value problem

dд

dt= vд(t ) , with д(0) = e ,

we can dene a curve φt (p) = τ (д(t ))p which satises

d

dtφt (p) = (DAp )д(t ) (

dд

dt) = (DAp )д(t ) ((Rд(t ) )∗(∇F (Φ(τ (д(t ))p)))).

e above expression can be rewrien as

d

dtφt (p) = δτ (∇F (Φ(τ (д(t ))p)))τ (д(t ))p = δτ (∇F (Φ(φt (p))))φt (p) ,

i.e. we have a solution for the initial value problem

d

dtφt (p) = δτ (∇F (Φ(φt (p))))φt (p) = XΦ∗ (F ) (φt (p)), with φ0(p) = p.

Now we observe the following fact. If F ∈ C∞(g∗)Ad∗ , it follows that

∇F (Ad∗(д)ξ ) = Ad(д)∇F (ξ ),

for every ξ ∈ g∗ and д ∈ G. us we obtain

vд = (Rд)∗(∇F (Φ(τ (д)p))) = (Lд)∗(∇F (Φ(p))),

where Lд : G → G denotes the le translation. From these the equation

35

dд

dt= vд(t ) , with д(0) = e ,

becomes exactly the equation of le invariant vector elds through the identity element. erefore,in this last case, we have д(t ) = exp(t∇F (Φ(p))) and the Hamiltonian ow associated to the col-lective Hamiltonian Φ∗(F ) is exactly the curve described in Proposition 2.4.4. Further discussionsabout the Hamiltonian ow of collective Hamiltonians can be found in [77, p. 241-242].

Let us illustrate the above ideas by means of an example which is the seing which we areinterested in

Example 2.4.2. Consider now the Hamiltonian G-space (O (λ),ωO (λ),G,Φ). If we take a col-lective Hamiltonian H = Φ∗(F ) ∈ C∞(O (λ)), from the above proposition we have

XH = ad∗((∇F ) Φ).

Since Φ in this case is just the inclusion map, we have the following expression for the trajec-tory of XH through the point ξ ∈ O (λ)

φt (ξ ) = Ad∗(exp(t∇F (Φ(ξ ))))ξ .

It follows that the dynamic dened by H = Φ∗(F ) ∈ C∞(O (λ)) can be understood through theequation which denes the le invariant vector eld associated to ∇F (ξ ) ∈ g.

2.5 imm’s trick

Now we will describe how to obtain quantities in involution when we consider Hamiltoniansystems dened by collective Hamiltonians.

As we have seen the Hamiltonian vector eld associated to a collective Hamiltonian Φ∗(F ) ∈C∞(M ), is given by

XFΦ = δτ ((∇F ) Φ).

If we consider other collective Hamiltonian Φ∗(I ) ∈ C∞(M ), for some I ∈ C∞(g∗), we haveF Φ, I Φ

M(p) = ω (δτ (∇F (Φ(p))p,δτ (∇I (Φ(p))p )

where ∇F (Φ(p)),∇I (Φ(p)) ∈ g, for every p ∈ M . Let us introduce a result which will beimportant for us

Proposition 2.5.1. Let (M,ω,G,Φ) be a HamiltonianG-space. Given p ∈ M and ξ = Φ(p) ∈ g∗.Let ι : G · p → M be the natural inclusion map, then

ι∗ω = Φ∗ωO (ξ ) ,

where (O (ξ ),ωO (ξ ) ) ⊂ g∗, denotes the coadjoint orbit of ξ ∈ g∗.

36

Proof. See [141, p. 497].

From the above result and the previous comments we obtain 5

F Φ, I Φ

M(p) =

F , Ig∗(Φ(p)),

for every p ∈ M and F , I ∈ C∞(g∗).

Now we will use the above results in order to describe imm’s trick [158]. Let (M,ω,G,Φ) bea HamiltonianG-space as before, if we take a closed and connected subgroup K ⊂ G, we havea natural Hamiltonian action of K on (M,ω) induced by restriction, it follows that we have aHamiltonian K-space (M,ω,K ,ΦK ), where the moment map

ΦK : (M,ω) → k∗ = Lie(K )∗,

is given by

ΦK = πK Φ,

where πK : g∗ → k∗ is the projection induced by the inclusion k → g.

If we take two collective Hamiltonians Φ∗(F ),Φ∗K (I ) ∈ C∞(M ), we obtain

F Φ, I ΦK

M=F Φ, I πK Φ

M

,

hence we have

F Φ, I ΦK

M=F , I πK

g∗ Φ.

From the last equality we have the following proposition

Proposition 2.5.2. Let (M,ω,G,Φ) be a Hamiltonian G-space, and let K ⊂ G be a closed andconnected subgroup. If we consider the Hamiltonian system (M,ω,Φ∗K (I )), then all collectiveHamiltonians obtained from the Casimir functions of (g∗, ·, ·g∗ ) and (k∗, ·, ·k∗ ) are quantities ininvolution for the system (M,ω,Φ∗K (I )).

Proof. is result is a consequence of the ideas developed in [158], [77] see also [76, prop. 3.1].In fact, from the above comments, for Φ∗(F ),Φ∗K (I ) ∈ C∞(M ) we have

F Φ, I ΦK

M(p) =

F , I πK

g∗(Φ(p)) = 0,

if F ∈ C∞(g∗) is a Casimir. Similarly, for Φ∗K (F ),Φ∗K (I ) ∈ C∞(M ), we haveF ΦK , I ΦK

M(p) =

F , Ik∗(ΦK (p)) = 0,

5Given a map between Poisson manifolds Φ : (M, ·, ·M ) → (N , ·, ·N ), we say that Φ is a Poisson map ifΦ∗ f ,Φ∗дM = Φ∗ f ,дN , ∀f ,д ∈ C∞ (N ).

37

if F ∈ C∞(k∗) is a Casimir.

e above construction and proposition are an application of the so called imm’s trick, see[158] for more details. We can use the previous proposition iteratively on a chain of closedand connected subgroups

G = K0 ⊃ K1 ⊃ . . . ⊃ Ks .

By denoting Φj : (M,ω) → k∗j the moment map associated to each Hamiltonian Kj-space, withj = 0, . . . , s , and by considering the Hamiltonian system

(M,ω,Φ∗s (F )),

for some F ∈ C∞(k∗s ). We obtain by imm’s trick a set of functions in involution composedby collective Hamiltonians dened by Casimir functions of each Poisson manifold (k∗j , ·, ·k∗j ).

When M is a coadjoint orbit of some compact Lie group, and the integrability condition holdsfor the set of quantities in involution described above, the integrable system is called Gelfand-Tsetlin system [76]. In the next section we will describe the construction of Gelfand-Tsetlinsystems in coadjoint orbits of U(N ) and SO(N). In appendix A of this work we provide acomplete discussion about the necessary condition for integrability of Hamiltonian systemsby means of collective Hamiltonians, our approach is according to [75], [73], [96] and [117].

For coadjoint orbits of the compact Lie group Sp(N ) we do not have Gelfand-Tsetlin systems.In spite of this in a recent work [79] M. Harada provided an alternative construction of inte-grable systems in regular adjoint orbits of Sp(N ). ese integrable systems were named byher as Gelfand-Tsetlin-Molev systems. We also provide a complete discussion about Harada’sconstruction which can be found in appendix B of this work.

2.6 Gelfand-Tsetlin integrable systems

Now we will apply all the ideas developed in the previous sections on the HamiltonianG-space(O (λ),ωO (Λ),G,Φ), where G = U(N ) or SO(N ). In what follows it will be important for us toconsider the isomorphism u(N ) iu(N ). In fact throughout of this section we will deal withcharacteristic polynomials and diagonalizantion of matrices, therefore we will not distinguishelements of u(N ) and iu(N ). We will proceed in this way in order keep an uniform notation,otherwise we would need to multiply by the imaginary unity the matrices in some expressions.

Unless otherwise stated, we will follow the conventions of [146, p. 141] for root systems asso-ciated to the classical Lie groups U(N ) and SO(N ). We will proceed describing Gelfand-Tsetlinintegrable systems for the unitary case following [76], once many steps of the construction ofGelfand-Tsetlin integrable systems for the special orthogonal case are the same of the unitarycase, we will give more details about the unitary case. For the special orthogonal case we willfollow the ideas developed in [136].

38

2.6.1 Gelfand-Tsetlin Systems for adjoint orbits of U(N)

e case which we will cover in this subsection corresponds to the complex Lie algebras ofA type. e underlying complex Lie algebra in this case is the complex reductive Lie algebragl (N ,C), which can be completely described by its semisimple part sl (N ,C). Below we havean illustration of the root system associated to sl (4,C).

Figure 2.1: e A3 root system with the simple roots represented in dark gray. is image wasextracted from the book [78].

We consider for this rst case the Hamiltonian U(N )-space (O (λ),ωO (λ),U(N ),Φ), whereO (λ) =Ad∗(U(N ))λ ⊂ u(N )∗. By taking a chain of closed and connected subgroups dened by diag-onal blocks

U(N ) ⊃ U(N − 1) ⊃ . . . ⊃ U(1),

for each k = 1, . . . ,N we have a Hamiltonian U(k )-space (O (λ),ωO (λ),U(k ),Φk ).

As we have mentioned in Section 2.1 we can replace O (λ) ⊂ u(N )∗ by O (Λ) ⊂ u(N ) iu(N ),where

Λ = diag(iλ1, . . . , iλN ),

with 0 ≥ λ1 ≥ λ2 ≥ . . . ≥ λN . Since the algebra of Ad-invariant polynomial functions of u(k ),for k = 1, . . . ,N , are generated by the coecients of the characteristic polynomial, i.e. givenX ∈ u(k ), we have

CX (t ) = det(X − t1k ) = P0(X )tk + P1(X )tk−1 + . . . + Pk−1(X )t + Pk (X ).

By applying imm’s trick iteratively on the chain of subgroups dened previously we get aset of not constant functions dened by collective Hamiltonians

Φ∗k (P1), . . . ,Φ∗k (Pk ),

k = 1, . . . ,N − 1, which Poisson commute with each other.

e above construction yields a set of N (N−1)2 functions which are in involution.

39

Remark 2.6.1. Notice that the Ad-invariant functions of u(N ) are constant when restricted toO (Λ) ⊂ u(N ), therefore we do not consider these functions.

e set of function obtained by imm’s trick denes in fact an integrable system in all coad-joint orbits associated to the compact connected Lie group U(N ), see [76] for more details.

Remark 2.6.2. e above construction was introduced in [76], and since then it has been widelystudied due to its nice properties and relations with geometric quantization representation theory[59] [121], and more recently with mirror symmetry (Landau–Ginzburg model), through of toricdegeneration [131].

Now we will look closely at the above construction in order to obtain a more concrete expres-sion for the quantities in involution.

As we have seen the above collective Hamiltonians are related to coecients of characteristicpolynomials, and these coecients are determined by polynomial functions of the form

X → Tr(X l ), l = 1, . . . ,k ,

for X ∈ u(k ). In fact, given X ∈ u(k ) we can rewrite

CX (t ) = (−1)k det(t1k − X ) = (−1)kk∑i=0

ai (X )t i ,

where Pi (X ) = (−1)kai (X ), with

ak−l (X ) =(−1)ll ! det

Tr(X ) l − 1 0 · · · 0

Tr(X 2) Tr(X ) l − 2 · · · 0...

......

. . ....

Tr(X l ) Tr(X l−1) Tr(X l−2) · · · Tr(X )

,

for l = 0, 1, . . . ,k , see [22, p. 53], [102, p. 298], [37, § 3].

Example 2.6.1. Given X ∈ u(3), we have

CX (t ) = −t3 + Tr(X )t2 −

12

[Tr(X )2 − Tr(X 2)

]t + det(X ),

and we can write

det(X ) =16

[Tr(X )3 + 2Tr(X 3) − 3Tr(X )Tr(X 2)

].

Besides the previous facts, since each element X ∈ u(k ) iu(k ), for k = 1, . . . ,N − 1, can bediagonalized, we can write

CX (t ) =∏

1≤l≤k

(iΛl (X ) − t

),

40

it follows that we can replace the functions that we have obtained previously by the functions

Φ∗k (Λ1), . . . ,Φ∗k (Λk ),

k = 1, . . . ,N − 1. e above functions are the imaginary part of the eigenvalues associatedto principal submatrices, or leading principal submatrices, dened by each moment map Φk ,where k = 1, . . . ,N − 1.

Remark 2.6.3. It is worthwhile to point out that the diagonalization process of X ∈ u(k ), interms of Lie theory, can be understood as being the process of taking the projection of X over theWeyl chamber of u(k ), for a suitable choice of maximal torus. For this reason when we considercollective Hamiltonians dened by the eigenvalues we obtain a set of continuous functions overall O (Λ), which are smooth in an open dense subset of O (Λ), see [76].

e interesting property of these functions dened by eigenvalues is that they satisfy theGelfand-Tsetlin paern, [59], [121], i.e. if we denote by

λ(l )k = Φ∗k (Λl ),

for 1 ≤ l ≤ k , 1 ≤ k ≤ N − 1, we have the following inequalities

λ(1)k (X ) ≥ λ(1)k−1(X ) ≥ λ(2)k (X ) ≥ . . . ≥ λ(k−1)k−1 (X ) ≥ λ(k )k (X ),

for every X ∈ O (Λ), 1 ≤ k ≤ N . Here we notice that

λ(l )N = λl ,

for 1 ≤ l ≤ N , i.e. the imaginary part of the spectrum of Λ ∈ u(N ).

Remark 2.6.4. e above inequalities obtained by diagonalizing submatrices follow from Cauchy’sinterlacing theorem, see [90], [52], [85, p. 242], applied iteratively at each Hermitian matrixiX ∈ iu(k ), 1 ≤ k ≤ N −1, which are obtained by the projections associated to the moment maps.In fact, given X ∈ u(X ), if we denote A = iX we obtain A∗ = A, it follows that A is a Hermitianmatrix. Now let B be a principal submatrix of A of order N − 1. If we denote, respectively, theeigenvalues of A and B by

λN (A) ≥ . . . ≥ λ1(A) and λN−1(B) ≥ . . . ≥ λ1(B),

then Cauchy’s interlacing theorem states that the eigenvalues of A and B satisfy the followinginequalities

λN (A) ≥ λN−1(B) ≥ λN−1(A) ≥ . . . ≥ λ2(A) ≥ λ1(B) ≥ λ1(A),

see [85, p. 242] for more details about this result. It is worth to observe that if we take X =UΛ(X )U ∗ ∈ u(N ), where U ∈ U(N ) and

Λ(X ) = diag(iλ1(X ), . . . , iλN (X )),

such that 0 ≥ λ1(X ) ≥ . . . ≥ λN (X ). From the above convention A = iX , we have

41

λk (A) = −λN−k+1(X ), for k = 1, . . . ,N .

e Gelfand-Tsetlin paern allows us to construct integrable systems for all coadjoint orbitsof U(N). In fact the previous inequalities tell us what functions we need to choose in the set ofPoisson commuting functions. We illustrate below how it can be done in two basic examples

Example 2.6.2. Consider the adjoint orbit O (Λ) ⊂ u(3), where

Λ = diag(iλ1, iλ2, iλ3),

and 0 > λ1 > λ2 > λ3. In this case we have dimR(O (Λ)) = 6, i.e. we have a regular orbit. Weobtain the following functions from the previous construction

λ1 ≥ λ(2)1 ≥ λ2 ≥ λ

(2)2 ≥ λ3,

λ(2)1 ≥ λ(3)1 ≥ λ

(2)2 ,

it follows that the functions λ(2)1 , λ(2)2 , λ

(3)1 : O (Λ) → R, dene an integrable system in an open

dense subset of O (Λ) U(3)/T 3.

Figure 2.2: Polytope dened by the image of the adjoint orbit of Λ = diag(−i,−2i,−3i ) underthe Gelfand-Tsetlin integrable system.

Example 2.6.3. Consider now the adjoint orbit O (Λ) ⊂ u(3), where

Λ = diag(iλ1, iλ2, iλ3),

and 0 > λ1 = λ2 > λ3. In this case we have dimR(O (Λ)) = 4, and the Gelfand-Tsetln paernbecomes

λ1 = λ(2)1 = λ2 ≥ λ

(2)2 ≥ λ3,

λ1 = λ2 = λ(2)1 ≥ λ

(3)1 ≥ λ

(2)2 ,

it follows that the functions λ(2)2 , λ(3)1 : O (Λ) → R, dene an integrable system in a open dense

subset of O (Λ) U(3)/U(2) × U(1) = CP2.

42

e construction which we have described so far for the unitary compact Lie groups is theprototype case to analyse. Actually, most of the previous construction can be reproduced forthe classical groups SO(N) and Sp(N). It follows from the fact that we have the following Liealgebra isomorphisms

so(N ) u(N ) ∩ so(N ,C) and sp(N ) u(2N ) ∩ sp(2N ,C).

Once we can consider so(N ) ⊂ u(N ) and sp(N ) ⊂ u(2N ), issues related to the diagonalizationprocess are essentially the same as the unitary case. In the next subsection we will discussthe construction of Gelfand-Tsetlin systems for coadjoint orbits of the special orthogonal Liegroup, which is quite similar to the unitary case. For the construction of integrable systemsin regular orbits of the compact symplectic Lie group we suggest Appendix B.2.

2.6.2 Gelfand-Tsetlin Systems for adjoint orbits of SO(N)

e case which we will cover in this subsection corresponds to complex Lie algebras of B typeand D type. e underlying complex Lie algebra in these cases is the complex semisimple Liealgebra so(N ,C), with N even for the D case and N odd for the B case. Below we have anillustration of the root system associated to so(7,C).

Figure 2.3: e B3 root system, with the simple roots represented in dark gray. is image wasextracted from the book [78].

e construction of integrable systems for coadjoint orbits of the special orthogonal groupscan be seen as an intermediate case between the unitary case [76] and the symplectic case[136]. In fact, as we will see in this subsection the construction of Gelfand-Tsetlin integrablesystems for special orthogonal groups does not come directly from the construction for theunitary case. Actually, there are some obstructions at the level of representation theory re-lated to restrictions of the representations and multiplicity-free modules, see [121, p. 111-116].

roughout this subsection we will follow [136], more details also can be found in [135]. Westart by taking the usual rst step in the main construction, i.e. we take a chain of closed andconnected subgroups dened by diagonal blocks

SO(N ) ⊃ SO(N − 1) ⊃ . . . ⊃ SO(2).

43

Here we have two possibilities, the rst one is to take a chain on which the parity changes ateach step, the second one is to take a chain of subgroups with the same parity of N .

For the rst case, consider N = 2n + 1, and take

SO(2n + 1) ⊃ SO(2n) ⊃ SO(2n − 1) ⊃ . . . ⊃ SO(2),

notice that in this case we need to deal with two dierent types of simple Lie algebras, i.e.B-type and D-type. As in the unitary case the above chain has an interpretation in terms ofthe construction of basis for irreducible representations [58].

It is worth to point out that the previous chain does not preserve the “naturality” as we havein the unitary case.

Remark 2.6.5. For us naturality in this context means the relation between the embedding ofeach subgroup and its respective Dynkin diagram, see [121, p. 137-164] and [58].

Even though the groups in the above chain are related to dierent kinds of simple Lie algebras,we can apply the same ideas as in the unitary case.

At rst, we need to observe that the odd special orthogonal Lie algebra so(2k + 1) and theeven special orthogonal Lie algebra so(2k ) have the same rank, i.e., up to isomorphism, themaximal torus of SO(2k + 1) and SO(2k ) are respectively generated by elements

Rθ1 · · · 02 0.... . .

......

02 · · · Rθk 0

0 · · · 0 1

, and

Rθ1 · · · 02.... . .

...

02 · · · Rθk

,

where the diagonal blocks are given by

Rθ =

cos(θ ) − sin(θ )

sin(θ ) cos(θ )

, with θ ∈ S1.

From these, given O (Λ) ⊂ so(2n + 1), with

Λ = diag(iλ1, . . . , iλn,−iλ1, . . . ,−iλn, 0),

and λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 0. Proceeding as in the unitary case, at each step we need todiagonalize a submatrix which lies in the odd special orthogonal Lie algebra or in the evenspecial orthogonal Lie algebra.

In terms of invariant functions we have the following generators for the algebra of Ad-invariantpolynomials:

• For so(2k + 1), X → Tr(X 2l ), 1 ≤ l ≤ k .

44

• For so(2k ), X → Tr(X 2l ), 1 ≤ l ≤ k − 1, and X → Pfa(X),

here Pfa(X)2 = det(X ), see [70, p. 97], [102, p. 302].

In the end of the process we obtain a set of n2 functions which Poisson commute with eachother. As in the unitary case we can replace these functions by the corresponding eigenvaluesassociated to each submatrix obtained by projections (moment maps).

We denote by

λ(l )2k+1 = Φ∗2k+1(Λl ),

with 1 ≤ l ≤ k , the functions associated to the odd special orthogonal subgroups which appearin the chain, and by

µ (l )2k = Φ∗2k (Λl ),

with 1 ≤ l ≤ k , the functions associated to the even special orthogonal subgroups whichappear in the chain. ese functions also satisfy an array of inequalities which we describebelow. Given X ∈ O (Λ) we have

λ(1)2k+1(X ) ≥ µ (1)2k (X ) ≥ . . . ≥ µ (k−1)2k (X ) ≥ λ(k )2k+1(X ) ≥ |µ (k )2k (X ) |,

for k = 1, . . . ,n, and

µ (1)2k (X ) ≥ λ(1)2k−1(X ) ≥ . . . ≥ µ (k−1)2k (X ) ≥ λ(k−1)

2k−1 (X ) ≥ |µ (k )2k (X ) |,

for k = 2, . . . ,n, for more details about these inequalities see for instance [121, p. 51], [136],[58].

Remark 2.6.6. We notice that we have used the identication

so(2k + 1) u(2k + 1) ∩ so(2k + 1,C) ⊂ u(2k + 1),and

so(2k ) u(2k ) ∩ so(2k,C) ⊂ u(2k ),

in order to apply the same ideas as in the unitary case on the diagonalization process of subma-trices.

e previous array of inequalities allows us to choose the functions necessary in the set ofPoisson commuting in order to obtain an integrable system for each adjoint orbit of SO(2n+1).

Example 2.6.4. Let O (Λ) ⊂ so(5) be a regular orbit, i.e.

Λ = diag(iλ1, iλ2,−iλ1,−iλ2, 0),

with λ1 > λ2 > 0. We have in this case dimR(O (Λ)) = 8, and the Gelfand-Tsetlin system isdetermined by the following array of inequalities

45

λ1 ≥ µ(1)4 ≥ λ2 ≥ |µ

(2)4 |,

µ (1)4 ≥ λ(1)3 ≥ |µ

(2)4 |,

λ(1)3 ≥ |µ(1)2 |.

From these we have an integrable system in an open dense subset ofO (Λ) SO(5)/T2, denedby the functions

µ (1)4 , µ(2)4 , λ

(1)3 , µ

(1)2 : O (Λ) → R.

Example 2.6.5. Let O (Λ) ⊂ so(7) be an adjoint orbit associated to

Λ = diag(iλ1, iλ2, iλ3,−iλ1,−iλ2,−iλ3, 0),

with λ1 > λ2 = λ3 > 0. We have in this case dimR(O (Λ)) = 16, and the Gelfand-Tsetlin systemis dened by the functions in the following array of inequalities

λ1 ≥ µ(1)6 ≥ λ2 = µ

(2)6 = λ3 ≥ |µ

(3)6 |,

µ (1)6 ≥ λ(1)5 ≥ µ

(2)6 ≥ λ

(2)5 ≥ |µ

(3)6 |,

λ(1)5 ≥ µ(1)4 ≥ λ

(2)5 ≥ |µ

(2)4 |,

µ (1)4 ≥ λ(1)3 ≥ |µ

(2)4 |,

λ(1)3 ≥ |µ(1)2 |.

From the above inequalities we obtain the following set of functions

λ(1)5 , λ(2)5 , λ

(1)3 : O (Λ) → R, µ (1)6 , µ

(3)6 , µ

(1)4 , µ

(2)4 , µ

(1)2 : O (Λ) → R,

this set of functions denes an integrable system in an open dense subset of the manifold

O (Λ) SO(7)/SO(3) × SO(2) × SO(2).

Now we consider the case N = 2n, i.e. the even special orthogonal case. If we take an adjointorbit O (Λ) ⊂ so(2n), where

Λ = diag(iλ1, . . . , iλn,−iλ1, . . . ,−iλn ),

with λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 0, for a chain of closed connected subgroups

SO(2n) ⊃ SO(2n − 1) ⊃ SO(2n − 2) ⊃ . . . ⊃ SO(2),

on which the parity changes, we can apply the same ideas as in the odd special orthogonalcase. Again we will deal with the Ad-invariant polynomial functions:

• X → Tr(X 2l ), 1 ≤ l ≤ k , for so(2k + 1).

• X → Tr(X 2l ), 1 ≤ l ≤ k − 1, and X → Pfa(X), for so(2k ).

In the end of the process we get n(n − 1) functions, and we can replace these functions by theeigenvalues associated to the submatrices dened by the moment map. Now we denote by

46

λ(l )2k = Φ∗2k (Λl ),

with 1 ≤ l ≤ k , the functions associated to the even special orthogonal subgroups whichappear in the chain, and by

µ (l )2k+1 = Φ∗2k+1(Λl ),

with 1 ≤ l ≤ k , the functions associated to the odd special orthogonal subgroups. e Gelfand-Tsetlin paern is given by the following array of inequalities

λ(1)2k (X ) ≥ µ (1)2k−1(X ) ≥ . . . ≥ µ (k−1)2k−1 (X ) ≥ |λ(k )2k (X ) |,

for k = 1, . . . ,n, and

µ (1)2k−1(X ) ≥ λ(1)2k−2(X ) ≥ . . . ≥ µ (k−1)2k−1 (X ) ≥ |λ(k−1)

2k−2 (X ) |,

for k = 2, . . . ,n, ∀X ∈ O (Λ), see [121, p. 51], [136], [58].

Example 2.6.6. Let O (Λ) ⊂ so(4) be a regular adjoint orbit where

Λ = diag(iλ1, iλ2,−iλ1,−iλ2),

with λ1 > λ2 > 0. In this case we have dimR(O (Λ)) = 4, and the Gelfand-Tsetlin paern is

λ ≥ µ (1)3 ≥ |λ2 |,µ (1)3 ≥ |λ

(1)2 |.

From these we obtain an integrable system dened by

µ (1)3 , λ(1)2 : O (Λ) → R,

in an open dense subset of O (Λ) CP1 × CP1.

Example 2.6.7. Let O (Λ) ⊂ so(8) be an adjoint orbit such that

Λ = diag(iλ1, iλ2, iλ3, iλ4,−iλ1,−iλ2,−iλ3,−iλ4),

with λ1 = λ2 = λ3 = λ4 > 0. Now we have dimR(O (Λ)) = 12, and the inequalities associatedto the Gelfand-Tsetlin paern can be described as follows

λ1 = µ(1)7 = λ2 = µ

(2)7 = λ3 = µ

(3)7 = |λ4 |,

µ (1)7 = λ(1)6 = µ

(2)7 = λ

(2)6 = µ

(3)7 ≥ |λ

(3)6 |,

λ(1)6 = µ(1)5 = λ

(2)6 ≥ µ

(2)5 ≥ |λ

(3)6 |,

µ (1)5 ≥ λ(1)4 ≥ µ

(2)5 ≥ |λ

(2)4 |,

λ(1)4 ≥ µ(1)3 ≥ |λ

(2)4 |,

µ (1)3 ≥ |λ(1)2 |.

47

From these inequalities we obtain a set of functions composed by

λ(3)6 , λ(1)4 , λ

(2)4 , λ

(1)2 : O (Λ) → R, µ (2)5 , µ

(1)3 : O (Λ) → R,

which denes an integrable system in an open dense subset in the manifold

O (Λ) SO(8)/SO(6) × SO(2).

It is worth to notice that if instead of the chain of closed connected subgroups on which appeartwo dierent kinds of Lie groups, we choose

SO(2n + 1) ⊃ SO(2(n − 1) + 1) ⊃ SO(2(n − 2) + 1) ⊃ . . . ⊃ SO(3),

in the end of the process we will not get enough collective hamiltonians in order to obtain anintegrable system. In fact, we will obtain in this case functions

λ(l )2k+1 = Φ∗2k+1(Λl ),

with 1 ≤ l ≤ k and k = 1, . . . ,n − 1.

e same is true for the even special orthogonal Lie group, if we take

SO(2n) ⊃ SO(2(n − 1)) ⊃ SO(2(n − 2)) ⊃ . . . ⊃ SO(2),

we will not get enough functions in order to obtain an integrable system, we have the functions

λ(l )2k = Φ∗2k (Λl ),

with 1 ≤ l ≤ k and k = 1, . . . ,n − 1.

As we have mentioned at the beginning, the construction of integrable systems for coadjointorbits of the special orthogonal Lie group is slightly dierent from the unitary case. As wehave seen we have two possibilities of choice for the chain of subgroups on which we can ap-ply imm’s trick in order to obtain a big set of quantities in involution. However, the chainon which we have integrability is not of the same nature as in the unitary case, namely, wedeal with two dierent kinds of classical Lie algebras. In appendix A we provide a completediscussion about the constraints which we have in order to obtain integrable systems denedby collective Hamiltonians in a more general seing.

48

Chapter 3Lax formalism and Gelfand-Tsetlinintegrable systems

e main purpose of this chapter is to establish a relation between the construction of Gelfand-Tsetlin integrable systems in coadjoint orbits of classical compact Lie groups with the Laxformalism. e goal of this chapter is to show that all the quantities in involution whichcompose the Gelfand-Tsetlin integrable systems can be recovered by means of a Lax equation

dL

dt+[L, P]= 0,

for a suitable choice of Lax pair L, P : O (Λ) → gl (r ,R). erefore, the main result which wewill provide in this chapter is the following theorem

eorem1. Let (O (Λ),ωO (Λ),G,Φ) be a HamiltonianG-space dened by an adjoint orbitO (Λ) =Ad(G )Λ, where G = U(N ) or SO(N ). en there exists a Lax pair L, P : O (Λ) → gl (r ,R) satisfy-ing

dL

dt+[L, P]= 0,


Although the integrability condition ensures the existence of a Lax pair for integrable systems,it is not clear what would be a suitable choice for such a pair, since we do not have unique-ness. However, as we will see the above result provides a canonical way to assign a Lax pair toGelfand-Tsetlin integrable systems. e ideas involved in our construction are quite naturalowing to the underlying matrix-nature which we have in the context of coadjoint orbits ofclassical compact Lie groups. Furthermore, as we will see all information about the Gelfand-Tsetlin paern are encoded in the set of spectral invariants of the matrix L which we willintroduce. Besides, we hope that the content which we will develop may help to establish newconnections between Gelfand-Tsetlin integrable systems and associated topics like quantumgroups, Yang-Baxter equation [12, p. 13-16] and geometric quantization.

e background required to a complete understanding of the content which we will developin this chapter are the concepts of collective Hamiltonians [77] and some ideas about imm’s

49

trick [158] which we have described in the previous sections.

It is worth to point out that, unless otherwise stated, throughout this chapter we will supposeG as being a connected, compact and semisimple Lie group.

3.1 Lax equation and collective Hamiltonians

Let us start by describing the relation between collective Hamiltonians and the Lax equation.As we have seen in Subsection 2.6 the functions which compose Gelfand-Tsetlin systems aregiven by collective Hamiltonians, i.e. if we consider a Hamiltonian G-space (M,ω,G,Φ), wecan take F ∈ C∞(g∗) and consider the smooth function given by

Φ∗(F ) = F Φ : M → R.

e Hamiltonian vector eld associated to a function of such a kind has the following expres-sion

XΦ∗ (F ) (p) = δτ (∇F (Φ(p))p ,

for every p ∈ M , see for instance [77, p. 241]. Here as before δτ denotes the innitesimalaction of G and ∇F (Φ(p)) ∈ g is obtained by the pairing (dF )Φ(p) = 〈· ,∇F (Φ(p))〉, see Section2.4 on page 31.

Consider the HamiltonianG-space (O (λ),ωO (λ),G,Φ) dened by a codjoint orbit of a compactLie group, if we take a collective Hamiltonian Φ∗(F ) ∈ C∞(O (λ)), we have

XΦ∗ (F ) (ξ ) = ad∗(∇F (Φ(ξ )))ξ ,

for every ξ ∈ O (λ). Since Φ in this case is just the inclusion map, we have

φt (ξ ) = Ad∗(exp(t∇F (ξ )))ξ ,

for every ξ ∈ O (λ). It follows that the dynamics dened by Φ∗(F ) ∈ C∞(O (λ)) through thepoint ξ ∈ O (λ) can be understood by means of the equation which denes the le invariantvector eld associated to ∇F (ξ ) ∈ g. We point out that the above curve is not in general theHamiltonian ow of an arbitrary collective Hamiltonain Φ∗(F ), see the discussion in 2.4.1 onpage 33.

By means of the Ad-invariant isomorphism g∗ g and the identication O (λ) O (Λ), weobtain from the ordinary dierential equation associated to XΦ∗ (F ) the following expression

d

dtφt (Z ) = XΦ∗ (F ) (φt (Z )) = ad(∇F (Φ(φt (Z ))))φt (Z )

for every initial condition Z ∈ O (Λ). Since the moment map Φ : O (Λ) → g is just the inclusionmap, we have the following equation for every Z ∈ O (Λ)

50

d

dtΦ(φt (Z )) =

[∇F (Φ(φt (Z )),Φ(φt (Z ))

].

Notice that if we denote by X = ∇F (Φ(φt (Z )) and Y = φt (Z ), we have

d

dtΦ(φt (Z )) = (DΦ)Y (ad(X )Y ) = ad(X )Φ(Y ) =

[X ,Φ(Y )

],

from these we obtain the following proposition

Proposition 3.1.1. Given a HamiltonianG-space (O (Λ),ωO (Λ),G,Φ), then the dynamic associ-ated to a collective Hamiltonian is completely determined by a zero curvature equation

dL

dt+[L, P]= 0,

where L, P : O (Λ) → g.

Proof. Let Φ∗(F ) ∈ C∞(O (Λ)) be a collective Hamiltonian associated to some F ∈ C∞(g).From the Hamiltonian ow of XΦ∗ (Φ) ∈ Γ(TO (Λ)) we have the following ordinary dierentialequation (ODE)

d

dtφt (Z ) = XΦ∗ (F ) (φt (Z )) = ad(∇F (Φ(φt (Z ))))φt (Z ),

for every Z ∈ O (Λ). We dene the following pair of Lie algebra valued functions

L : Z ∈ O (Λ) 7→ Φ(Z ) ∈ g and P : Z ∈ O (Λ) 7→ ∇F ((Φ(Z ))) ∈ g.

From the previous comments we obtain

d

dtL(φt (Z )) =

d

dtΦ(φt (Z )) =

[∇F (Φ(φt (Z )),Φ(φt (Z ))

],

but from our denition of L and P the last expression is exactly

d

dtL(φt (Z )) =

[P (φt (Z )),L(φt (Z ))

],

for every Z ∈ O (Λ). Now we notice that since the adjoint action of G on its Lie algebra is aproper action we have thatO (Λ) ⊂ g is an embedded compact submanifold of g, it follows thatC∞(O (Λ)) = Φ∗(C∞(g)), see for more details [165, p. 29] and [141, p. 181-283].

Givenψ ∈ C∞(O (Λ)) we have I ∈ C∞(g) such thatψ = Φ∗(I ), therefore the equation of motion

d

dtψ (φt (Z )) =

Φ∗(F ),ψ

O (Λ)

(φt (Z )),

can be rewrien as follows

d

dtI (Φ(φt (Z ))) =

F , Ig(Φ(φt (Z ))).

Notice that in the last equation we have used that Φ : (O (Λ),ωO (Λ) ) → g is equivariant, see[141, p. 497] and [145, p. 330]. Since the le side of the last equation is given by

51

d

dtI (Φ(φt (Z ))) =

d

dtI (L(φt (Z ))) = (dI )L(φt (Z )) (

d

dtL(φt (Z ))),

it follows that the dynamics of the Hamiltonian system (O (Λ),ωO (Λ),Φ∗(F )) is completely

described by the zero curvature equation (Lax equation)

d

dtL(φt (Z )) +

[L(φt (Z )), P (φt (Z ))

]= 0,

for every Z ∈ O (Λ), where L = Φ and P = ∇F (Φ).

Once we have described the dynamics associated to any collective Hamiltonian in terms of azero curvature equation, our next task will be to understand how we can use the zero curvatureequation described above to recover the quantities in involution obtained by means of imm’strick.

3.2 imm’s trick and spectral invariants

In this section we will establish a relation between imm’s trick and the zero curvature equa-tion described in the previous section.

Let (O (Λ),ωO (Λ),G,Φ) be a HamiltonianG-space as before and letK ⊂ G be a closed connectedsubgroup of G. By restriction we can consider the Hamiltonian K-space (O (Λ),ωO (Λ),K ,ΦK ),where

ΦK : O (Λ) → k, with ΦK = πK Φ,

here we denote by πK : g → k the projection map. If we take F ∈ C∞(k) we can consider thecollective Hamiltonian Φ∗K (F ) ∈ C

∞(O (Λ)), here we notice that

ΦK (F ) = Φ∗(F πK ),

from this we denote F = F πK and consider Φ∗K (F ) = Φ∗(F ) also as a collective Hamiltonianassociated to the Hamiltonian G-space (O (Λ),ωO (Λ),G,Φ). From the last section we have thedynamics associated to Φ∗(F ) completely described by the Lax equation

d

dtL(φt (Z )) +

[L(φt (Z )), P (φt (Z ))

]= 0,

for every Z ∈ O (Λ), where L = Φ and P = ∇F (Φ).

Lemma 3.2.1. Given F = F πK ∈ C∞(g), such that F ∈ C∞(k) and πK : g → k as before, then

we have

∇F (Z ) = ∇F (πK (Z )) ∈ k,

for every Z ∈ g.

Proof. We rst choose a basis Xi for g. By denition of the ∇F we have

52

∇F (Z ) =∑i

〈(dF )Z ,Xi〉Xi ,

since (dF )Z = (dF )πK (Z ) (DπK )Z and (DπK )Z = πK , for every Z ∈ g, if we choose a basisobtained from a completion of a basis for k ⊂ g, we have ∇F (Z ) = ∇F (πK (Z )).

From the above lemma we see that for the Lax pair L = Φ and P = ∇F (Φ) associated toF = F πK , we have

P = ∇F (Φ) = ∇F (ΦK ) and XΦ∗ (F ) = XΦ∗K (F ),

furthermore, we have the following equation

d

dtΦK (φt (Z )) =

[∇F (ΦK (φt (Z )),ΦK (φt (Z ))

],

where φt (Z ) is the Hamiltonian ow of XΦ∗ (F ) , in fact if we denote W = φt (Z ) ∈ O (Λ) andY = ∇F (ΦK (φt (Z ))) ∈ k, we have

d

dtΦK (φt (Z )) = (DΦK )W (ad(Y )W ) = ad (Y )ΦK (W ).

Notice that the equality in the right side in the above equation follows from the fact that ΦK

is equivariant and Y = ∇F (ΦK (φt (Z )) ∈ k.

Now if we take I ∈ C∞(k) and consider the collective Hamiltonian

ψ = Φ∗K (I ) ∈ C∞(O (Λ)),

from the equation of motion associated to the Hamiltonian system (O (Λ),ωO (Λ),Φ∗K (F )) we

have

d

dtψ (φt (Z )) =

Φ∗K (F ),Φ

∗K (I )

O (Λ)

(φt (Z )),

which can be rewrien as follows

d

dtI (ΦK (φt (Z ))) =

F , Ik(ΦK (φt (Z ))).

e le side on the above equation can be wrien also as

d

dtI (ΦK (φt (Z ))) = (dI )ΦK (φt (Z )) (

d

dtΦK (φt (Z ))),

from these we see that the equation of motion

d

dtψ (φt (Z )) =

Φ∗K (F ),Φ

∗K (I )

O (Λ)

(φt (Z )),

53

is completely described by the zero curvature equation

d

dtΦK (φt (Z )) +

[ΦK (φt (Z )),∇F (ΦK (φt (Z ))

]= 0. (3.2.1)

erefore, if we take LK = ΦK and Pk = ∇F (ΦK ), we obtain the following proposition

Proposition 3.2.2. Let (O (Λ),ωO (Λ),G,Φ) be a HamiltonianG-space and let K ⊂ G be a closedand connected Lie subgroup. Given F ∈ C∞(k), the equation of motion of the Hamiltonian system(O (Λ),ωO (Λ),Φ

∗K (F )) is completely described by the Lax equation

dL

dt+[L, P]= 0,

where L = Φ and P = ∇F (ΦK ). Moreover, ifψ ∈ Φ∗K (C∞(k)) we have the equivalence

dψ

dt=Φ∗K (F ),ψ

O (Λ)

⇐⇒dLKdt+[LK , PK

]= 0,

where LK = ΦK and PK = ∇F (ΦK ).

Proof. e rst equation follows directly from Proposition 3.1.1 and from the Lemma 3.2.1.e second statement follows from the fact that if ψ ∈ Φ∗K (C

∞(k)), we have ψ = Φ∗K (I ) forsome I ∈ C∞(k), hence we can write

d

dtψ (φt (Z )) =

d

dtI (ΦK (φt (Z ))) = (dI )ΦK (φt (Z )) (

d

dtLK (φt (Z ))),

and dLKdt = [PK ,LK ].

Now we illustrate how we can employ the results developed so far to recover some familiarfacts which are used in the construction of the Gelfand-Tsetlin integrable systems

Example 3.2.1. Consider the Hamiltonian U(4)-space (O (Λ),ωO (Λ),U(4),Φ) and let U(3) ⊂U(4) be the closed connected subgroup dened by the block diagonal matricesU 0

0 1

, such that UU ∗ = 13,

whereU ∈ GL(3,C). By restriction we have a Hamiltonian U(3)-space (O (Λ),ωO (Λ),U(3),ΦU(3) ),in order to simplify the notation in what follows we will denote ΦU(3) = Φ3.

If we consider the Hamiltonian system (O (Λ),ωO (Λ),Φ∗3 (F )), for some F ∈ C∞(u(3)), the pre-

vious proposition provides an alternative way to study the dynamic of the Hamiltonian vectoreld XΦ∗3 (F ) in terms of the equation

dL

dt+[L, P]= 0,

for L = Φ and P = ∇F (Φ3). As we have seen we can take

54

L(Z ) = Φ(Z ) ∈ u(4) and P (Z ) = ∇F (Φ3(Z )) ∈ u(3) ⊂ u(4)

for all Z ∈ O (Λ). Now we consider the collective Hamiltonian ψ = Φ∗3 (I ) ∈ C∞(O (Λ)), where

I ∈ C∞(u(3)) is given by

I (X ) = det(X ), for every X ∈ u(3),

here we used the identication u(3) iu(3) in order to get a real valued function. Notice thatwe can also writeψ = det(Φ3). Now we look at the equation of motion

dψ

dt=Φ∗3 (F ),ψ

O (Λ)

⇐⇒dL3dt+[L3, P3

]= 0,

where L3 = Φ3 and P3 = ∇F (Φ3). We observe that in this case we have

L3(Z ) = Φ3(Z ) ∈ u(3) and P3(Z ) = ∇F (Φ3(Z )) ∈ u(3).

From the previous comments we obtain

d

dtψ (φt (Z )) =

d

dtI (L3(φt (Z ))) = (dI )L3 (φt (Z )) (

d

dtL3(φt (Z ))),

from the Jacobi’s formula we have

d

dtdet(A(t )) = Tr

(adj(A(t ))dA

dt

),

for every matrix-curve A(t ), it follows that

d

dtI (L3(φt (Z ))) =

d

dtdet(L3(φt (Z ))) = Tr

(adj(L3(φt (Z ))))

d

dtL3(φt (Z ))

),

from this we can use the equation ddtL3 = [P3,L3] on the last expression to obtain

dψ

dt= Tr

(adj(L3)

dL3dt

)= Tr

(adj(L3)

[P3,L3

]).

Since L3adj(L3) = det(L3)1 it follows that

dψ

dt= Tr

([L3, adj(L3)

]P3

)= 0 =⇒ d

dtψ =

Φ∗3 (F ),ψ

O (Λ)= 0,

it follows thatψ = det(Φ3) is a constant of motion of the Hamiltonian system (O (Λ),ωO (Λ),Φ∗3 (F ))

as expected.

It is worthwhile to point out that the last equation shows us how the pair L3, P3 : O (Λ) → u(3)can be employed to understand the equation of motion associated to collective Hamiltonians.

e conclusion of the above example is actually a more general fact. Under the hypothesis ofthe last proposition, forψ ∈ Φ∗K (C∞(k)) we associate the equations

55

dψ

dt=Φ∗K (F ),ψ

O (Λ)

⇐⇒dLKdt+[LK , PK

]= 0.

If we denote by ψ = Φ∗K (I ) a straightforward calculation shows that the above equations arerelated by

dψ

dt(t ) = (dI )Lk (t ) (ad(PK (t ))LK (t )),

it follows that if I ∈ C∞(k)Ad, we have

dψ

dt=Φ∗K (F ),ψ

O (Λ)= 0,

i.e. ψ is a constant of motion of the Hamiltonian system (O (Λ),ωO (Λ),Φ∗K (F )). e above ideas

can be seen as an alternative way to recover imm’s trick which is used in the constructionof quantities in involution through of collective Hamiltonians.

Now we denote by

IΦ∗K (F )=ψ ∈ C∞(O (Λ))

∣∣∣ Φ∗K (F ),ψO (Λ)= 0

,

the subspace of functions which commute with Φ∗K (F ) ∈ C∞(O (Λ)), we have the followingcharacterization for this subspace

Proposition 3.2.3. Let (O (Λ),ωO (Λ),Φ∗K (F )) be a Hamiltonian system as before, then the sub-

space IΦ∗K (F )⊂ C∞(O (Λ)) is given by the pullback by the moment map Φ : O (Λ) → g of the

following subspace

I(L,P ) =I ∈ C∞(g)

∣∣∣ ddt (I L) = 0

,

where P = ∇F , with F = F πK and L = Φ.

Proof. e proof goes as follows, if we takeψ = Φ∗(I ) ∈ IΦ∗K (F )we obtain

d

dtψ (φt (Z )) =

d

dtI (L(φt (Z ))),

it follows that IΦ∗K (F )= Φ∗(I(L,P ) ).

As we have seen so far all about the study of quantities in involution obtained through ofimm’s trick and collective Hamiltonians can be recovered by means of the Lax pair whichwe have introduced in this chapter. In fact the above proposition shows that for I ∈ C∞(k)Ad,we have

ψ = Φ∗K (I ) ∈ IΦ∗K (F )= Φ∗(I(L,P ) ),

from the last results and comments we have the following theorem

56

eorem 3.2.4. Let (O (Λ),ωO (Λ),G,Φ) be a HamiltonianG-space. Given a chain of closed con-nected subgroups

G = K0 ⊃ K1 ⊃ . . . ⊃ Ks ,

if we denote by Φl the moment map associated to the Hamiltonian action of each subgroup Kl byrestriction, then for every F ∈ C∞(ks ) we can associate to the Hamiltonian system (O (Λ),ωO (Λ),Φ

∗s (F ))

the following set of Lax equations

dLkdt+[Lk , Pk

]= 0,

where Lk = Φk and Pk = ∇(F πks ) (Φk ), for k = 0, 1, . . . , s and πk

s : tk → ks the projection map.

Proof. e result follows from the fact that Φs = πks Φk , for all k = 0, 1, . . . , s − 1, therefore

we have

d

dtLk (φt (Z )) =

d

dtΦk (φt (Z )) = (DΦk )φt (Z ) (

d

dtφt (Z )).

Now we notice that F Φs = (F πks ) Φk , moreover a straightforward calculation shows us

that

∇(F πks ) (Z ) = ∇F (π

ks (Z )) ∈ ks ⊂ kk ,

for every Z ∈ kk . From the above comments the equation

d

dtφt (Z ) = ad(∇F (Φs (φt (Z ))))φt (Z ),

becomesd

dtφt (Z ) = ad(∇(F πk

s ) (Φk (φt (Z ))))φt (Z ).

Since ∇(F πks ) (Φk (φt (Z ))) ∈ kk and Φk is equivariant, we have

d

dtΦk (φt (Z )) = ad(∇(F πk

s ) (Φk (φt (Z ))))Φk (φt (Z )),

but the above last expression is exactly

d

dtLk (φt (Z )) = ad(Pk (φt (Z )))Lk (φt (Z )),

where Lk = Φk and Pk = ∇(F πks ) (Φk ), it follows that

d

dtLk (φt (Z )) +

[Lk (φt (Z )), Pk (φt (Z ))

]= 0,

for every Z ∈ O (Λ) and for every k = 0, 1, . . . , s .

Now we observe the following fact, under the hypotheses of the last theorem, given I ∈C∞(kk )

Ad if we considerψ = Φ∗k (I ), we have

57

d

dtψ (φt (Z )) =

d

dtI (Lk (φt (Z ))) = 0,

since we have (dI )Lk (t ) (ad(Pk (t ))Lk (t )) = 0. We obtain the following result

Corollary 3.2.5. Under the hypotheses of the previous theorem, if we suppose that rank(Kl ) = rl ,then the Hamiltonian system (O (Λ),ωO (Λ),Φ

∗s (F )) admits at least N = r1 + . . . + rs functions in

involution.

Proof. It follows from the following fact, if we x rk generators forC∞(kk )Ad, from the previouscomments we have

Φ∗k(C∞(kk )

Ad) ⊂ IΦ∗s (F ) ,

for every k = 1, . . . s .

We notice that from the above corollary if we take ψ1 = Φ∗k (I1), with I1 ⊂ C∞(kk )Ad, and

ψ2 = Φ∗l (I2), with I2 ⊂ C∞(kl )Ad, we have

Φ∗s (F ),ψ1ψ2O (Λ)=Φ∗s (F ),ψ1

O (Λ)

ψ2 +Φ∗s (F ),ψ2

O (Λ)

ψ1 = 0,

it follows thatψ1ψ2 ∈ IΦ∗s (F ) , furthermore we can suppose k ≥ l , thus we obtain

ψ1,ψ2

O (Λ)= Φ∗k

I1, I2 π

kl

kk= 0,

since I1 is a Casimir function. It allows us to dene the following Poisson subalgebra of(C∞(O (Λ)), ·, ·O (Λ) )

Denition 3.2.6. Under the hypotheses of the previous theorem, we dene the Gelfand-Tsetlincommutative Poisson subalgebra ΓΦ∗s (F ) ⊂ C∞(O (Λ)) as being

ΓΦ∗s (F ) :=⟨Φ∗k(S (kk )

Ad) ∣∣∣ k = 1, . . . , s

⟩,

where S (kk )Ad denotes the subalgebra of Ad-invariant polynomial functions. In particular we haveΓΦ∗s (F ) ⊂ IΦ∗s (F ) .

Our motivation for the above denition of Gelfand-Tsetlin Poisson subalgebra is the conceptof the Gelfand-Tsetlin subalgebras of the universal enveloping algebras, which are examplesof Harish-Chandra subalgebras [44, p. 87], see also [56].

In order to illustrate the content of the last theorem we consider the following example

Example 3.2.2. Consider the Hamiltonian U(4)-space (O (Λ),ωO (Λ),U(4),Φ) and take the fol-lowing chain of closed connected subgroups

U(4) ⊃ U(3) ⊃ U(2) ⊃ U(1),

for U(k ) ⊂ U (4) given by the group of matrices of the form

58U 0

0 1

, such that UU ∗ = 1,

where U ∈ GL(k,C), k = 1, 2, 3. Associated to this chain we can consider the HamiltonianU(k )-spaces (O (Λ),ωO (Λ),U(k ),Φk ), if we take F ∈ C∞(u(1)) and consider the Hamiltoniansystem (O (Λ),ωO (Λ),Φ

∗1 (F )), from the previous theorem we have

Φ∗k(C∞(u(k ))Ad) ⊂ IΦ∗1 (F ) ,

for k = 1, 2, 3. From these we look at the generators of the S (u(k ))Ad in order to understandΓΦ∗(F ) .

For S (u(3))Ad we have the following generators

• I (3)1 : X → Tr(X ), for every X ∈ u(3),

• I (3)2 : X → −12

[Tr(X )2 − Tr(X 2)

], for every X ∈ u(3),

• I (3)3 : X → det(X ), for every X ∈ u(3).

For S (u(2))Ad we have the following generators

• I (2)1 : X → Tr(X ), for every X ∈ u(2),

• I (2)2 : X → det(X ), for every X ∈ u(2).

For S (u(1))Ad, we observe that

C∞(u(1))Ad = C∞(u(1)).

Since u(1) = iR, thus we can take the function I (1)1 : X → iX , for every X ∈ u(1). As expectedthe functions

Φ∗k (I(k )l ), for 1 ≤ l ≤ k and 1 ≤ k ≤ 3,

provide a complete integrable system in O (Λ) ⊂ u(4), it is in fact the Gelfand-Tsetlin system,see for instance [76]. e important fact to notice here is that the above set of functions gen-erates ΓΦ∗ (F ) .

As we have seen in eorem 3.2.4, associated to the Hamiltonian system (O (Λ),ωO (Λ),Φ∗1 (F ))

we have a set of equations

dLkdt+[Lk , Pk] = 0,

where Lk = Φk and Pk = ∇(F πk1 ) (Φk ), for k = 1, 2, 3. Now if we take the matrices

59

L = diagLk

∣∣∣ k = 1, 2, 3

, and P = diagPk

∣∣∣ k = 1, 2, 3

,

we obtain from this a pair of matrix valued functions

L, P : O (Λ) → u(6),

the interesting point is that a straightforward calculation shows us that

d

dtL(φt (Z )) +

[L(φt (Z )), P (φt (Z ))

]= 0,

for every Z ∈ O (Λ). Furthermore, we can recover the Gelfand-Tsetlin system by means of justone equation of the form

det(L − t16) = det(L1 − t ) det(L2 − t12) det(L3 − t13), (3.2.2)in fact we have

• det(L1 − t ) = L1 − t ,

• det(L2 − t12) = t2 − Tr(L2)t + det(L2),

• det(L3 − t13) = −t3 + Tr(L3)t

2 −12

[Tr(L3)

2 − Tr(L23)]t + det(L3).

From these we obtain

• det(L1 − t ) = −iΦ∗1 (I

(1)1 ) − t

• det(L2 − t12) = t2 − Φ∗2 (I(2)1 )t + Φ∗2 (I

(2)2 )

• det(L3 − t13) = −t3 + Φ∗3 (I

(3)1 )t2 + Φ∗3 (I

(3)2 )t + Φ∗3 (I

(3)3 ).

us the Equation 3.2.2 dened by L encodes all the quantities in involution which dene theGelfand-Tsetlin integrable system.

e last comments in the above example are more general. In fact, if we consider a Hamiltoniansystem (M,ω,H ) which admits a Lax pair (L, P ) : M → gl (r ,R), then the coecients of thecharacteristic polynomial of L, namelly

det(L − t1r ) = f0(L)tr + f1(L)t

r−1 + . . . + fr−1(L)t + fr (L),

provide a set of quantities in involution for the Hamiltonian system (M,ω,H ). In order to seethis, if we take Fl = L∗( fl ) it follows that

d

dtFl (t ) =

H , Fl

M(t ),

but

d

dtFl (t ) = (d fl )L(t ) (

d

dtL(t )) = (d fl )L(t ) (ad(P (t ))L(t )) = 0,

60

since fl : gl (r ,R) → R is Ad-invariant. It is worth to point out that the coecients of thecharacteristic polynomial of L can be expressed in terms of functions of the form Tr(Lk ), k =0, 1, 2 . . .. erefore, if L = UΛU −1, with Λ ∈ gl (r ,R) being a diagonal matrix of the form

Λ = diag(Λ1, . . . ,Λr ),

it follows that the functions dened by the eigenvalues of L are quantities in involution forthe Hamiltonian system (M,ω,H ). e constants of motion obtained from the characteristicpolynomial of L are called spectral invariants. We will denote the set of spectral invariantsassociated to a Lax pair by σ (L) ⊂ C∞(M ).

Once we have a Lax pair (L, P ) for a Hamiltonian system (M,ω,H ), we can take the solutionfor the zero curvature equation dL

dt + [L, P] = 0 as being

L(t ) = д(t )L(0)д(t )−1, where dд

dt= P (t )д(t ),

with д(0) = 1, see [141, p. 578-579] for more details. us if L : M → gl (r ,R) is diagonalizablethe spectrum of L remains invariant by the Hamiltonian ow of H ∈ C∞(M ).

Now we come back to the context of Hamiltonian G-spaces (O (Λ),ωO (Λ),G,Φ). Since we areconcerned with the classical Lie groups, from now we consider

G = U(N ), SO(N ) or Sp(N ).

We have a natural chain of closed connected subgroups

U(N ) ⊃ U(N − 1) ⊃ . . . ⊃ U(1),SO(N ) ⊃ SO(N − 1) ⊃ . . . ⊃ SO(2),Sp(N ) ⊃ Sp(N − 1) ⊃ . . . ⊃ Sp(1),

given by block diagonal matrices. Applying the previous results we have the following propo-sition

Proposition 3.2.7. Let (O (Λ),ωO (Λ),G,Φ) a Hamiltonian G-space. If we take a chain of closedconnected subgroups given by diagonal blocks

G = K0 ⊃ K1 ⊃ . . . ⊃ Ks ,

where rank(Kl ) = rl for each elementKl . en for F ∈ C∞(ks ), the Hamiltonian system (O (Λ),ωO (Λ),Φ∗s (F ))

admits a pair of matrix valued functions (L, P ) : O (Λ) → gl (r ,R) which satises the zero curva-ture equation

d

dtL(φt (Z )) +

[L(φt (Z )), P (φt (Z ))

]= 0,

for all Z ∈ O (Λ), where φt (Z ) denotes the Hamiltonian ow of Φ∗s (F ).

61

Proof. First we consider the set of Lax equations associated to the chain of closed connectedsubgroups (eorem 3.2.4)

dLkdt+[Lk , Pk

]= 0,

where Lk = Φk and Pk = ∇(F πks ) (Φk ), for k = 1, . . . , s . Now we dene

L = diagLk

∣∣∣ k = 1, . . . , s

, and P = diagPk

∣∣∣ k = 1, . . . , s

,

from this we have (L, P ) : O (Λ) → gl (r ,R), such that r ≥ r1 + . . .+ rs , notice that kk is a matrixLie algebra for each k = 1, . . . , s . A straightforward calculation shows us that

d

dtL(φt (Z )) +

[L(φt (Z )), P (φt (Z ))

]= 0,

for all Z ∈ O (Λ), from these we have the desired result.

Remark 3.2.1. In the proof of the last proposition we can also denote the block diagonal matricesL and P by

L =

s∑k=1

Lk and P =

s∑k=1

Pk ,

furthermore we can choose r ∈ N, such that

s⊕k=1kk ⊂ gl (r ,R).

We have the following direct consequences from the previous result

Corollary 3.2.8. Under the hypotheses of the above proposition the spectral invariants of L denea set of r1 + . . . + rs quantities in involution for the Hamiltonian system (O (Λ),ω,Φ∗s (F )).

Corollary 3.2.9. Under the hypotheses of the previous theorem, the Gelfand-Tsetlin Poisson sub-algebra ΓΦ∗s (F ) is generated by the spectral invariants of the Lax matrix L, i.e.

ΓΦ∗s (F ) =⟨σ (L)

⟩.

Inspired by the last corollary we can make the following denition

Denition 3.2.10. Under the hypotheses of the above corollary, we dene σ (L) as being theGelfand-Tsetlin spectrum of O (Λ).

By means of the content which we have established so far we have the following theorem

eorem 3.2.11. Let (O (Λ),ωO (Λ),G,Φ) be a Hamiltonian G-space dened by an adjoint orbitO (Λ) = Ad(G )Λ, whereG = U(N ) or SO(N ). en there exists a Lax pair L, P : O (Λ) → gl (r ,R)satisfying

62

dL

dt+[L, P]= 0,


Proof. e proof follows from the following facts, we rst take the chain of closed connectedsubgroups given by block diagonal matrices

U(N ) ⊃ U(N − 1) ⊃ . . . ⊃ U(1),SO(N ) ⊃ SO(N − 1) ⊃ . . . ⊃ SO(2),

then we apply the previous proposition. Now given Z ∈ O (Λ), since Lk (Z ) = Φk (Z ) belongsto u(k ) or so(k ), with 1 ≤ k < N in the rst case and 2 ≤ k < N in the second case, wecan diagonalize the matrix L, which is dened by diagonal blocks Lk , from this the spectralinvariants of L denes the Gelfand-Tsetlin integrable system, i.e. all the information about theGelfand-Tsetlin integrable system are codied in σ (L).

e denition of the Gelfand-Tsetlin spectrum σ (L) (3.2.10) which we set in this work gathertogether all the information about the Gelfand-Tsetlin integrable in a single object. In thenext section we will examine the geometrical aspects of the ideas developed in this section bymeans of Liouville’s theorem.

3.3 Liouville’s theorem and Lax formalism for Gelfand-Tsetlin integrable systems

In this last section of the chapter we will perform some calculations by using the content de-veloped in the previous section. e idea is to describe by means of concrete examples theDarboux’s coordinates provided by the Gelfand-Tsetlin integrable systems through Liouville’stheorem. For a complete proof of Liouville‘s theorem see Appendix A.5.

Consider the compact Lie group G = U(N ), by xing an element

Λ = diag(iλ1, . . . , iλN ),

with λ1 ≥ · · · ≥ λN we take its adjoint orbitO (Λ) = Ad(U(N ))Λ ⊂ u(N ). As stated in eorem3.2.11 the Gelfand-Tsetlin integrable system on (O (Λ),ωO (Λ) ) is completely described by theLax pair L, P : O (Λ) → u(N (N−1)

2 ), where

dL

dt+[L, P]= 0,

for L = diagLk

∣∣∣ k = 1, . . . ,N − 1

and P = diagPk

∣∣∣ k = 1, . . . ,N − 1

, such that

Lk = Φk and Pk = ∇(F πk1 ) (Φk ),

63

where k = 1, . . . ,N − 1, πk1 : u(k ) → u(1) as being the projection map and F ∈ C∞(u(1)), see

eorem 3.2.4 for more details. By looking at the spectral invariants of L, a straightforwardcalculation shows us that

det(L − t1N (N−1)2

) =

N−1∏k=1

det(Lk − t1k ),

from this we have

det(Lk − t1k ) = I (k )0,k (Lk )tr + I (k )1 (Lk )t

r−1 + . . . + I (k )k−1(Lk )t + I(k )k (Lk ). (3.3.1)

ese two last equations provide a big set of quantities in involution dened by the functions

H (k )l = I (k )l (Lk ), 1 ≤ k ≤ N − 1, 1 ≤ l ≤ k ,

hence we have a Lagrangian foliation dened in open dense subset ofO (Λ) which is generatedby the vector elds

XH

(k )l= ad(∇I (k )l (Lk )), 1 ≤ k ≤ N − 1, 1 ≤ l ≤ k ,

see 2.4 to remember the properties of collective Hamiltonians and their Hamiltonian vectorelds. As we can see the above quantities in involution are exactly the elements of the wellknown Gelfand-Tsetlin integrable system introduced in [76].

We notice that since Lk = A(Lk )Λ(Lk )A(Lk )∗, with A(Lk ) ∈ U(k ) and

Λ(Lk ) = diag(iΛ1(Lk ), . . . , iΛk (Lk )),

the Equation 3.3.1 becomes

det(Lk − t1k ) =k∏j=1

(iΛj (Lk ) − t ) = (iΛ1(Lk ) − t ) · · · (iΛk (Lk ) − t ),

thus we have

det(L − t1N (N−1)2

) =

N−1∏k=1

(iΛ1(Lk ) − t ) · · · (iΛk (Lk ) − t ),

therefore if we denote by

λ(j )k = Λj (Lk ),

for 1 ≤ j ≤ k , 1 ≤ k ≤ N − 1, we have the following inequalities

λ(1)k (X ) ≥ λ(1)k−1(X ) ≥ λ(2)k (X ) ≥ . . . ≥ λ(k−1)k−1 (X ) ≥ λ(k )k (X )

64

for every X ∈ O (Λ), 1 ≤ k ≤ N − 1, these inequalities are exactly the Gelfand-Tsetlin paern,compare with the description which we have done in the Subsection 2.6.1.

Now we take a look in the content of Corollary 3.2.9. We have the Gelfand-Tsetlin spectrumof L : O (Λ) → u(N (N−1)

2 ) given by

σ (L) =H (k )l = I (k )l (Lk )

∣∣∣ 1 ≤ k ≤ N − 1, 1 ≤ l ≤ k

.

As we have seen we have a close relationship between the above set of functions and theGelfand-Tsetlin paern, in fact the functions which dene the spectrum of the maitrixL : O (Λ) →

u(N (N−1)2 ) are the actions coordinates for the integrable system dened by the elements of the

above set 1.

As we have mentioned before, inside of σ (L) we have a set of quantities in involution, namely

H (k1)l1, . . . ,H (kd )

ld,

here we suppose dimR(O (Λ)) = 2d , such that

H := (H (k1)l1, . . . ,H (kd )

ld) : (O (Λ),ωO (Λ) ) → Rd

denes an integrable system on (O (Λ),ωO (Λ) ). Now we consider the following result

eorem3.3.1 (Liouville). Let (M,ω,H ) be an integrable system with integrals of motionH1, . . . ,Hn,and let x ∈ M be a regular point of the map H = (H1, . . . ,Hn ). en there exists an open neigh-bourhoodW ⊂ M of x and smooth functions q1, . . . , qn onW complementing p1 = H1, . . . , pn =

Hn to Darboux coordinates. In these coordinates, the ow ϕXHit of the Hamiltonian vector eldXHi

is (locally) given by

ϕXHit (q, p) = (q1, . . . , qi + t , qi+1, . . . , qn, p1, . . . , pn ).

for every i = 1, . . . ,n.

If we denote by O (Λ)H the open dense subset of O (Λ) where the previous map H denes asubmersion we can apply Liouville’s theorem in order to get (canonical transformation) coor-dinates (q, p,W ), whereW ⊂ O (Λ)H denotes an open subset, such that

qi (q,p) = qi −H ∗(αi ) and pi (q,p) = pi , (3.3.2)

with∂qi = X

H(ki )li

= ad(∇I (ki )li(Lki )) and pi = H (ki )

li= I (ki )li

(Lki ), (3.3.3)

for i = 1, . . . ,d , and α1, . . . ,αd ∈ C∞(H (W )) determined for some suitable 1-form

1See [141, p. 589] for the denition of action coordinates and [76, p. 113, eorem 3.4] to see the importantproperty satised by the Hamiltonian vector eld of the functions dened by the eigenvalues of L.

65

∑i

αidxi ∈ Ω1(H (W )).

It is worth to observe that the above description is essentially the content of the proof ofLiouville’s theorem, see Appendix A.5. e important features of these last coordinates arethat

ωO (Λ) |W =

d∑i=1

dpi ∧ dqi ,

and if we denote by ϕ (li ,ki )t (q, p) the Hamiltonian ow of X

H(ki )li

through the point (q, p) ∈ W ,we have

ϕ (li ,ki )t (q, p) = (q1, . . . , qi + t , qi+1, . . . , qn, p1, . . . , pn ).

Now let us perform some calculations by using the above ideas on the basic example whichwe have examined in the previous sections.

Example 3.3.1. Consider the Hamiltonian U(4)-space (O (Λ),ωO (Λ),U(4),Φ) as in the Ex-ample 3.2.2. As we have seen, by taking F ∈ C∞(u(1)) we obtain a Hamiltonian system(O (Λ),ωO (Λ),Φ

∗1 (F )) and an associated set of equations

dLkdt+[Lk , Pk

]= 0,

where Lk = Φk and Pk = ∇(F πk1 ) (Φk ), for k = 1, 2, 3. ese equations allow us to dene a

Lax pair (L, P ) : O (Λ) → u(6) by

L = diagLk

∣∣∣ k = 1, 2, 3

, and P = diagPk

∣∣∣ k = 1, 2, 3

.

us we have an associated zero curvature equation ddtL + [L, P] = 0 which encodes the well

known facts related to Gelfand-Tsetlin integrable system. In fact as we have described in thissection we can recover the Gelfand-Tsetlin system by means of the characteristic polynomial

det(L − t16) = det(L1 − t ) det(L2 − t12) det(L3 − t13).

e above equation provides the following description for the quantities in involution involvedin the construction of the Gelfand-Tsetlin integrable system

• det(L1 − t ) = L1 − t

• det(L2 − t12) = t2 − Tr(L2)t + det(L2)

• det(L3 − t13) = −t3 + Tr(L3)t

2 −12

[Tr(L3)

2 − Tr(L23)]t + det(L3).

erefore, the relation between the eigenvalues of L and the quantities in involution given bythe coecients of the characteristic polynomial can be described from the following expres-sions

66

• L1 = iΛ1(L1);

• Tr(L2) = i (Λ1(L2) + Λ2(L2)), det(L2) = −Λ1(L2)Λ2(L2);

• Tr(L3) = i (Λ1(L3) + Λ2(L3) + Λ3(L3)), Tr(L3)2 = −(Λ1(L3) + Λ2(L3) + Λ3(L3))

2;

• Tr(L23) = −(Λ1(L3)

2 + Λ2(L3)2 + Λ3(L3)

2), det(L3) = −Λ1(L3)Λ2(L3)Λ3(L3).

Now as we know the functions on the right side of the above equations satisfy the followingrelations (Gelfand-Tsetlin paern)

λ1 ≥ Λ1(L3) ≥ λ2 ≥ Λ2(L3) ≥ λ3 ≥ Λ3(L3) ≥ λ4,Λ1(L3) ≥ Λ1(L2) ≥ Λ2(L3) ≥ Λ2(L2) ≥ Λ3(L3),

Λ1(L2) ≥ Λ1(L1) ≥ Λ2(L2),

Here, as before, we suppose that Λ = diag(iλ1, iλ2, iλ3, iλ4), with λ1 ≥ λ2 ≥ λ3 ≥ λ4. If weconsider for instance λ1 > λ2 > λ3 > λ4, namely if we take O (Λ) regular, the commentsat the beginning of this section tell us that the function H : O (Λ) → R6 given by H =

(H (1)1 ,H

(2)1 ,H

(2)2 ,H

(3)1 ,H

(3)2 ,H

(3)3 ), such that

H =(L1,−Tr(L2), det(L2),Tr(L3),−

12

[Tr(L3)

2 − Tr(L23)], det(L3)

),

denes the Gelfand-Tsetlin integrable system. We can apply Louville’s theorem as we haveexplained previously in order to get coordinates (q, p,W ), where W ⊂ O (Λ)H denotes anopen subset, such that

ql (q,p) = ql −H ∗(αl ) and pl (q,p) = pl ,

where

∂ql = XH

(k )l= ad(∇I (k )l (Lk )) and pl = H (k )

l = I (k )l (Lk ),

for 1 ≤ l ≤ n and 1 ≤ k ≤ 3, here we are using the following convention

det(Lk − t1k ) = I (k )0,k (Lk )tr + I (k )1 (Lk )t

r−1 + . . . + I (k )k−1(Lk )t + I(k )k (Lk )

for k = 1, 2, 3. is example is a meaningful exercise which shows us how the ideas involvedin the construction of the Gelfand-Tsetlin system [76] and our approach via Lax matrix ttogether in the same framework.

67

Chapter 4Kahler structure on coadjoint orbits

In the rst part of this work we have studied some questions related to the construction ofHamiltonian integrable systems in coadjoint orbits. As we have seen the main features ofmanifolds dened by coadjoint orbits are their natural connections with Lie theory. eseconnections allow us to approach general problems concerned with symplectic geometry withtools which we have available in the environment of Lie groups and Lie algebras. e mainpurpose of this chapter is to provide an extensive discussion about the Lie-theoretical natureof coadjoint orbits.

It is worth to observe that in this chapter we will work in a more general seing of compactsimple Lie groups, therefore, the content which we will describe in what follows goes beyondthe seing of classical compact Lie groups. Our main references for results concerned withLie algebras are [87] and [144], for results in Lie group theory see for instance [98] and [145].

4.1 Generalities about coadjoint orbits

Unless otherwise stated, we will x a pair (G, g), where G is a compact and connected Liegroup with Lie algebra g.

From the coadjoint action of G on g∗, we have a manifold associated to each element λ ∈ g∗,dened by its coadjoint orbit

O (λ) =

Ad∗(д)λ ∈ g∗∣∣∣ д ∈ G,

if we consider the isotropy subalgebra gλ ⊂ g, where

gλ =X ∈ g

∣∣∣ ad∗(X )λ = 0

,

we can realize the coadjoint orbit of λ ∈ g∗ as a homogeneous space by using the followingcorrespondence

Ad∗(д)λ ∈ O (λ) ↔ дHλ ∈ G/Hλ,

68

here we denote by Hλ ⊂ G the isotropy subgroup associated to λ ∈ g∗, i.e.

Hλ =д ∈ G

∣∣∣ Ad∗(д)λ = λ

.

In what follows we will x the above basic data, requiring additionally in some cases that g tobe a simple Lie algebra orG to be simply connected. Our main goal through the next sectionswill be to describe the relationship between the complex and symplectic geometry underlingthis kind of manifold.

4.2 Compact real form of simple Lie algebras

In this section we consider g = Lie(G ) as being a simple Lie algebra. Let gC = g ⊗ C be thecomplexication of g, since this complex Lie algebra is a simple Lie algebra we can x a Cartansubalgebra h ⊂ gC, and take its triangular decomposition with respect to some choice of simpleroot system Σ ⊂ h∗

gC = n+ ⊕ h ⊕ n−,

where

n+ =∑α∈Π+

gα and n− =∑α∈Π−

gα .

Now we denote the root system and simple root system respectively by

Π = Π+ ∪ Π−,

by xing a Chevalley basis for gC correspondent to the above decomposition, we have

gC = SpanC

xα ,yα ,hβ

∣∣∣ α ∈ Π+, β ∈ Σ,

where xα and yα are weight vectors correspondent respectively to α ,−α ∈ Π, and hβ , β ∈ Σ,generate h, besides we have the relations

[hβ ,hβ ′] = 0, [hβ ,xα ] = α (hβ )xα , [hβ ,yα ] = −α (hβ )yα ,

notice that we can use the Killing form κC : gC × gC → C of gC and write the relations aboveas

[hβ ,hβ ′] = 0, [hβ ,xα ] = κC(hβ ,hα )xα , [hβ ,yα ] = κC(hβ ,h−α )yα .

We can use the above facts to dene the compact real form of gC as the real Lie algebra

gσ = ihR ⊕∑α∈Π+

uα ,

69

here we denote by

hR = SpanZ

hβ

∣∣∣ β ∈ Σ ⊗ R,

the realication of h ⊂ gC, and each uα , α ∈ Π+, denotes a real two dimensional vectorsubspace generated by

Aα =12 (xα − yα ), and iSα =

i

2 (xα + yα ).

e compact real form gσ ⊂ g can be seen as the set of xed point of a involutive anti-automorphism σ : gC → gC dened on the basis by

σ (xα ) = −yα , σ (yα ) = −xα , σ (hβ ) = −hβ ,

see [2, p. 15] or [98, p. 353]. A straightforward calculation shows us that

gσ =X ∈ gC

∣∣∣ σ (X ) = X

,

further more we have gC = gσ ⊗ C. Since a compact real form of a complex simple Lie algebrais unique up to automorphisms of gC, see for example [98, p. 359], we will suppose that g = gσ ,where σ is the map dened above.

From the above discussion we x, once for all, compatible pairs of Lie algebras

(gC, h) ←→ (g, t),

where gC = g ⊗ C, i.e. g = gσ is a compact real form, h ⊂ gC is a Cartan subalgebra satisfyingt = ihR, where t ⊂ g is the Lie algebra of a maximal torus.

4.3 Generalities about Lie group decompositions

In this section we will cover some basic results related to Lie group decompositions. LetGC bea connected, simply connected and complex simple Lie group, we denote by gC its Lie algebra.We have a decomposition of GC given by

GC = GN +A,

this decomposition is called Iwasawa decomposition, see for instance [98, p. 374]. e com-ponents in the Iwasawa decomposition are given by the maximal compact and connected Liegroup G ⊂ GC associated to the compact real form of gC, by

N + = exp ∑α∈Π+

zαxα

∣∣∣ zα ∈ C

,

and by

70

A = exp∑

α∈Σ

aαhα

∣∣∣ aα ∈ R

.

Now we consider the maximal torus T ⊂ G with Lie(T ) = ihR, denoting by t = ihR, we obtainthe following characterization for the compact Lie group G and its Lie algebra g, respectively

G =⋃д∈G

дTд−1, and g =⋃д∈G

Ad(д)t,

for more details see for instance [145, p. 236] or [98, p. 255].

Remark 4.3.1. We can use the Ad-invariant inner product of g, induced by the Killing form κC

of gC, given by

(X ,Y ) = −κ (X ,Y ) = −Re(κC(X ,Y )),

∀X ,Y ∈ g, in order to get an isomorphism between g and g∗, from these we can write

g∗ =⋃д∈G

Ad∗(д)t∗.

In most cases, in order to simplify the notation, we will not distinguish between κ and κC.

From the above remark, given a coadjoint orbit O (λ) ⊂ g∗, we can always suppose λ ∈ t∗.We can also use the induced Ad-invariant inner product in order to get a bijection betweenadjoint and coadjoint orbits, we just need to use the fact that

λ = −κ (Λ, ·),

for some Λ ∈ t = ihR.

It will be useful for us also to consider other decomposition of theGC called the polar decom-position. It is induced by the Cartan decomposition

gC = q ⊕ g,

where q = ig. At the level of the Lie groups we have a dieomorphism

G × q → GC, (д,X ) → д exp(X ),

see [29, p. 205], [98, p. 355-362]. is dieomorphism allows us to study the topology of GC

in terms of the topology of G × q. Notice that GC has the same homotopy type of G since q isa vector space.

Remark 4.3.2. It is worth to point out that from the last comment, if we take the complex simplyconnected Lie groupGC which integrates the simple Lie algebra gC, the compact real formG ⊂ GC

on the above Cartan decomposition is a compact simply connected Lie group.

71

4.4 Symplectic structure on coadjoint orbits

In this section we will give a complete description of the symplectic structure on coadjointorbits, as we will see such a structure arises in a very natural way.

Let G be a Lie group with Lie algebra g, we have a identication between le invariant dier-ential forms on G and exterior powers of g∗, i.e.

Ω•(G )G ∧•(g∗),

here Ω•(G )G denotes the space of le invariant forms on G, see [141, p. 249].

e above isomorphism allows us to establish an isomorphism of cochain complex

(Ω•(G )G ,d ) (∧•(g∗),δ ),

where in the le side of the above equivalence we have the cochain complex associated tothe exterior derivative (Ω•(G )G ,d ), and on the right side we have the so called Chevalley-Eilenberg complex when we consider R as a trivial g-module, see [144, p. 123], [32].

If we denote the cohomology group of le invariant dierential forms by H •L (G,R), we havethe following result

eorem 4.4.1. LetG be a Lie group with Lie algebra g, then we have the following isomorphism

H •L (G,R) H •((∧•(g∗),δ )),

where H •((∧•(g∗),δ )) denotes the Chevalley-Eilenberg cohomology group.

Proof. See for example [32] or [51, p. 16].

We denote by H •(g) = H •((∧•(g∗),δ )) the Lie algebra cohomology associated the Chevalley-

Eilenberg complex as described before.

For the case which we are concerned, i.e. whenG is compact, we have the following importantresult.

eorem 4.4.2. If M is a compact connected manifold on which we have a le G-action, whereG is a compact and connected Lie group, then

H •G (M,R) H •DR (M ),

where H •G (M,R) denotes the cohomology of le invariant dierential forms and H •DR (M ) denotesthe de Rham cohomology group.

Proof. See for example [51, p. 13].

From the above theorem we have

72

Corollary 4.4.3. Let G be a compact and connected Lie group, then H •L (G,R) H •DR (G ).

Remark 4.4.1. As we have mentioned in the previous section, as a consequence of the polardecomposition for complex simple Lie groups, we can study the topology of such a group GC

in terms of its compact real form G. From the above results the cohomology of a compact andconnected Lie group is completely determined by its Lie algebra cohomology.

Now we come back to study the symplectic geometry of coadjoint orbits. Let O (λ) ⊂ g∗ be acoadjoint orbit as before. As we have seen we can use an Ad-invariant inner produt on g, inorder to write

λ = −κ (Λ, ·),

for some Λ ∈ t = ihR. We can extend λ in a natural way in order to obtain λ ∈ g∗ =∧1(g∗),

remember that from Remark 4.3.1 we have λ ∈ t∗. Hence we get a le invariant 1-form λ ∈Ω1(G )G , if we consider its exterior derivative dλ ∈ Ω2(G )G , a straightforward calculationshows us that

dλ(X ,Y ) = −λ([X ,Y ]),

for every pair of le invariant vector elds X ,Y ∈ g.

Now we consider the kernel of dλ ∈ Ω2(G )G , i.e.

ker(dλ) =X ∈ g

∣∣∣ dλ(X ,Y ) = 0,∀Y ∈ g

,

since dλ(X ,Y ) = −λ([X ,Y ]), we can rewrite the above space as

ker(dλ) =X ∈ g

∣∣∣ (ad∗(X )λ)Y = 0,∀Y ∈ g

,

it follows that ker(dλ) = gλ, the isotropy subalgebra of λ ∈ g.

Remark 4.4.2. It is worthwhile to point out that if we denote by Gλ ⊂ Hλ the connected compo-nent of the identity of Hλ, the map

G/Gλ → G/Hλ,

is a covering map. If we require thatG to be simply connected, then from the homotopy long exactsequence for the bration Gλ → G → G/Gλ, see [83, p. 376]

· · · → πn (Gλ) → πn (G ) → πn (G/Gλ) → πn−1(Gλ) → · · · ,

we have π1(G/Gλ) = 0, it follows that the above covering space is an universal covering ofO (λ).

From these the le invariant 2-formdλ ∈ Ω2(G )G descends to a non-vanishing invariant closed2-form ωO (λ) ∈ Ω

2(G/Hλ)G , such that

73

π ∗ωO (λ) = dλ, where π : G → G/Hλ.

e ideas described so far are based on the following general result

Proposition 4.4.4. Every coadjoint orbit O (λ) associated with a connected Lie group is a sym-plectic manifold (O (λ),ωO (λ) ), the 2-form ωO (λ) is called Kirillov-Kostant-Souriau symplecticform.

Proof. For the proof and a complete discussion about this theme see for instance [17].

Remark 4.4.3. We are interested in studying compact connected simple Lie groups. In this con-text, by Whitehead’s lemma we have H 1(g) = H 2(g) = 0, therefore we have a correspondence, upto covering space, between symplectic homogeneous spaces and elements of Z 2(g) = ker(δ ).

Actually, if G/H is a simply connected symplectic homogeneous space of a compact connectedand simple Lie group G, with invariant symplectic structure ωG/H , since H 2(g) = 0, we have

π ∗ωG/H = dθ ∈ Z2(g) = B2(g), for some θ ∈ g∗,

from the last remark, we have the universal covering map

G/H → Ad∗(G )θ G/Hθ ,

notice that in this case H = Gθ ⊂ Hθ , see [17] fot more details.

4.5 Complex and Kahler structures on coadjoint orbits

In the previous section we have shown how to equip a coadjoint orbit with a symplectic struc-ture. In this section we will look more carefully its symplectic structure in order to obtain anatural compatible complex structure. For this purpose we will explore some basic propertiesof the root space decomposition which we have for complex simple Lie algebras.

Let us start with some basic denitions

Denition 4.5.1. A Borel subalgebra b is dened as being a maximal solvable subalgebra of gC.

Since all Borel subalgebras are conjugate with the standard Borel subalgebra b = n+ ⊕ h, see[87, p. 84], we will just work with the standard Borel subalgebra.

Denition 4.5.2. A subalgebra p ⊂ gC is called parabolic if it contains a Borel subalgebra of gC.

Denition 4.5.3. A parabolic subgroup is dened by the normalizer P = NGC (p) ⊂ GC, wherep ⊂ gC is a parabolic subalgebra.

Denition 4.5.4. LetGC be a complex simply connected and simple Lie group, given a parabolicsubgroup P ⊂ GC, a complex ag manifold assocaited to P is dened by the complex homogeneousspace GC/P .

74

Associated to a choice of Θ ⊂ Σ we have a standard parabolic subalgebra dened by

pΘ = n+ ⊕ h ⊕

∑α∈〈Θ〉−

gα .

Actually, every parabolic subalgebra is conjugated with some standard parabolic subalgebra,i.e. given a parabolic subalgebra p ⊂ gC, without loss of generality we can suppose p = pΘ, forsome Θ ⊂ Σ.

Let O (λ) ⊂ g∗ be a coadjoint orbit as before, we have seen that the symplectic structure onO (λ) is completely determined by λ ∈ t∗, where t = ihR. Now we take the extension λ : h → C,and consider the subset

Θλ =α ∈ Σ

∣∣∣ λ(hα ) = 0

.

We can associate to this roots subset Θλ ⊂ Σ a parabolic subalgebra

pΘλ = n+ ⊕ h ⊕

∑α∈〈Θλ〉−

gα ,

here 〈Θλ〉− = 〈Θλ〉 ∩ Π−, see [98, p. 325] or [29, p. 291-292] for more details about parabolic

subalgebras.

Now we consider the parabolic subgroup Pλ = NGC (pΘλ ), i.e. the normalizer of pΘλ in GC. Inwhat follows we will show that

Hλ = Pλ ∩G.

First we notice that as λ = −κ (Λ, ·), thus we have

ad∗(X )λ = 0 ⇐⇒ [X ,Λ] = 0,

now we observe that

ad(Λ)Aα = −iα (Λ)iSα , and ad(Λ)iSα = iα (Λ)Aα .

It follows that

ad(Λ)Aα = 0 or ad(Λ)iSα = 0,

if, and only if, α ∈ 〈Θλ〉+, from these we obtain

gλ = ihR ⊕∑

α∈〈Θλ〉+

uα .

A straightforward calculation shows us that the above expression is exactly the intersectionpΘλ ∩ g.

75

Remark 4.5.1. From the above idea we can also conclude that

gλ = Zg (Λ) =X ∈ g

∣∣∣ [X ,Λ] = 0

,

from this we have Hλ = ZG (Λ), see for instance [98, p. 338].

erefore from the Iwasawa decomposition we obtain

GC/Pλ GN +A/Pλ G/(Pλ ∩G ),

since Hλ = Pλ ∩G, it follows that

GC/Pλ O (λ).

We have just proved the following proposition

Proposition 4.5.5. LetO (λ) be a coadjoint orbit of a compact, connected, simply connected andsimple Lie group G, then there exists a parabolic subgroup Pλ ⊂ GC, such that

GC/Pλ O (λ),

i.e. O (λ) is dieomorphic to a complex ag manifold.

Now we will show that there exists a natural complex structure on O (λ) which is compatiblewith the Kirillov-Kostant-Souriau symplectic form.

Since [Λ,Λ] = 0, we can consider the following endomorphism at the tangent space TλO (λ)

LΛ : TλO (λ) → TλO (λ),

such that for all X ∈ g = TeG

LΛ(π∗(X )) = π∗([Λ,X ]),

here we used that π∗(TeG ) = TλO (λ). Since the kernel of π∗ : TeG → TλO (λ) is gλ, by theprevious characterization of gλ, we have

TλO (λ) ∑

α∈Π+\〈Θλ〉+

uα ,

actually, since ker(ad(Λ)) = Zg (Λ), it follows that LΛ = ad(Λ).

By means of the above isomorphism we can understand how the matrix associated to theendomorphism ad(Λ) is dened. A straightforward computation shows us the following

ad(Λ)∣∣uα=

0 iα (Λ)

−iα (Λ) 0

=iα (Λ) 0

0 iα (Λ)

0 1

−1 0

,

76

for α ∈ Π+\〈Θλ〉+, notice that each uα = SpanRAα , iSα ⊂ TλO (λ), is an invariant subspace of

ad(Λ). From these we have the following decomposition for the matrix associated to ad(Λ)

ad(Λ) = Mλ J0,

such that

Mλ = diagiα (Λ) 0

0 iα (Λ)

∣∣∣∣∣ α ∈ Π+\〈Θλ〉+

,

and J0 satises

J0(Aα ) = −iSα , J0(iSα ) = Aα ,

for all α ∈ Π+\〈Θλ〉+. It follows that J0 : TλO (λ) → TλO (λ) denes an almost complex structure

on TλO (λ).

In order to see the compatibility with the symplectic structure we proceed as follows. Noticethat

dλ(J0X , J0Y ) = −λ([J0X , J0Y ]) = κ (Λ, [J0X , J0Y ]),

since κ (X , [Y ,Z ]) = κ ([X ,Y ],Z ), we have

dλ(J0X , J0Y ) = κ (ad(Λ) J0X , J0Y ]) = −κ (MλX , J0Y ),

where we used J 20 = −1. However, once the basis which diagonalizes Mλ is orthogonal with

respect to −κ (·, ·), see for instance [144, p. 336], it follows that

dλ(J0X , J0Y ) = −κ (MλX , J0Y ) = −κ (X ,Mλ J0Y ) = −κ (X , ad(Λ)Y ).

erefore we obtain

dλ(J0X , J0Y ) = −κ (X , ad(Λ)Y ) = κ (Λ, [X ,Y ]) = dλ(X ,Y ).

Now we consider the complexication TλO (λ)C = TλO (λ) ⊗ C, by taking the extension

JC0 : TλO (λ)C → TλO (λ)C,

we have the following eingenspace spliing

TλO (λ)C = T (1,0)λ O (λ) ⊕ T (0,1)

λ O (λ),

where

T (1,0)λ O (λ) =

X ∈ TλO (λ)C

∣∣∣ J0(X ) = iX

,

77

is the eigenspace associated do i ∈ C, and

T (0,1)λ O (λ) =

X ∈ TλO (λ)C

∣∣∣ J0(X ) = −iX

,

is the eigenspace associated do −i ∈ C. From the previous description of the tangent space interms of the root spaces, we have

TλO (λ)C =∑

α∈Π\〈Θλ〉

gα ,

a simple calculation shows us that

JC0 (xα ) = −ixα , and JC

0 (yα ) = iyα ,

for all α ∈ Π+\〈Θλ〉+, i.e.

T (1,0)λ O (λ) =

∑α∈Π−\〈Θλ〉−

gα , and T (0,1)λ O (λ) =

∑α∈Π+\〈Θλ〉+

gα .

Now we consider the extension πC∗ : gC → TλO (λ)C, if we denote by

a+ = (πC∗ )−1(T (1,0)

λ O (λ)), and a− = (πC∗ )−1(T (0,1)

λ O (λ)),

we can show that these two subspace are in fact subalgebras of gC, furthermore we have thefollowing relations

gC = a+ + a−, a+ ∩ a− = gCλ .

It is not dicult to see that a− = pΘλ , and a+ = σ (a−). In fact we have

a− = (πC∗ )−1(T (0,1)

λ O (λ)) =∑

α∈Π+\〈Θλ〉+

gα + gCλ .

From the last comments we have the following result

Proposition 4.5.6. Let G be a compact, connected and simply connected simple Lie group withLie algebra g, then there exists an one-to-one correspondence betweenG-invariant complex struc-tures on O (λ) and subalgebras a+ ⊂ gC = g ⊗ C satisfying the above relations.

Proof. See for instance [5, p. 965], [18, p. 497-499] and [130, p. 23].

If we denote by lд : O (λ) → O (λ) the dieomorphism induced by the le translation, i.e.

lд (xHλ) = дxHλ,

for all xHλ ∈ O (λ), and д ∈ G, the complex structure associated to J0 is dened by

JxHλ = (lx )∗ J0 (lx−1 )∗.

78

From the above comments we have the following result

Corollary 4.5.7. Under the hypothesis of the last proposition, every coadjoint orbit O (λ) admitsa Kahler structure such that the dieomorphism

GC/Pλ O (λ),

is in fact a biholomorphism when we consider the natural complex structure ofGC/Pλ induced byGC.

erefore, we have that the tiple (O (λ),ωO (λ), J0) denes a Kahler manifold. It is worthwhileto observe that from the identication

GC/Pλ O (λ),

the complex structure J0 coincides with the complex structure induced on GC/Pλ by GC, i.e.we have a biholomorphism

GC/Pλ (O (λ), J0).

Now we look closely the invariant Kahler metric associated to the Kirillov-Kostant-Souriausymplectic form. We have a metric dened by

ds2(X ,Y ) = ωO (λ) (X , J0Y ) = −λ([X , J0Y ]),

∀X ,Y ∈ TλO (λ), it follows that

ds2(X ,Y ) = κ (Λ, [X , J0Y ]) = −κ ([Λ, J0Y ],X ),

therefore we obtain

ds2(X ,Y ) = κ (X ,MλY ),

∀X ,Y ∈ TλO (λ). As we have seen this last G-invariant Kahler structure dened on O (λ) isencoded in the decomposition LΛ = ad(Λ) = Mλ J0.

As we have seen so far two important hypotheses allows us to establish the result stated inCorollary 4.5.7, namely gC to be a complex simple Lie algebra andGC to be a simply connectedLie group. Now we will explain why this particular case is in fact the basic and more importantcase.

Let G be a compact and connected Lie group with Lie algebra Lie(G ) = g, from the theory ofcompact Lie groups we have the following decomposition

g =[g, g]⊕ Zg,

where [g, g]

is semisimple and Zg is the center of g, see for instance [98, p. 249]. e abovedecomposition induces a Lie group decomposition in the following commuting product

79

G = Gss (ZG )0,

whereGss is a closed, connected and semisimple Lie subgroup with Lie(Gss ) = [g, g], and (ZG )0is the closed connected subgroup dened by the connected component of the identity of thecenter, see for example [98, p. 250]. Given X ∈ g, consider its adjoint orbit

O (X ) =

Ad(д)X ∈ g∣∣∣ д ∈ G,

we notice that for д ∈ G we have д = д′z, with д′ ∈ Gss and z ∈ (ZG )0, furthermore, we canalso write X = X ′ + X ′′, with X ′ ∈ [g, g

]and X ′′ ∈ Zg, therefore we obtain

O (X ) = Ad(G )X = Ad(Gss )X′ + X ′′,

it follows that O (X ) Ad(Gss )X′. Hence without loss of generality we can suppose G = Gss

in the study of adjoint orbits.

Taking into account the above comments, now we consider G as being a semissimple Liegroup. From the theory of Lie groups, see for instance [98, p. 90], we have the universal coverhomomorphism p : G → G, such that

G G/ ker(p),

with ker(p) ⊂ ZG being a discrete subgroup. From this we can use (Dp)e : TeG → TeG to setan isomorphism g Lie(G ), therefore from the identity

Ad(p (д)) (Dp)e = (Dp)e Ad(д),

we obtain

O (X ) = Ad(G )X = Ad(p (G ))X = (Dp)eAd(G ) (Dp)−1e X ,

thus O (X ) = Ad(G )X Ad(G )X ′, where X = (Dp)eX′. us we can consider without loss of

generality G = G in the study of adjoint orbits.

It is worthwhile to observe that the construction which we have described for simple Lie alge-bras also can be done for semisimple Lie algebras, i.e. we can do the same construction usingroot systems and parabolic subalgebras in order to provide a biholomorphism between adjointorbits and complex ag manifolds.

Remark 4.5.2. Based on the previous ideas we will x the following notation, given a parabolicsubgroup P ⊂ GC we denote by

XP = GC/P and O (λP ),

the associated complex ag manifold and the associated coadjoint orbit, respectively. In most caseswe will consider λP ∈ h∗ as being an integral weight which denes a subset

80

Θ =α ∈ Σ

∣∣∣ λP (hα ) = 0

,

such that P = PΘ. us from Corollary 4.5.7 we have XP = O (λP ).

We nish this chapter by giving a basic example on which we illustrate all the ideas that wehave covered about symplectic and complex geometry of coadjoint orbit.

Example 4.5.1. Consider the classical compact Lie group SU(2), this Lie group can be de-scribed as follows

SU(2) =a −b

b a

∣∣∣∣∣ a,b ∈ C, |a |2 + |b |2 = 1

,

furthermore its Lie algebra is given by the following matrices

su(2) =iθ −z

z −iθ

∣∣∣∣∣ θ ∈ R, z ∈ C

.

If we take λ ∈ su(2)∗ such that

λ = −κ (Λ, ·), where Λ =

iR 0

0 −iR

,

for some R < 0, we obtain a manifold dened by

O (Λ) =дΛд−1

∣∣∣ д ∈ SU(2)⊂ su(2).

Given X ∈ O (Λ) we have det(X − t12) = 0 ⇐⇒ t = ±√

det(X ), therefore the elements ofO (Λ) satisfy the following equation

X =

iθ −z

z −iθ

∈ O (Λ) ⇐⇒ θ 2 + |z |2 = R2.

It means thatO (Λ) ⊂ su(2) is the 2-sphere of radius |R |. We have dened onO (Λ) a canonicalsymplectic structure given by

ωO (Λ) (ad(v )X , ad(w )X ) = κ (X ,[v,w

])

∀ad(v )X , ad(w )X ∈ TXO (Λ). Now we will describe how the complex and symplectic structureof O (Λ) are encoded in Λ ∈ su(2). We start by xing the following canonical basis for su(2)

Aα =12

0 1

−1 0

, iSα =i

2

0 1

1 0

, ihα =i

2

1 0

0 −1

,

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for this base we have the relations[ihα ,Aα

]= iSα ,

[ihα , iSα

]= −Aα ,

[Aα , iSα

]= ihα .

from these we analyse the linear map ad(Λ) : TΛO (Λ) → TΛO (Λ). Since we have

TΛO (Λ) = SpanR

Aα , iSα

,

it follows that

ad(Λ)Aα = 2Rad(ihα )Aα = 2RiSα and ad(Λ)iSα = 2Rad(ihα )iSα = −2RAα ,

thus we get the following expression for the matrix ad(Λ) with respect to the base Aα , iSα

ad(Λ) =

0 −2R

2R 0

=−2R 0

0 −2R

0 1

−1 0

,

therefore we have the Kahler structure of O (Λ) completely determined by the decompositionad(Λ) = Mλ J0, where

Mλ =

2|R | 0

0 2|R |

and J0 =

0 1

−1 0

,

moreover, from Proposition 4.5.5 we obtain

SL(2,C)/B = SU(2)AN +/B = SU(2)/SU(2) ∩ B = (O (Λ), J0),

notice that SU(2) ∩ B = U(1). Now let us look closely the U(1)-principal bundle

U(1) → SU(2) → SU(2)/U(1) = O (Λ),

at rst we notice that we have a natural identication between S3 ⊂ C2 and SU(2). In fact, ifwe denote

S3 =

(z0, z1) ∈ C2 ∣∣ |z0 |

2 + |z1 |2 = 1

,

we can set the following correspondence

(z0, z1) ∈ S3 ←→

z0 −z1

z1 z0

∈ SU(2),

since we also have a identication S1 U(1), the above correspondence provides a equivalencebetween the following actions

82

eiϕ · (z0, z1) = (eiϕz0, eiϕz1) ←→ eiϕ ·

z0 −z1

z1 z0

=z0 −z1

z1 z0

eiϕ 0

0 e−iϕ

,

for every eiϕ ∈ S1 and (z0, z1) ∈ S3. e quotient space obtained from the action on the leside is S3/S1 = S2, the quotient projection h : S3 → S2 is given by

h(z0, z1) = (2z0z1, |z0 |2 − |z1 |

2),

where we consider S2 ⊂ C× R ⊂ C2. We have the following correspondence between h(S3) =S2 and O (Λ) ⊂ su(2)

(2z0z1, |z0 |2 − |z1 |

2) ←→

iR ( |z0 |2 − |z1 |

2) 2iRz0z1

2iRz1z0 −iR ( |z0 |2 − |z1 |

2)

.

us can we also regard U(1) → SU(2) → SU(2)/U(1) = O (Λ) as the Hopf bration.

Figure 4.1: e image of two distinct Hopf bers by the stereographic projection s : S3\N →R3, notice that P ,Q ∈ S2\N , S . is gure was extracted from [115]

.

e above ideas boil down to the following commutative diagram

U(1) SU(2) O (Λ)

S1 S3 S2

As we have seen this basic example is an illustrative exercise which shows us how the Lietheoretical approach can be useful to understand some important geometric constructionsfrom a dierent standpoint.

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Chapter 5Generalities on Kahler-Einsteinmanifolds

e purpose of this chapter is to provide a brief exposition of some results related to holomor-phic vector bundles and Kahler-Einstein metrics. In what follows we consider X = (M, J ) asbeing a complex manifold, where M denotes the underlying smooth manifold and J denotesthe underlying complex structure. We also will denote by OX the structural sheaf of holo-morphic functions of X . e basic idea is to collect some denitions and results related to thestudy of the Einstein equation in the context of Kahler manifolds. e main references for theresults which we will describe in this chapter can be found in [126], [63], [166], [16] and [94].

5.1 Holomorphic line bundles

LetX be a complex manifold, by denition a holomorphic line bundle π : L → X is a holomor-phic complex vector bundle of rank one. By taking an open cover

X =⋃i∈I

Ui ,

which trivializes L we have biholmorphic maps

φi : L|Ui → Ui × C,

∀i ∈ I , here L|Ui = π−1(Ui ). Relative to the trivialization (U ,φi )i∈I , on the overlaps we have

(φi φ−1j ) : Ui ∩Uj × C→ Ui ∩Uj × C, (φi φ

−1j ) (p, z) = (p,дij (p)z),

for Ui ∩Uj , ∅, with i, j ∈ I , where

дij : Ui ∩Uj → C×,

are holomorphic functions which satisfy the cocycle condition

дijдji = 1, дikдklдl j = 1.

84

Here we denote by OX the structure sheaf of holomorphic functions and by O∗X the sheafof multiplicative group of nonzero holomorphic functions, from the above comments for aholomorphic line bundle we have дij ∈ O∗X (Ui ∩Uj ), i, j ∈ I , see for example [63, p. 132].

Example 5.1.1. Given a complex manifoldX , consider the holomorphic vector bundle denedby

KX = det(T (1,0)X ∗) =∧(1,0) (X ).

As we will see this line bundle plays an important role in the study of the geometry of X , it iscalled canonical bundle of X .

Since the cocycle дij ∈ O∗X (Ui ∩Uj ) denes L completely up to isomorphism, we can regardL → X as a cohomology class

L ∼ дij ∈ H1(X ,O∗X ).

Conversely, every cohomology class in H 1(X ,O∗X ) denes a holomorphic line bundle over Xwhich is uniquely determined up to isomorphism, see for instance [63, p. 133]. erefore,we have a natural structure of abelian group in H 1(X ,O∗X ) where the product and the inverseelement can be characterized, respectively, by

L ⊗ L′ ∼ дijд′ij , L−1 ∼ д−1

ij ,

and the identity element is given by the trivial line bundle. We also can denote the inverseelement by −L ∼ д−1

ij . From these the group of line bundles over X , called Picard group, canbe characterized by the following identication

Pic(X ) H 1(X ,O∗X ),

see [63] for more details. Given L ∈ Pic(X ) a Hermitian structure on L is a Hermitian scalarproduct Hx on each ber Lx = π−1(x ) which depends dierentiably on x ∈ X . e pair(L,H ) is called a Hermitian line bundle [89, p. 166]. e notion of compatible connection on aHermitian vector bundle can be described as follows, consider

∇ : Γ(L) → Ω1(L),

as being a connection on (L,H ), here Ω1(L) denotes the space of L-valued 1-forms on X , i.e.the sections of

∧1(X ) ⊗ L, see for example [126, p. 77] and [89, p. 173]. If we take a localtrivialization U ⊂ X , the local expression of ∇ is given by

∇ = d +AU ,

where AU ∈ Ω1(U ) ⊗ C. Since X is a complex manifold the connection decomposes in twopieces, i.e. ∇ = ∇(1,0) + ∇(0,1) , where

∇(1,0) = ∂ +A(1,0)U and ∇(0,1) = ∂ +A(0,1)

U .

We have two important notions of compatibility associated to ∇, namely:

85

1. We say that ∇ is compatible with the holomorphic structure ∂ of L if ∇(0,1) = ∂, see forinstance [89, p. 177];

2. We say that ∇ is compatible with the Hermitian structure H of L if

d (H (σ1,σ2)) = H (∇σ1,σ2) + H (σ1,∇σ2),

see for instance [89, p. 176]. If the above equality holds we call ∇ a H -connection.

e question related to the existence of a Hermitian structure on L ∈ Pic(X ) can be solvedby gluing with a partition of unity the standard Hermitian structure which we have denedlocally [126, p. 78], from this the important result, see [89, p. 177], which we have related tothe previous comments is the following

Proposition 5.1.1. Let (L,H ) be a Hermitian holomorphic line bundle, then there exists a uniqueH -connection ∇ on L such that ∇(0,1) = ∂.

e nice feature of the above proposition is that for a Hermitian holomorphic line bundle (L,H )the compatible connection ∇ is completely determined by the Hermitian structure, such thatits local expression is given by

∇ = d + ∂ logH = d + ∂HH

.

We notice that on the above expression we have H = H (σ ,σ ), for some locally dened non-vanishing holomorphic section σ .

e curvature form for a connection dened as above is given locally by F∇ = dAU , see forexample [89, p. 182]. Actually, by denition we have F∇ = ∇ ∇, and given a locally denedholomorphic section σ ′ : U → L, by taking U ⊂ X as being an open set which trivializes L, astraightforward calculation shows us that

(∇ ∇)σ ′ = (dAU )σ′,

since AU = ∂ logH , we obtain locally F∇ = ∂∂ logH = −∂∂ logH . We notice that for anyother open set V ⊂ X which trivializes L → X , on U ∩V we have

AU = AV − d log(дUV ),

where дUV : U ∩ V → C× denotes the associated cocycle. us F∇ = dAU is globally welldened.

From the previous discussion, given L ∈ Pic(X ), by choosing a Hermitian structure on L wecan dene its rst Chern class by

c1(L,H ) =[ i

2π F∇]∈ H 2

DR (X ,R),

86

we can show that the above denition does not depend on the choice ofH , thus we just denotethe Chern class of L ∈ Pic(X ) by c1(L) ∈ H

2DR (X ,R), see for more details [126, p. 115] or [89,

p. 193].

We can also dene the rst Chern class of L ∈ Pic(X ) in an alternative way which can bedescribed as follows. Consider the short exact (exponential) sequence of sheaves

0 Z OX O∗X 0,

the second arrow on the right side above is just the inclusion of the locally constant sheaf Z →OX , the third arrow corresponds to the map f → exp(2πi f ), ∀f ∈ OX , see for instance [89,p. 70]. Associated to this short exact sequence we have a long exact sequence of cohomologygroups

· · · H 1(X ,OX ) H 1(X ,O∗X ) H 2(X ,Z) · · · ,δ

from these, given L ∈ Pic(X ) H 1(X ,O∗X ) we can dene its rst Chern class as being δ (L) ∈H 2(X ,Z), where

δ : H 1(X ,O∗X ) → H 2(X ,Z),

denotes the connecting homomorphism, also called Bockstein operator. When we comparethese two approaches from the following commutative diagram

H 1(X ,O∗X ) H 2(X ,Z)

H 2DR (X ,R)

δ

c1j

we have c1(L) = j (δ (L)), see for example [166, p. 106, eorem 4.5], here the map j : H 2(X ,Z) →H 2DR (X ,R) denotes the canonical homomorphism obtained by identifying the de Rham coho-

mology with some suitable 1 cohomology. In the most cases we will deal with X simply con-nected, thus we can consider H 2(X ,Z) ⊂ H 2

DR (X ,R) and we can in fact denote c1(L) = δ (L).

5.2 Kahler manifolds and Einstein metrics

Let us establish some basic facts about the study of Einstein equation on Kahler manifolds.We start by collecting some basic denitions and general facts. Let (M,д) be a Riemannianmanifold, if we consider the associated Levi-Civita connection

∇ : Γ(TM ) → Γ(TM ⊗ TM∗)

1e word “suitable” means a compatible cohomology theory, which in this case is the Cech cohomology.Issues related to the homology and cohomology language as well as the conventions which we are following canbe found in D.1.1.

87

from curvature tensor eld R∇ we can dene the Ricci tensor by

Ric(X ,Y ) := TrZ → R∇(Z ,X )Y

,

where Tr denotes the trace of the linear map Z → R∇(Z ,X )Y . e Ricci tensor is a symmetrictensor which allows to make the following denition

Denition 5.2.1. A Riemannian manifold (M,д) is called an Einstein manifold if there exists areal constant λ such that

Ric(X ,Y ) = λд(X ,Y ),

for each X ,Y ∈ TpM and each p ∈ M .

A metric which satises the equation of the above denition is called an Einstein metric, seefor instance [16]. e above denition is motivated by the Einstein eld equation (EFE), infact the equation which denes Einstein manifolds is a particular case of the Einstein equa-tion called the vacuum Einstein eld equation, it is also denoted by Ric(д) = λд 2 .

In general nding Einstein metrics is a highly nontrivial problem, since the equation Ric(д) =λд is equivalent to a nonlinear partial dierential equation. We are interested in a seingwhere besides of a Riemannian metric we also have a complex structure, let us make thecontext which we will consider more precise

Denition 5.2.2. A Hermitian metric on a complex manifold (M, J ) is a Riemannian metric дsuch that д(X ,Y ) = д(JX , JY ), ∀X ,Y ∈ TM . e fundamental 2-form of a Hermitian metric isdened by ω (X ,Y ) := д(JX ,Y ).

A complex manifold with a Hermitian metric is called Hermitian manifold [126], we will re-quire an extra condition which is the content of the following denition

Denition 5.2.3. A Hermitian manifold (M, J ,д) is a Kahler manifold if dω = 0, where ωdenotes the fundamental 2-form associated to д.

Henceforth we will consider X = (M, J ,д) as being a Kahler manifold and we will denote ωX

(Kahler form) the fundamental 2-form associated toд (Kahler metric). Let us collect some basicfacts about Kahler manifolds.

If we take holomorphic coordinates z1, . . . , zn on an open set U ⊂ X , dimC(X ) = n, we havethe following local expression for ωX

ωX = i∑i,s

дisdzi ∧ dzs ,

where дis = д( ∂∂zi ,∂∂zs

), i, s = 1, . . . ,n. From this, by the i∂∂-lemma [126, p. 68] we obtain

д(∂

∂zi,∂

∂zs) =

∂2φ

∂zi∂zs,

2Notice that in particular every Einstein manifold has scalar curvature S = Trд (r ) constant, namely S =dim(M )λ, see [16, p. 44, eorem 1.97] for more details.

88

where φ : U ⊂ X → R satises ωX = i∂∂φ. A straightforward calculation [126, p. 87] showsthat

ρ (X ,Y ) = Ric(JX ,Y ),

denes a 2-form on X , this form is called the Ricci form of X .

e Ricci form is one of the most important objects on Kahler geometry. Let us briey describesome of its properties

• the Ricci form ρ is closed;

• the cohomology class of ρ is equal (up to some real multiple) to the Chern class of thecanonial bundle of X ;

• in local coordinates, ρ can be expressed by ρ = −i∂∂ log det(д), here д = (дis ) denotesthe Hermitian matrix dened by the Hermitian metric.

From the above comments we see that the Einstein condition on Kahler manifolds can bewrien as ρ = λωX (or Ric(ωX ) = λωX ) which leads to the topological condition λ[ωX ] =2πc1(X ), where c1(X ) denotes the Chern class of X . In the 1950’s, Calabi initiated the studyof Kahler-Einstein metrics [27], we have the following famous result conjectured in 1950’s byCalabi and proved two decades later by Yau:eorem 5.2.4. (Calabi [27], Yau [168]) Let X be a compact Kahler manifold with Kahler formωX and Ricci form ρ. en for every closed real (1, 1)-form ρ1 in the cohomology class 2πc1(X ),there exists a unique Kahler metric with Kahler form ω1 ∈ [ωX ], whose Ricci form is exactly ρ1.In particular, if the rst Chern class of a compact Kahler manifold X vanishes, then X carries aRicci-at metric.

e above result gives us an answer for the existence of solutions of the equation Ric(ωX ) = 0when X is a compact Kahler manifold such that c1(X ) = 0, in this case X is called Calabi-Yaumanifold [94, p. 122].

Since Ric(cд) = Ric(д) for every positive constant c > 0, for the case λ , 0 we can alwayssuppose λ = ±1 in the Einstein condition namely we can write Ric(д) = ±д or Ric(ωX ) = ±ωX .For the case λ = −1 we have the following resulteorem 5.2.5. (Aubin [7], Yau [169]) A compact complex manifold with negative rst Chernclass admits an unique Kahler-Einstein metric with Einstein constant λ = −1 [94, p. 122].

Remark 5.2.1. Notice that since the rst Chern class of X is given by ρ2π , we see that a necessary

condition for the existence of a Kahler-Einstein metric on a given compact Kahler manifold is thatits rst Chern class is denite 3 (positive or negative), see for instance [63, p. 148].

According to [42], when λ = 1 the existence of a Kahler-Einstein metric, may hold or may not.Moreover, although enormous progress has been made towards to understand precisely whena solution exists, the problem remains without a solution.

In this work we will deal with the concept of Calabi-Yau manifolds in the sense of the followingtheorem [126, p. 121, eorem 17.5]

3We say that ω ∈ Ω1,1 (X ) is denite positive (negative) 2-form if −iω (v,v ) > 0 (−iω (v,v ) < 0), ∀v ∈ T (1,0)X .

89

eorem 5.2.6. Let X be a complex simply connected Kahler manifold, then the ve statementsbelow are equivalent

1. X is Ricci-at.

2. e Chern connection of the canonical bundle KX is at.

3. ere exists a ∇-parallel complex volume form, that is, a parallel section of∧(n,0) (X ).

4. X has a SU(n)-structure.

5. e Riemannian holonomy group of X is a subgroup of SU(n).

Henceforth we will consider a Calabi-Yau manifold as being a complex simply connected Kahlermanifold which satises one of the conditions of the above theorem.

90

Chapter 6Calabi ansatz technique and complexag manifolds

In this chapter we will provide a complete description of how to construct Ricci-at metrics oncanonical bundles of complex ag manifolds. e main tool which we will apply in the contextof complex ag manifolds is Calabi’s technique [26], which provides a constructive method toobtain Kahler-Einstein metrics on the total space of holomorphic vector bundles over Kahlermanifolds. e main result which we will use can be found in [143, p. 108, eorem 8.1], seealso [100], [60] and [24].

Our main motivation for the content which will be developed in this chapter is [60]. e coreidea behind our approach is to provide a complete description of Calabi-Yau metrics on thetotal space of the canonical bundle of complex ag manifolds. Our approach can be seen as arst step to understand the relations between special Lagrangian brations on total spaces ofcanonical bundles and integrable systems on the base manifold.

It is worth to point out that the general recipe that we will establish here for complex agmanifolds is new in the literature. Many of the well known examples for the construction ofCalabi-Yau metrics on the total space of the canonical bundle by means of Calabi’s technique,for instance complex projective spaces, are in fact particular examples of ag manifolds. ekey point in our approach is to describe how it works in the general seing of complex agmanifolds. erefore the content covered in Appendix D.1.1 will be essential to understandthe Lie-theoretical features of our description, we also recommend Appendix D.2.

e main result which we will establish here is the following

eorem 6.0.1. Let (XP ,ωXP ) be a complex ag manifold associated to some parabolic subgroupP = PΘ ⊂ GC, such that dimC(XP ) = n. en the total space KXP admits a complete Ricci-atKahler metric with Kahler form given by

ωCY = (2πu +C )1

n+1π ∗ωXP −i

n + 1 (2πu +C )− nn+1∇ξ ∧ ∇ξ , (6.0.1)

where C > 0 is some positive constant and u : KXP → R≥0 is given by u ([д, ξ ]) = |ξ |2, ∀[д, ξ ] ∈KXP . Furthermore, the above Kahler form is completely determined by the quasi-potentialφ : GC →

91

R dened by

φ (д) =1


| |дv+ωα | |2〈δP ,h∨α 〉

),


e theorem above can be seen as a combination of the Calabi ansatz technique [26] with thedescription of quasi-potentials provided by H. Azad and I. Biswas in [8]. In the section belowwe will provide a proof for the above result and give explicit examples of how to apply it inconcrete cases. In this way, besides recovering many well known results related to XP = CPnwe will also establish a new class of examples.

Our motivations to set up the above result are [26], [143, p. 108], [100] and [60], see also[138], [55], [112] and [23]. We hope to use the above description to study special Lagrangiansubmanifolds in the total space of the canonical bundle over complex ag manifolds, and alsoanalyse its relations with integrable systems dened on ag manifolds, e.g. Gelfand-Tsetlinintegrable systems, these ideas are based on [60] and [95].

6.1 Calabi-Yau metrics on canonical bundles of ag man-ifolds

Let GC be a connected, simply connected and complex Lie group with simple Lie algebra gC.By choosing a Cartan subalgebra h ⊂ gC and xing a simple root system Σ ⊂ h∗, we have thetriangular decomposition

gC = n+ ⊕ h ⊕ n−.

From these, given a parabolic subgroup P ⊂ GC, without loss of generality we can suppose

P = PΘ, for some Θ ⊂ Σ.

By denition we have PΘ = NGC (pΘ), where Lie(PΘ) = pΘ ⊂ gC is given by

pΘ = n+ ⊕ h ⊕ n(Θ)−, with n(Θ)− =

∑α∈〈Θ〉−

gα .

For our purposes it will be useful to consider the following basic chain of subgroups

TC ⊂ B ⊂ P ⊂ GC,

for each element in the above chain of subgroups we have the following facts

• TC = exp(h), (complex torus)

• B = N +TC, where N + = exp(n+), (Borel subgroup)

• P = PΘ = NGC (pΘ), form some Θ ⊂ Σ ⊂ h∗. (parabolic subgroup)

92

Associated to the above basic data we will be concerned with the study of the complex agmanifold

XP = GC/P .

An important holomorphic line bundle to consider overXP in our approach is its anticanonicalbundle

−KXP = det(T (1,0)XP

)=∧(n,0) (T (1,0)XP

),

here we suppose dimC(XP ) = n. In the context of complex ag manifolds the above line bundlecan be described as follows. Consider the identication

m =∑

α∈Π+\〈Θ〉+

g−α = T(1,0)x0 XP ,

where x0 = eP ∈ XP , we have the following characterization 1 for T (1,0)XP

T (1,0)XP = GC ×P m,

here the twisted product above on the right side is obtained from Ad: P → GL(m) as anassociated holomorphic vector bundle. erefore, we can check 2 that


)= det

(GC ×P m

)= GC ×χ C,

where χ ∈ Hom(TC,C×) is a holomorphic character dened by

χ (exp(h)) = e−δP (h) ,

for every h ∈ h, where δP ∈ h∗ is given by

δP =∑

α∈Π+\〈Θ〉+

α .

e proof of the above fact can be found in Appendix C.4 on page 233 of this work.

Remark 6.1.1. In order to perform some local calculations it will be convenient for us to considerthe open set dened by the “opposite” big cell on XP . is open set is a distinguished coordinateneighbourhood U ⊂ XP of x0 = eP ∈ XP dened by the maximal Schubert cell which can bebriey described as follows. Let Π = Π+ ∪ Π+ be the root system associated to the simple rootsystem Σ ⊂ h∗, from this we can dene the opposite big cell U ⊂ XP by

1is characterization can be obtained as follows. First we consider the Maurer–Cartan form of GC, i.e. themap θд : TдGC → gC, this form allows us to dene a map Ψ : GC ×P m → T (1,0)XP such that Ψ([д,X ]) =π∗ (θ

−1д (X )),∀[д,X ] ∈ GC×Pm, here π : GC → XP denotes the canonical projection. A straightforward calculation

show us that Ψ denes an isomorphism of holomorphic vector bundles.2It is worthwhile to point out that on the twisted product GC ×χ C we consider the representation χ : P →

GL(1,C) obtained from an extension of χ ∈ Hom(TC,C×), notice that TC ⊂ P . erefore, −KXP can be regardedas a homogeneous holomorphic line bundle associated to the principal bundle GC → GC/P .

93

U = B−x0 = Ru (PΘ)−x0 ⊂ XP ,

where B− = exp(h ⊕ n−) and

Ru (PΘ)− =

∏α∈Π−\〈Θ〉−

N −α ,

with N −α = exp(gα ), ∀α ∈ Π−\〈Θ〉−. Further comments about Schubert cells and Schubert vari-eties can be found in Appendix C.4 on page 198 of this work.

Now let us collect some facts about the Kahler geometry of complex ag manifolds. Let P ⊂ GC

be a parabolic subgroup, such that P = PΘ for some Θ ⊂ Σ. Associated to the complex agmanifold XP = G

C/P , we have the following results

• ere exists a G-invariant Kahler form ωXP ∈ c1(XP ) completely determined by the(quasi-potential) φ : GC → R dened by

φ (д) =1


| |дv+ωα | |2〈δP ,h∨α 〉

).

By “completely determined” we mean that for every locally dened smooth sectionsU : U ⊂ XP → GC, we have

ωX |U = i∂∂(s∗U (φ)),

this result is a consequence of eorem D.1.4, see page 220 of this work. e proof ofthe fact that ωXP ∈ c1(XP ) can be found in Proposition D.1.9, see page 234 of this workor see for instance [8];

• (XP ,ωXP ) is a compact Kahler-Einstein Fano manifold with Ric(ωXP ) = 2πωXP , this factfollows as a corollary of the above result, see Corollary D.1.10 on page 235;

• Consider an open cover XP =⋃

i∈I Ui which trivializes both P → GC → XP and KXP →

XP . We take a collection of local sections (si )i∈I , such that si : Ui → GC, from these wedene qi : Ui → R+ by

qi = e2πφsi =∏

α∈Σ\Θ

| |siv+ωα | |

2〈δP ,h∨α 〉, (6.1.1)

for every i ∈ I . e collection of functions (qi )i∈I satisesqj = |χδP ψij |2qi onUi∩Uj , ∅,here we have used that sj = siψij onUi ∩Uj , ∅

3, and pv+ωα = χωα (p)v+ωα for every p ∈ P

and α ∈ Σ\Θ. From the above collection of smooth functions we can dene a Hermitianstructure H on KXP by taking on each trivialization ϕi : KXP → Ui × C a metric denedby

H ((x ,v ), (x ,w )) = qi (x )vw, (6.1.2)3It is worthwhile to observe that the maps ψi j : Ui ∩ Uj → P , ∀i, j ∈ I , are the cocycle which dene the

principal bundle P → GC → XP , namely GC = (Ui )i ∈I ,ψi j : Ui ∩Uj → P . Moreover, in terms of Cech cocyclewe have KXP = дi j = χδP ψi j , where дi j : Ui ∩Uj → C×, ∀i, j ∈ I .

94

for (x ,v ), (x ,w ) ∈ KXP |Ui Ui × C. e above Hermitian metric induces a Chern con-nection ∇ = d + ∂ logH with curvature F∇ satisfying

F∇ = 2π√−1ωXP .

Notice that c1(KXP ) = −[ωXP ] = −c1(XP ), see Corollary D.1.10 on page 235 for moredetails about this last comment.

In order to prove the main result of this chapter we need the following important theorem forwhich the proof can be found in Appendix C.4 on page 243 of this work

eorem 6.1.1. [26] Let (X ,ωX ) be a compact Kahler-Einstein manifold such that c1(X ) > 0,i.e. a Kahler-Einstein Fano manifold, then there exists a complete Ricci-at metric on its canonicalbundle KX =

∧(n,0) (X ).

From the above facts we have the following main result of this chapter

eorem 6.1.2. Let (XP ,ωXP ) be a complex ag manifold associated to some parabolic subgroupP = PΘ ⊂ GC, such that dimC(XP ) = n. en the total space KXP admits a complete Ricci-atKahler metric with Kahler form given by

ωCY = (2πu +C )1

n+1π ∗ωXP −i

n + 1 (2πu +C )− nn+1∇ξ ∧ ∇ξ , (6.1.3)

where C > 0 is some positive constant and u : KXP → R≥0 is given by u ([д, ξ ]) = |ξ |2, ∀[д, ξ ] ∈KXP . Furthermore, the above Kahler form is completely determined by the quasi-potentialφ : GC →

R dened by

φ (д) =1


| |дv+ωα | |2〈δP ,h∨α 〉

),


Proof. e fact that (KXP ,ωCY ) is a Ricci-at manifold follows from eorem 6.1.1 above (Cal-abi’s technique), thus we just need to verify that Hol(∇ωCY ) ⊆ SU(n + 1), where ∇ωCY denotesthe Levi-Civita connection induced on

∧(n+1,0) (KXP ) by the Kahler metric ωCY , and check thatωCY is completely determined by φ.

In order to see that Hol(∇ωCY ) ⊆ SU(n+1) we observe that since KXP → XP is a vector bundle,it follows that KXP has the same homotopy type of XP . Once XP is simply connected, we havethat KXP is also simply connected. Hence, we have

Hol(∇ωCY ) = Hol0(∇ωCY ) ⊆ SU(n + 1),

see for instance [94, p. 27, Proposition 2.2.6].

Now we will check thatωCY is completely determined byφ. First, notice that on the trivializingopen set given by the opposite big cellU = Ru (PΘ)

−x0 ⊂ XP we have holomorphic coordinates(n, ξU ) on π−1(U ) = KXP |U , where n ∈ Ru (PΘ)− and ξU : C→ C. From this we can take a localsection sU : U ⊂ XP → GC dened by sU (nP ) = n ∈ G

C, ∀nP ∈ U .

95

Once on the open setU ⊂ XP we haveωXP |U =√−1∂∂(s∗U (φ)), the (1, 1)-horizontal component

of ωCY is (locally) dened by

ωXP |U =

√−1

2π ∂∂ log( ∏α∈Σ\Θ

| |nv+ωα | |2〈δP ,h∨α 〉

).

Now, according to [143] we have ∇ξU = dξU + ξUπ∗AU , for some Chern connection locally

described by ∇|U = d +AU . We claim that on the (1, 1)-vertical component of ωCY we have

∇ξU = dξU + ξU ∂ log( ∏α∈Σ\Θ

| |nv+ωα | |2〈δP ,h∨α 〉

).

In fact, if we take the Hermitian structure on KXP dened by the collection of functions (qi )i∈Ias in 6.1.1, namely

H ((x ,v ), (x ,w )) = e2πφ (si (x ))vw,

see 6.1.2. On coordinates (n, ξU ) in π−1(U ) = KXP |U we have H = e2πφ (n)ξU ξU . erefore, if wetake a local section σU : U ⊂ XP → KXP , dened by σU (nP ) = (sU (nP ), 1) = (n, 1) ∈ KXP |U , wehave H (σU ,σU ) = e2πφsU , thus we obtain

∇|U = d + ∂ logH (σU ,σU ) = d + 2π∂(s∗U (φ)).

It follows that ∇ξU = dξU + 2πξU ∂(s∗U (φ)). Hence the Kahler form ωCY is (locally) determinedby the forms

• ωXP |U =

√−1

2π ∂∂ log( ∏α∈Σ\Θ

| |nv+ωα | |2〈δP ,h∨α 〉

), (Horizontal)

• ∇ξU = dξU + ξU ∂ log( ∏α∈Σ\Θ

| |nv+ωα | |2〈δP ,h∨α 〉

). (Vertical)

us the Ricci-at structure dened byωCY can be completely described by means of the quasi-potential φ : GC → R, from these we have the desired result.

One of the most important feature of the above theorem is that it allows us to assign to eachsubset Θ ⊂ Σ a complete non-compact Calabi-Yau manifold for which we have the Calabi-Yaumetric completely determined by elements of the Lie theory.

6.2 Examples of complete non-compact Calabi-Yau man-ifolds via Lie theory

Now we will apply eorem 6.1.2 in a more concrete situation in order to provide a huge classof examples of non-compact Calabi-Yau manifolds obtained via Calabi’s technique. We startby describing the construction on the building block of the general seing

96

Example 6.2.1. Consider GC = SL(2,C) and P = B as in the Example D.1.2. In this case wehave

XB = SL(2,C)/B = CP1 and ωCP1 =i

π∂∂ log(1 + |z |2),

furthermore we have

−KCP1 = T (1,0)CP1 = TCP1.

It follows that KCP1 = (TCP1)∗. From the last theorem we can equip KCP1 = O (−2) with acomplete Ricci-Flat metric induced by the Kahler form

ωCY = (2πu +C ) 12π ∗ωCP1 − 1

2 (2πu +C )− 1

2 i∇ξ ∧ ∇ξ .

If we take the tivializing open set U = N −x0 ⊂ CP1 and consider local coordinates (z, ξ ) onO (−2) |U , we have the following local expression for ωCY

ωCY =

√2π |ξ |2 +C

π (1 + |z |2)2 idz ∧ dz −1

2√

2π |ξ |2 +Ci(dξ +

ξzdz

1 + |z |2)∧

(dξ +

ξzdz

1 + |z |2). (6.2.1)

Once this example is quite simple, let us verify the Ricci-atness condition.

Consider the tautological form θ ∈ Ω(1,0) (O (−2)), if we take a nonvanishing unitary localsection σ : U → O (−2) the local expression of Ω = dθ ∈ Ω(2,0) (O (−2)) is given by

Ω = dξ ∧ σ ,

see the proof of eorem 6.1.2. A straightforward calculation shows us that

2!Ω ∧ Ω = i22| |Ω | |2ωCYω

2CY = | |Ω | |

2ωCYω

2CY ,

from the Equation 6.2.1 we have ω2CY =

−i2 ωCP1 ∧ dξ ∧ dξ , thus we obtain

2!Ω ∧ Ω =−i

2 | |Ω | |2ωCYωCP1 ∧ dξ ∧ dξ .

On the other hand we have

Ω ∧ Ω =(dξ ∧ σ

)∧(dξ ∧ σ

)= −i | |σ | |2ωCP1ωCP1 ∧ dξ ∧ dξ ,

where we used σ ∧ σ = i | |σ | |2ωCP1ωCP1 . Since | |σ | |2ωCP1 = 1 we obtain

| |Ω | |2ωCY = 4 =⇒ ∇ωCYΩ = 0.

97

erefore, Hol(∇ωCY ) ⊂ SU(2) and it follows that (O (−2),ωCY ,Ω) is a noncompact Calabi-Yaumanifold.

e above prototype is quite interesting on its own right. In fact, it is worthwhile to point outthat besides of interesting geometric features of the metric described above, the constructionof Ricci-at metrics on O (−2) is also interesting for its applications in mathematical physics.According to [143, p. 109] the Ricci-at metric described above is the Eguchi-Hanson metricon O (−2), see [48] [47]. Moreover, this manifold also denes an example of toric Calabi-Yausurface which has many applications in mirror symmetry, see for example [30]. We also pointout that since SU(2) = Sp(1) it follows that the metric described above is also a hyperkahlermetric.

Our next examples will be an illustration of how to compute Calabi’s metric directly fromeorem 6.1.2. us we will not be worried about the description of the holomorphic volumeform. Actually, it is just the exterior derivative of the tautological form, see for more detailsAppendix D.2.

Example 6.2.2. Consider now the case GC = SL(3,C), here we will use the same choice ofCartan subalgebra and conventions for the simple root system as in the Example D.1.3 on page239, therefore, the simple root system can be described as follows

Σ =α1 = ϵ1 − ϵ2,α2 = ϵ2 − ϵ3

,

the set of positive roots in this case is given by Π+ = α1,α2,α1 + α2. By taking P = Pα2

XP = CP2.

In the above case we have the following quasi-potential φ : SL(3,C) → R

φ (д) =〈δP ,h

∨α1〉

2π log | |дv+ωα1| |2.

Since V (ωα1 ) = C3 and v+ωα1= e1, on the open set dened by the opposite big cell U = B−x0 ⊂

XP we have ωCP2 ∈ c1(CP2) given by

ωCP2 =〈δP ,h

∨α1〉

2π i∂∂ log(

1 + |z1 |2 + |z2 |

2)

.

Here we have used the parameterization of U = B−x0 ⊂ XP in complex coordinates given bythe matrices

n =

1 0 0

z1 1 0

z2 0 1

,

98

with z1, z2 ∈ C. As in the Example D.1.3 we have 〈δP ,h∨α1〉 = 3, therefore by applying the resultof eorem 6.1.2 we obtain the following (local) expression for the Kahler form associated tothe Ricci-at metric on KCP2 = O (−3) provided by Calabi’s technique

ωCY = (2π |ξ |2 +C ) 13ωCP2 − 1

3 (2π |ξ |2 +C )−

23 i∇ξ ∧ ∇ξ ,

for some constant C > 0. Here we consider the local coordinates (n, ξ ) ∈ O (−3) |U , thus wehave in the above expression the following description

ωCP2 =3i2π ∂∂ log

(1 +

2∑k=1|zk |

2)

and ∇ξ = dξ + 3ξ∂ log(

1 +2∑

k=1|zk |

2). (6.2.2)

Hence (O (−3),ωCY ) denes an example of noncompact complete Calabi-Yau manifold. As inthe previous example Calabi-Yau manifold obtained from the holomorphic line bundle O (−3) →CP2 is a very interesting example of noncompact Calabi-Yau threefold which has many appli-cations in mathematical physics, more precisely string theory, see for example [1, p. 429-431]and [34].

e next example is a brief generalization of the description which we did above forXP = CP2.

Example 6.2.3. Let us briey describe the application of eorem 6.1.2 forXP = CPn, here weconsider the same basic Lie-eoretical data which we have used in the Example D.1.3 on page239. As we have seen, in this case we have P = PΣ\α1 and the expression of ωCPn ∈ c1(CPn )over the opposite big cell is

ωCPn =(n + 1)

2π i∂∂ log(

1 +n∑l=1|zl |

2)

.

us from eorem 6.1.2 we have the following local description for the Calabi-Yau metric onKCPn = O (−n − 1)

ωCY = (2π |ξ |2 +C ) 1n+1ωCPn −

1n+1 (2π |ξ |

2 +C )−n

n+1 i∇ξ ∧ ∇ξ ,

for some constant C > 0, where ωCPn is locally described as above, and ∇ξ is given by

∇ξ = dξ + (n + 1)ξ∂ log(

1 +n∑l=1|zl |

2). (6.2.3)

It follows that (O (−n−1),ωCY ) denes a complex (n+1)-dimensional noncompact Calabi-Yaumanifold. is example inspired us to use elements of Lie theory to describe Calabi’s methodfor the class of Kahler-Einstein Fano manifolds dened by complex ag manifolds, see [26, p.284-285 ].

An important fact concerned with the above well known examples is that the complex agmanifold CPn is also a toric manifold. Since the Kahler structure of toric manifolds are com-pletely determined by combinatorial elements, see for example [66], in some sense it makes

99

the application of Calabi’s technique somewhat manageable.

e result which we established in eorem 6.1.2 allows us to describe explicitly a huge classof non-compact complete Calabi-Yau manifolds beyond the toric cases. In what follows wedescribe some results which can be obtained from eorem 6.1.2.

First, notice that if we consider the Wallach spaceW6 dened by

W6 = SU(3)/T 2,

we also can writeW6 = SL(3,C)/B, thus we have the following result

Proposition 6.2.1. e total space of the canonical bundle KW6 over the Wallach space W6 =SL(3,C)/B admits a Calabi-Yau metric ωCY (locally) dened by

ωCY = (2π |ξ |2 +C ) 14ωXB −

14 (2π |ξ |

2 +C )−34 i∇ξ ∧ ∇ξ ,


ωXB =1π

[i∂∂ log

(1 +

2∑k=1|zk |

2)+ i∂∂ log

(1 + |z3 |

2 + |z1z3 − z2 |2)], (6.2.4)

and

∇ξ = dξ + 2ξ[∂ log

(1 +

2∑k=1|zk |

2)+ ∂ log

(1 + |z3 |

2 + |z1z3 − z2 |2)]. (6.2.5)

Proof. Since the metric

ωCY = (2π |ξ |2 +C ) 14ωXB −

14 (2π |ξ |

2 +C )−34 i∇ξ ∧ ∇ξ ,

is the Ricci-at metric which we have from Calabi’s technique, we just need to apply eorem6.1.2 to verify its local expression.

Here again we consider for GC = SL(3,C) the same choice of Cartan subalgebra and conven-tions for the simple root system as in the Example D.1.3 on page 239. erefore the simpleroot system can be described as follows

Σ =α1 = ϵ1 − ϵ2,α2 = ϵ2 − ϵ3

,

and the set of positive roots in this case is given by Π+ = α1,α2,α1 +α2. We st observe thatfor XB = SL(3,C)/B the quasi-potential φ : SL(3,C) → R is given by

φ (д) =〈δB,h

∨α1〉

2π log | |дv+ωα1| |2 +

〈δB,h∨α2〉

2π log | |дv+ωα2| |2.

100

Here we have V (ωα1 ) = C3 and V (ωα2 ) =∧2(C3), and the highest-weight vectors are, respec-

tively, given by

v+ωα1= e1 and v+ωα2

= e1 ∧ e2,

we also consider the canonical basis ej for C3 and the basis ei ∧ ej i<j for∧2(C3).

In order to compute the local expression of ωXB ∈ c1(XB ), we take the open set dened by theopposite big cell U = B−x0 ⊂ XB , on which we have coordinates

n =

1 0 0

z1 1 0

z2 z3 1

,

with z1, z2, z3 ∈ C. A straighforward calculation shows us that (locally) we have

ωXB =〈δB,h

∨α1〉

2π i∂∂ log(

1 + |z1 |2 + |z2 |

2)+〈δB,h

∨α2〉

2π i∂∂ log(

1 + |z3 |2 + |z1z3 − z2 |

2)

,

from the Cartan matrix of sl (3,C) we obtain 〈δB,h∨α1〉 = 〈δB,h∨α2〉 = 2, notice that in this case

we have δB = 2α1 + 2α2. erefore, from eorem 6.1.2 we obtain a Calabi-Yau metric on KXB

(locally) given by

ωCY = (2π |ξ |2 +C ) 14ωXB −

14 (2π |ξ |

2 +C )−34 i∇ξ ∧ ∇ξ ,

for some C > 0, here we consider coordinates (n, ξ ) ∈ KXB |U in order to obtain the followingexpressions

ωXB =1π

[i∂∂ log

(1 +

2∑k=1|zk |

2)+ i∂∂ log

(1 + |z3 |

2 + |z1z3 − z2 |2)], (6.2.6)

and

∇ξ = dξ + 2ξ[∂ log

(1 +

2∑k=1|zk |

2)+ ∂ log

(1 + |z3 |

2 + |z1z3 − z2 |2)]. (6.2.7)

From these we have a noncompact complete Calabi-Yau manifold (KXB ,ωCY ) with complexdimension 4.

e interesting feature of the above result is to show how the hidden Lie-theoretical structuresallows us to compute explicitly the expression of the Ricci-at metric obtained by Calabi’smethod.

e next result can be seen as a prototype for the application of eorem 6.1.2 on Kahlermanifolds dened by complex Grassmannians.

Proposition 6.2.2. e total space of the canonical bundleKGr(2,C4) over the complex Grassman-nian Gr(2,C4) admits a complete Calabi-Yau metric ωCY (locally) described by

101

ωCY = (2π |ξ |2 +C ) 15ωGr(2,C4) −

15 (2π |ξ |

2 +C )−45 i∇ξ ∧ ∇ξ ,



(1 +

4∑k=1|zk |

2 +∣∣ det

z1 z3

z2 z4

∣∣2), (6.2.8)

and

∇ξ = dξ + 4ξ∂ log(

1 +4∑

k=1|zk |

2 +∣∣ det

z1 z3

z2 z4

∣∣2), (6.2.9)

Proof. Consider GC = SL(4,C), here we use the same choice of Cartan subalgebra and con-ventions for the simple root system as in Example D.1.3 on page 239. Since our simple rootsystem is given by

Σ =α1 = ϵ1 − ϵ2,α2 = ϵ2 − ϵ3,α3 = ϵ3 − ϵ4

,

by taking Θ = Σ\α2 we obtain XP = Gr(2,C4), for P = PΘ. Notice that in this case we have

Pic(Gr(2,C4)) = Z[ηα2],

see Proposition D.1.6 on page 227, thus from Proposition D.1.9 on page 234 it follows that

−KGr(2,C4) = L⊗〈δP ,h

∨α2 〉

χωα2.

For this case by considering our Lie-theoretical conventions, we have

Π+\〈Θ〉+ =α2,α1 + α2,α2 + α3,α1 + α2 + α3

,

hence

δP =∑

α∈Π+\〈Θ〉+

α = 2α1 + 4α2 + 2α3.

By means of the Cartan matrix of sl (4,C) we obtain

〈δP ,h∨α2〉 = 4 =⇒ −KGr(2,C4) = L⊗4

χωα2.

Here we will use the following notation

L⊗kχωα2= Oα2 (k ),

for every k ∈ Z, therefore we can write KGr(2,C4) = Oα2 (−4).

In order to calculate the local expression of ωGr(2,C4) ∈ c1(Gr(2,C4)), we observe that thequasi-potential φ : SL(4,C) → R in this case is given by

φ (д) =〈δP ,h

∨α2〉

2π log | |дv+ωα2| |2 =

2π

log | |дv+ωα2| |2,

102

whereV (ωα2 ) =∧2(C4) andv+ωα2

= e1∧e2, here we x the basis ei∧ej i<j forV (ωα2 ) =∧2(C4).

Similarly to the previous examples we consider the open set dened by the opposite big cellU = B−x0 ⊂ Gr(2,C4). In this case we have local coordinates n ∈ U given by

n =

1 0 0 0

0 1 0 0

z1 z3 1 0

z2 z4 0 1

,

with zi ∈ C, i = 1, 2, 3, 4, thus we obtain

φ (n) =2π

log(

1 +4∑

k=1|zk |

2 +∣∣ det

z1 z3

z2 z4

∣∣2).

From the above calculations we have the following local expression forωGr(2,C4) ∈ c1(Gr(2,C4))


(1 +

4∑k=1|zk |

2 +∣∣ det

z1 z3

z2 z4

∣∣2). (6.2.10)

Now we can apply eorem 6.1.2 in order to get a Ricci-at metric on Oα2 (−4) locally describedby

ωCY = (2π |ξ |2 +C ) 15ωGr(2,C4) −

15 (2π |ξ |

2 +C )−45 i∇ξ ∧ ∇ξ ,

for some constant C > 0, where ωGr(2,C4) can be wrien as above in 6.2.10 and ∇ξ is locallygiven by

∇ξ = dξ + 4ξ∂ log(

1 +4∑

k=1|zk |

2 +∣∣ det

z1 z3

z2 z4

∣∣2). (6.2.11)

us (Oα2 (−4),ωCY ) denes a complete noncompact Calabi-Yau manifold of complex dimen-sion 5.

e ideas of the above example can be easily extended for any complex Grassmannian Gr(k,Cn ).Actually, by xing the notations as in Example D.1.3 on page 239 we can describe the completeRicci-at metric on the total space of the canonical bundle

KGr(k,Cn ) → Gr(k,Cn ),

by means of the Lie-theoretical objects like fundamental representations and simple root sys-tem, notice that we have in this case Gr(k,Cn ) = SL(n,C)/PΣ\αk .

Now we describe a more general result related to full ag manifolds of SL(n + 1,C)

eorem 6.2.3. ConsiderGC = SL(n+1,C) and B ⊂ GC = SL(n+1,C) (Borel susbgroup). enthe total space of the canonical bundleKXB over the complex full ag manifoldXB = SL(n+1,C)/Badmits a complete Calabi-Yau metric ωCY (locally) described by

103

ωCY = (2π |ξ |2 +C )2

n (n+1)+2ωXB −2

n(n+1)+2 (2π |ξ |2 +C )−

n (n+1)n (n+1)+2 i∇ξ ∧ ∇ξ


• ωXB =

n∑k=1

〈δB,h∨αk〉

2π i∂∂ log(

1 +∑I,I0,k

∣∣∆(k )I (n−(z))


• ∇ξ = dξ +n∑

k=1〈δB,h

∨αk〉ξ∂ log

(1 +

∑I,I0,k

∣∣∆(k )I (n−(z))


Proof. Consider GC = SL(n + 1,C), from this we x the same Lie-theoretical data as in theExample D.1.3. By taking Θ = ∅, we have the complex ag manifold XB = SL(n + 1,C)/B,where B ⊂ SL(n + 1,C) is the standard Borel subgroup. From this by Calabi’s technique andour eorem 6.1.2 it follows that the metric

ωCY = (2π |ξ |2 +C )2

n (n+1)+2ωXB −2

n(n+1)+2 (2π |ξ |2 +C )−

n (n+1)n (n+1)+2 i∇ξ ∧ ∇ξ ,

denes a Ricci-at metric on KXB . In order to verify the second assertion let us introduce somenotations. Let U = N −B ⊂ XB be the opposite big cell, where N − = exp(n−), this open set isparameterized by the holomorphic coordinates

n =

1 0 0 · · · 0

z21 1 0 · · · 0

z31 z32 1 · · · 0...

......

. . ....

zn+1,1 zn+1,2 zn+1,3 · · · 1

,

where n ∈ N − e z = (zij ) ∈ Cn (n+1)

2 , therefore we can denote by n = n−(z) ∈ N −.

Consider now the following ideas, given д ∈ SL(n + 1,C) we have

д =

д11 д12 д13 · · · д1,n+1

д21 д22 д23 · · · д2,n+1

д31 д32 д33 · · · д3,n+1...

......

. . ....

дn+1,1 дn+1,2 дn+1,3 · · · дn+1,n+1

.

We dene for each subset I = i1 < · · · < ik ⊂ 1, · · · ,n + 1, with 1 ≤ k ≤ n, the followingfunctions ∆(k )

I : SL(n + 1,C) → C

104

∆(k )I (д) = det

дi11 дi12 · · · дi1k

дi21 дi22 · · · дi2k...

.... . .

...

дik1 дik2 · · · дikk

.

By seing I0,k = 1, 2, . . . ,k , we have

д · (e1 ∧ . . . ∧ ek ) = ∆(k )I0,k

(д)e1 ∧ . . . ∧ ek +∑I,I0,k

∆(k )I (д)ei1 ∧ . . . ∧ eik , (6.2.12)

for ei1 ∧ . . . ∧ eik ∈ V (ωαk ) =∧k (Cn+1) and I = i1 < · · · < ik ⊂ 1, · · · ,n + 1.

From eorem 6.1.2 we obtain the following expression for the quasi-potential φ : SL(n +1,C) → R

φ (д) =1

2π log( n∏

k=1| |д(e1 ∧ . . . ∧ ek ) | |

2〈δB ,h∨αk 〉)

,

notice that we can also write

φ (д) =

n∑k=1

〈δB,h∨αk〉

2π log(| |д(e1 ∧ . . . ∧ ek ) | |

2)

,

remember that δB =∑α∈Π+

α .

Now by taking a local section sU : U ⊂ XB → SL(n + 1,C), where U = N −B ⊂ XB , dened by

sU (n−(z)B) = n−(z) ∈ SL(n + 1,C), ∀n−(z)B ∈ U ,

by means of the above section we obtain the following local expression

ωXB |U = i∂∂(s∗U (φ)).

Hence we need to determine s∗U (φ) in order to compute the local expression of ωXB and ωCY .By denition we have

s∗U (φ) (n−(z)B) = φ (sU (n

−(z)B)) = φ (n−(z)),

from Equation 6.2.12 and the functions ∆(k )I : SL(n + 1,C) → C we get

φ (n−(z)) =

n∑k=1

〈δB,h∨αk〉

2π log(

1 +∑I,I0,k

∣∣∆(k )I (n−(z))

∣∣2),

thus we have

ωXB |U =

n∑k=1

〈δB,h∨αk〉

2π i∂∂ log(

1 +∑I,I0,k

∣∣∆(k )I (n−(z))

∣∣2).

105

Now we consider local coordinates (n(z), ξ ) ∈ KXB |U . On these coordinates we have

∇ξ |U = dξ + 2πξ∂(s∗U (φ)),

therefore we obtain

∇ξ |U = dξ +

n∑k=1〈δB,h

∨αk〉ξ∂ log

(1 +

∑I,I0,k

|∆(k )I (n−(z)) |2

).

From eorem 6.1.2 we have ωCY on the open set KXB |U completelly determined by the forms

• ωXB |U =

n∑k=1

〈δB,h∨αk〉

2π i∂∂ log(

1 +∑I,I0,k

∣∣∆(k )I (n−(z))


• ∇ξ |U = dξ +n∑

k=1〈δB,h

∨αk〉ξ∂ log

(1 +

∑I,I0,k

∣∣∆(k )I (n−(z))


Notice that we can calculate 〈δB,h∨αk 〉, 1 ≤ k ≤ n, by means of the Cartan matrix which denessl (n + 1,C).

Remark 6.2.1. It is worth to point out that the above description of Calabi’s metric on canonicalbundles of complex ag manifolds can be reproduced for complex ag manifolds associated toGC = SO(n,C) and Sp(2n,C).

e diculty which we have in obtaining explicit formulas in the above cases lies in the descrip-tion of fundamental representations. Actually, in the case GC = SO(n,C) we have fundamentalrepresentations of the form

∧j (Cn ) and spinor representations. In the case of GC = Sp(2n,C),unlike the case SL(n + 1,C), the fundamental representations are not the whole space

∧j (C2n )when j , 1, see [144] for more details about fundamental representations of classical simple Liealgebras.

We nish our discussion with a basic description of how the result of eorem 6.1.2 can beapplied in some complex ag manifolds on which the underlying complex simple Lie algebrais not of the classical type.

Example 6.2.4. Let us give some brief idea of how eorem 6.1.2 can be applied in some agmanifolds associated to nonclassical complex simple Lie algebras.

• e Cayley plane: Consider the exceptional complex Lie group GC = E6. Our conven-tions for the Lie algebra structure are according to [144] and our approach of the Cayleyplane are according to [91].

Figure 6.1: Dynkin diagram associated to the Lie algebra E6.

106

e Cayley plane is the complex ag manifold OP2 obtained from Θ = Σ\α5, namely

OP2 = E6/PΘ.

for this manifold we have dimC(OP2) = 16. From propositions D.1.6 and D.1.6 we have

Pic(OP2) = Z[ηα5],

thus from Proposition D.1.9 we obtain

−KOP2 = L⊗〈δPΘ ,h

∨α5 〉

χωα5.

e Kahler form ωOP2 ∈ c1(OP2) can be described by means of the quasi-potentialφ : E6 → R given by

φ (д) =〈δPΘ,h

∨α5〉

2π log | |дv+ωα5| |2.

erefore, we can apply eorem 6.1.2 and obtain a complete Ricci-at metric on KOP2

with associated Kahler form ωCY described by the expression

ωCY = (2πu +C ) 117π ∗ωOP2 − 1

17 (2πu +C )− 16

17 i∇ξ ∧ ∇ξ ,

for some constant C > 0. us (KOP2,ωCY ) denes a complete noncompact Calabi-Yaumanifold with complex dimension 17. e computation of the local expression of ωCY

in this case is highly non-trivial since the matrix realization of E6 is quite complicated.

• Freudenthal variety: Consider the exceptional complex Lie group GC = E7. As beforeour conventions for the Lie algebra structure are according to [144].

Figure 6.2: Dynkin diagram associated to the Lie algebra E7.

e Freudenthal variety is dened by the 27-dimensional complex ag manifold associ-ated to Θ = Σ\α6, namely E7/PΘ. By the same arguments of the previous example wehave

Pic(E7/PΘ) = Z[ηα6],

thus we obtain −KE7/PΘ = L⊗〈δPΘ ,h

∨α6 〉

χωα6. From these the quasi-potential φ : E7 → R which

denes ωE7/PΘ ∈ c1(E7/PΘ) is given by

107

φ (д) =〈δPΘ,h

∨α6〉

2π log | |дv+ωα6| |2.

Now if we apply eorem 6.1.2 on KE7/PΘ we obtain a complete Ricci-at metric withassociated Kahler form

ωCY = (2πu +C ) 128π ∗ωE7/PΘ −

128 (2πu +C )

− 2728 i∇ξ ∧ ∇ξ ,

for some constant C > 0. From these we obtain by eorem 6.1.2 a complete noncom-pact Calabi-Yau manifold (KE7/PΘ,ωCY ). As the previous case the computation of thelocal expression of the Kahler form ωCY above is highly non-trivial since the underlyingcomplex Lie algebra in this context has dimension 133.

• Full ag manifold associated to G2 : Consider the exceptional complex Lie groupGC = G2. e full ag manifold associated to G2 is obtained by taking Θ = ∅, thus wehave P = B and our complex ag manifold in this case is G2/B.

Figure 6.3: Geometric representation of the G2-root system.

As we can see from the above gure we have dimC(G2/B) = 6, furthermore, in this casethe Picard group has two generators, namely

Pic(G2/B) = Z[ηα1] + Z[ηα2],

thus from Proposition D.1.9 we obtain

−KG2/B = L⊗〈δB ,h

∨α1 〉

χωα1⊗ L

⊗〈δB ,h∨α2 〉

χωα2.

Our Kahler form ωG2/B ∈ c1(G2/B) is dened by the quasi-potential φ : G2 → R whichcan be wrien as

φ (д) =〈δB,h

∨α1〉

2π log | |дv+ωα1| |2 +

〈δB,h∨α2〉

2π log | |дv+ωα2| |2.

108

For the complex Lie group G2 we have that V (ωα1 ) is a 14-dimensional representationdened by the adjoint representation, and V (ωα2 ) is a 7-dimensional representation de-ned by the action ofG2 on the imaginary octonions. We can also compute 〈δB,h∨α1〉 and〈δB,h

∨α2〉 by means of the Cartan matrix associated to G2.

From the above comments and from eorem 6.1.2 we obtain a complete Ricci-at metricon KG2/B associated to the Kahler form

ωCY = (2πu +C ) 17π ∗ωG2/B −

17 (2πu +C )

− 67 i∇ξ ∧ ∇ξ ,

for some constant C > 0. us (KG2/B,ωCY ) denes a complete noncompact Calabi-Yaumanifold. Compared with the two previous exceptional cases the local expression of theabove Kahler form can be computed if we considerG2 ⊂ SO(7,C), see [3] for a completedescription of this matrix realization.

As we have seen in the last examples, for ag manifolds associated to exceptional complexsimple Lie algebras the application of eorem 6.1.2 becomes more complicated when we tryto compute the local expression of the Ricci-at metric obtained from Calabi’s method. Sincefor these types of Lie algebras we do not have a manageable matrix realization, we can notdirectly derive a suitable local expression for the Kahler metric on the open set dened by theopposite big cell. Moreover, the fundamental representations associated to these algebras arehighly non-trivial.

109

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119

Appendices

120

Appendix ANecessary condition for integrability ofcollective Hamiltonian systems

In this chapter we will establish the necessary condition for integrability of Hamiltonian sys-tems dened by collective Hamiltonians. All the results which we will cover throughout thissection can be found in [74], further results related to the subjects that we will deal in this sec-tion also can be found in [75], [73], [96] and [117]. Our purpose is to investigate some issuesconcerned with collective Hamiltonians and provide a unied approach by gathering togetherimportant results related to the study of integrable systems in coadjoint orbits of compact Liegroups.

We would like to point out that although the results which we will cover here are well known,there is a lack of texts with detailed proofs for some results. erefore, besides our own inter-est we hope to contribute with an introductory guide text in this topic. It is worth to observethat in the Subsection A.4 we will provide an alternative proof for the following main result

eorem 2. Let (M,ω,G,Φ) be a Hamiltonian G-space which admits an integrable system de-ned by collective Hamiltonians Hj = Φ∗(Ij ), with Ij ∈ C

∞(g∗) for j = 1, . . . ,n. en the algebraof G-invariant functions on M is commutative under the Poisson bracket, i.e. M is a multiplicityfree space.

Besides the proof which can be found in [74] and [75, 359-362] we also will provide a slightlymore direct proof for the above result.

A.1 Basic notations and conventions

Unless otherwise stated, we will x a Hamiltonian G-space (M,ω,G,Φ) with an equivariantmoment map Φ : M → g∗. Given X ∈ g, from the equation

d〈Φ,X 〉 + ιδτ (X )ω = 0,

remember that δτ : g → Γ(TM ) denotes the innitesimal action, we get a Lie algebra anti-homomorphism

Φ∗ : g → C∞(M ),

121

such that Φ∗(X ) = 〈Φ,X 〉, ∀X ∈ g. erefore, we obtain

Φ∗([X ,Y ]) =Φ∗(X ),Φ∗(Y )

M

,

for all X ,Y ∈ g, see for instance [141, p. 496, Proposition 10.1.14].

Remark A.1.1. It is worthwhile to point out that if we consider the natural inclusion

g → C∞(g∗),

the map Φ∗ : g → C∞(M ) is exactly the pullback map restricted to g ⊂ C∞(g∗).

We will follow the notation used in [74], so if we consider the natural isomorphism

ω[ : TM → TM∗,

dened by

〈ω[ (v ),w〉 := ωp (w,v ),

for every v,w ∈ TpM , here 〈·, ·〉 denotes the natural pairing between TpM∗ and TpM . At each

point p ∈ M we can dene the map

up : g → TpM , such that up (X ) = δτ (X )p ,

from this we have the map

Ψp : g → TpM∗,

such that Ψp (X ) = ω[ (up (X )), for every X ∈ g.

A.2 Moment map and some of its properties

Following the notations and conventions of the previous subsection, the main purpose here iscovering some results of [74] fullling the details in the proof of some important properties ofthe moment map.

Proposition A.2.1. Let (M,ω,G,Φ) be a Hamiltonian G-space, then for every p ∈ M we have

(DΦ)∗p = Ψp ,

where (DΦ)∗p : g → T ∗pM denotes the adjoint map induced by (DΦ)p .

Proof. Given v ∈ TpM , we have (DΦ)p (v ) ∈ TΦ(p)g∗ g∗. By regarding g ⊂ C∞(g∗), we can

consider for each X ∈ g the induced linear smooth function

FX : g∗ → R, such that FX (ϕ) = 〈X ,ϕ〉.

We notice that

122

〈Ψp (X ),v〉 = ωp (v,δτ (X )p ) = d〈Φ,X 〉p (v ),

however 〈Φ,X 〉 = Φ∗(FX ), thus we have

d〈Φ,X 〉p (v ) = Φ∗(dFX )p (v ) = (dFX )Φ(p) ((DΦ)p (v )).

Since FX is the linear map induced by X , we obtain

(dFX )Φ(p) ((DΦ)p (v )) = 〈X , (DΦ)p (v )〉,

it follows that

〈Ψp (X ),v〉 = 〈X , (DΦ)p (v )〉 = 〈(DΦ)∗p (X ),v〉,

for every X ∈ g and v ∈ TpM .

Now we will suppose thatW = Φ(M ) ⊂ g∗ is a submanifold, furthermore we will also supposethat

(DΦ)p (TpM ) = TξW ,

where ξ = Φ(p) ∈W . We have the following proposition

Proposition A.2.2. Under the previous hypotheses, for each p ∈ M we have

ker(Ψp ) = (TξW )0,

where Φ(p) = ξ , and (TξW )0 ⊂ g denotes the annihilator of TξW ⊂ g∗.

Proof. e proof follows from the fact that (Im((DΦ)p ))0 = ker((DΦ)∗p ). Actually, we have

X ∈ ker(Ψp ) ⇐⇒ 〈Ψp (X ),v〉 = 〈(DΦ)∗p (X ),v〉 = 0, ∀v ∈ TpM ,

which is equivalent to X ∈ (Im((DΦ)p ))0 = ker((DΦ)∗p ) = ker(Ψp )

Now we denote by

gp =X ∈ g

∣∣∣ δτ (X )p = 0

,

the isotropy subalgebra associated to p ∈ M . From the previous comments we have

Proposition A.2.3. Under the previous hypotheses, we have

gp = ker(Ψp ),

for every p ∈ M .

Proof. e proof for this fact follows from the denition of Ψp . In fact, we have

〈Ψp (X ),v〉 = ωp (v,δτ (X )p ),

123

for every X ∈ g and v ∈ TpM . erefore, if δτ (X )p = 0 we obtain

〈Ψp (X ),v〉 = 0,

for each v ∈ TpM , which implies that X ∈ ker(Ψp ).

Given ξ ∈ g∗ we denote the isotropy subalgebra of ξ with respect to the coadjoint action by

hξ =X ∈ g

∣∣∣ ad∗(X )ξ = 0

.

From the equivariance of Φ : M → g∗, a straightforward calculation shows us that

gp ⊂ hξ ,

where ξ = Φ(p), furthermore we have the following result

PropositionA.2.4. e isotropy subalgebra gp is an ideal of hξ , where ξ = Φ(p), for everyp ∈ M .

Proof. Under the previous hypothesis we have W = Φ(M ) as a submanifold of g∗, and astraightforward calculation shows us that

W =⋃q∈M

O (Φ(q)),

where O (Φ(q)) ⊂ g∗ denotes the coadjoint orbit of Φ(q). It follows that G leavesW invariantby the coadjoint action. We denote the isotropy subgroup of ξ ∈ g∗ by

Hξ =д ∈ G

∣∣∣ Ad∗(д)ξ = ξ

,

from the above comments we have that Hξ leaves W invariant. If we consider ξ = Φ(p), forp ∈ M , we have that Hξ also leavesTξW invariant. Now we notice that forX ∈ (TξW )0, д ∈ Hξ

and η ∈ TξW ⊂ g∗, we have

〈Ad(д)X ,η〉 = 〈X ,Ad∗(д−1)η〉 = 0,

it follows that Hξ also leaves (TξW )0 ⊂ g invariant. However, since gp = (TξW )0, we have[hξ , gp

]⊂ gp ,

then it follows that gp ⊂ hξ is an ideal.

Now given p ∈ M we consider the subset Vξ = Φ−1(O (ξ )) ⊂ M , where ξ = Φ(p). We have thefollowing technical result

Proposition A.2.5. ere exists an open dense subset M0 ⊂ M such that for every q ∈ M0 wecan nd a neighbourhoodU of q such that VΦ(q) ∩U is a submanifold. Furthermore, the tangentspace at x ∈ VΦ(q) ∩U is given by

TxVΦ(q) = Tx(VΦ(q) ∩U

)= (DΦ)−1

x

(TΦ(x )O (Φ(q))

).

124

Proof. e proof for this proposition and further results in this context can be found in [96].We also suggest the exposition about this fact presented in [75, p. 196-200].

In what follows given p ∈ M , we denote ξ = Φ(p), and we will assume that Vξ = Φ−1(O (ξ )) isa submanifold of M , such that

TxVξ = (DΦ)−1x

(TΦ(x )O (ξ )

),

for every x ∈ Vξ .

RemarkA.2.1. In the previous proposition we have seen that at least locally for generic points theconcept of cleanly intersection for Vξ = Φ−1(O (ξ )) holds. e denition of this kind of intersectionstates that given a smooth map between smooth manifolds F : M → N andW ⊂ N an embeddedsubmanifold, then we say that F intersectsW cleanly if the two conditions below are satised

• F−1(W ) is a submanifold of M ,

• At each x ∈ F−1(W ), TxF−1(W ) = (DF )−1x (TF (x )W ),

see for instance [75, p. 189] for more details. For the case thatO (ξ ) ⊂ g∗ is an embedded subman-ifold our hypothesis above coincides with the cleanly intersection denition.

Proposition A.2.6. e submanifold Vξ ⊂ M is a co-isotropic submanifold and we have thefollowing characterization for its tangent spaces

TxVξ = ux (hΦ(x ) )⊥,

for every x ∈ Vξ , where ux (hΦ(x ) )⊥ denotes the ω-orthogonal space of ux (hΦ(x ) ).

Proof. We rst observe that for every Z ∈ g, we have

(DΦ)x (ux (Z )) = (DΦ)x (δτ (Z )x ) = ad∗(Z )Φ(x ).

It follows that

(DΦ)x (ux (g)) = TΦ(x )O (ξ ),

for every x ∈ Vξ , thus we obtain

(DΦ)−1x (TΦ(x )O (ξ )) = ux (g) + ker((DΦ)x )).

Now we have that if (DΦ)x (v ) = 0 for v ∈ TxM , it implies that

0 = 〈(DΦ)x (v ),Z 〉 = d〈Φ,Z 〉x (v ) = ωx (v,δτ (Z )x ),

for every Z ∈ g, it follows that ker((DΦ)x )) ⊂ ux (g)⊥. Conversely, if v ∈ ux (g)⊥, we have

0 = ωx (v,δτ (Z )x ) = 〈(DΦ)x (v ),Z 〉

for every Z ∈ g, it follows that

ker((DΦ)x )) = ux (g)⊥.

125

erefore we obtain

(DΦ)−1x (TΦ(x )O (ξ )) = ux (g) + ux (g)

⊥.

Now a straightforward calculation shows that

(ux (g) + ux (g)

⊥)⊥= ux (g)

⊥ ∩ ux (g),

see for instance [141, p. 318], it follows that Vξ ⊂ M is a co-isotropic submanifold. Now wenotice the following, given X ,Z ∈ g, we obtain

ωx (ux (Z ),ux (X )) = d〈Φ,X 〉x (ux (Z )) = 〈(DΦ)x (ux (Z )),X 〉 = 〈ad∗(Z )Φ(x ),X 〉,

it follows from the right and le side of the above equation that

ux (g)⊥ ∩ ux (g) = ux (hΦ(x ) ),

hence we have (DΦ)−1x (TΦ(x )O (ξ )) = ux (hΦ(x ) )

⊥.

From the previous result if we denote by (Gp )0 and (Hξ )0 the connected components of theisotropy subgroups associated to p ∈ M and ξ = Φ(p) ∈ g∗, respectively, then the leaf of thenull foliation

x ∈ M → (TxVξ )⊥ = ux (hΦ(x ) ),

through the point p ∈ Vξ is given by the orbit (Hξ )0 · p. It follows that if x ∈ (Hξ )0 · p, thereexists h ∈ Hξ such that x = τ (h)p, here τ : G → Di(M ) denotes the Hamiltonian action.erefore, we have

(Gx )0 = h(Gp )0h−1,

moreover, since (Gp )0 ⊂ (Hξ )0 is a normal subgroup, it follows from A.2.4, we have

(Gx )0 = (Gp )0,

for every x ∈ (Hξ )0 · p. All the results we have covered so far boil down to the followingtheorem

eorem A.2.7. (Kazhdan, Kostant, and Sternberg) If Φ : M → g∗ intersects an adjointorbit O (ξ ) ⊂ g∗ cleanly, then the submanifold manifold Φ−1(O (ξ )) is coisotropic and the leaf ofthe null foliation through a point x ∈ Φ−1(O (ξ )) is the orbit of x by (HΦ(x ) )0.

126

A.3 Moment map, symplectic slice and transversality

Now we establish some basic results related to moment maps and cross sections. Consider thefollowing denition

Denition A.3.1. Let τ : G → Di(M ) be a smooth action of a Lie group on a smooth manifoldM . A slice at a point p ∈ M for the action τ is a Gp-invariant submanifold S ⊂ M for which themap

G ×Gp S → M , [д, s]→ τ (д)s ,

is a dieomorphism onto its image such that the image is an open neighbourhood of the orbitG ·p.

Now we come back to our context of Hamiltonian G-spaces. As before we suppose thatW =Φ(M ) ⊂ g∗ is a submanifold such that (DΦ)p (TpM ) = TΦ(p)W , ∀p ∈ M . Let N ⊂ W be asubmanifold that is a transverse section to the G orbit ofW , i.e. for each ξ ∈W we have thedecomposition

TξW = TξN ⊕ TξO (ξ ).

For every p ∈ Φ−1(ξ ) we take

U = (DΦ)−1p (TξN ),

since (DΦ)p (TpM ) = TξW , we have that Φ−1(N ) ⊂ M is a submanifold hence its tangent spaceat p ∈ Φ−1(N ) is given by U ⊂ TpM . Actually, we have the following result

eoremA.3.2. Suppose that N ⊂W is a submanifold that is a transverse section of theG orbitsofW = Φ(M ), i.e. for every ξ ∈W we have the decomposition

TξW = TξN ⊕ TξO (ξ ).

en Φ−1(N ) ⊂ M is a symplectic submanifold.

Proof. Given p ∈ Φ−1(N ), we have that (DΦ)p (TpM ) = TΦ(p)W , thus

(DΦ)p (TpM ) +TΦ(p)N = TΦ(p)W .

It follows that Φ−1(N ) ⊂ M is a submanifold [69, p. 28]. In order to show that Φ−1(N ) is asymplectic submanifold, we denote ξ = Φ(p), for p ∈ Φ−1(N ), and

U = (DΦ)−1p (TξN ) = TpΦ

−1(N ).

From these we will show now that U ∩U ⊥ = 0. First we notice that ker((DΦ)p ) ⊂ U , sinceker((DΦ)p ) = up (g)⊥, we have

U ⊥ ⊂ up (g).

erefore, if up (X ) ∈ U ⊥, we have

(DΦ)p (up (X )) = ad∗(X )ξ ,

127

it follows that (DΦ)p (up (X )) ∈ TξO (ξ ). en if up (X ) ∈ U ∩U ⊥, we have

(DΦ)p (up (X )) ∈ TξN ∩TξO (ξ ) = 0,

it follows that (DΦ)p (up (X )) = 0. Now will show that in fact up (X ) = 0 for up (X ) ∈ U ∩U ⊥.In order to show this consider Y ∈ g, we have

ωp (up (X ),up (Y )) = d〈Φ,Y 〉p (up (X )) = 〈(DΦ)p (up (X )),Y 〉 = 0,

thus 〈(DΦ)p (up (Y )),X 〉 = 〈ad∗(Y )ξ ,X 〉 = 0, ∀Y ∈ g, it implies that X ∈ (TξO (ξ ))0.

We conclude the proof as follows, since (DΦ)p (TpM ) = TξW , it follows that given η ∈ TξNthere exists v ∈ U such that (DΦ)p (v ) = η, from this since we have up (X ) ∈ U ⊥ we obtain

0 = ωp (v,up (X )) = d〈Φ,X 〉p (v ) = 〈X , (DΦ)p (v )〉 = 〈X ,η〉,

it implies that X ∈ (TξN )0. erefore X ∈ (TξW )0 = gp =⇒ up (X ) = 0, hence U ∩U ⊥ = 0,that is Φ−1(N ) ⊂ M is a symplectic submanifold.

As we have seen in the last theorem the existence of a “global” transverse section of the Gorbits in W is a fundamental hypothesis in its statement. Actually a submanifold N ⊂ Wwhich satises the following properties

• TξW = TξN ⊕ TξO (ξ ), ∀ξ ∈W ,

• If ϕ ∈ N and Ad∗(д)ϕ ∈ N , for some д ∈ G, then д ∈ Hϕ ,

denes a slice at each point ofW , see for example [68, p. 32], further results can be found in[20, p. 82-84].

For the context of Hamiltonian G spaces which we are interested in we do not need to beworried about issues related to the existence of slice at each point, in fact we have the followingresult

eorem A.3.3. Let M be a smooth G-space, where G is a compact Lie group. en there existsa slice Sp ⊂ M at each point p ∈ M .

Proof. e proof can be found in [20, p. 86], see also [159, p. 40].

For a HamiltonianG-space (M,ω,G,Φ) on whichG is a compact Lie group, when we considerg∗ asG-space with respect to the coadjoint action, we have a well known suitable kind of sliceat each point ξ ∈ g∗ which is transversal to O (ξ ) ⊂ g∗.

Let us briey outline the construction of the slice mentioned above at ξ ∈ g∗, more details canbe found in [68, p. 32-34]. At rst we x a (closed) positive Weyl chamber t∗+, where t ⊂ gdenotes the Lie algebra of a xed maximal torus. Given ξ ∈ g∗, since the coadjoint orbit of ξintersects t∗+, without loss of generality we can suppose that ξ ∈ t∗+. From these we set

Sξ = Hξ ·ϕ ∈ t∗+

∣∣ hϕ ⊂ hξ, (A.3.1)

128

this submanifold of g∗ denes a slice at ξ ∈ g∗ which is transversal to the orbit O (ξ ) ⊂ g, wecall Sξ ⊂ g∗ the natural slice at ξ ∈ g∗.

eorem A.3.4. (Cross-section) Let (M,ω,G,Φ) be a Hamiltonian G-space such that G is acompact connected Lie group. Given ξ ∈ g∗ and a natural slice Sξ ⊂ g∗, then the cross-sectionR = Φ−1(Sξ ) is a Hξ -invariant symplectic submanifold of M . Furthermore, the restriction Φ|R is amoment map for the action of Hξ on R.

Proof. e proof for this result can be found in [68, p. 34-37].

So far we have described two dierent kinds of submanifolds which will be important for usin the next subsection and in the next chapter.

In the next section we will provide a proof of the necessary condition for integrability ofHamiltonian systems dened by functions dened by collective Hamiltonians. For this pur-pose we will deal with the coisotropic submanifold Φ−1(O (ξ )) = Vξ ⊂ M , where ξ = Φ(p) forsome p ∈ M .

In the next chapter besides of the coisotropic submanifold Vξ ⊂ M we also will deal withthe symplectic submanifold R = Φ−1(Sξ ), where ξ = Φ(p) for some p ∈ M . As we will seeaerwards these two kinds of submanifolds play an important role in the symplectic geometryinterpretation of the construction of Gelfand-Tsetlin basis for irreducible representations.

A.4 Collective Hamiltonians and necessary condition forintegrability

In this subsection we provide a proof for the main result of this section. All the results we haveexplored in the previous subsections can be found in [75], [73], [96] and [117]. e proof ofthe necessary condition for integrability of Hamiltonian systems dened by collective Hamil-tonians which we will describe in this section is according to [74].

Suppose we have a Hamiltonian G-space (M,ω,G,Φ) such that dim(M ) = 2n, and supposethat I1, . . . , In are smooth functions on g∗ such that Hj = Φ∗(Ij ), for j = 1, . . . ,n, satisfy theintegrability condition. Given p ∈ M we denote by ξ = Φ(p) ∈ g∗ and cj = Ij (ξ ), for j =1, . . . ,n. Now consider the map

H = (H1, . . . ,Hn ) : M → Rn,

since this map denes a submersion in an open dense subset of M , we can suppose that p ∈ Mbelongs to this open dense subset. en we have that the submanifold Φ−1(ξ ) is contained inthe level submanifold H −1(c ) ⊂ M , here c = (c1, . . . , cn ) ∈ Rn.

Remark A.4.1. Here it is worthwhile to point out that we are working under the assumption thatVξ ⊂ M is a submanifold which satisesTpVξ = (DΦ)−1

p (TξO (ξ )). Furthermore, for every x ∈ Vξwe have

• (DΦ)−1x (TΦ(x )O (ξ )) = ux (g) + ux (g)

⊥,

129

• (DΦ)x (up (g)) = TΦ(x )O (ξ ),

thus if we consider the restriction Φ|Vξ : Vξ → O (ξ ), by transversality we have that Φ−1(Φ(x )) ⊂Vξ is a submanifold, for every x ∈ Vξ . We notice that under the above assumption every x ∈ Vξis necessarily a regular point of Φ : M → g∗, it follows from the fact that ker((DΦ)x ) ⊂ TxVξ .

From the previous comments we have Φ−1(ξ ) ⊂ H −1(c ) thus Φ−1(ξ ) is an isotropic subman-ifold, it follows that TpΦ−1(ξ ) = ker((DΦ)p ) = up (g)

⊥ is an isotropic subspace. erefore wehave

up (g)⊥ ⊂ (up (g)

⊥)⊥ = up (g), (A.4.1)

thus we obtain

TpVξ = (DΦ)−1p (TξO (ξ )) = up (g) + up (g)

⊥ = up (g). (A.4.2)

However, it implies thatG acts transitively on Vξ = Φ−1(O (ξ )). It follows that we can consider

Vξ = Φ−1(O (ξ )) = G · p, (A.4.3)

where ξ = Φ(p) and G · p denotes the orbit through the point p ∈ M associated to the Hamil-tonian action τ : G → Di(M ). From these given x ∈ Φ−1(ξ ) ⊂ Φ−1(O (ξ )) there exists д ∈ Gsuch that

τ (д)p = x ⇐⇒ Ad∗(д)ξ = ξ .

e equivalence above follows from the equivariance of the moment map, therefore we havethat the spaces

Φ−1(O (ξ ))/G and Φ−1(ξ )/Hξ , (A.4.4)

are both just a point. Notice that in general the above two reduced spaces are symplectomor-phic, provided that ξ ∈ g∗ is a regular value. For general Hamiltonian G-spaces the reducedspaces Φ−1(ξ )/Hξ are symplectic manifolds called Marsden-Weinstein reduced spaces [117].We briey discuss some issues related to symplectic reduction below.

Remark A.4.2. Since we are concerned to study Hamiltonian G-spaces associated to compactconnected Lie groups, the reduced spaces which we will deal in this context can be divided in thefollowing cases that we briey describe:

• Regular case: In this case besides the requirement ofG to be compact we also require thatthe Hamiltonian action of G to be a free action. If ξ ∈ g∗ is a regular value of Φ, sincethe action of Hξ on the embedded level manifold Φ−1(ξ ) is free and proper, the orbit spaceMξ = Φ−1(ξ )/Hξ is a smooth manifold. We also have a symplectic form ωξ dened on Mξ

which satises

130

ι∗ξω = π∗ξωξ ,

where ιξ : Φ−1(ξ ) → M (inclusion) and πξ : Φ−1(ξ ) → Φ−1(ξ )/Hξ (projection). From thesewe obtain the so-called Marsden-Weinstein reduced space (Mξ ,ωξ ), see for instance [117],[141, p. 509] and [38, p. 333].

• Weakly regular case: We consider for this case a Hamiltonian G-space on which the Liegroup G is compact and connected. For a regular value ξ ∈ g∗ of Φ : M → g∗ we havethat the action of Hξ on the embedded level manifold Φ−1(ξ ) is locally free, from these theorbit space Mξ = Φ−1(ξ )/Hξ can be endowed with the structure of orbifold. We also have asymplectic form ωξ dened on Mξ which satises

ι∗ξω = π∗ξωξ ,

where ιξ : Φ−1(ξ ) → M (inclusion) and πξ : Φ−1(ξ ) → Φ−1(ξ )/Hξ (projection), see [38,337-338] for the proof of the facts mentioned above.

• Singular case: For this last case we just suppose that we have a Hamiltonian G-space onwhich the Lie group is compact and connected. Given p ∈ M we denote ξ = Φ(p) ∈ g∗, wehave an isotropy type decomposition for M given by

M =⋃H<G

M (H ) ,

where each M (H ) ⊂ M is a submanifold (which may have components with dierent di-mensions, see for instance [159, p. 42], [152, p. 56] and [118, p. 14]) dened by

M (H ) =x ∈ M

∣∣∣ Gx = дHд−1, for some д ∈ G

.

If we denote by M the set of the connected components of M (H ) , where H < G varies overall compact subgroups of G, we obtain a stratication of M . is stratication induces astratication of Φ−1(ξ ) on which the strata is given by the smooth pieces

Φ−1(ξ ) ∩ N , for N ∈ M.

e stratication of Φ−1(ξ ) induces a structure of stratied space on the reduced spaceMξ = Φ−1(ξ )/Hξ such that the projection πξ : Φ−1(ξ ) → Φ−1(ξ )/Hξ is a morphism ofstratied spaces. In fact, each piece of the strata of Mξ is a smooth symplectic manifold.Furthermore, for each N ∈ M the symplectic stratum (πξ (Φ

−1(ξ ) ∩ N ),ωξ ,N ) of the Mξ

satises

ω |Φ−1 (ξ )∩N = π∗ξωξ ,N ,

hereω |Φ−1 (ξ )∩N denotes the restriction over the stratum. More details about the results whichwe have discussed above can be found in [38, p. 114-123], see also [150].

131

As we have described previously, under the assumption of the existence of an integrable sys-tem composed by collective Hamiltonians Hj = Φ∗(Ij ), for Ij ∈ C∞(g∗) with j = 1, . . . ,n, ifp ∈ M satises the property that Vξ ⊂ M , here Φ(p) = ξ ∈ g∗, is a submanifold on which A.4.1holds, the reduced spaces

Φ−1(O (ξ ))/G and Φ−1(ξ )/Hξ ,

are just a point. Now we observe that

TxΦ−1(ξ ) = ux (g)

⊥,

from this, given aG-invariant function f ∈ C∞(M )G , we have that its Hamiltonian vector eldX f ∈ Γ(TM ) satises

(d f )x (ux (Z )) = ωx (ux (Z ),X f (x )) = 0,

for every Z ∈ g. It follows that

SpanR

X f (x ) ∈ TxM

∣∣∣ f ∈ C∞(M )G⊂ ux (g)

⊥ = TxΦ−1(ξ ),

for every x ∈ Φ−1(ξ ). As we have seen previouslyΦ−1(ξ ) ⊂ H −1(c ), with c = (c1, . . . , cn ) ∈ Rn

and cj = Ij (ξ ), for j = 1, . . . ,n, i.e. Φ−1(ξ ) is an isotropic submanifold. erefore givenf ,h ∈ C∞(M )G we have

f ,hM(x ) = ωx (X f (x ),Xh (x )) = 0,

for every x ∈ Φ−1(ξ ). As a consequence of Proposition A.2.5, by following the ideas that wehave discussed so far and according to [74] and [75, 359-362], we have the following result

eorem A.4.1. Let (M,ω,G,Φ) be a Hamiltonian G-space which admits an integrable systemdened by collective Hamiltonians Hj = Φ∗(Ij ), with Ij ∈ C∞(g∗) for j = 1, . . . ,n. en thealgebra of G-invariant functions on M is commutative under the Poisson bracket, i.e. M is amultiplicity free space.

Proof. Here we will provide two dierent proofs for this theorem. e rst one is based onthe results that we have described in this section, the main purpose is to fulll some details inthe proof described in [74] and [75, 359-362].

e second approach which we will present to prove this theorem is dierent from the rstone, the main dierence is that we do not need to deal with clean intersections. In fact ourapproach is in some sense a direct application of Liouville’s theorem, which has the integra-bility condition as an input.

Proof 1. According to Proposition A.2.5, given p ∈ M0 ⊂ M there exists an open neighbour-hood U ⊂ M of p ∈ M such that U ∩ Vξ is a submanifold of M , here we can suppose thatξ = Φ(p) ∈ g∗. Once we have

Tp(U ∩ Vξ

)= (DΦ)−1

p (TξO (ξ )) = up (g) + up (g)⊥,

132

if we consider the restrictionΦ|U∩Vξ : U∩Vξ → O (ξ ), we have by transversality thatΦ−1(ξ )∩Uis a submanifold ofU ∩Vξ , notice that ξ ∈ g∗ does not need to be a regular value of Φ : M → g∗.Now under the hypothesis that H = (H1, . . . ,Hn ) : M → Rn denes an integrable systemcomposed by collective Hamiltonians Hj = Φ∗(Ij ), with j = 1, . . . ,n, since

Φ−1(ξ ) ∩U ⊂ H −1(c ),

for c = (c1, . . . , cn ) ∈ Rn such that cj = Ij (ξ ), j = 1, . . . ,n, we have that Φ−1(ξ ) ∩ U is anisotropic submanifold. erefore, once we have

SpanR

X f (p) ∈ TpM

∣∣∣ f ∈ C∞(M )G⊂ up (g)

⊥ = Tp(Φ−1(ξ ) ∩U

),

if we take f ,h ∈ C∞(M )G , from the last comments we obtainf ,hM(p) = ωp (X f (p),Xh (p)) = 0.

Since p ∈ M0 and M0 ⊂ M is an open dense subset, it follows by continuity that the algebra ofG-invariant functions is commutative under the Poisson bracket.

Proof 2. Our proof for this theorem is a direct application of the Liouville’s theorem, it goes asfollows. Suppose we have a completely integrable system dened by collective HamiltoniansH1 = Φ∗(I1), . . . ,Hn = Φ∗(In ), where Ij ∈ C∞(g∗), for j = 1, . . . ,n. Now we remember, seeSection 2.4, that the Hamiltonian vector eld associated to each Hj is given by

XHj (p) = δτ (∇Ij (Φ(p)))p ,

for j = 1, . . . ,n and any p ∈ M . By the integrability condition, we have an open dense subsetMr ⊂ M , on which for every p ∈ Mr there exists Darboux coordinates (ψi ,Hi ) in an openneighbourhood p ∈ U ⊂ Mr . Actually, it is a consequence of the Liouville’s theorem, see A.5of this work for more details. e main feature of these coordinates is that

∂

∂ψj

∣∣∣q= XHj (q),

for every q ∈ U and j = 1, . . . ,n. erefore, given f ∈ C∞(M )G we obtain

∂ f

∂ψj(p) = (d f )p (XHj (p)) =

d

dt

∣∣∣t=0

f (γ (t )),

for j = 1, . . . ,n, where γ : (−ϵ, ϵ ) → U ⊂ M is a smooth curve which satises γ (0) = p andγ (0) = XHj (p), here we take ϵ > 0 suciently small. By the characterization of the tangentspace by means of equivalence classes of curves, we can take

γ (t ) = τ (exp(t∇Ij (Φ(p))))p.

Since f ∈ C∞(M )G we have ddt

∣∣t=0 f (γ (t )) = 0 =⇒ ∂ f

∂ψj(p) = 0, notice that it is in fact true for

every j = 1, . . . ,n and for any p ∈ U . erefore, on U for f ,h ∈ C∞(M )G we have

f ,hM=

n∑k=1

(∂ f

∂ψk

∂h

∂Hk−∂ f

∂Hk

∂h

∂ψk

)= 0,

133

it follows that f ,hM (p) = 0, for every p ⊂ Mr ⊂ M . Since Mr is open and dense, by continu-ity we have that C∞(M )G is commutative under the Poisson bracket.

Remark A.4.3. Here it is worth to observe that the concept of multiplicity free space which wehave mentioned in the previous theorem was rst introduced in [74]. is concept can be seenas a “classical” analogue of the denition of multiplicity free spaces in Lie group representationtheory.

Figure A.1: Geometric quantization scheme.

As mentioned in [72], problems in classical mechanics can oen be reduced to the study of linearoperators on Hilbert spaces H by means of the geometric quantization procedure. erefore the“quantum” analogue of multiplicity free space is equivalent to say that the quantum action of aLie groupG on a Hilbert spaceH is multiplicity free if and only if the algebra of theG-invariantoperators is commutative. For more details about geometric quantization in HamiltonianG-spacessee for example [72] and [67].

As in the previous subsection, consider Φ(M ) = W ⊂ g∗ as a submanifold which satises(DΦ)p (TpM ) = TΦ(p)W , for every p ∈ M . Now we will look at the following case, let K ⊂ G bea closed connected Lie subgroup, and consider the natural HamiltonianK-space (M,ω,K ,ΦK ).Suppose we have a set of functions dened by the following collective Hamiltonians

Φ∗(I1), . . . ,Φ∗(In−k ),Φ

∗K (P1), . . . ,Φ

∗K (Pk ),

where Ij ∈ C∞(W ) and Pl ∈ C

∞(k∗), for j = 1, . . . ,n − k and l = 1, . . . ,k , here dim(M ) = 2n.Since the moment map ΦK : (M,ω) → k∗ is dened by

ΦK = πK Φ,

where πK : g∗ → k∗ denotes the projection map, we can write

Φ∗K (Pl ) = Φ∗(Pl πK ),

for l = 1, . . . ,k . Now we notice that for every ξ ∈ g∗, the restriction map πK |O (ξ ) : O (ξ ) → k∗

denes the moment map associated to the natural Hamiltonian action of K on the coadjointorbit (O (ξ ),ωO (ξ ) ), therefore the set of functions

P1 πK , . . . , Pk πK ,

denes a set of collective Hamiltonians when restricted to O (ξ ) ⊂ g∗, notice that here weconsider the Hamiltonian K-space (O (ξ ),ωO (ξ ),K ,πK |O (ξ ) ), for every ξ ∈ g∗. From these if wesuppose that the restrictions

134

(Pl πK ) |O (Φ(q)) ,

for l = 1, . . . ,k , dene an integrable system on each coadjoint orbit O (Φ(q)) ⊂ g∗, for everyq ∈ M . Here we point out that this assumption imposes the multiplicity free condition overall orbits through the elements of W ⊂ g∗, it is a consequence of eorem A.4.1. If we alsosuppose that the orbits O (Φ(q)) ⊂ g∗, for every q ∈ M , are the common level surface ofthe set of algebraically independent functions I1, . . . , In−k , then the necessary condition forintegrability of eorem A.4.1 becomes sucient, i.e. the set of functions

Φ∗(I1), . . . ,Φ∗(In−k ),Φ

∗K (P1), . . . ,Φ

∗K (Pk ),

dene an integrable system on (M,ω). Now we observe some consequences of the last com-ments. At rst we notice that we have

k = max

12 dim(O (ξ ))

∣∣∣ ξ ∈W,

furthermore

n − k = dim(W ) − dim(O (ξ )) ⇐⇒ n = dim(W ) − 12 dim(O (ξ )),

whereO (ξ ) ⊂W is any orbit with maximal dimension. Since we are supposing (DΦ)p (TpM ) =TΦ(p)W , we have

dim(M ) = dim(W ) + dim(ker((DΦ)p )),

now we consider the following facts

ker((DΦ)p ) = up (g)⊥ and dim(M ) = dim(up (g)) + dim(up (g)⊥),

it follows from the above facts that dim(W ) = dim(up (g)) = dim(G · p), therefore

12 dim(M ) = dim(G · p) + 1

2 dim(O (Φ(p))),

hence we have

dim(M ) = 2 dim(G · p) + dim(O (Φ(p))),

where O (Φ(p)) ⊂ W is any orbit of maximal dimension. erefore, under the previous hy-potheses the sucient condition to get integrability by means of the collective Hamiltoniansobtained from imm’s trick, applied on the chain K ⊂ G, can be summarized as follows:

1. e Hamiltonian G-space (M,ω,G,Φ) needs to be multiplicity free;

2. Each Hamiltonian K-space (O (ξ ),ωO (ξ ),K ,πK |O (ξ ) ), where ξ ∈ W = Φ(M ), needs to bemultiplicity free;

3. e set of functions P1 πK , . . . , Pk πK needs to dene an integrable system whenrestricted to each orbit O (ξ ) ⊂W , i.e. each coadjoint orbit O (ξ ) ⊂W needs to be com-pletely integrable by means of collective Hamiltonians obtained from the Hamiltonianaction of K ⊂ G;

135

4. It needs to be possible to nd a set of algebraically independent functions Ij ∈ C∞(W ),with j = 1, . . . ,n−k , such that all coadjoint orbitsO (ξ ) ⊂W are common level manifoldsof I1, . . . , In−k .

As we have seen on the previous sections, the above conditions actually hold for HamiltonianG-spaces dened by coadjoint orbits of G = U(N ) or SO(N ), i.e. the necessary condition forintegrability of eorem A.4.1 is in fact sucient in these two cases.

Based on our analyses developed in this section over the above conditions, the next chapterwill be devoted to study the case G = Sp(N ). All the content which we have covered in thissection it will be useful to understand the constraints on the integrability for Hamiltonianssystems composed by collectives in coadjoint orbits of the compact Lie group Sp(N ).

A.5 Liouville’s theorem

In this section we will provide a proof for the well known Liouville’s theorem. We start re-membering some basic consequences of the Liouville integrability for Hamiltonian systems.

Let (M,ω,H ) be an Hamiltonian integrable system. By denition of the Liouville integrabilitycondition, there exists H1, . . . ,Hn : (M,ω) → R, such that Hi ∈ C

∞(M ), for each i = 1, . . . ,n =12 dim(M ), satisfying

• Hi ,Hj M = 0, for all i, j = 1, . . . ,n,

• dH1 ∧ . . . ∧ dHn , 0 in an open dense subset of M ,

we usually take H = H1. Under the above condition, we can set

H : (M,ω) → Rn,

where xi H = Hi , for every i = 1, . . . ,n, here we denote by xi : Rn → R, the standard coor-dinate system of Rn.

If we denote by Mr ⊂ M the open dense subset where the second condition of integrabilityholds, for every x ∈ Mr , we have that

(DH )x =

(dH1)x...

(dHn )x

,

denes a surjective map (DH )x : TxM → Rn. It follows that if we denote by

Hr : Mr → Rn,

136

the restriction of H over Mr , we get a submersion map.

From now we assume the above hypothesis, i.e. we will suppose that the Liouville integrabil-ity holds for a Hamiltonian system (M,ω,H ). By considering the submersion Hr : Mr → Rn,associated to the system, the subset B =Hr (Mr ) is an open subset of Rn.

For each point b ∈ B, we have a submanifold

H −1r (b) ⊂ Mr ,

which we denote by Fb = H −1r (b). Now, if we consider the foliation dened by the Hamilto-

nian vector elds associated to the components functions of Hr , we obtain from the Liouvilleintegrability conditions an integrable regular distribution

DHr : Mr → TMr ,

such that

DHr : x →⟨XH1 (x ), . . . ,XHn (x )

⟩⊂ TxMr .

It is not dicult to see that

TxFb = 〈XH1 (x ), . . . ,XHn (x )⟩,

in fact, from the Liouville integrability conditions, we have

ker(DHr )x = 〈XH1 (x ), . . . ,XHn (x )⟩.

It follows that each level manifold Fb ⊂ Mr , is an union of connected leaves of DHr .

Since the functions Hi , i = 1, . . . ,n, are in involution, we have

ω (XHi ,XHj ) =Hi ,Hj

M= 0,

hence all submanifolds Fb ⊂ Mr are Lagrangian. If we denote by OHrx , the connected integral

submanifold (leaf) of DHr through the point x ∈ Mr , we can write

Fb =⋃x∈Fb

OHrx .

Around each point of Fb we have coordinates adapted to DH , actually given x ∈ Fb , we take

Ψx : Bεx (0) ⊂ Rn → Fb ,

such that

Ψx (t1, . . . , tn ) = (ϕXH1t1 . . . ϕ

XH1tn ) (x ),

137

whereϕXHi is the ow ofXHi , i = 1, . . . ,n. From these we obtain a coordinate system (q,Ψx (Bεx (0) ))around x ∈ Fb which satises

∂qi = XHi , for i = 1, . . . ,n.

We complete these coordinates in order to obtain local coordinates on Mr by taken

pi = Hi , for i = 1, . . . ,n,

it provides a coordinate system (q,p) in an open neighbourhoodW ⊂ Mr of x ∈ Fb , such that

Ψx (Bεx (0)) =y ∈W

∣∣ p (y) = cte

,

these laer results follow from the fact that DHr is a regular foliation and its connected leaves

are immersed submanifolds, see for instance [137].

We have constructed a coordinate system (q,p,W ) in Mr around x ∈ Fb which satises

∂qi = XHi , and pi = Hi ,

for every i = 1, . . . ,n.

Now we will look at the local expression of ω on these coordinates. We have on the localframes

ω (∂qi , ∂qj ) = ω (XHi ,XHj ) = 0,

and

ω (∂pi , ∂qj ) = ω (∂pi ,XHj ) = dHj (∂pi ) = δij ,

it follows that

ω |W =∑i

dpi ∧ dqi +∑ij

hijdpi ∧ dpj ,

here we denote by hij = ω (∂pi , ∂pj ), the smooth functions dened on the open set W ⊂ Mr ,with hij = −hji .

Notice that the equation of motion of the Hamiltonian system dened by H = H1 has thefollowing expression on these local coordinates

dqjdt= dqj (XH ) = δ1j ,

dpjdt= dpj (XH ) = 0,

thus we have the ow of XH locally given by

ϕXHt (q,p) = (q1 + t ,q2, . . . ,qn,p1, . . . ,pn ).

138

More generally, a straightforward calculation shows us that the Hamiltonian ow of XHi islocally given by

ϕXHit (q,p) = (q1, . . . ,qi + t ,qi+1, . . . ,qn,p1, . . . ,pn ).

So far we have seen how to construct a suitable coordinate system on Mr in order to obtain asimple expression for the ow of each Hamiltonian vector eld XHi , i = 1, . . . ,n.

In symplectic geometry we have a distinguished coordinate system (u,v,U ) called Darbouxcoordinate system, on which the expression of ω is given by

ω |U =∑i

dui ∧ dvi .

Our next step will be to obtain a Darboux coordinates with the same properties of the previouscoordinates (q,p) of simplify the expression of ϕXHi

t , for every i = 1, . . . ,n.

Now we remember that for each smooth function F : (M,ω) → R, we have

dF + ιXFω = 0,

from the Cartan’s magic formula, we have

LXFω = dιXFω + ιXFdω.

Since dω = 0 and dιXFω = −d2F = 0, we have that ω is invariant under the ow of XHi . By

using the local expression for the ows it is not dicult to see that dpi ∧ dqi and dpi ∧ dpjare invariant. It follows that the functions hij are also invariant under the ow of XHi , fori = 1, . . . ,n.

From the above facts, we can dene the functions

kij : Hr (W ) → R,

such that

kij (Hr (q,p)) = hij (q,p).

ese functions are well dened since the functions hij are constant in the variables q, noticethat pi = xi Hr , where xi : Rn → R are the standard coordinates given by the projections.

As a consequence of the above comments, we have

hijdpi ∧ dpj =H ∗r (kij )d (xi Hr ) ∧ d (xj Hr ),

it follows that

139

hijdpi ∧ dpj =H ∗r (kijdxi ∧ dxj ),

where kijdxi ∧ dxj ∈ Ω2(Hr (W )).

Since ∑ij

H ∗r (kijdxi ∧ dxj ) = ω |W −

∑i

dpi ∧ dqi ,

we have

dH ∗r (∑ij

kijdxi ∧ dxj ) = 0,

from the fact that dH ∗r =H ∗

r d and H ∗r is injective, we obtain

d (∑ij

kijdxi ∧ dxj ) = 0.

By applying the Poincare’s lemma, see for example [19], we get

d (∑i

αidxi ) =∑ij

kijdxi ∧ dxj ,

for some∑i

αidxi ∈ Ω1(Hr (W )).

Now we dene new coordinates from the following coordinate transformation

qi (q,p) = qi −H ∗r (αi ), and pi (q,p) =H ∗

r (xi ) = pi .

From the previous local expression

ω |W =∑i

dpi ∧ dqi +∑ij

hijdpi ∧ dpj ,

we have

dpi ∧ dqi = dpi ∧ d (qi +H ∗r (αi )).

Now we notice that∑i

dpi ∧ dH ∗r (αi ) =

∑i

H ∗r (dxi ) ∧H ∗

r (dαi ) = −∑i

H ∗r (kijdxi ∧ dxj ),

therefore we obtain

ω |W =∑i

dpi ∧ dqi .

140

From these we have shown that there exists a distinguished coordinate system around eachpoint of Mr , i.e. besides of these coordinates to be Darboux they also satisfy

d

dtqj (ϕ

XHt (q, p)) =

d

dtqj (ϕ

XHt (q, p)) −

d

dtH ∗

r (αj ) (ϕXHt (q, p)).

Since the components functions of Hr are constant along the ow of XH , the above equationbecomes

d

dtqj (ϕ

XHt (q, p)) =

d

dtqj (ϕ

XHt (q, p)),

it follows that the equation of motion associated to the Hamiltonian system (M,ω,H ) is givenby

d

dtqj = δ1j , and d

dtpj = 0.

us we have the ow of XH locally given by

ϕXHt (q, p) = (q1 + t , q2, . . . , qn, p1, . . . , pn ).

More generally, a straightforward calculation shows us that the Hamiltonian ow of XHi islocally given by

ϕXHit (q, p) = (q1, . . . , qi + t , qi+1, . . . , qn, p1, . . . , pn ),

thus we have just shown the following result

eorem A.5.1 (Liouville). Let (M,ω,H ) be an integrable system and let x ∈ M be a regu-lar point of H = (H1, . . . ,Hn ). en there exists an open neighbourhood W ⊂ M of x andsmooth functions q1, . . . , qn on W complementing H1, . . . ,Hn to Darboux coordinates. In thesecoordinates, the ow ϕ

XHit of the Hamiltonian vector eld XHi is given by

ϕXHit (q, p) = (q1, . . . , qi + t , qi+1, . . . , qn, p1, . . . , pn ).

Proof. It follows directly from the above ideas.

I this section we have described the main ideas behind the proof of the Liouville’s theorem,in fact, under the integrability condition, we have coordinates which allow us to solve theequation of motion by quadrature methods. In the next section we will describe Arnold’stheorem, which provide a good description for the regular level manifolds of H .

141

A.6 Arnold’s theorem

Now we start to describe the main ideas behind the proof of Arnold’s theorem. Again we willassume the integrability condition for a Hamiltonian system (M,ω,H ), from these conditionwe can consider the submersion

Hr : Mr → B ⊂ Rn.

As we have seen in the last section, the bers Fb = H −1(b) are Lagrangian submanifolds ofM , and its connected components are the leaves OHr

x associated to the regular Lagrangianfoliation dened by the Hamiltonian vector elds XH1, . . . ,XHn , i.e.

Fb =⋃x∈Fb

OHrx .

For our purposes we will suppose that the ow ofXH1, . . . ,XHn , are complete on Fb , from thesewe can dene an Lie group action

Ψ : Rn × Fb → Fb

such that

Ψ(t ,x ) = t · x := (ϕXH1t1 . . . ϕ

XH1tn ) (x ),

here we denote t = (t1, . . . , tn ) ∈ Rn.

Notice that if we x x ∈ Fb , the map Ψ(.,x ) : Rn → Fb , denes a local dieomorphim from anopen neighbourhood of 0 ∈ Rn to an open neighbourhood of x ∈ Fb . In fact this map denesa coordinate system adapted to DHr .

Now we will verify that the above action is transitive on each connected component of Fb . Inorder to do this, we x x ∈ Fb and consider its connected component OHr

x . Now we considerthe following set

Nx =y ∈ OHr

x

∣∣ y = t · x , for some t

.

Given y ∈ Nx , we have Ψ(.,y) : Rn → Fb , thus we obtain a dieomorphism from a openneighbourhood of 0 ∈ Rn onto an open neighbourhood of y ∈ OHr

x . e image of this neigh-bourhood is entirely contained in Nx , it implies that Nx ⊂ OHr

x is a open set.

Now we can apply the same idea for y ∈ OHrx \Nx , if the open neighbourhood of y dened by

the image of Ψ(.,y) intersects Nx , we have

s · y = t · x ⇐⇒ y = (t − s ) · x ,

which implies that y ∈ Nx . It follows that for every y ∈ OHrx \Nx we have a open neighbour-

hood entirely contained in OHrx \Nx , which implies that Nx is closed, thus we have Nx = O

Hrx .

From the above comments we have shown that

142

OHrx Rn/Γx ,

where Γx ∈ Rn is the isotropy subgroup of x ∈ Fb . Now we notice that 0 ∈ Γx , and Ψ(.,x )is a dieomorphism in an open neighbourhood of 0 ∈ Rn. If we denote by Bεx (0) ⊂ Rn thisneighbourhood, we have

Bεx (0) ∩ Γx = 0.

if we take t ∈ Γx , we have an open neighbourhood

t + Bεx (0) ⊂ Rn.

Now we claim that

(t + Bεx (0)) ∩ Γx = t .

Actually, if s ∈ (t + Bεx (0)) ∩ Γx , we can write

s = t + r ⇐⇒ r = s − t ,

once s − t ∈ Γx and r ∈ Bεx (0)), we have r = 0. erefore the isotropy subgroup Γx ⊂ Rn isdiscrete.

Since the discrete subgroups of Rn are laices, we conclude that

OHrx Rn−k ×T k ,

where T k is the compact torus, and k = rankZ(Γx ). In the case that OHrx is compact we have

the Liouville torus

OHrx Tn.

Under the assumption that the ow of the Hamiltonian vector eldsXH1, . . . ,XHn , are completeon Fb , we have described all connected component of Fb . is result is summarized in thefollowing theorem

eorem A.6.1 (Arnold). Let (M,ω,H ) be an integrable system and let Fb = H −1(b) be aregular level manifold of H = (H1, . . . ,Hn ). If Fb is compact then its connected componentsare dieomorphic to the compact torus Tn If Fb is not compact but the ow of XH1, . . . ,XHn arecomplete on Fb , then the connected components of Fb are dieomorphic to Rn−k ×T k , where 0 ≤k < n.

143

Appendix BIntegrable systems in regular adjointorbits of compact symplectic Lie group

In this chapter we will provide a complete description of the Gelfand-Tsetlin-Molev integrablesystems dened on regular orbits of Sp(N ) by means of the technique which was introducedin [79] as well as to perform some computations for low dimensional examples.

e dicult of construct integrable systems on the coadjoint orbits associated the symplecticLie group Sp(N ) = U(N ,H) composed by collective Hamiltonians, at a rst glance, is that wedo not have a suitable intermediate closed and connected Lie subgroup between Sp(k ) andSp(k − 1). Hence, when we consider the nested chain of Lie subgroups

Sp(N ) ⊃ Sp(N − 1) ⊃ . . . ⊃ Sp(2) ⊃ Sp(1),

where Sp(k ) U(2k )∩Sp(2k,C), for k = 1, . . . ,N , and consider a regular orbitO (Λ) ⊂ sp(N ),dened by

Λ = diag(−iλN , . . . ,−iλ1, iλ1, . . . , iλN ),

with 0 > λ1 > . . . > λN1, we do not get enough quantities in involution to obtain integrability.

In other words, if we apply the construction by using imm’s trick, we will obtain functions

λ(l )k = Φ∗k (Λl ),

with 1 ≤ l ≤ k , and k = 1, . . . ,N − 1. ese functions are not enough to set a completeintegrable system on O (Λ), since dimR(O (Λ)) = 2N 2 and we have from the above chain a setwith N (N−1)

2 functions.

As we have seen, the key points in the construction of integrable systems for the unitary andthe special orthogonal groups, besides the existence of a suitable chain of Lie subsgroups, isthe fact that the orbits associated to these groups are multiplicity free spaces, see eoremA.4.1 on page 131. Other important point is that the inequalities satised by the functions in

1Here we are following [79, p. 17-27], see also [121, p. 137-139], for the notations and conventions

144

the unitary and special orthogonal cases allows us to extend the construction from regular or-bits to orbits associate to non regular elements. It means that the inequalities satised by thefunctions tells us what functions we need to eliminate from the big set obtained from imm’strick in order to get integrability.

e problem of construct an integrable system for regular orbits of the symplectic Lie groupwas solved in [79]. e technique employed involves deformation of Poisson Hopf algebrastogether with new ideas which come from the construction of the Gelfand-Tsetlin-Molev basisfor irreducible representations of the Lie algebra sp(2N ,C), see for instance [121], [123] and[124].

Although we have that the regular orbits of Sp(N ) are integrable, the problem of extending theconstruction for non-regular orbits remains. In the previous sections we have described thenecessary condition for the existence of an integrable system composed by collective Hamil-tonians on Hamiltonian G-spaces with equivariant moment map. We also have seen that thecondition is in fact sucient for HamiltonianG-spaces dened by coadjoint orbits of the clas-sical compact lie groups U(N ) and SO(N ). In what follows we will give an overview aboutthe integrable system constructed in [79] for regular orbits of Sp(N ). Furthermore, we willprovide a complete description for the concrete cases dened by regular orbits of Sp(2) andSp(3).

For these two concrete cases cited above we have the following results

Proposition 1. Let O (Λ) ⊂ sp(2) be a regular adjoint orbit. en the functions which composethe Gelfand-Molev integrable system [79] on O (Λ) can be wrien as

• µ (1)2 : X → rTr(Φ2(X )2E22) =2∑

k=1X2kXk2,

• µ (2)2 : X → rTr(Φ2(X )iE22) = ReH(X22i ),

• λ(1)1 : X → Λ1(Φ1(X )) = ±√

det(X11),

• µ (1)1 : X → ReH(Φ1(X )iE11) = ReH(X11i ),

∀X ∈ O (Λ), where Φk : O (Λ) → sp(k ), k = 1, 2, denotes the moment map associated to theHamiltonian action of Sp(k ) on O (Λ), and Λ = diag(iλ1, iλ2).

Proposition 2. Let O (Λ) ⊂ sp(3) be a regular orbit. en the Gelfand-Tsetlin-Molev system isdened by the following functions

• µ (1)3 (X ) =

3∑k=1

X3kXk3 = −2∑

k=1(λ2

k − λ23) | |q3k (X ) | |2 − λ2

3,

• µ (2)3 (X ) =

3∑k1,k2=1

ReH(X3k1Xk1k2Xk23) =2∑

k=1(λ4

k − λ43) | |q3k (X ) | |2 + λ4

3,

• µ (3)3 (X ) = rTr(XiE33) = ReH(X33i ) = −Im(z (X33)),

145

• λ(1)2 (X ) = Λ1(Φ2(X )),

• λ(2)2 (X ) = Λ2(Φ2(X )),

• µ (1)2 (X ) =

2∑k=1

X2kXk2 = −(λ(1)2 (X )2 − λ(2)2 (X )2) | |q21(Φ2(X )) | |2 − λ(2)2 (X )2,

• µ (2)2 (X ) = rTr(Φ1(X )iE11) = ReH(X11i ) = −Im(z (X22)),

• λ(1)1 (X ) = Λ1(Φ1(X )) = ±√

det(X11),


∀X ∈ O (Λ), where Φk : O (Λ) → sp(k ), k = 1, 2, 3, denotes the moment map associated to theHamiltonian action of Sp(k ) on O (Λ), and Λ = diag(iλ1, iλ2, iλ3).

ese two results which we have obtained can be seen as the rst step in the study of therelations between the functions which compose the Gelfand-Tsetlin-Molev system and theGelfand-Tsetlin paern for irreducible modules of the Lie algebra sp(2N ,C). We hope that thecomputations established here for Sp(2) and Sp(3) can be used in other concrete applications,e.g. on the study of the Lagrangian submanifolds dened by the regular values associated tothe above integrable Hamiltonian systems. We have the following geometric illustration ofthe root system associated to sp(3,C).

Figure B.1: e C3 root system consists of the vertices of an octahedron, together with themidpoints of the edges of the octahedron. is image was extracted form the book [78].

B.1 Generalities about H and sp(N )

We start our analysis by describing the structure of quaternions and some basic facts aboutthe symplectic Lie algebra sp(2N ,C) and its compact real form sp(N ).

Let H be the non-commutative quaternions algebra, given q ∈ H we can write

146

q = q1 + q2i + q3j + q4k = z (q) + jw (q),

where z (q) = q1 + q2i ∈ C, and w (q) = q3 − q4i ∈ C, i.e. we have an identication

H C ⊕ jC.

It will be useful to see H as a matrix algebra by the embedding

q = z (q) + jw (q) →

z (q) w (q)

−w (q) z (q)

,

we denote the above map by ψ : H → gl (2,C). Now we consider the Lie algebra sp(N ) =u(N ,H), this algebra is dened by

sp(N ) =X ∈ gl (N ,H)

∣∣∣ X + X? = 0

,

here X? = XT , where the bar denotes the quaternions conjugation. We also consider the

following identication of HN with C2N

(q (1), . . . ,q (N ) ) → (w (q (1) ), . . . ,w (q (N ) ), z (q (1) ), . . . , z (q (N ) )),

this isomorphism provides an embedding gl (N ,H) → gl (2N ,C), as follows

X11 · · · X1N.... . .

...

XN 1 · · · XNN

→

z (XNN ) · · · z (XN 1) w (XN 1) · · · w (XNN )...

. . ....

.... . .

...

z (X1N ) · · · z (X11) w (X11) · · · w (X1N )

−w (X1N ) · · · −w (X11) z (X11) · · · z (X1N )...

. . ....

.... . .

...

−w (XNN ) · · · −w (XN 1) z (XN 1) · · · z (XNN )

.

We have the following description for the complex symplectic Lie algebra

sp(2N ,C) =X ∈ gl (2N ,C)

∣∣∣ XTA +AX = 0

,

where the matrix A is given by

A =

0 1N

−1N 0

, (B.1.1)

here we denote the N × N matrix 1N

147

1N =

0 · · · 0 1

0 · · · 1 0...

.........

1 · · · 0 0

.

From these we can realize the Lie algebra sp(N ) as the Lie algebra

sp(N ) = u(2N ) ∩ sp(2N ,C),

for more details about this realization see for example [79, p. 18-24].

We will set some basic properties associated to the trace functions. We denotebyΨ : gl (N ,H) →gl (2N ,C) the embedding dened before, from this we can dene the following trace function

rTr(X ) =12Tr(Ψ(X )),

for X ∈ gl (N ,H). A straightforward calculation shows us that

Tr(Ψ(X )) =

N∑i=1

Tr(Xii ) =

N∑i=1

2ReC(z (Xii )) =

N∑i=1

2ReH(Xii ),

we also notice that for every q ∈ H, we have

ReH(q) =12Tr(q),

and

| |q | |2 = det(q),

where | |q | |2 = q21 + q

22 + q

23 + q

24. ese basic facts will be important for us in order to make

an analysis in low dimensional cases. We nish this section by describing a basic example ofhow the above facts and generalities can be used to describe low dimensional adjoint orbits asHamiltonian spaces associated to the symplectic Lie group action.

Example B.1.1. e manifold dened by the regular adjoint orbit

O (Λ) =QΛQ? ∈ sp(3)

∣∣∣ Q ∈ Sp(3)

,

where Λ = diag(iλ1, iλ2, iλ3), with 0 > λ1 > λ2 > λ3, has a natural symplectic structure

ωO (Λ) (ad(v )X , ad(w )X ) = rTr(X [v,w]),

for every X ∈ O (Λ), and v,w ∈ sp(3). In this context we consider the adjoint action of Sp(3)in O (Λ) given by

148

(Q,X ) −→ QXQ?,

for every Q ∈ Sp(3), and X ∈ O (Λ). As we have seen this action is Hamiltonian with anequivariant moment map

Φ : (O (Λ),ωO (Λ) ) → sp(3)∗.

If we consider the identication sp(3)∗ sp(3), provided by some bi-invariant inner prod-uct, e.g. Cartan-Killing form, we can show that the above moment map is in fact the naturalinclusion map

Φ : (O (Λ),ωO (Λ) ) → sp(3).

erefore, we have the following commutative diagram of inclusions

sp(1) sp(2) sp(3)

sp(2,C) sp(4,C) sp(6,C)

e sequence of inclusions represented by the rst line of the above diagram induces a se-quence of projections

sp(3) sp(2) sp(1),

in terms of matrix it can be seen as follows. Given X ∈ sp(3) we denote its entries by Xij =

zij + jwij , for 1 ≤ i, j ≤ 3, from this we have

z33 z32 z31 w31 w32 w33

z23 z22 z21 w21 w22 w23

z13 z12 z11 w11 w12 w13

−w13 −w12 −w11 z11 z12 z13

−w23 −w22 −w21 z21 z22 z23

−w33 −w32 −w31 z31 z32 z33

−→

z22 z21 w21 w22

z12 z11 w11 w12

−w12 −w11 z11 z12

−w22 −w21 z21 z22

−→ z11 w11

−w11 z11

.

Remark B.1.1. It is worthwhile to observe that for X ∈ sp(N ) we have X = −X?, it implies thatXij = −X

∗ji , where

Xij = z (Xij ) + jw (Xij ) →

z (Xij ) w (Xij )

−w (Xij ) z (Xij )

.

erefore z (Xij ) = −z (X ji ) and w (Xij ) = w (X ji ), i.e. the matrix (z (Xij )) is skew-Hermitian and(w (Xij )) is skew-symmetric.

149

From the above comments, if we consider the following chain of subgroups

Sp(3) ⊃ Sp(2) ⊃ Sp(1),

we have a natural Hamiltonian action associated to each group in the above chain on thesymplectic manifold (O (Λ),ωO (Λ) ). e moment map for each one of these actions are obtainedby composition of the natural inclusion of O (Λ) in sp(3) and projections over its subalgebrasdescribed previously. We summarize the idea in the following diagram

(O (Λ),ωO (Λ) ) → sp(3) sp(2) sp(1),

the rst arrow corresponds to the moment map associated to the Hamiltonian action of Sp(3),the composition of the rst arrow with the second arrow provides the moment map associ-ated to the Hamiltonian action of Sp(2) and the composition of the three arrows provides themoment map associated to the Hamiltonian action of Sp(1).

e above example will be important for us aerwards to describe the ideas developed in [79].

B.2 Anoverviewabout theGelfand-Tsetlin-Molev integrablesystem

e main purpose of this section is to provide an overview about the construction of integrablesystems in regular adjoint orbits of the compact Lie group Sp(N ) following basically [79]. Ourexposition will not be complete, since it demands to deal with some issues which are beyondthe purpose of this work. In what follows we will provide a big picture of the main ingredientswhich were used to ensure integrability for regular orbits of Sp(N ).

B.2.1 Multiplicity free action and representation theory

As we have seen so far the construction of integrable systems in adjoint orbits by meansof imm’s trick and collective Hamiltonians can be summarized in the following way. Forsimplicity we will work on U(N ). LetO (Λ) ⊂ u(N ) be a regular adjoint orbit dened by somediagonal matrix

Λ = diag(iλ1, . . . , iλN ).

From this we take a nested chain of closed connected subgroups

U(N ) ⊃ U(N − 1) ⊃ · · · ⊃ U(1), (B.2.1)

associated to each Hamiltonian U(k )-space (O (Λ),ωO (Λ),U(k ),Φk ), we get a set of k functionsdened by the “eigenvalues” pulled back by Φk : O (Λ) → u(k ), i.e. we have the collectiveHamiltonians

λ(1)k = Φ∗k (Λ1), . . . , λ(k )k = Φ∗k (Λk ).

150

By applying it iteratively we obtain a set of N (N−1)2 Poisson commuting functions given by

λ(l )k = Φ∗k (Λl ), 1 ≤ l ≤ k , 1 ≤ k < N ,

notice that for a general regular orbit O (Λ) ⊂ u(N ) we have

12 dim(O (Λ)) =

N (N − 1)2 .

Since the functions described above satisfy a set of implicit equations involving the coe-cients of characteristic polynomials, which are algebraically independent functions, we canshow that these functions are in fact algebraically independent in an open dense subset ofO (Λ), therefore we have integrability, see [76] for more details.

Now let us describe the action of imm’s torus [79] which comes from the above construction.We observe that we have a chain of maximal torus induced by the previous nested chain B.2.1of closed and connected subgroups, i.e.

TN ⊃ TN−1 ⊃ · · · ⊃ T2 ⊃ T1,

where Tk ⊂ U(k ), k = 1, . . . ,N . We denote by tk the Lie algebra of each Tk and x a (closed )positive Weyl chamber tk+ ⊂ tk , for k = 1, . . . ,N . erefore the construction of the Gelfand-Tsetlin integrable system for O (Λ) ⊂ u(N ) can be summarized in the following diagram

(O (Λ),ωO (Λ) ) u(N ) u(N − 1) u(N − 1) · · ·

tN−1+ tN−2

+ · · ·

e Hamiltonian action of the Gelfand-Tsetlin torus is dened as follows. At rst we considerthe following commutative diagram for every k = 1, . . . ,N − 1,

(O (Λ),ωO (Λ) ) u(k )

tk+

Λ(k )

Φk

π(k )+

where π (k )+ : u(k ) → u(k )/U(k ) tk+ denotes the natural projection given by the intersection

of adjoint orbits and the Weyl chamber, and Λ(k ) : (O (Λ),ωO (Λ) ) → tk+ is dened by

Λ(k ) (X ) = diag(iΛ1(Φk (X )), . . . , iΛk (Φk (X ))),

for every X ∈ O (Λ). Now we consider the following facts:

• ere exists an unique open face σ 0k ⊂ t

k+ such that Rk = Φ−1

k (σ 0k ) ⊂ O (Λ) denes a

Tk-invariant symplectic submanifold.

151

• e restriction of the moment map Φk |Rk is the moment map for the Hamiltonian actionof Tk ⊂ U(k ) on Rk ⊂ O (Λ) given by the adjoint action.

• e function Λ(k ) : (O (Λ),ωO (Λ) ) → tk+ is smooth on the open dense subset

Vk = (Λ(k ) )−1(σ 0k ) = Ad(U(k ))Rk .

• ere exists an action of Tk on Vk dened in the following way. For X ∈ Vk ⊂ O (Λ) wehave X = Ad(д)Y , for some Y ∈ Rk and д ∈ U(k ), therefore we can set

t ∗ X = Ad(д)Ad(t )Ad(д−1)X = Ad(д)Ad(t )Y , (B.2.2)

for every t ∈ Tk .

• e above new action dened by Tk is Hamiltonian with moment map given by Λ(k ) .

e rst and the second facts listed above can be found in [113, p. 249], see also [135, p. 35-43],here is important to observe that we are supposing O (Λ) ⊂ u(N ) to be regular it follows thatσ 0k = (tk+)

o , i.e. the interior of the slice dened in A.3.1. e third fact is a direct consequenceof the decomposition Λ(k ) = π (k )

+ Φk . e fourth and h facts above are described in moredetails in [135, p. 35-43].

From the above facts and comments, the action of imm’s torus

TN (N−1)

2 = TN−1 × · · · × T1,

on the open dense subsetV = ∩N−1k=1 Vk ⊂ O (Λ) is dened through the action of each Tk ⊂ U(k ),

k = 1, . . . ,N − 1, described in B.2.2.

Now we will provide an outline of the ideas involved in the construction of the Gelfand-Tsetlinbasis for representation of gl (N ,C) and show how this construction can be seen as the repre-sentation theory side of the Gelfand-Tsetlin integrable system.

e correspondence between adjoint orbits and representation theory is given by geometricquantization, which in this case basically states that for an (integral) adjoint orbit O (Λ) ⊂u(N ), such that

Λ = diag(iλ1, . . . , iλN ), with λj − λj+1 ∈ Z>0,

for j = 1, . . . ,N −1, we can assign an irreducible representationV (λ) of u(N ) which allows usto establish a dictionary between “classical observables” and “quantum observables”. As beforewe will x the integral adjoint orbit O (Λ) ⊂ u(N ) as being also regular, i.e. λj − λj+1 ∈ Z>0,for j = 1, . . . ,N − 1.

Remark B.2.1. We point out that will not to provide an extensive description about geometricquantization procedure. Our references are [106, p. 206, eorem 5.7.1] for the integrality con-dition, [45, p. 300, eorem 4.12.5] for the description of V (λ) and [72] for the relations betweenHamiltonian action and representation theory via quantization.

152

Now we consider ρ : gl (N ,C) → gl (V (λ)) the irreducible representation induced by O (Λ) ⊂u(N ). From this we take the sequence of subalgebras

gl (N ,C) ⊃ gl (N − 1,C) ⊃ · · · ⊃ gl (1,C),

and consider the iterative restrictions of (ρ,V (λ)) over each element of the above chain. Atthe rst step we have the following decomposition

V (λ) |gl (N−1,C) =⊕µ

Homgl (N−1,C)

(W (µ );V (λ)

)⊗W (µ ), (B.2.3)

where the elements on the above equality can be described as follows

• W (µ ) = nite dimensional irreducible gl (N − 1,C)-module with highest weight µ;

• Homgl (N−1,C)

(W (µ );V (λ)

)= space of linear maps fromW (µ ) to V (λ) which commutes

with the gl (N − 1,C) action;

• Homgl (N−1,C)

(W (µ );V (λ)

)⊗W (µ ) = µ-isotypic component, also denoted by V (λ)µ .

If we denote by V (λ)+µ ⊂ V (λ) the subspace dened by

V (λ)+µ =v ∈ V (λ)+

∣∣∣ v = highest weight vector with weight µ

,

where

V (λ)+ =v ∈ V (λ)

∣∣∣ v = highest weight vector of gl (N − 1,C)

,

from these we have

V (λ)+µ Homgl (N−1,C)

(W (µ );V (λ)

),

see for instance [29, p. 188-189] or [41, p. 299]. Since the multiplicity ofW (µ ) ⊂ V (λ) is givenby c (µ ) = dim(V (λ)+µ ), it follows that the decomposition B.2.3 also can be wrien as

V (λ) |gl (N−1,C) =⊕µ

c (µ )W (µ ),

where

c (µ )W (µ ) :=W (µ ) ⊕ · · · ⊕W (µ )

c (µ )−times

.

e main feature of the above description for restriction of the representation of gl (N ,C) togl (N − 1,C) is that c (µ ) = 0 or 1, see for example [121, p. 117], [170, p. 187]. It meansthat the restriction is multiplicity free. erefore V (λ) splits in a direct sum of irreduciblerepresentationsW (µ ) of gl (N − 1,C) parameterized by µ = (µ1, . . . , µN−1) ∈ ZN−1 such that

153

λ1 ≥ µ1 ≥ λ2 ≥ · · · ≥ λN−1 ≥ µN−1 ≥ λN .

By proceeding inductively in this way for each pair (gl (k,C), gl (k −1,C)), we obtain a decom-position of V (λ) parameterized by an array of inequalities

λ1 λ2 · · · λN−1 λN−1λ(N−1)

1 · · · λ(N−1)N−1

· · · · · ·

λ(2)1 λ(2)2λ(1)1

these inequalities are called Gelfand-Tsetlin paern. From these inequalities we can constructa basis for V (λ) with explicit formulas for the basis elements. is canonical basis was earlyintroduced by I. M. Gelfand and M. L. Cetlin in 1950s, see for instance [59]. Now we will de-scribe the analogy of the construction of the Gelfand-Tsetlin basis and the action of imm’storus which we have described previously.

In the same way we have constructed a Hamiltonian action of imm’s torus in an open densesubset of the symplectic manifold O (Λ) ⊂ u(N ), we will provide an action at the representa-tion theory level of the so called Gelfand-Tsetlin torus, our approach is according to [79].

Consider V (λ) as an irreducible U(N )-module and again we take a nested chain of closedconnected subgroups as in B.2.1. By the previous comments, if we consider the restrictionV (λ) |U(N−1) we have the following decomposition

V (λ) |U(N−1) =⊕µ

c (µ )W (µ ),

where eachW (µ ) is an irreducible U(N − 1)-module and c (µ ) = 0 or 1. From these, we denea new action of TN−1 on each irreducible componentW (µ ) by

t ∗w = µ (t )w, (B.2.4)

for each t ∈ TN−1 andw ∈W (µ ). Proceeding inductively on the decomposition parameterizedby the Gelfand-Tsetlin paern, we obtain by means of B.2.4 an action of the Gelfand-Tsetlintorus

TN (N−1)

2 = TN−1 × · · · × T1,

on the space V (λ).

RemarkB.2.2. Here it is worth to point out that once the irreducible representations of T1 = U(1)are one dimensional, and the action of each torus Tk commutes with each other, we obtain acomplete decomposition ofV (λ) in one dimensional subspaces with respect to the Gelfand-Tsetlintorus action.

All the ideas which we have described so far are in the framework of the geometric quantiza-tion scheme

154

Unitary representations antum mechanical systems

Coadjoint orbit cover Classical mechanical systems

We can obtain the same results for adjoint orbits of the special orthogonal compact Lie group.Since we have also on the symplectic geometry side the action of imm’s torus provided bythe Gelfand-Tsetlin systems, see for instance [136], and on the representation theory side theaction of the Gelfand-Tsetlin torus related to the Gelfand-Tsetlin basis, see for example [170,p. 377-380]. Further discussions about the above scheme can be found in [76], [162, p. 9-13],[72] and [106].

We nish this subsection by explaining why the construction of the Hamiltonian action ofimm’s torus fails when we try to use the same construction of integrable systems via imm’strick and collective Hamiltonians in adjoint orbits of the compact group Sp(N ).

In order to analyse all the previous issues associated with the construction of imm’s torusaction and the Gelfand-Tsetlin action, we gather in B.2 the main ingredients which we havein the construction of both actions described previously in this subsection

Figure B.2: Correspondence between the action of imm’s torus and the action of the Gelfand-Tsetlin torus. is table is inspired by [79, p. 9]

Let O (Λ) ⊂ sp(N ) be a regular adjoint orbit such that

Λ = diag(iλ1, . . . , iλN ),

with 0 > λ1 > λ2 > · · · > λN . Proceeding as before we take a nested chain of closed andconnected subgroups

Sp(N ) ⊃ Sp(N − 1) ⊃ · · · ⊃ Sp(1), (B.2.5)

we have a chain of maximal torus induced by the above chain, i.e.

TN ⊃ TN−1 ⊃ · · · ⊃ T2 ⊃ T1,

where Tk ⊂ Sp(k ), k = 1, . . . ,N . Let tk be the Lie algebra of each Tk and x a (closed ) positiveWeyl chamber tk+ ⊂ tk , for k = 1, . . . ,N . e collective Hamiltonians obtained by imm’strick can be schematically described by the following diagram

155

(O (Λ),ωO (Λ) ) sp(N ) sp(N − 1) sp(N − 2) · · ·

tN−1+ tN−2

+ · · ·

e collective Hanmiltonians at each step are the components of Λ(k ) = π (k )+ Φk , k =

1, . . . ,N − 1, namely,

(O (Λ),ωO (Λ) ) sp(k )

tk+

Λ(k )

Φk

π(k )+

therefore we obtain a set of N (N−1)2 functions which are algebraically independent and Poisson

commute with each other. However, in this case we have

12 dim(O (Λ)) = N 2,

it follows that the collective Hamiltonians obtained by imm’s trick are not enough to getcomplete integrability onO (Λ). In order to understand the reason for this fact we look closelythe elements listed in B.2 in the context of the Hamiltonian spaces (O (Λ),ωO (Λ), Sp(k ),Φk ).

Consider at rst the Hamiltonian Sp(N − 1)-space (O (Λ),ωO (Λ), Sp(N − 1),ΦN−1), let µ ∈sp(N − 1) be a regular element, i.e.

µ = diag(iµ1, . . . , iµN−1),

such that 0 > µ1 > · · · > µN−1. Since the action of Sp(N − 1) on O (Λ) ⊂ sp(N ) is a properaction, we have a structure of stratied manifold on Φ−1

N−1(µ ) ⊂ O (Λ), see our Remark A.4.2.Given Z ∈ Φ−1

N−1(µ ) we have

(DΦN−1)Z : TZO (Λ) → sp(N − 1),

since dim(O (Λ)) = 2N 2 and dim(sp(N −1)) = 2N 2− (3N −1), by Sard’s theorem [69, p. 39-45]we can suppose that µ ∈ sp(N − 1) is a regular value of ΦN−1. erefore Φ−1

N−1(µ ) is in fact aconnected manifold [113], i.e. it can be seen as a stratied manifold with just one connectedstrata piece with dimension

dim(Φ−1N−1(µ )) = (ker(DΦN−1)Z ) = dim(O (Λ)) − dim(sp(N − 1)).

It follows that dim(Φ−1N−1(µ )) = 3N − 1, since Hµ = TN−1, see the last line of B.2, we obtain

dim(Φ−1N−1(µ )/TN−1) = 3N − 1 − (N − 1) = 2N .

156

e above calculation shows us that the symplectic reduced spaces of the Hamiltonian Sp(N −1)-space (O (Λ),ωO (Λ), Sp(N − 1),ΦN−1) are in general non-trivial. If we do the same calcula-tions for the unitary and the special orthogonal Lie groups we will obtain that the symplecticreduced spaces are just a point.

What we can conclude is that (O (Λ),ωO (Λ), Sp(N − 1),ΦN−1) is not a multiplicity free space.In fact, by keeping the previous data we have that

Vµ = Φ−1N−1(O (µ )),

is a submanifold of O (Λ), see for example [38, p. 341]. Once we have

ker((DΦN−1)Z ) =XF (Z ) ∈ TZO (Λ)

∣∣∣ F ∈ C∞(O (Λ))Sp(N−1)

,

see for instance [38, p. 342], from Remark A.4.1 and A.4.1-A.4.4, if we suppose that theHamiltonian Sp(N − 1)-space (O (Λ),ωO (Λ), Sp(N − 1),ΦN−1) is multiplicity free it implies thatker((DΦN−1)Z ) is an isotropic subspace and it also implies that Sp(N − 1) acts transitively onVµ . From these we can show that the reduced space Φ−1

N−1(µ )/TN−1 is just a point, i.e. TN−1

acts transitively on Φ−1N−1(µ ). However, as we have seen the reduced spaces are not in general

trivial spaces. erefore the Hamiltonian Sp(N − 1)-space (O (Λ),ωO (Λ), Sp(N − 1),ΦN−1) cannot be multiplicity free, which violates the necessary condition for integrability by means ofcollective Hamiltonians, see eorem A.4.1 on page 131.

e above ideas are in fact a geometric manifestation of an important feature of the irreducibleSp(N )-modules. Now we will look at the right side of the table B.2 in order to understand whathappens on the representation theory side when we try to apply the ideas involved in the con-struction of the Gelfand-Tsetlin torus action.

We start by taking a regular adjoint orbitO (Λ) ⊂ sp(N ) as before with the additional require-ment that

Λ = diag(iλ1, . . . , iλN ),

with 0 > λ1 > λ2 > · · · > λN , λj − λj+1 ∈ Z>0 and −λ1 ∈ Z>0. From these we can associatedto O (Λ) ⊂ sp(N ) an irreducible representation ρ : sp(2N ,C) → gl (V (λ)) with highest weightλ. Since sp(N ) ⊂ sp(2N ,C) is a compact real form we can also regard V (λ) as irreduciblesp(N )-module, see for instance [99, p. 89, eorem 4.28].

As before we take the sequence of subalgebras

sp(2N ,C) ⊃ sp(2(N − 1),C) ⊃ · · · ⊃ sp(2,C),

and consider the iterative restrictions of (ρ,V (λ)) over each element of the above chain. Like-wise in B.2.3 at the rst step we have a decomposition

V (λ) |sp(2(N−1),C) =⊕µ

Homsp(2(N−1),C)

(W (µ );V (λ)

)⊗W (µ ),

157

where the elements in the above decomposition are the isotypic components

V (λ)µ = Homsp(2(N−1),C)

(W (µ );V (λ)

)⊗W (µ ),

we also can write every µ-isotypic component as

V (λ)µ = c (µ )W (µ ) :=W (µ ) ⊕ · · · ⊕W (µ )

c (µ )−times

,

where c (λ) is the multiplicity thatW (µ ) occurs inside of V (λ).

As was showed in [170, p. 380-383], dierent from the unitary and special orthogonal Lie alge-bras, we may have c (µ ) > 1 in this case, i.e. the restriction V (λ) |sp(2(N−1),C) is not multiplicityfree. erefore, we can not construct an action of the Gelfand-Tsetlin torus on V (λ). Recallfrom the Equation B.2.4 that the action is dened on each isotypic component and in somenon-multiplicty free µ-istotypic component we have the subspace

Homsp(2(N−1),C)

(W (µ );V (λ)

).

In [72, p. 532, eorem 6.2] we have the relation between the symplectic reductions and iso-typic components. In the same way that we can associate to O (Λ) ⊂ sp(N ) its “quantum”analogue [

O (Λ)]

quantum:= V (λ),

we can also associate a “quantum” analogue to the symplectic reductions[Φ−1N−1(µ )/TN−1

]quantum

:= Homsp(2(N−1),C)

(W (µ );V (λ)

). (B.2.6)

e Equation B.2.6 shows us how the symplectic geometry of O (Λ) can be understood bymeans of the representation theory, and how the elements in the table B.2 can be used as aguide to understand the interplay between Gelfand-Tsetlin integrable systems and Gelfand-Tsetlin basis.

Remark B.2.3. In the last two equations we keep the notation [· · · ]quantum used in [72] to denotethe “quantum” analogues for the classical phase spaces. Details about the construction of the“quantum” analogue for symplectic reduction can be found in [72].

As we have seen so far the obstructions for the integrability of the Gelfand-Tsetlin systems inregular adjoint orbits of Sp(N ) are summarized in the Equation B.2.6. e strategy to constructthe Gelfand-Tsetlin-Molev integrable system [79] comes from the method of to construct theso called Gelfand-Tsetlin-Molev basis for irreducible representations of sp(2N ,C), which isdescribed in [122], see also [121, p. 137-158].

e basic elements which were used in [121] to construct the Gelfand-Tsetlin-Molev basis arethe following. According to [41, p. 299, eorem 9.1.12] there exists a natural representationof the centralizer

U (sp(2N ,C))sp(2(N−1),C) =a ∈ U (sp(2N ,C))

∣∣∣ ab = ba, ∀b ∈ sp(2(N − 1)), (B.2.7)

158

on the space Homsp(2(N−1),C)

(W (µ );V (λ)

), whereU (sp(2N ,C)) denotes the universal envelop-

ing algebra of sp(2N ,C). We will denote CN = U (sp(2N ,C))sp(2(N−1),C) in order to simplifythe notation.

e new elements in this seing used in [121] are the twisted Yangian Y−(2) ( in the nextsubsection we will describe this algebra in more details) and the following sequence of homo-morphisms

Y−(2) → CN → Z (sp(2N ,C), sp(2(N − 1),C)). (B.2.8)

e algebra Z (sp(2N ,C), sp(2(N − 1),C)) is called the Mickelsson–Zhelobenko algebra, see[121, p 121-124] for a description of its construction. e rst arrow in B.2.8 allows us to equipB.2.6 with a structure of Y−(2) module [122, p. 606], the second arrow allows us to obtainsuitable expressions for the elements of Y−(2) as operators on B.2.6, see for example [122, p.607, eorem 5.1]. Using the structure of Y−(2)-module on the space B.2.6, a basis parameter-ized by the Gelfand-Tsetlin paern of sp(2N ,C) was provided in [122], see also [121, p. 143,eorem 3.2].

In [79] the concept of the twisted Yangian Y−(2) is used to solve the problem of constructintegrable systems on regular adjoint orbits O (Λ) ⊂ sp(N ). In the next subsection we willprovide a brief introduction of the algebras Yangians, and its twisted versions, as well as toprovide an outline of their geometric interpretation in the construction of the Gelfand-Tsetlin-Molev integrable systems [79].

B.2.2 Generalities about Yangians and twisted Yangians

e concept of Yangians was rst introduced on the study of the quantum inverse scaeringmethod, late 1970s and early 1980s, in the work of Ludvig D. Faddeev as a convenient tool togenerate the solutions of the quantum Yang–Baxter equation [31, p. 124, Proposition 4.2.7].e name Yangian was introduced by V. Drinfeld (1985) in honor of Chen-Ning Yang. ForDrinfeld’s approach of Yangians we suggest [31, p. 375-377], see also [43]. As mentioned in[124], the Yangians form a remarkable family of quantum groups related to rational solutionsof the classical Yang-Baxter equation, see for instance [31, p. 95] and [43, p. 814-816].

e approach of Yangians which we will describe bellow is dierent from Drinfeld’s approach,in fact we will work with the concept of YangiansY (gl (N ,C)) introduced in the work of LudvigD. Faddeev. e main distinguishing feature of these two approaches is that Y (gl (N ,C)) arenot a deformation of U (gl (N ,C)[z]) in the category of Hopf algebras, here

gl (N ,C)[z] = s∑

k=0Xkz

k∣∣∣ Xk ∈ gl (N ,C), ∀ 0 ≤ k ≤ s , s ∈ Z≥0

,

denotes the polynomial current Lie algebra of gl (N ,C). Moreover, for the YangianY (gl (N ,C))we have a homomorphism

Y (gl (N ,C)) → U (gl (N ,C)),

159

called evaluation homomorphism. is homomorphism plays an important role on applica-tions of the Yangians to the conventional representation theory [124]. In what follows we willdescribe how the Yangian and twisted Yangian, according to [124], are dened and discussthe geometric characterization used in [79] in order to apply it to the study of Hamiltoniansystems dened in regular adjoint orbits O (Λ) ⊂ sp(N ).

A brief description of the ideas developed in [79] can be done in the following way. ConsiderY~(N ) as being the non-commutative, associative, unital algebra free generated over C by theset of symbols t (l )ij , with l ≥ 0, 1 ≤ i, j ≤ N , where t (0)ij = δij , subjected to the followingcommutator relations

[t (r )ij , t

(s )kl

]~= ~

min(r ,s )∑a=1

(t (a−1)kj t (r+s−a)il − t (r+s−a)kj t (a−1)

il

), (B.2.9)

here we consider ~ ∈ C as being a xed formal parameter. e algebra Yangian Y (gl (N ,C))is dened by Y (gl (N ,C)) = Y1(N ), see for instance [124] and [123], the Yangian associated togl (N ,C) is also denoted by Y (N ) = Y (gl (N ,C)).

Now we consider the matrix of generators dened by

T (u) = (Tij (u)) ∈ Y (N )[[ 1u ]] ⊗ End(CN ),

such that

Tij (u) = δij +

∞∑l=1

t (l )ij

ul∈ Y (N )[[ 1

u ]].

From this matrix the dening relations B.2.9 can be wrien as a single equation

R12(u −v )T1(u)T2(v ) = T2(v )T1(u)R12(u −v ), (B.2.10)

where T1(u) =∑

ij Eij ⊗ 1 ⊗ Tij (u), T2(v ) =∑

ij 1 ⊗ Eij ⊗ Tij (v ) and

R12(u −v ) =(

1 ⊗ 1 +P

u −v

)⊗ 1 = R (u −v ) ⊗ 1,

with P ∈ End((CN )⊗2) being the permutation operator P =∑

ij Eij ⊗ Eji , and R (z) = 1 ⊗ 1+ Pz

the Yang R-matrix. e Equation B.2.10 above is called the ternary (or RTT) relation, see [124]for more details.

Remark B.2.4. We observe that the Equation B.2.10 is performed as an equation in the algebra[End(CN )⊗2 ⊗ Y (N )

]((u,v )).

e ternary relation allows us to write the dening relations of Y (N ) in a suitable way. More-over, we can use the ternary relation to show that the map σ : Y (2N ) → Y (2N ) given by

σ : T (u) → A(T (−u)T )−1A−1, (B.2.11)

160

denes an involution in Y (2N ), see for example [124, p. 5-7, Proposition 1.3.1 and 1.3.3] and[79, p. ], here we consider A as in B.1.1.

Remark B.2.5. As we have mentioned at the beginning, our exposition about Yangians will notbe a complete exposition. erefore we will not give more details about the Hopf algebra structureof Y (N ), your main references for the background on this maer are [124, p. 9-12] and [31].

Now we will provide a brief description of the twisted Yangian, in order to do this we considersp(2N ,C) ⊂ gl (2N ,C). Following [124] we consider the following change of enumeration forthe lines and columns of elements of gl (2N ,C)

1, . . . , 2N←→

− N , . . . ,−1, 1, . . . ,N

.

Aer this change we can consider sp(2N ,C) ⊂ gl (2N ,C) as being the Lie algebra generatedby the elements

Fij = Eij − θijE−j−i ,

with −N ≤ i, j ≤ N , where θij = sgn(i )sgn(j ), here “sgn” denotes the sign of i and j. By usingthis realization for sp(2N ,C) we have

X =

α β

γ −α ′

,

for every X ∈ sp(2N ,C), here α ′ is obtained from α by transposing the matrix with respectthe anti-diagonal, furthermore, we have β = β′ and γ = γ ′.

Now we describe how the twisted Yangian Y−(2N ) associated to sp(2N ,C) is constructed.Consider the set of symbols s (l )ij , −N ≤ i, j ≤ N , l ≥ 0 with s (0)ij = δij . We dene Y−(2N ) asbeing the non-commutative, associative, unital algebra free generated over C, subjected to thefollowing dening relations

R (u −v )S1(u)RT12(−u −v )S2(v ) = S2(v )R

T12(−u −v )S1(u)R12(u −v ), (B.2.12)

and the symmetric relation

S (−u)T = S (u) −S (u) − S (−u)

2u (B.2.13)

Likewise the case Y (N ) we have the matrix of generators given by

S (u) = (Sij (u)) ∈ Y−(2N )[[ 1

u ]] ⊗ End(CN ),

such that

Sij (u) = δij +

∞∑l=1

s (l )ij

ul∈ Y−(2N )[[ 1

u ]].

161

We notice that on the Equation B.2.12 we have

RT (z) = 1 ⊗ 1 +P

z, with P =

∑ij E

Tij ⊗ Eji .

ere are two important homomorphisms used in [121] on the construction of the Geland-Tsetlin basis for irreducible representations of sp(2N ,C) which are fundamental on the con-struction of the Gelfand-Tsetlin-Molev integrable system. ese homomorphisms are givenby

• Y−(2) → Y−(2N ), s±1±1 → s±N±N ,

• Y−(2) → Z (sp(2N ,C), sp(2(N − 1),C)), see for instance [121, p. 153].

By means of these homomorphisms, according to [121] the space dened by Homsp(2(N−1),C)

(W (µ );V (λ)

)can be identied with a tensor product of irreducible representations of Y−(2), see [121] formore details.

Geometric realization of the Yangian Y (N )

Let us briey describe the geometric realization of Y (N ) used in [79, p. 50]. Let U ⊂ CP1

be an open neighbourhood of +∞ ∈ CP1, consider the trivial holomorphic vector bundleE = U × CN → U . We have the following characterization for the Gauge group associated toE

G (E) =ϕ : U → GL(N ,C)

∣∣∣ ϕ is holomorphic

.

If we considerU ⊂ CP1 as being a coordinate neighbourhood around +∞ ∈ CP1 with coordi-nates 1

u , where u denotes the standard coordinate of C, for every ϕ ∈ G (E) we have

ϕ = (ϕij ), with ϕij : U → C,

for 1 ≤ i, j ≤ N , such that each component ϕij denes a holomorphic germ in +∞ ∈ U . uswe can write

ϕij (u) =∑l≥0

ϕ (l )ij

ul,

where ϕ (l )ij ∈ C, ∀1 ≤ i, j ≤ N . erefore each element ϕ ∈ G (E) can be represented by a

power series of the form

ϕ (u) =∑l≥0

ϕlul

,

with ϕl = (ϕ (l )ij ) ∈ gl (N ,C) = End(CN ). An important subgroup of G (E) which we will

consider is the pointed Gauge group G0(E) ⊂ G (E), this subagroup is dened by

162

G0(E) =ϕ ∈ G (E)

∣∣∣ ϕ (+∞) = 1

.

From the denition above and from the previous comments, for every ϕ ∈ G0(E) we have thefollowing expression

ϕ (u) = 1 +∑l≥1

ϕlul

.

Now we consider Fun(G0(E)) as being the algebra of polynomials on the countably manyvariables z (l )ij , with l ≥ 0 and 1 ≤ i, j ≤ N , where

z (l )ij : G0(E) → C, z (l )ij (ϕ (u)) = ϕ(l )ij ,

for every ϕ ∈ G0(E). According to [124, p. 8, Remark 1.4.4] this polynomial algebra can beendowed with a Poisson bracket dened by

z (r )ij , z(s )kl

G0 (E)

=

min(r ,s )∑a=1

(z (a−1)kj z (r+s−a)il − z (r+s−a)kj z (a−1)

il

).

From the above facts we can consider Fun~(G0(E)) = Y~(N ) as a deformation quantization 2

of Fun(G0(E)), such that z (r )ij , z

(s )kl

G0 (E)

≡1~

[z (r )ij , z

(s )kl

]~

mod(~),

therefore (G0(E), ·, ·G0 (E) ) can be seen as the classical limit of Y~(N ).

rough of a similar idea we can associate to the twisted Yangian Y−(2N ) a classical limitwhich we briey describe now. Consider the following involution dened in GL(2N ,C)

σ : GL(2N ,C) → GL(2N ,C), σ : M → A−1(MT )−1A

here the matrix A is the same in B.1.1. is involution induces a involution σ on G0(E), hereE = U × C2N , dened by

σ (ϕ) (u) = σ (ϕ (−u)),

see [79] for more details. e set of xed points of σ is precisely the subgroup Sp(2N ,C) andtheset of xed points of σ we denote by K σ . From these we have the following resulteorem B.2.1. e classical limit of the twisted Yangian Y−(2N ) is Fun(G0(E)/K σ )

Proof. e proof of this result can be found in [79, p. 54].

e algebra Fun(G0(E)/K σ ) is a subalgebra of Fun(G0(E)) compose by the functions whichdescend to G0(E)/K σ via the natural projection G0(E) → G0(E)/K σ , see [79, p. 53]. We willnot make a extensive discussion about the details of the previous results since it goes beyondour purpose for this section. e important thing here is to keep in mind the objects whichwe have briey described above. Our next task will be to use the previous ideas to describethe construction of Gelfand-Tsetlin-Molev integrable systems.

2Given a Poisson algebra (A , ·, ·) we consider “deformations” (A~, ·, ·~) of A , where these A~ are non-commutative, associative algebras equipped with a “star product”, and ~ is a formal deformation parameter. ezero-th order term in the star product is the original commutative algebra structure, and the rst-order term in~ is given by the original Poisson bracket of A , for more details see [31, p. 177] and references therein.

163

B.2.3 antities in involution via classical limit procedure

Now we are in position to give a basic explanation of how to obtain quantities in involutiondened on regular orbitsO (Λ) ⊂ sp(N ) by means of classical limit. At rst we need to considerthe following scheme of deformation quantization

“antum group” Classical limit Poisson space

Y (2N ) Fun(G0(E)) G0(E)

Y−(2N ) Fun(G0(E)/K σ ) G0(E)/K σ

U (sp(2N ,C))sp(2(N−1),C) Fun(sp(2N ,C)/Sp(2(N − 1),C)) sp(2N ,C)/Sp(2(N − 1),C)

Table B.1: Deformation quantization scheme. Here the term “antum group” is used just torefer to the deformed object. Actually the term “antum group” is usually used to designatedeformations of Hopf algebras in the category of Hopf algebras, and it is not our case onceY−(2N ) is not even a Hopf algebra.

In the third line of the above table we consider Fun(sp(2N ,C)/Sp(2(N − 1),C)) as being thealgebra of polynomial functions dened in sp(2N ,C)/Sp(2(N − 1),C), therefore the quatumalgebra U (sp(2N ,C))sp(2(N−1),C) is obtained from the star product

X ?~ Y − Y ?~ X = ~[X ,Y ],

dened in U~(sp(2N ,C)), notice that U~(sp(2N ,C)) U (sp(2N ,C)) via the correspondenceX → 1

~X , more details can be found in [31, p. 180].

e key point in [79] is to use the previous data to show that the Molev’s map

Y−(2) → CN = U (sp(2N ,C))sp(2(N−1),C) ,

employed in [121] to construct the Gelfand-Tsetlin basis for irreducible representations ofsp(2N ,C) has an associated “classical limit” Poisson map

∆ : sp(N )/Sp(N − 1) → G0(U × C2)/K σ , (B.2.14)here we consider G0(U × C2) and K σ ⊂ G0(U × C2) just like a particular case (N = 1) of theprevious constructions, for more details about how to obtain the classical limit map from theMolev’s map see [79, p. 46-74].

e map B.2.14 allows us pulling back commutative subalgebras of Y−(2), let us make moreclear this last statement, as showed in [79] the map B.2.14 can be wrien as

∆(X ) (u) =

(1 − Xu

)−1−N−N

(1 − X

u

)−1−NN(

1 − Xu

)−1N−N

(1 − X

u

)−1NN

here we notice that (1 − X

u )−1ij denotes the (i, j )-entry of the matrix

164

(1 −

X

u

)−1= 1 +

X

u+X 2

u2 + . . .,

now as by taking the trace of ∆(X ) (u) we get

Tr(∆(X ) (u)) = 1 +∞∑l=1

(X l )−N−N + (X l )NN

ul,

on the above expression if we consider the inclusion Ψ : gl (N ,H) → gl (2N ,C) described inthe Section B.1, we can rewrite the above expression as

Tr(∆(X ) (u)) = 1 +∞∑l=1

2ReH((Xl )NN )

ul, (B.2.15)

now as mentioned in [122, p. 616, Remark 6.3], the coecients of the series

Tr(S (u)) = s−N−N (u) + sNN (u),

here S (u) ∈ Y−(2)[[ 1u ]] ⊂ Y−(2N )[[ 1

u ]], generates a commutative subalgebra of Y−(2), fromthese we have that

∆∗(s (l )−N−N + s(l )NN ) (X ) = 2ReH((X

l )NN ), ∀l ≥ 1.

dene a family of quantities in involution dened in sp(N )/Sp(N − 1).

erefore by taken the restriction over Φ−1N−1(O (µ ))/Sp(N − 1), for some regular value of

ΦN−1 : O (Λ) → sp(N − 1), see B.2.1 to remember the details, we obtain a set of Poisson com-muting functions dened on the reduced space Φ−1

N−1(O (µ ))/Sp(N − 1). e nice feature ofthese functions is that they can be naturally lied to O (Λ), thus from the previous ideas weget new functions

Φ∗N∆∗(s (l )−N−N + s

(l )NN ) : O (Λ) → R, ∀l ≥ 1,

which we can incorporate to the set functions obtained from imm’s trick in order to getintegrability. All about the above ideas can be done at each step of the standard chain B.2.5of closed subgroups. We would like to emphasize that our approach is intended to give just abasic idea of the techniques employed in [79], namely there are many details which we haveomied in order to spend more time with some calculations and examples.

Our next step will be to provide some concrete description of the functions obtained via clas-sical limit procedure which we have described above. We notice that the expression obtainedin the Equation B.2.15 can be rewrien in the following way

Tr(∆(ΦN (X )) (u)) = 1 +∞∑l=1

2ReH((ΦN (X )l )NN )

ul= 1 +

∞∑l=1

2rTr(ΦN (X )lENN )

ul,

165

thus we have

∆∗(s (l )−N−N + s(l )NN ) (ΦN (X )) = 2rTr(ΦN (X )lENN ). (B.2.16)

In the next section we will examine how the above expression are related with the functionsobtained by imm’s trick as well as explain the construction of the Gelfand-Tsetlin-Molevsystem.

B.3 e Gelfand-Tsetlin-Molev integrable systems

In what follows we will provide a complete description of the Gelfand-Tsetlin-Molev inte-grable system [79], the main idea is to explain the relations between the quantities in involu-tion which we have available by classical limit procedure and by imm’s trick.

Given X ∈ sp(N ), from the intersection of the adjoint orbit of X with the Weyl chamber ofsp(N ), or equivalently by diagonalization process, we can write

X = Q (X )Λ(X )Q (X )?,

where Q (X ) ∈ Sp(N ), and Λ(X ) ∈ sp(N ) is a diagonal matrix. According to [79] the Gelfand-Tsetlin-Molev integrable system can be obtained essentially by means of four kinds of func-tions which we describe below

1. e rst kind are dened by the following map

X = Q (X )Λ(X )Q (X )? → ||qNk (X ) | |2,

here we used the notation Q (X ) = (qij (X )). Notice that from the above correspondencewe get N − 1 functions, since each column of Q (X ) corresponds to an unitary vector, sowe have k ∈ 1, . . . ,N − 1.

2. e second kind are given by the imaginary part of the eigenvalues associated to X , wehave the following map

X = Q (X )Λ(X )Q (X )? → Λk (X ),

here we denote Λ(X ) = diag(iΛ1(X ), . . . , iΛN (X )), from this we have N functions. ecorrespondence above is exactly the same used in imm’s trick.

3. e third kind of functions are given by the following correspondence

X → rTr(X 2lENN ) = ReH((X2l )NN ).

notice that the functions above are up to scale factor exactly the functions obtained byclassical limit procedure, see the Equation B.2.16. is third kind are related with theprevious two kinds by the following ideas, since X = Q (X )Λ(X )Q (X )? we have

166

Xij =

N∑k=1

qik (X )iΛk (X )qjk (X ),

now once X 2l = Q (X )Λ(X )2lQ (X )?, we can write

(X 2l )ij =N∑k=1

(−1)lΛk (X )2lqik (X )qjk (X ),

therefore we obtain

(X 2l )ii =N∑k=1

(−1)lΛk (X )2l | |qik (X ) | |2,

it follows that ReH((X2l )ii ) = (X 2l )ii . From the above comments we obtain the following

relation between the functions

rTr(X 2lENN ) =

N∑k=1

(−1)lΛk (X )2l | |qNk (X ) | |2.

Now since on the previous expression we have Q (X ) = (qij (X )) ∈ Sp(N ) it follows that

| |qNN (X ) | |2 = 1 −N−1∑k=1| |qNk (X ) | |2,

therefore we obtain the nal expression

rTr(X 2lENN ) =

N−1∑k=1

(−1)l (Λk (X )2l − ΛN (X )2l ) | |qNk (X ) | |2 + (−1)lΛN (X )2l .

Remark B.3.1. is last expression allows us to relate the functions obtained by imm’strick and the functions introduced in [79] via classical limit.

Before we describe the fourth kind of functions, we point out that we can still write thisthird kind of function in the following way, from the expression

(X 2l )ii =N∑

k1,...,k2l−1=1Xik1Xk1k2 . . .Xk2l−1i ,

we can write

rTr(X 2lENN ) =

N∑k1,...,k2l−1=1

ReH(XNk1Xk1k2 . . .Xk2l−1N ),

167

It follows that this third kind of function also can be seen as real part of a polynomial inthe entries of the matrix X ∈ sp(N ).

Remark B.3.2. It is worthwhile to observe that the above polynomial are in fact obtainedfrom the generators of the subalgebra of the polynomial algebra P (sp(2N ,C)) which areinvariant by the adjoint action of sp(2(N − 1),C), see [123] for more details.

4. e forth kind of functions are given by the map

X → rTr(XiEkk ),

as we will see aerwards the above map is related with the k-component of the momentmap associated to Hamiltonian action of the maximal torus of Sp(k ) on O (Λ) ⊂ sp(N ).We can also rewrite the above expression as follows

X → ReH(XNN i ),

notice that since X +X? = 0, we have ReH(Xkk ) = 0, for all k = 1, . . . ,N , but ReH(Xkki ),k = 1, . . . ,N , is not necessary zero.

Now we explain how these functions can be used in the construction of the Gelfand-Tsetlin-Molev integrable system.

Let O (Λ) ⊂ sp(N ) be a regular adjoint orbit associated to

Λ(X ) = diag(iλ1, . . . , iλN ),

where 0 > λ1 > . . . > λN , we have dimR(O (Λ)) = 2N 2. Now we apply imm’s trick on thechain

Sp(N ) ⊃ Sp(N − 1) ⊃ . . . ⊃ Sp(2) ⊃ Sp(1),

this procedure yields a set of functions given by the following collective hamiltonians

λ(l )k = Φ∗k (Λl ),

with 1 ≤ l ≤ k , and k = 1, . . . ,N − 1, i.e. we have N (N−1)2 functions. Now, for k = 1, . . . ,N ,

we consider the following functions, given X ∈ O (Λ) we set

µ (l )k (X ) = rTr(Φk (X )2lEkk ),

for l = 1, . . . ,k − 1 and we also consider

µ (k )k (X ) = rTr(Φk (X )iEkk ),

notice that the last function which we consider above are in fact the k-component of themoment map associated to the Hamiltonian torus action ofT k ⊂ Sp(k ). From these we have aset of N (N+1)

2 functions. We have the following diagram that summarizes the main idea

168

RN RN−1 RN−2 · · ·

(O (Λ),ωO (Λ) ) sp(N ) sp(N − 1) sp(N − 1) · · ·

tN−1+ tN−2

+ · · ·

µ (N ) µ (N−1)

Λ(N−1)

µ (N−2)

Λ(N−2)

Let us observe that on the above diagram we denote

µ (k ) = (µ (k )1 , . . . , µ(k )k ) and Λ(k ) = (λ(k )1 , . . . , λ

(k )k ),

from these we can state the main result of this section

eorem B.3.1. LetO (Λ) ⊂ sp(N ) be a regular orbit, then the functions obtained from imm’strick

λ(l )k : O (Λ) → R, for 1 ≤ k ≤ l , 1 ≤ l ≤ N − 1,

together with the functions obtained from the classical limit procedure and the torus moment mapcomponent described above

µ (l )k : O (Λ) → R, for 1 ≤ k ≤ l , 1 ≤ l ≤ N ,

dene a completely integrable Hamiltonian system in an open dense subset of O (Λ), this Hamil-tonian system is called Gelfand-Tsetlin-Molev integrable system.

Proof. e complete proof for this result can be found in [79].

Before we pass to more concrete calculations performed in low dimensional examples it isworthwhile to point out that an important step in the construction of the above integrablesystem boils down to analyse the equation

rTr(Φr (X )2lErr ) =r−1∑k=1

(−1)l (Λk (Φr (X ))2l − Λr (Φr (X ))2l ) | |qrk (Φr (X )) | |2 + (−1)lΛr (Φr (X ))2l ,

at each step sp(N ) ⊃ sp(r ), 1 ≤ r ≤ N . In fact this equation involves the functions obtainedfrom imm’s trick and the new functions introduced in [79] by means of the classical limitprocedure.

Now consider the formal power series obtained by classical limit procedure in the previoussection, namely

Tr(∆(ΦN (X )) (u)) = 1 +∞∑k=1

2rTr(ΦN (X )kENN )

ul,

169

since we are not worried about convergence issues of the above power series, we can use ananalogy with the residue formula to obtain the following expression

2rTr(ΦN (X )2lENN ) =1

2πi

∮∂B (0;δ )

u2l−1Tr(∆(ΦN (X )) (u))du (B.3.1)

thus at each step sp(r ) ⊂ sp(N ) the quantities in involution provided by imm’s trick andthe classical limit procedure are encoded in the integrals

2rTr(Φr (X )2lErr ) =1

2πi

∮∂B (0;δ )

u2l−1Tr(∆(Φr (X )) (u))du (B.3.2)

where the maps involved on the above expression can be schematically described through thefollowing diagram

(O (Λ),ωO (Λ) ) sp(r ) sp(r )/Sp(r − 1)

G0(U × C2)/K σ

Φr

∆

Our next step will be to describe explicitly how the above ideas work in concrete examples,thus the next section will be devoted to perform some calculations in low dimensional exam-ples. We hope that our calculations can be useful for future developments on the study ofintegrable systems in adjoint orbits and related topics.

B.4 Applications in lowdimensional symplectic Lie groups

Now we will apply all about the construction describe in the previous section in some basicexamples, the main purpose is to derive suitable expressions for the quantities in involutionwhich compose the Gelfand-Tsetlin-Molev integrable system for regular adjoint orbits of Sp(2)and Sp(3).

B.4.1 Integrable system in regular orbits of Sp(2)

Consider the symplectic Lie algebra sp(2) ⊂ gl (2,H) and x a regular element

Λ =

iλ1 0

0 iλ2

∈ sp(2),with 0 > λ1 > λ2, we denote its adjoint orbit by

O (Λ) =QΛQ? ∈ sp(2)

∣∣∣ Q ∈ Sp(2)

.

From the Hamiltonian action dened by the chain Sp(2) ⊃ Sp(1), we have two moment maps

Φ2 : O (Λ) → sp(2), Φ1 : O (Λ) → sp(1),

170

which can be seen respectively as followsX11 X12

X21 X22

→X11 X12

X21 X22

,X11 X12

X21 X22

→ X11,

for every X = (Xij ) ∈ O (Λ). e Gelfand-Tsetlin-Molev integrable system in this case can bedescribed as follows, given X ∈ O (Λ) we have

X = Q (X )ΛQ (X )?,

for some Q (X ) ∈ Sp(2), from the above relation we can write

X =

2∑

k=1q1k (X )iλkq1k (X )

2∑k=1

q1k (X )iλkq2k (X )

2∑k=1


2∑k=1


At the rst step of the chain Sp(2) ⊃ Sp(1) we have two kinds of functions, the rst one isgiven by

µ (1)2 (X ) = rTr(Φ2(X )2E22) = −(λ21 − λ

22) | |q21(Φ2(X )) | |2 − λ2

2,

as we have seen, we can write

rTr(Φ2(X )2E22) =2∑

k=1ReH(X2kXk2),

since X + X? = 0, i.e. Xij = −X ji , it follows that

ReH(X2kXk2) = ReH(−||X2k | |2) = X2kXk2,

where k = 1, 2. We obtain from these

2∑k=1

X2kXk2 = −(λ21 − λ

22) | |q21(Φ2(X )) | |2 − λ2

2.

We have the following relation between the functions dened in the previous section for thiscase

• X → ||q21(Φ2(X )) | |2 = −1

λ21 − λ

22

2∑k=1

X2kXk2 −λ2

2λ2

1 − λ22,

• X → rTr(Φ2(X )2E22) =2∑

k=1X2kXk2,

171

Still at the rst step we have the following function

µ (2)2 (X ) = rTr(Φ2(X )iE22) = ReH(X22i ),

which is the second component of the moment map associated to the Hamiltonian action ofT2 ⊂ Sp(2).

erefore, the rst step of the chain provides the functions

µ (1)2 (X ) = rTr(Φ2(X )2E22) = −(λ21 − λ

22) | |q21(Φ2(X )) | |2 − λ2

2,

and

µ (2)2 (X ) = rTr(Φ2(X )iE22) = ReH(X22i ).

Now we analyze the second step. Associated to the Hamiltonian action of Sp(2) on O (Λ), wehave a moment map Φ1 : O (Λ) → sp(1). Notice that sp(1) su(2), actually

Φ1(X ) = X11 =

z (X11) w (X11)

−w (X11) z (X11)

,

where ReC(z (X11)) = 0. Now given X ∈ O (Λ), we can write

Φ1(X ) = q(Φ1(X ))Λ(Φ1(X ))q(Φ1(X ))?,

by denition of the functions of the rst kind introduced in the rpevious section, we have

X → ||q(Φ1(X )) | |2 = q(Φ1(X ))q(Φ1(X ))?,

since q(Φ1(X )) ∈ Sp(1) ⊂ H, it follows that q(Φ1(X ))q(Φ1(X ))? = 1, i.e. the function of therst kind is constant.

Now we look at the functions of the second kind, which are exactly the functions obtained byimm’s trick. We have the following map

X → Λ1(Φ1(X )),

since that Φ1(X ) = X11 ∈ su(2), and TrC(X11) = 0, we obtain

det(X11 − t12) = t2 + det(X11),

it follows that

Λ1(Φ1(X )) = ±√

det(X11).

From these we can write

172

λ(1)1 (X ) = Λ1(Φ1(X )) = ±√

det(X11),

notice that by the identication sp(1) su(2), we have det(X11) = | |X11 | |2.

Now we look what happens when we take the functions of the third kind. ese functions aregiven by the correspondence

X → rTr(Φ1(X )2lE11),

since

Φ1(X )2l = (−1)lΛ1(Φ1(X ))2l ,

here we used q(Φ1(X ))q(Φ1(X ))? = 1, and Λ(Φ1(X )) = iΛ1(Φ1(X )).

erefore from the above comments we obtain

X → rTr(Φ1(X )2lE11) = (−1)lΛ1(Φ1(X ))2l = (−1)l det(X11)l ,

it follows that the third kind of functions do not provide nothing new, in fact the above ex-pression shows us that we have just a variation of the function obtained by imm’s method

Now we have the last function dened by the component of the moment map associated tothe Hamiltonian action of T 1 ⊂ Sp(1). Given X ∈ O (Λ), we have

µ (1)1 (X ) = ReH(Φ1(X )iE11),

which also can be rewrien as

µ (1)1 (X ) = ReH(X11i ).

In the nal of the second step we obtain the functions

• X → Λ1(Φ1(X )) = ±√

det(X11),

• X → ReH(Φ1(X )iE11) = ReH(X11i ),

for all X ∈ O (Λ). erefore, the second step of the chain provides the functions

λ(1)1 (X ) = Λ1(Φ1(X )) = ±√

det(X11),

and

µ (1)1 (X ) = ReH(X11i ),

for every X ∈ O (Λ). From the above calculations we obtain the following result

Proposition B.4.1. Let O (Λ) ⊂ sp(2) be a regular adjoint orbit, then the functions which com-pose the Gelfand-Tsetlin-Molev integrable system [79] on O (Λ) can be wrien as

173

• µ (1)2 : X → rTr(Φ2(X )2E22) =2∑

k=1X2kXk2,

• µ (2)2 : X → rTr(Φ2(X )iE22) = ReH(X22i ),

• λ(1)1 : X → Λ1(Φ1(X )) = ±√

det(X11),

• µ (1)1 : X → ReH(Φ1(X )iE11) = ReH(X11i ),

for every X ∈ O (Λ).

in the next subsection we will perform the calculations for an arbitrary regular adjoint orbitO (Λ) ⊂ sp(3), notice that in this case we have dimR(O (Λ)) = 18, thus the description will bemore complicated.

B.4.2 Integrable system in regular orbits of Sp(3)

Now we will develop the ideas involved on the construction of the integrable system for ad-joint orbits of Sp(3) by means of the recipe of the Gelfald-Tsetlin-Molev systems. In the sameway we have done for the Sp(2) case, the main purpose here is to provide suitable expressionsfor the functions which compose the Gelfand-Tsetlin-Molev integrable system .

We star by xing a regular element Λ ∈ sp(3) dened by

Λ =

iλ1 0 0

0 iλ2 0

0 0 iλ3

,

with 0 > λ1 > λ2 > λ3. Associated to this element we have a regular adjoint orbit

O (Λ) =QΛQ? ∈ sp(3)

∣∣∣ Q ∈ Sp(3)

,

as we mentioned previously for this manifold we have dimR(O (Λ)) = 18, thus our integrablesystem in this case will be composed by 9 functions. We will work on the following chain ofclosed and connected subgroups

Sp(3) ⊃ Sp(2) ⊃ Sp(1),

for each element in the above chain we have a Hamiltonian action on O (Λ), we denote theassociated moment maps by

Φ3 : O (Λ) → sp(3), Φ2 : O (Λ) → sp(2), and Φ1 : O (Λ) → sp(1),

these maps are given respectively by the following correspondence, given

174

X =

X11 X12 X13

X21 X22 X23

X31 X32 X33

∈ O (Λ),

we have the following sequence of projections which denes the above moment mapsX11 X12 X13

X21 X22 X23

X31 X32 X33

→X11 X12

X21 X22

→ X11,

we notice that givenX ∈ O (Λ), we haveX = Q (X )ΛQ (X )?, thus a straightforward calculationshows us that

X =

3∑k=1


3∑k=1


3∑k=1


3∑k=1


3∑k=1


3∑k=1


3∑k=1


3∑k=1


3∑k=1


,

here we used for Q (X ) ∈ Sp(3) the following notation

Q (X ) =

q11(X ) q12(X ) q13(X )

q21(X ) q22(X ) q23(X )

q31(X ) q32(X ) q33(X )

, Q (X )? =

q11(X ) q21(X ) q31(X )

q12(X ) q22(X ) q32(X )

q13(X ) q23(X ) q33(X )

,Now proceeding as in the previous section we start by taken the following functions at therst step

µ (1)3 (X ) = rTr(Φ3(X )2E33), µ (2)3 (X ) = rTr(Φ3(X )4E33), µ (3)3 (X ) = rTr(Φ3(X )iE33),

for all X ∈ O (Λ). ese functions have the following description, for the rst function we canwrite

µ (1)3 (X ) = rTr(X 2E33) = −2∑

k=1(λ2

k − λ23) | |q3k (X ) | |2 − λ2

3,

similar for the second one we have

µ (2)3 (X ) = rTr(X 4E33) =2∑

k=1(λ4

k − λ43) | |q3k (X ) | |2 + λ4

3,

we can also write these two functions in the following way

175

µ (1)3 (X ) = ReH((X2)33) and µ (2)3 (X ) = ReH((X

4)33)

from these we obtain

µ (1)3 (X ) =

3∑k=1

ReH(X3kXk3), and µ (2)3 (X ) =

3∑k1,k2=1

ReH(X3k1Xk1k2Xk23).

For the function µ (3)3 : O (Λ) → R, we have the following alternative expression

µ (3)3 (X ) = rTr(XiE33) = ReH(X33i ) = −Im(z (X33)),

here we used X33 = z (X33) + jw (X33). Now in the second step we have the following relation

Φ2(X ) = Q (Φ2(X ))Λ(Φ2(X ))Q (Φ2(X ))?,

where Q (Φ2(X )) ∈ Sp(2), and

Λ(Φ2(X )) =

iΛ1(Φ2(X )) 0

0 iΛ2(Φ2(X ))

.

We have the following functions at the second step

λ(1)2 (X ) = Λ1(Φ2(X )), λ(2)2 (X ) = Λ2(Φ2(X )),

µ (1)2 (X ) = rTr(Φ2(X )2E22), µ (2)2 (X ) = rTr(Φ2(X )iE22),

furthermore, likewise in the previous section we can write

µ (1)2 (X ) = −(Λ1(Φ2(X ))2 − Λ2(Φ2(X ))2) | |q21(Φ2(X )) | |2 − Λ2(Φ2(X ))2,

and

µ (2)2 (X ) = ReH(X22i ).

At the third step we have the following functions

λ(1)1 (X ) = Λ1(Φ1(X )) = ±√

det(X11),

and

µ (1)1 (X ) = rTr(Φ1(X )iE11) = ReH(X11i ) = −Im(z (X11)),

from the above description we obtain the following result

Proposition B.4.2. Let O (Λ) ⊂ sp(3) be a regular orbit, then the Gelfand-Tsetlin-Molev systemis dened by the following functions

176

• µ (1)3 (X ) =

3∑k=1

X3kXk3 = −2∑

k=1(λ2

k − λ23) | |q3k (X ) | |2 − λ2

3,

• µ (2)3 (X ) =

3∑k1,k2=1

ReH(X3k1Xk1k2Xk23) =2∑

k=1(λ4

k − λ43) | |q3k (X ) | |2 + λ4

3,

• µ (3)3 (X ) = rTr(XiE33) = ReH(X33i ) = −Im(z (X33)),

• λ(1)2 (X ) = Λ1(Φ2(X )),

• λ(2)2 (X ) = Λ2(Φ2(X )),

• µ (1)2 (X ) =

2∑k=1

X2kXk2 = −(λ(1)2 (X )2 − λ(2)2 (X )2) | |q21(Φ2(X )) | |2 − λ(2)2 (X )2,


• λ(1)1 (X ) = Λ1(Φ1(X )) = ±√

det(X11),


for all X ∈ O (Λ).

As we have seen for the low dimensional case given byO (Λ) ⊂ sp(3) the calculations are morecomplicated, nevertheless we can obtain explicit expression for the functions which denethe Gelfand-Tsetlin-Molev system. We hope that our calculations can be used to understandthe behavior of the Gelfand-Tsetlin-Molev integrable system as well as its relations with theGelfand-Tsetlin paern.

177

Appendix CProjective algebraic realization ofcoadjoint orbits

e purpose of this chapter is to describe how the transition from the complex analytic to thecomplex algebraic geometry works for complex ag manifolds. Our approach will not takeaccount of deep questions related to algebraic geometry, in fact our interest here is to providea unied text on which many interest results related to complex geometry, algebraic geometryand representation theory can be used to describe the GAGA principle.

It is worthwhile to point out that we will deal just with basic content in algebraic geometry,thus our exposition will require the introductory basic language used to study algebraic vari-eties. erefore we will assume some basic results which can be found in classical text booksas [82], [148] and [127].

As we will see aerwards, one of the main features of work with complex ag manifolds isthe underlying tools of Lie theory which we have disposable to approach geometric questions.Even though many results that we will cover here are well known, there is a lack of introduc-tory texts which combine the avor of these three branches of mathematics, thus we hope tocontribute with the literature and bibliography.

C.1 Representation theory of simple Lie algebras

In this section we will establish some basic results about representation theory of complexsimple Lie algebras.

Let gC be a complex simple Lie algebra, by xing a Cartan subalgebra h and a simple rootsystem Σ ⊂ h∗, we have a decomposition of gC given by

gC = n− ⊕ h ⊕ n+,

where n− =∑

α∈Π− gα and n+ =∑

α∈Π+ gα , here we denote by Π = Π+ ∪ Π− the root systemassociated to the simple root system Σ = α1, . . . ,αl ⊂ h

∗.

Given α ∈ Π+ we have hα ∈ h such that α = κC(·,hα ), from this we can choose xα ∈ gα and

178

yα ∈ g−α such that [xα ,yα ] = hα , by taken the normalization

h∨α =2

κ (hα ,hα )hα ,

we obtain a subalgebra sl (α ) isomorphic to sl (2,C), generated by yα ,h∨α ,xα , see for instance[87, p. 37]. We will x a basis for gC dened byyα ,h∨α ,xα , α ∈ Π+, this basis is called Chevalleybasis [154, p. 6].

ConsiderV as being gC-module, namely,V is complex vector space which denes a represen-tation of gC. If we consider the restriction of this representation over h, it follows that h actsdiagonally on V . us for each λ ∈ h∗ we can dene

Vλ =v ∈ V

∣∣∣ hv = λ(h)v,∀h ∈ h.

When Vλ , 0 the functional λ ∈ h is called weight of V and Vλ is called the weight space ofλ. Although the denitions that we have done so far are independent of the dimension of V ,we will be concerned to the case when V is nite dimensional, [87, p. 107].

We denote by V′ the sum of the weight spaces of V , i.e. V ′ =

∑Vλ, from this we have the

following result

Lemma C.1.1. Let V be a gC-module, then we have

1. gαVλ ⊂ Vλ+α , ∀λ ∈ h∗ and ∀α ∈ Π;

2. V′

=⊕

Vλ, and V′

is a gC-submodule of V ;

3. If dim(V ) < ∞, then V = V′

.

Proof. e proof can be found in [87, p. 107].

We say that λ ∈ h∗ is a highest weight if there exists v+ ∈ Vλ, v+ , 0, such that n+v+ = 0,in this case v+ is called highest weight vector, and we denote v+ = v+λ . Notice that if V is agC-module which admits a highest weight vector v+λ associated to some weight λ ∈ h∗, wehave that V (λ) = U(gC)v+λ ⊂ V denes gC-submodule, here U(gC) denotes the universal en-veloping algebra associated to gC, see for example [87, p. 89]. erefore, since our purposeis to establish some results about irreducible modules, the study of the case V = U(gC)v+λ isfundamentally necessary.

When V = U(gC)v+λ we say that V is a standard cyclic gC-module with highest weight λ ∈ h∗.We have the following result relatedo to this kind of gC-module

eorem C.1.2. Let V be a standard cyclic gC-module with highest weight vector v+λ ∈ Vλ anddenote Π+ = β1, . . . , βm. en:

1. V is generated by elements of the form yi1β1. . .yimβmv

+λ , ik ∈ Z; in particularV is given by the

direct sum of its weight spaces;

2. e weights of V are given by λ −∑l

s=1 ksαs , ks ∈ Z+;

179

3. For every µ ∈ h∗, we have dim(Vµ ) < ∞, and dim(Vλ) = 1;

4. Each submodule of V is a sum of its weight spaces;

5. V is a gC-module indecomposable, with a unique maximal submodule and a unique irre-ducible quotient;

6. Every nonzero homomorphic image of V is also standard cyclic of weight λ ∈ h∗.


Corollary C.1.3. Let V as in the previous theorem. Suppose that V is irreducible, then v+λ ∈ Vλis the unique highest weight vector of V , up to nonzero scalar multiples.

Proof. See [87, p. 109].

e next results are very important results on the study of the representation theory of com-plex (semi)simple Lie algebras. Actually from these results we can obtain the classication ofall nite dimensional irreducible representations associated to a complex simple Lie algebras.

e rst result is related to the uniqueness, up to isomorphism, of irreducible standard cyclicgC-modules.

eorem C.1.4. LetV andW be standard cyclic gC-modules with highest weight λ. IfV andWare irreducible, then V andW are isomorphic.


e next result is related to the existence of irreducible standard cyclic gC-modules.

eorem C.1.5. Given λ ∈ h∗, then there exists an irreducible standard cyclic gC-module V (λ)with highest weight λ.


It is worth to observe that even though the previous theorem ensures the existence of irre-ducible standard cyclic gC-module with predetermined highest weight at a rst glance we donot have information about the dimension of the associated representation. It follows from thefact that V (λ) is given by a quotient of an innite dimensional module, called Verma module,by a maximal submodule, see for example [87, p 110] for more details.

Now let us describe the basic abstract language of weights in order to deal with the case whendim(V (λ)) < ∞. We say that λ ∈ h∗ is a dominant weight if λ(hα ) ≥ 0, ∀α ∈ Σ, and we saythat λ ∈ h∗ is an integral weight if λ(hα ) ∈ Z, ∀α ∈ Σ. We denote by Λ∗Z ⊂ h

∗ the laice ofintegral weights. By taking ωα = (h∨α )

∗, α ∈ Σ, i.e. the dual basis of h∨α ∈ h | α ∈ Σ, we get aZ-basis for the laice of integral weights. e elements of the basis ωα = (h∨α )

∗ ∈ h∗ | α ∈ Σare called fundamental weights.

180

We denote by Λ∗Z>0the set of integral dominant weights, and by Π(λ) the set of weights as-

sociated to the standard cyclic gC-module obtained from eorem C.1.5. From these we havethe following theorem

eorem C.1.6. If λ ∈ h∗ is an integral dominant weight, then the standard cyclic gC-moduleV (λ) is nite dimensional and its set of weights Π(λ) is permuted by the Weyl group W .

Proof. e proof can be found in [87, p. 112-113].

C.2 Analytic projective subvarieties and algebraic realiza-tion

In what follows we will provide some basic ideas related to complex manifolds and analyticvarieties. e main purpose is to set up a basic language and notation in order to deal withag manifolds as projective varieties in the analytic and algebraic context.

We start by seing the basic objects in complex geometry, our approach is based on [103]and [63]. Let (M, J ) be a n-dimensional complex manifold and z1, . . . , zn local holomorphiccoordinates, where zk = xk + iyk , k = 1, . . . ,n. On this coordinate system we have dzk =dxk + idyk and dzk = dxk − idyk . By taking

∂zk =12 (∂xk − i∂yk ), ∂zk =

12 (∂xk + i∂yk ), k = 1, . . . ,n,

we have that ∂z1, . . . , ∂zn (resp. ∂z1, . . . , ∂zn ) denes a basis for T (1,0)p M (resp. T (0,1)

p M), for ev-ery p in the coordinate neighbourhood, and dz1, . . . ,dzn (resp. dz1, . . . ,dzn) denes a basis for∧(1,0) (M ) (resp.

∧(0,1) (M )).

From the above comments we x the following notation for a complex manifold (M, J ) withlocal coordinates z1, . . . , zn:

• Zi = ∂zi , Zi = ∂zi , i = 1, . . . ,n.Given a Hermitian metric д on M , we can extend the inner product dened by д on eachTpM to a complex symmetric bilinear form on TpM

C, which we also denote by д. Wehave the following

• дAB = д(ZA,ZB ), A,B = 1, . . . ,n, 1, . . . ,n.

• дis = дis = 0, i = 1, . . . ,n,where (дis ) is a n × n Hermitian matrix. We also denote the metric д by

ds2 =∑

i,s дisdzi ⊗ dzs .

Now we remember that the fundamental 2-form associated to a Hermitian metric д isdened by ω (X ,Y ) = д(JX ,Y ), ∀X ,Y ∈ Γ(TM ). is 2-form can be wrien locally as

ω = i∑

i,s дisdzi ∧ dzs .

181

e Hermitian structure induced by a Hermitian metric д is dened by the sesquilinearform

H (X ,Y ) = д(X ,Y ) − iω (X ,Y ),

∀X ,Y ∈ Γ(TMC).

Since we are concerned to study complex manifolds dened by ag manifolds endowed withits canonical complex structure, which are also Kahler manifolds (see 4.5.7). In the previouscontext we will assume in the most cases that dω = 0.

Example C.2.1. e complex Grassmannian Gr(p,Cp+q ), dened by the spaces which param-eterizes all linear subspaces of a vector space Cp+q of given dimensionp, is a complex manifold.Let us describe its structure of complex manifold. Consider z1, . . . , zp+q as being the canonicalcoordinates in Cp+q , where each zk : Cp+q → C is regarded as a linear projection map. For eachset of integers i = i1, . . . , ip such that

1 ≤ i1 < . . . < ip ≤ p + q,

letUi be the subset of Gr(p,Cp+q ) formed by thep-dimensional subspaces S such that zi1 |S , . . . , zip |Sare linearly independent. Consider now ip+1, . . . , ip+q the set dened by the complementof i1, . . . , ip in 1, . . . ,p + q, arranged in ascending order. Since for every S ∈ Ui the setzi1 |S , . . . , z

ip |S denes a basis for S∗, we can write

zip+k |S =

p∑l=1

skl (zil |S ), k = 1, . . . ,q.

erefore, we can dene

φi : Ui → Mp×q (C) Cpq , φi (S ) = (skl ) ∈ Mp×q (C) Cpq ,

∀S ∈ Ui . A straightforward calculation shows us that φi : Ui → Mp×q (C) Cpq denes a set oflocal coordinates in Gr(p,Cp+q ), this the family of

(p+qp

)charts (Ui ,φi ) denes an holomorphic

atlas on Gr(p,Cp+q ) on which dimC(Gr(p,Cp+q )) = pq.

Remark C.2.1. It is worth to observe that all complex Grassmannian Gr(p,Cp+q ) can be pre-sented as an adjoint or coadjoint orbit of the unitary group (see Example 2.1.4). us these mani-folds also can be realized as complex ag manifolds.

Although the previous construction of the complex Grassmannian was did by taking the vectorspace Cp+q , we can apply the same ideas for any complex vector space, since the transitionsfunctions obtained by the previous construction are biholomorphic maps between open setsof Cpq . e next example provide a basic description of geometric aspects from the Kahlergeometry standpoint of the ambient space where ag manifolds can be embedded.

Example C.2.2. Consider the particular case Gr(1,Cn+1) = CPn of the previous example. Asbefore for each i = 1, . . . ,n + 1, we have a local chart (Ui ,φi ), where φi : Ui ⊂ CPn → Cn

is dened as follows. Given S ∈ Ui , we have S = SpanCv, where v ∈ Cn+1 − 0, sincezi |S , 0⇒ zi (v ) , 0. erefore, for v =

∑n+1k=1 t

kek we have

182

φi (S ) =(t1

t i, . . . ,

t i−1

t i,t i+1

t i, . . . ,

tn+1

t i)∈ Cn,

for S ∈ Ui . We denote by w1i , . . . , w

ii , . . . ,w

n+1i the local coordinates in Ui , where wk

i =tk

t i ,k = 1, . . . ,n + 1 e i = 1, . . . ,n + 1.

Now for each coordinate neighbourhood Ui we can dene a function fi : Ui ⊂ CPn → R, suchthat fi =

∑n+1k=1w

ki w

ki , then we have

fi = fswsiw

si , on Ui ∩Us .

Related to the above dened functions we have the following facts:

• By construction the functions wsi are holomorphic in Ui , particularly they are holomor-

phic in Ui ∩Us ;

• We have log fi = log fs+log(wsiw

si ) ⇒ ∂∂ log fi = ∂∂ log fs inUi∩Us , in fact ∂wl

i∂wl

i(log(ws

iwsi )) =

0, for l = 1, . . . ,n + 1, in Ui ∩Us

erefore we can dene

ω = i∂∂ log fi on Ui .

It follows from the previous comments that ω is a globally dened 2-form. Furthermore, ω ∈Ω(1,1) (CPn ) and dω = 0, since d = ∂ + ∂. We can dene a bilinear symmetric form

д(X ,Y ) = ω (JX ,Y ).

In order to see that the above expression denes a Hermitian metric we look at its local expres-sion. Let w1

i , . . . , wii , . . . ,w

n+1i be holomorphic coordinates in Ui , we denote by hi = log fi =

log(∑

k wki w

ki ), from this we have

∂wsi∂wl

i(hi ) =

δsl (∑

k wki w

ki ) −w

liw

si

(1 +∑

k,i wki w

ki )

2 ,

where s, l = 1, . . . ,n. us we obtain

ω = i∂wsi∂wl

i(hi )dw

si ∧ dw

li = i∂ws

i∂wl

i(hi )dw

si ∧ dw

li ,

notice that on the above expression we omied the sum on the index s and l . Since дsl =2∂ws

i∂wl

i(hi ), where s, l = 1, . . . ,n, we can show that д in fact denes a Hermitian metric, see

for instance [148, p. 188-189]. erefore, locally our metric can be expressed by

ds2 = ∂wsi∂wl

i(hi )dw

si ⊗ dw

li =

δsl (∑

k wki w

ki ) −w

liw

si

(1 +∑

k,i wki w

ki )

2 dwsi ⊗ dw

li ,

this metric is called Fubini–Study Kahler metric. With the above metric the complex projectivespace CPn denes a Kahler manifold.

183

As before the construction above can be done in an arbitrary complex vector space. erefore,for an arbitrary complex vector space V we denote by P(V ) the complex manifold dened bythe spaces with parameterizes all 1-dimensional subspace of V , i.e. the space of all complexlines through the origin of V . If we x a basis v1, . . . ,vn+1 for V , we obtain an identicationwith Cn+1, this identication induces a biholomorphism between P(V ) and CPn.

Now we will establish some basic results related to the concept of analytic subvariety. Acomplex manifold (M, J ) is a smooth analytic variety, since it admits an atlas with holomorphictransitions. erefore we can talk about the concept of analytic subvariety of (M, J ).

Denition C.2.1. An analytic subvariety N ⊂ (M, J ) is a subset dened locally as the zero-locusof a nite set of holomorphic functions.

Notice that if (N , J ′ ) is a complex submanifold of (M, J ), i.e. J ′ = J |N , it implies that N ⊂ Mis an analytic subvariety. Moreover, if (M, J ) admits a Kahler structure then (N , J

′

) is also aKahler submanifold when we consider the induced Kahler structure.

Let N ⊂ (M, J ) be an analytic subvariety, given p ∈ N we say that p is a smooth point ifthere exists an open neighbourhood U ⊂ M such that U ∩ N denes a complex submanifoldof M , we denote by N ∗ the set of smooth points of N . A point p ∈ N \N ∗ is called singularpoint of N and we denote by Ns the set of singular points of N . Notice that when (N , J

′

) isa complex submanifold of (M, J ), namely, when the underlying submanifold is embedded, wehave (N , J

′

) as an analytic subvariety without singularities, in this case we say that (N , J ′ ) isa smooth subvariety.

An analytic subvariety N ⊂ (M, J ) is called reducible if N = N1 ∪ N2, where N1,N2 ( N areanalytic subvarieties. From this an analytic subvariety N ⊂ (M, J ) is called irreducible if it isnot reducible.

From the last comments it is worth to point out that since our approach is concerned to studycomplex manifolds we will not give more details about analytic varieties with singularities.Actually, all we have mentioned above about analytic varieties is intended to make more con-sistent the passage from the complex analytic geometry to the complex algebraic geometry.

Denition C.2.2. A projective algebraic variety N ⊂ P(V ) is dened as being the zero-locus ofa nite set of homogeneous polynomials P1(z), . . . , Ps (z).

Notice that from the above denition every projective algebraic variety denes a closed an-alytic subvariety 1 of P(V ), thus we can also talk about smoothness and reducibility of thesespecial kind of analytic varieties.

e next result gives account of the correspondence between closed analytic subvarieties ofP(V ) and projective algebraic varieties

1It is worthwhile to observe that in general in the denition of the projective algebraic varieties it is alsorequired irreducibility, i.e. from the viewpoint of algebraic geometry our denition of projective algebraic varietycorresponds to the denition of closed algebraic set in P(V ), see for example [82, p. 10] and [127, p. 21].

184

Figure C.1: Correspondence provided by Chow’s theorem.

eorem C.2.3 (Wei-Liang Chow). Every closed analytic subvariety of P(V ) is a projectivealgebraic variety.


Based on the content developed in this section our next task will be to construct a projectiveembedding for a ag manifold which is associated to an adjoint orbit of a compact simple Liegroup. We point out that the last eorem C.2.3 will be essential to prove the main result ofthis chapter.

C.3 Projective embedding of ag manifolds

We consider again gC as being a simple complex Lie algebra. As before, by xing a Cartansubalgebra h and a simple root system Σ ⊂ h∗, we have a decomposition of gC given by

gC = n− ⊕ h ⊕ n+,

where n− =∑

α∈Π− gα and n+ =∑

α∈Π+ gα , with Π = Π+ ∪ Π− as root system associated toΣ = α1, . . . ,αl ⊂ h

∗.

Let P ⊂ GC be a parabolic subgroup, as we have seen in 4.5.2, we can associate to P a compactKahler manifold dene by

XP = O (λP ),

see Corollary 4.5.7. Furthermore, we have P = PΘ, where Θ ⊂ Σ can be characterized by

Θ =α ∈ Σ

∣∣∣ λP (hα ) = 0

,

where λP ∈ h∗ can be chosen as integral weight of the form

λP =∑α∈Σ\Θ

nαωα ,

where nα ∈ Z>0. erefore, from eorem C.1.6 we can associate to XP a standard cyclic irre-ducible gC-module V (λP ), with dim(V (λP )) < ∞.

We denote by ρ : gC → gl (V (λP )) the irreducible representation with highest weight λP ∈ Λ∗Zdescribed above, and by v+λP the highest weight vector associated to λP . We will also denoteby ρ (x )v = xv ,∀x ∈ gC and ∀v ∈ V (λΘ), the action of gC in V (λP ).

From the above considerations we have the following result

185

eorem C.3.1. e parabolic subalgebra p = pΘ ⊂ gC is the subalgebra that stabilizes thecomplex line Cv+λP .

Proof. At rst we notice that since

pΘ = n+ ⊕ h ⊕

∑α∈〈Θ〉−

gα = b ⊕∑

α∈〈Θ〉−

gα .

By denition of highest weight vector we have that b stabilizes the complex line Cv+λP . ere-fore in order to obtain the desired result it is enough to show that

∑α∈〈Θ〉− gα stabilizes the

complex line Cv+λP .

Given α ∈ 〈Θ〉+, suppose that

yαv+λP, zv+λP , para todo z ∈ C,

here yα generates the root space g−α . It follows that yαv+λP , 0, thus we have that

U(sl (−α ))yαv+λP⊂ V (λP ),

denes an irreducible sl (−α )-module with highest weight vector given by yαv+λP .

Now given β ∈ Σ we have

h∨βyαv+λP= ([h∨β ,yα ] + yαh∨β )v+λP ,

it follows that h∨βyαvλP = (−α +λP ) (h∨β )yαvλP , therefore we have −α +λP ∈ Π(λP ). From these

if we take the simple reexion rα ∈ W , we obtain

rα (α + λP ) = α + λP −2κ (α ,α + λP )

κ (α ,α )α = −α + λP .

Since W permutes the elements of Π(λP ), see eorem C.1.6, we have α+λP ∈ Π(λP ), howeverα ∈ 〈Θ〉+ ⊂ Π+ which contradicts the fact that λP is the highest weight. us we have that pΘstabilizes the complex line Cv+λP .

If we denote by p′ ⊂ gC the subalgebra which stabilizes the complex line Cv+λP , from our pre-vious calculation we have shown that b = n+ ⊕ h ⊂ p′. us p′ denes a parabolic subalgebra.Since h ⊂ p′we have that p′ is given by a direct sum of b and some root spaces gα , with α ∈ Π−,i.e. p′ is a standard parabolic subalgebra.

Consider α ∈ Π−, we have that

x−αy−αv+λP= λP (h−α )v

+λP

,

it follows that gα estabilizes the complex line CvλΘ if and only if λP (h−α ) = 0, however it isequivalent to −α ∈ 〈Θ〉+. erefore we conclude that pΘ = p′.

Now we observe that ρ : gC → gl (V (λP )) induces a holomorphic representation η : GC →

GL(V (λP )), such that (dη)e = ρ. From this we have a holomorphic action of GC on P(V (λP )),dened by д[v] = [дv], ∀д ∈ GC and ∀v ∈ V (λP ).

186

Proposition C.3.2. Under the previous hypothesis we have

P =д ∈ GC

∣∣∣ д[v+λP ] = [v+λP ]

,

i.e. the parabolic subgroup P = PΘ is the stabilizer group of [v+λP ] ∈ P(V (λP )).

Proof. e fact that

PΘ ⊂д ∈ GC

∣∣∣ д[vλP ] = [vλP ]

,

is a consequence of the of the following idea. Given x ∈ pΘ we have

exp(x )[v+λP ] = [eρ (x )v+λλP ] = [v+λP ].

Now given д ∈ GC such that д[v+λP ] = [v+λP ], for every x ∈ pΘ we have

Ad(д)xvλλΘ = дxд−1vλΘ = µ (x )vλΘ ,

for some µ (x ) ∈ C. Actually the rst equality above follows from the following fact. Sinceρ = (dη)e given y ∈ gC we obtain for д = exp(y) the following expression

dηe (Ad(exp(y))x ) = dηe (ead(y)x ) = ead(dηey)dηex = Ad(edηey )dηex .

erefore, once edηey = η(exp(y)) it follows that ρ (Ad(д)x ) = Ad(η(д))ρ (x ), notice that thislast equality holds form every element д ∈ GC. From the above comments we conclude that

д[vλP ] = [vλP ] =⇒ Ad(д)x ∈ pΘ =⇒ д ∈ NGC (pΘ) = PΘ,

i.e. P = PΘ is in fact the stabilizer of [v+λP ] ∈ P(V (λP )).

From the last results we have the following theorem

eorem C.3.3. Every coadjoint orbit O (λP ) = XP associated to some parabolic subgroup P ⊂GC is an irreducible projective algebraic variety.

Proof. In order to show the result, we consider the following map

Ψ : дP 7→ д[v+λP ],

hence we have Ψ : XP → P(V (λP )) which denes a biholomorphis between XP and the GC-orbit of [v+λP ] ∈ P(V (λP )). SinceXP = O (λP ) is a compact manifold and P(V (λΘ)) is a Hausdorspace, it follows that

Ψ : XP → P(V (λP )) ,

denes an embedding ofXP in P(V (λP )), thus we conclude thatXP is a closed nonsingular ana-lytic subvariety of P(V (λP )). From eorem C.2.3 it follows thatXP = O (λP ) is a (nonsingular)projective algebraic variety. e irreducibility of XP follows from the fact that closed sets inthe Zariski topology are also closed in the induced analytic topology. Once XP is connected inthe induced analytic topology it can not be wrien as the disjoint union of two proper alge-braic subsets.

Let us illustrate all we have done so far by means of two basic examples

187

ExampleC.3.1. LetGC = SL(3,C) and consider P = PΘ, forΘ = Σ\α1, here we are followingthe conventions used in [29, p. 186] for the choice of Cartan subalgebra and simple root system,in this case we have Σ = α1,α2. From these we obtain

XP = SL(3,C)/P = CP2,

with λP = ωα1 . Since V (ω1) = C3, see for instance [29, p. 186], we have v+ωα1= e1, here we

consider the canonical basis for C3, thus from eorem C.3.3 we have

Ψ : XP → P(C3) = CP2, Ψ : дP → [дe1].

Since P is the stabilizer of the line [v+ωα1], we can denote P = Pωα1

. In this basic example wesee that since SL(3,C)e1 = C3\0 we have in fact XPωα1

= P(V (ωα1 )).

Example C.3.2. Consider the case GC = SL(3,C) and P = B, we have

O (λB ) = XB = SL(3,C)/B,

where λB = ωα1 + ωα2 . Here we are following the conventions used in [29, p. 186] for thechoice of Cartan subalgebra and simple root system. In this case we have Σ = α1,α2 with

α1 = λ1 − λ2, and α2 = λ2 − λ3,

for the diagonal projectors λi : sl (3,C) → C, j = 1, 2, 3. e representation with highest weightλB satises

V (λB ) ⊂ V (ωα1 ) ⊗ V (ω2) = C3 ⊗∧2(C3),

thus its highest weight vector is given by

v+λB = v+ωα1⊗ v+ωα2

= e1 ⊗ (e1 ∧ e2),

here ei ⊂ C3, i = 1, 2, 3, denotes the standard basis and ei ∧ ej , 1 ≤ i < j ≤ 3, denotes a weightbasis for

∧2(C3). erefore, according to eorem C.3.3 we have

XB → P(V (λB )) ⊂ P(C3 ⊗∧2(C3)).

Let us describing the map which denes the above embedding. Given дB ∈ XB , we have

д =

z11 z12 z13

z21 z22 z23

z31 z32 z33

∈ SL(3,C),

thus we obtain

Ψ(дB) = [дv+λB ] = [(дe1) ⊗((дe1) ∧ (дe2)

)].

188


[(дe1) ⊗((дe1) ∧ (дe2)

)] = [

(Z1(д),Z2(д),Z3(д)

)⊗(W12(д),W13(д),W23(д)

)],

where Zi (д) = zi1, for i = 1, 2, 3, and the componentsWij (д) are given by the minors of size 2

W12(д) = det

z11 z12

z21 z22

, W13(д) = det

z11 z12

z31 z32

, W23(д) = det

z21 z22

z31 z32

.

From the above expression for the coordinate functions which denes the projective embed-ding of XB we can show that

Z1(д)W23(д) − Z2(д)W13(д) + Z3(д)W12(д) = 0.

Actually we have the following algebraic presentation for XB ⊂ P(V (λB ))

XB =Z1W23 − Z2W13 + Z3W12 = 0

,

here we have [(Z1,Z2,Z3

)⊗(W12,W13,W23

)] ∈ P(V (λB )).

From eorem C.3.3 we have that XP ⊂ P(V (λP )) can be wrien as

XP = Z (I (XP )),

where Z (I (XP )) denotes the zero-locus dened by a homogeneous prime ideal I (XP ).

Let us provide a brief outline of how we can nd I (XP ), our approach is based on [61], [108],[57] [139, p. 367-369] see also [114].

As we have seen in the proof of eorem C.3.3 the embedding of XP is constructed by meansof the projection of the orbit GC · v+λP ⊂ V (λP )\0 over P(V (λP )). What we will show bellowis that GC · v+λP is an algebraic set in V (λP ) dened by homogeneous polynomials, i.e. we willshow that GC · v+λP ∪ 0 is the ane cone of XP ⊂ P(V (λP )).

RemarkC.3.1. Let AnC be the complex ane space, and consider the natural projectionπ : An

C\0 →Pn−1. Given and algebraic subset Y ⊂ Pn−1, the ane cone C (Y ) over Y is dened by

C (Y ) = π−1(Y ) ∪ 0 ⊂ AnC.

e ane coneC (Y ) ⊂ AnC is an algebraic set in An

C such that I (C (Y )) = I (Y ), furthermoreC (Y )is irreducible if and only if Y is irreducible.

189

Figure C.2: is gure illustrate the ane cone over a curve Y in P2, this image was extractedfrom the book [82, p. 12, Figure 1].

If we consider P(V (λP )) as an algebraic variety we have its homogeneous coordinate ringgiven by

C[P(V (λP ))

]= S (V (λP )

∗) =

∞⊕l=1

Sl (V (λP )∗),

here Sl (V (λP )∗) denotes the l-symmetric tensor power of V (λP )

∗, ∀l ≥ 0. From these we have

C[XP

]= S (V (λP )

∗)/I (XP ).

We denote by w0 ∈ W the unique element which satises w0Σ = −Σ, see for example [144, p.256], it follows that we can identifyV (λP )

∗ = V (µP ), where µP = −w0λP , see for instance [132,p. 95-96] or [139, p. 347]. By means of this identication we can show that

Sl (V (µP )) = V (lµP ) ⊕(Ω − clµP 1

)Sl (V (µP )),

here Ω ∈ U(gC) denotes the universal Casimir element [87, p. 118] and

clµP = κ (lµP + 2δ , lµP ), for δ =12∑α∈Π+

α .

In fact in the above decomposition we have V (lµP ) ⊂ Sl (V (µP )) with multiplicity 1 and thecomponent

(Ω−clµP 1

)Sl (V (µP )) collects the irreducible gC-modulesV (ξ ) with highest weight

ξ ≺ lµ. We will denote by Cl =(Ω − clµP 1

)Sl (V (µP )), ∀l ≥ 0.

From the above decomposition of Sl (V (µP )), we have that the irreducible highest weight com-ponents V (ξ ) in Sl (V (µP )) ·Cq ⊂ Sl+q (V (µP )) satises ξ ≺ (l + q)µ2, thus we have that

2Here the symbol “ ≺ ” means the partial order that we have in the set of roots, for more details see forexample [87, p. 47].

190

C =⊕l≥0

Cl ,

denes a homogeneous ideal in S (V (µ )). e important consequences of the above ideas are:

1. C =⟨(Ω − c2µP 1

)S2(V (µP ))

⟩=⟨C2⟩;

2. x ∈ GC · v+λP ⇐⇒ Ω(x ⊗ x ) = c2µP (x ⊗ x );

the proof for the two facts above can be found in [61], see also [108, p. 337-347] and [57]. Asimple calculation shows us that c2µP = c2λP , furthermore the two last conditions implies that

XP = Z (I (XP )) ⊆ Z (C ) ⇐⇒ C ⊆ I (XP ).

If we suppose C $ I (XP ), since XP ⊂ P(V (λP )) is an invariant set it implies that I (XP ) ⊂S (V (µP )) is a GC-invariant subspace, moreover from the decomposition

I (XP ) =⊕l≥0

(I (XP ) ∩V (lµP )

)⊕(I (XP ) ∩Cl

).

We have that V (l0µP ) ∩ I (XP ) , 0 for some l0 0, since V (l0µP ) is irreducible we haveV (l0µP ) ⊂ I (XP ). us V (lµP ) ⊂ I (XP ) for all l ≥ l0, it follows that

C[XP

]= S (V (µP ))/I (XP ) =

⊕0≤l<l0

V (lµP ).

However we have I (XP ) =√I (XP ) =⇒ S (V (µP ))/I (XP ) is reduced [49, p. 33], i.e. ∀f ∈

C[XP

], f ⊗r = 0 ⇐⇒ f = 0, here we can take for example f = v+µP and r l0 to get a

contradiction with our hypothesis C $ I (XP ). erefore, we obtain

I (XP ) = C =⟨(Ω − c2λP 1

)S2(V (λP )

∗)⟩, (C.3.1)

furthermore we have

C[XP

]= S (V (µP ))/I (XP ) =

⊕l≥0

V (lλP )∗, (C.3.2)

here we used the identication V (lµP ) = V (lλP )∗, ∀l ≥ 0, remember that µP = −w0λP .

Let us compute the generators of I (XP ). In order to do this we need to x a weight-vectorbasis for V (λP ), namely

B =vϕ

∣∣∣ ϕ ∈ Π(λP ) ⊂ V (λP ).

We also will x a basis for gC, here we will follow the notation of [108, p. 340], we consider abasis given by

191x (1)α , . . . ,x

(dα )α

∣∣∣ α ∈ Π ∪ 0,

where dα = dim(gα ) = 1, ∀α ∈ Π, and d0 = dim(g0) = dim(h). Associated to this basis we cantake its dual with respect to the Cartan-Killing form κ, we denoted by

z (1)α , . . . , z(dα )α

∣∣∣ α ∈ Π ∪ 0,

here we point out that we can take z (i )α ∈ g−α , such that κ (x (i )α , z

(i )α ) = 1, ∀i = 1, . . . ,nα and

∀α ∈ Π, and for α = 0 we can take the usual basis associated to Σ and its dual, see for example[87, p. 118]. By taking the dual basis

B∗ =v∗ϕ

∣∣∣ ϕ ∈ Π(λP ) ⊂ V (λP )∗,

a straightforward calculation which can be found in [108, p. 340] shows us that the generatorsof I (XP ) are given by the elements Pϕ,ϕ ′ ∈ S2(V (λP )

∗) dened by

Pϕ,ϕ ′ := κ (λP , λP )v∗ϕ ⊗ v∗ϕ ′ −∑

α∈Π∪0

dα∑j=1

x (j )α (v∗ϕ ) ⊗ z

(j )α (v∗ϕ ′ ), (C.3.3)

where ϕ,ϕ′ ∈ Π(λP ). erefore, we obtain

I (XP ) =⟨Pϕ,ϕ ′

∣∣∣ ϕ,ϕ′ ∈ Π(λP )⟩, (C.3.4)

thus we have XP =

z ∈ P(V (λP ))

∣∣ Pϕ,ϕ ′ (z) = 0, ϕ,ϕ′ ∈ Π(λP )

.

Remark C.3.2. It is worth to point out that in the construction of a presentation for the idealI (XP ) we have xed a basis for gC and V (λP ). In the choice of basis for gC we do not have am-biguity due the properties of the universal Casimir element Ω ∈ U(gC) which are independent ofthe basis that we chose to represent it. Actually there is a canonical choice which was exactly thechoice that we did, see for example [108].

On the other hand the choice of weight basis forV (λP ) which we did carries such a kind of ambi-guity. In fact we do not have a canonical basis for irreducible gC-modules, for the case of classicalLie algebras see [121, p. 111] for more details and historical remarks. As we will see in the nextsection this problem has the geometric meaning of how XP is embedded in the projective space.

C.4 Borel-Weil theorem and Kodaira embedding for agmanifolds

e purpose of this section is to provide a proof for the Borel-Weil theorem, which provides ageometrical characterization for the irreducible modules of simple Lie algebras.

192

Basic facts about complex algebraic Lie groups

Let us collect some basic data, actually, all the elements listed below we have used throughoutthe last section

• Again GC denotes a connected and simple connected complex Lie group which inte-grates a complex simple Lie gC.

• We x a triangular decomposition of (gC, h) with respect to some choice of Cartan sub-algebra and simple root system, namely

gC = n+ ⊕ h ⊕ n−.

• We denote by B ⊂ GC the standard Borel subgroup, and consider its decompositonB = N + o TC, where TC = exp(h).

• Let G ⊂ GC be the compact real form associated to g ⊂ gC as in Section 4.2.

• We denote by

Λ∗Z≥0=⊕α∈Σ

Z≥0ωα ,

the set of integral dominant weights of gC.

For us will be important the following result related to complex simple Lie groups

eorem C.4.1. Every connected and simply connected complex simple Lie group GC is thecomplex Lie group associated to an algebraic group.

Proof. e proof for the above theorem can be found in [157, p. 453].

e basic idea behind the above theorem is that we can use a faithful nite dimensional holo-morphic representation to embed GC as an irreducible ane variety in an high-dimensionalane space. erefore we also have a structure of algebraic variety which underlies GC.

Now we will describe the coordinate ring of C[GC]. At rst we observe that for every µ ∈ Λ∗Z≥0

we can aach a linear map as follows

θµ : V (µ )∗ ⊗ V (µ ) → OanGC (G

C), θµ (ϕ ⊗ v ) (д) = ϕ (дv ),

where OanGC (G

C) denotes the space of global sections of the structure sheaf OanGC of GC, i.e. the

space of globally dened holomorphic functions on GC.

Since V (µ )∗ ⊗ V (µ ) is an irreducible G ×G-module, it follows that

ker(θµ ) = 0,

193

we denote by Aµ = Im(θµ ) V (µ )∗ ⊗ V (µ ). From these we have the following result

eoremC.4.2. LetGC be a connected, simply connected and complex simple Lie group equippedwith a compatible structure of algebraic group. en its algebra of regular functions C[GC] isgiven by

C[GC] =⊕µ

Aµ ⊕µ

V (µ )∗ ⊗ V (µ ),

here the direct sum runs over all integral dominant weights.

Proof. is result can be found in [157, p. 455].

We have the following consequence of the previous facts, see for example [157, p. 456] formore details

Corollary C.4.3. Under the hypotheses of the last theorem, the structure of ane variety

C[GC] =⊕µ

Aµ ,

is the unique compatible with the structure of complex Lie group of GC.

Remark C.4.1. In the above results we have used the fact that for an irreducible ane varietythe space of globally dened regular functions coincides with its coordinate ring, see for example[82, p. 17, eorem 3.2]. us we use the notation C[GC] to denote both spaces.

e Borel-Weil theorem

Let XB = GC/B be the complex ag manifold associated to the standard Borel subgroup B ⊂GC. In this case we have Θ = ∅ and B = P∅, see Remark 4.5.2. e irreducible representationassociate to XB is dened by an integral dominant weight which satises

λB (hα ) > 0, ∀α ∈ Σ,

i.e. λB ∈ h is a strongly dominant weight, see for example [87, p. 67]. From these as wehave seen in eorem C.3.3 we can see XB as an irreducible projective variety by means of thefollowing projective embedding

XB → P(V (λB )), дB 7→ [дv+λB ].

Now we will collect some basic facts about characters in order to describe the relation betweenthe above projective embedding and the Kodaira embedding.

Consider the following results related to the character group

Ch(TC) = Hom(TC,C×).

eorem C.4.4. Let GC be a connected, simply connected and complex simple Lie group withLie algebra gC. en λ ∈ h∗ is the dierential of a character of TC if and only if λ is a dominantintegral weight.

194

Proof. is result can be found in [157, p. 466].

We notice that since B = N + o TC, we have the following result

Lemma C.4.5. Every character χ : TC → C× can be uniquely extended to a multiplicative char-acter of B.

From the last results we have the following construction. Let λB ∈ Λ∗Z≥0⊂ h∗ be the weight

associate to XB , we denote by χλB ∈ Hom(TC,C×) the character which satises the conditionstated in the last theorem, namely

(dχλB )e = λB .

Since χλB ∈ Hom(TC,C×) can be extended to B, we obtain

χλB : B → C× = GL(1,C),

by means of the above representation we can construct a holomorphic line bundle over XB asan associated bundle to the B-principal bundle

B → GC → GC/B,

the construction goes as follows. We consider C as a B-space equipped with the followingholomorphic action

b · z = χλB (b−1)z,

∀b ∈ B and ∀z ∈ C, we will denote the underlying B-space C endowed with this structure byC−λB . From this we consider the action of B on GC × C−λB dened by

b · (д, z) = (дb,b−1z) = (дb, χλB (b)z),

∀b ∈ B, ∀z ∈ C and д ∈ GC. is action allows us to dene the holomorphic line bundle

LχλB = GC × C−λB/B = G

C ×χλBC−λB .

RemarkC.4.2. Notice that the above construction can be done for every µ ∈ Λ∗Z≥0. In this chapter

we will just work with the case µ = λB or λP , for some P ⊂ GC. In the next chapter we will providea more complete discussion about this subject.

Now we consider the space of globally dened holomorphic sections associated to the aboveholomorphic line bundle, i.e.

Γhol.(XB,LχλB ) = H 0(XB,LχλB ).

Before we state and prove the main result of this section, let us briey to specify the categorywhich we will work on. Since XB is a projective algebraic variety we can use the GAGA prin-ciple to replace holomorphic sections by algebraic sections.

195

Remark C.4.3. In fact if we denote by OXB the structure sheaf of regular functions of XB and byOanXB

its sheaf of holomorphic functions, then by Serre’s theorem we have a isomorphism betweenthe category of coherent algebraic sheaves and coherent analytic sheaves.

erefore, in the proof below we will consider the sheaf of sections of LχλB as an algebraic sheaf ofOXB -modules instead of an analytic sheaf of Oan

XB-modules 3, for more details about Serre’s GAGA

theorem see [157, p. 344-351] and [129, p. 190-207]. From these in the most cases in the context ofprojective algebraic varieties we will not distinguish the algebraic and analytic structure sheaves.

In what follows we will provide a proof for the main theorem of this section, the proof thatwe will cover bellow is based in [107]

eorem C.4.6 (Borel-Weil). Under the previous hypotheses we have

H 0(XB,LχλB ) V (λB )∗

Proof. Given σ ∈ H 0(XB,LχλB ) we have

σ (дB) = [д, fσ (д)], where fσ : GC → C−λB .

Since σ does not depend on of the representative for the class дB ∈ XB , it follows that

σ (дB) = σ (дbB) = [дb, fσ (дb)] = [д, χ−1λB(b) fσ (дb)],

thus fσ (д) = χ−1λB(b) fσ (дb) =⇒ (Rb )

∗ fσ = χλB (b) fσ , ∀b ∈ B.

Now consider the action of B on C[GC] ⊗ C−λB dened by

b ( f ⊗ z) = (Rb )∗ f ⊗ χλB (b)

−1z, (C.4.1)

from this we have that the map σ 7→ fσ ⊗1 provides an identication of complex vector spaces

H 0(XB,LχλB ) [C[GC] ⊗ C−λB

]B.

erefore from eorem C.4.2 we obtain

H 0(XB,LχλB ) [⊕

µ

V (µ )∗ ⊗ V (µ ) ⊗ C−λB]B

.

If we consider the action of GC on C[GC] dened by

д f = (Lд−1 )∗ f ,

∀f ∈ C[GC] and ∀д ∈ GC, a straightforward calculation shows us that this action is exactlythe restriction to the rst factor of the action ofGC ×GC on

⊕µ

V (µ )∗ ⊗V (µ ). In fact we have

3We proceed in this way in order to use the characterization of the coordinate ring ofGC provided by eoremC.4.2. is theorem makes possible an interplay between (algebraic) geometry and representation theory.

196

((дϕ) ⊗ v ) (x ) = (дϕ) (xv ) = ϕ (д−1xv ),

and on the other hand we also have

(Lд−1 )∗((ϕ ⊗ v ) (x ) = ϕ ⊗ v (д−1x ) = ϕ (д−1xv ),

for every д,x ∈ GC, and ∀ϕ ⊗ v ∈ V (µ )∗ ⊗ V (µ ).

We can extend the last dened action in a natural way to C[GC] ⊗ C−λB by taking

д( f ⊗ z) = (Lд−1 )∗ f ⊗ z, (C.4.2)

since the action C.4.1 commutes with the action C.4.2, it follows that 4

H 0(XB,LχλB ) =⊕µ

[V (µ )∗ ⊗ V (µ ) ⊗ C−λB

]B.

Now we notice that

b (ϕ ⊗ v ⊗ z) = ϕ ⊗ bv ⊗ χλB (b)−1z,

∀ϕ ⊗ v ⊗ z ∈ V (µ )∗ ⊗ V (µ ) ⊗ C−λB , and ∀b ∈ B, it follows that[V (µ )∗ ⊗ V (µ ) ⊗ C−λB

]B= V (µ )∗ ⊗

[V (µ ) ⊗ C−λB

]B,

thus we obtain

H 0(XB,LχλB ) ⊕µ

V (µ )∗ ⊗[V (µ ) ⊗ C−λB

]B. (C.4.3)

By means of the natural isomorphism C[GC] ⊗ C−λB C[GC] we have

ϕ ⊗ v ⊗ z ∈ V (µ )∗ ⊗[V (µ ) ⊗ C−λB

]B⇐⇒ zϕ (v − χλB (b)

−1bv ) = 0,

since χλB : B → C× is the extension of a character of TC, the last equivalence becomes

ϕ ⊗ v ⊗ z ∈ V (µ )∗ ⊗[V (µ ) ⊗ C−λB

]B⇐⇒ tv = χλB (t )v ,

∀t ∈ TC. Now by taking the derivative of the equation tv = χλB (t )v , we get

ϕ ⊗ v ⊗ z ∈ V (µ )∗ ⊗[V (µ ) ⊗ C−λB

]B⇐⇒ hv = λB (h)v ,

∀h ∈ h. erefore the unique component which remains in the sum C.4.3 is µ = λB , thus weobtain

4Notice that C[GC] ⊗ C−λB has a structure of GC × B-module, the fact that the two cited actions commuteallows us to see H 0 (XB ,LχλB ) also as a GC × B-module, therefore we can write H 0 (XB ,LχλB ) as a direct sum ofits intersection with the components V (µ )∗ ⊗ V (µ ) ⊗ C−λB .

197

H 0(XB,LχλB ) V (λB )∗ ⊗ 〈v+λB 〉 ⊗ C−λB V (λB )

∗,

from this we have the desired result.

Now we will describe how the above result can be extended to any complex ag manifolds XP

associated to an arbitrary parabolic subgroups P ⊂ GC. Let P ⊂ GC be a parabolic subgroup,as we have seen we can associate toXP an integral dominant weight λP ∈ Λ∗Z≥0

. If we considerthe character χλP ∈ Hom(TC,C×), induced by the weight λP , the sequence of inclusions

TC ⊂ B ⊂ P ,

allows us to take an extension χλP : P → C× and dene a holomorphic line bundle

LχλP = GC × C−λP /P → XP ,

exactly as we did forXB5. We will explore this construction in more details in the next chapter.

erefore, we can show that

H 0(XP ,LχλP ) V (λP )∗, (C.4.4)

the proof for the above fact is essentially the same as in the case when P = B, just replace Pto B in the proof of the previous theorem.

Before we discuss the ideas involved in the Kodaira embedding let us make some remarksabout Schubert cells and Schubert varieties. e results that we will present bellow can befound in [110, p. 22-24].

Let gC be a complex simple Lie algebra, and consider the basic data xed at the beginning ofthis section. e Weyl group associated to the root system Π is given by

W =⟨rα∣∣ α ∈ Π⟩,

where

rα (λ) = λ −2κ (λ,α )κ (α ,α )

α ,

with α ∈ Π and λ ∈ h∗. Given Θ ⊂ Σ we can assign to this subset a subgroup of W given by

WΘ =⟨rα∣∣ α ∈ ΠΘ

⟩,

ΠΘ = 〈Θ〉+ ∪ 〈Θ〉−, here 〈Θ〉± = 〈Θ〉 ∩ Π±. From these we can take the quotient space W /WΘ

and dene the following set of minimal length representatives of W /WΘ

W minΘ =

w ∈ W

∣∣∣ `(ww′) = `(w ) + `(w′), ∀w′ ∈ WΘ

,

5As we have pointed out in C.4.2 the statement of the Borel-Weil is more general in the sense that all we havedone also works for any integral dominant weight.

198

where the length of w ∈ W is dened by

`(w ) := #α ∈ Π+

∣∣∣ w−1(α ) < 0

.

Now we consider the isomorphism between W and NGC (TC)/TC, where TC = exp(h) andNGC (TC) ⊂ GC denotes its normalizer in GC, dened by the correspondence

rα 7→ Ad∗(nα ), ∀α ∈ Π,

where nα = exp(xα ) exp(−yα ) exp(xα ), for xα ∈ gα and yα ∈ g−α , such that α ([xα ,yα ]) = 2, seefor instance C.1. By means of this identication we can considerw ∈ W as an element ofGC 6.

Now consider the projective algebraic variety dened by XP , where P = PΘ ⊂ GC. Givenw ∈ W we dene the Schubert cell associated to w as being the B-orbit CP (w ) = BwP ⊂ XP ,from this we can show that

CP (w ) ∏

α∈β∈Π+\〈Θ〉+ | w−1 (β )<0Nα , (C.4.5)

where Nα = exp(gα ), ∀α ∈ Π+, and the above identication is in the category of alge-braic varieties. e Schubert variety associated to w ∈ W is dened by the Zariski closureXP (w ) = CP (w ).

By taking P = B, for w ∈ W from the above identication and comments we have

CB (w ) C`(w ) . (C.4.6)

We notice that given Θ ⊂ Σ we have an induced decomposition at each element of W whichcan be described as follows. Every w ∈ W can be uniquely wrien as w = uv , such thatu ∈ W min

Θ and v ∈ WΘ, therefore for P = PΘ the Schubert cell associated to w = uv satises

CP (w ) = CP (u) C`(u) . (C.4.7)

In fact the rst equality is a consequence that BuvP = BuP , and a straightforward calculationshows us that

`(u) = #α ∈ Π+\〈Θ〉+

∣∣∣ u−1(α ) < 0

.

From the last ideas if we consider the natural projection π : GC/B → GC/P , we can show thatthis projection maps XB (u) birrationally 7 onto XP (w ), for w = uv ∈ W . us we have abirrational equivalence

π |XB (u) : XB (u) 99K XP (w ). (C.4.8)6Here it is worthwhile to point out that every element [n] ∈ NGC (TC)/TC denes and linear map

Ad∗ (n) : h∗ → h∗. In our identication above we actually have rα = Ad∗ (nα ), thus each reexion rα is regardedas nα ∈ GC, ∀α ∈ Π.

7For the denition of rational maps see for instance [82, p. 24].

199

Now we observe the following fact, if we take w0 ∈ W as being the element with maximallength, from the previous comments we have w0 = u0v0, such that u0 ∈ W min

Θ and v0 ∈ WΘ.e maximal length of w0 implies that

`(u0) = dim(XP ) =⇒ CP (w0) ⊂ XP , (C.4.9)

is an ane open set, thus we have XP (w0) = XP . Here it is worth to observe that if we takethe “opposite” Borel subgroup

B− = N − o TC, (C.4.10)

a straightforward calculation shows us that B− = w0Bw0. It follows that the B−-orbit B−P =w0CP (w0) is also an ane open set of XP , this ane open set is called the “opposite” big cell,see for example [110, p. 23].

e concept of opposite big cell will be important for us, in fact this cell provides a distin-guished coordinate neighbourhood for the origin x0 = eP ∈ XP .

Remark C.4.4. By means of the language of Schubert varieties there exists an alternative wayto understand the Borel-Weil theorem for a general ag variety XP . Actually, if we consider thebirrational equivalence

π |XB (u0) : XB (u0) 99K XP (w0) = XP ,

where w0 = u0v0 ∈ W is the element with maximal length, we can show that

H 0(XP ,LχλP ) H 0(XB (u0),π∗LχλP ).

e space in the right side in the above equation is exactly V (λP )∗, in order to show this we need

to use the concept of Demazure modules, see [21] and [110] for more dtails.

We nish this chapter by gathering all the results we have collected so far in order to discussthe relations between the Kodaira embedding for ag manifolds and the representation theorylanguage which is behind of the embedding.

Let X be a projective algebraic variety and L → X be a very ample holomorphic line bundle,from Kodaira embedding theorem we have that X → P(E∗), where E = H 0(X ,L). Associatedto L we have its ring of section dened by by the graded algebra

R (X ,L) :=⊕k≥0

H 0(X ,L⊗k ),

see for instance [89, p. 73], here we denote L⊗0 = OX , notice that since X is compact wehave H 0(X ,OX ) = C. Furthermore, in the context of the Kodaira embedding, if we denote theKodaira map by ΨL : X → P(E∗) we can also show that

Ψ∗L(OP(E∗) (1)

)= L and Ψ∗L

(H 0(P(E∗),OP(E∗) (1))

)= H 0(X ,L),

200

where OP(E∗) (−1) → P(E∗) is the tautological line bundle. Now since

R (P(E∗),OP(E∗) (1)) = C[z1, . . . , zN ],

where N = dim(E∗) − 1, it follows that R (X ,L) is nite generated, see for example [89, p.91]. We denote by R =

⊕k≥0 Rk a graded subalgebra of R (X ,L), in the context of the Kodaira

embedding we have that the coordinate ring of X → P(E∗) is given by such a kind of gradedsubalgebra R =

⊕k≥0 Rk ⊂ R (X ,L).

Now let us come back to the context of ag manifolds. From C.3.2 we have

C[XP

]= S (V (µP ))/I (XP ) =

⊕k≥0

V (kλP )∗,

a simple calculation shows us that LχkλP = L⊗kχλP, ∀k ≥ 0, therefore by applying the Borel-Weil

theorem we obtain

C[XP

]=⊕k≥0

V (kλP )∗ =⊕k≥0

H 0(X ,L⊗kχλP),

i.e. we have R (XP ,LχλP ) = C[XP

].

As we have seen the underlying structure of projective variety which we have for a ag man-ifold is described by means of its projective embedding, furthermore such a embedding iscompletely described in terms of the representation theory language. In the next chapter wewill provide a more extensive discussion about the relation between holomorphic line bundlesand representation theory. e main purpose is to understand the geometric information en-coded in the objects that we can associate to a ag manifold by using characters and integralweights.

201

Appendix DHolomorphic line bundles and Calabiansatz technique

As we have seen in the last chapter the Borel-Weil theorem provides a very rich interplaybetween representation theory and complex/algebraic geometry of ag manifolds. In orderto study the complex geometry of a projective variety X , when we consider the Kodaira mapΨL : X → P(E∗), where L → X is a very ample holomorphic line bundle and E = H 0(X ,L),the study of X as a projective subvariety of P(E∗) is completely dependent on the choice ofL ∈ Pic(X ). erefore, issues related to the global geometry of X require a more dicult ex-planation. It follows from the fact that the presentation of the homogeneous coordinate ringof X is not canonical, namely in the context of the Kodaira embedding we have

C[X]= S (H 0(X ,L)∗)/I (X ),

where I (X ) ⊂ S (H 0(X ,L)∗) is a homogeneous prime ideal.

Other important question which arises in the study of projective varieties is that we do nothave globally dened regular functions. In terms of the analytic geometry it can be seen asa consequence of the fact that smooth analytic projective varieties are compact, thus they donot admit globally dened analytic (holomorphic) functions. erefore, the study of the birra-tional geometry of projective varieties requires a more elaborated machinery than the anecase.

In order to deal with the problem which we briey described above, mainly the denition of asuitable “coordinate ring” for projective varieties, Hu and Keel proposed in [86] as a candidatethe Cox Ring of a projective variety, which is a generalization of the toric case [35] and can beloosely described by

Cox(X ) =⊕

L∈Pic(X )

H 0(X ,L).

e nice feature of the above denition is to gather together all information about the possibleprojective algebraic realizations ofX by means of the Kodaira embedding. Besides of the abovefacts related to the algebraic geometry of projective varieties there are other important ques-tions related to the study of holomorphic line bundles. In the context of the complex geometry

202

which underlies these varieties, we have many interesting geometric objects like connectionsand curvature tensors.

Inspired by these facts which we have briey described, the main purpose of this chapter is toprovide a complete description of holomorphic line bundles over ag manifolds as well as toprovide some results related to Chern classes and Kahler-Einstein metrics.

D.1 Local Kahler potential and Chern class of line bun-dles

e main purpose of this section is to provide an overview about the calculation of the Chernclass associated to holomorphic line bundles over ag manifolds, our approach is based on[8], [18] and [11]. e main feature of this overview is to establish a complete descriptionwith respect to some details related to the homological and cohomological questions whichare behind the construction described in [8].

Even though the results which will be covered here are well known, there is a lack of intro-ductory texts on this subject in the literature. In this way, besides our interest to use the ideasdescribed here, the material developed through the next subsections also can be seen as anuseful guide to understand [8], [18] and [11].

D.1.1 Preliminary generalities

roughout the next subsections we will deal with homological and cohomological languages,thus it is worthwhile to point out some basic conventions which will be important for us inorder to keep an uniform notation.

Since our approach involves C∞-manifolds, for homology theory we denote by H•(X ,Z) thecorresponding homology group associated to any of the standard homology theory with inte-gers coecients on a manifold X , i.e. simplicial homology theory, singular homology theoryor cellular homology theory [83], [128, p. 190], see also [125, p. 101, eorem 3.3].

For cohomology theory we denote by H •(X ,F ) the corresponding cohomology group asso-ciated to any classical cohomology theory with coecients in sheaves of K-modules. In mostcases we consider K = Z or R, see for instance [165, p. 162], and the equivalence with Cechcohomology

H •(X ,F ) = H •(X ,F ).

For the case when F = K denotes the sheaf of locally constant smooth functions on X withvalues in K, we also consider the equivalence between the classical cohomology theory, sim-plicial or singular, and the sheaf cohomology, i.e.

H •(X ,K) = H •(X ,K),

203

see for example [165, p. 191-200]. Finally, for the particular case when F = R denotes thesheaf of locally constant smooth functions onX with values in R, we consider the equivalencebetween the de Rham cohomology and the sheaf cohomology, i.e.

H •(X ,R) = H •DR(X ,R),

see for instance [165, p. 189-191], see also [125, p. 113-116]. In what follows unless otherwisestated we will consider the equivalences described above and some basic results which can befound in the references which we have mentioned.

Parabolic subgroup decompositions

Let GC be the connected, simply connected and complex Lie group associated to a complexsimple Lie algebra gC. Fixed a Cartan subalgebra h ⊂ gC and a simple root system Σ ⊂ h∗, wehave the triangular decomposition

gC = n+ ⊕ h ⊕ n−.

Given a parabolic subgroup P ⊂ GC, without loss of generality we can suppose that

P = PΘ, for some subset Θ ⊂ Σ,

from this we can write PΘ = NGC (pΘ), where

pΘ = n+ ⊕ h ⊕ n(Θ)−, with n(Θ)− =

∑α∈〈Θ〉−

gα .

e structure of pΘ yields the following decomposition for PΘ

PΘ = LΘRu (PΘ), (D.1.1)

where we have

LΘ = G (Θ)CT (Σ\Θ)C and Ru (PΘ) =∏

α∈Π+\〈Θ〉+

Nα . (D.1.2)

e subgroups involved in the above decomposition can be described in the following way

G (Θ)C = 〈exp(g(Θ)C)〉, T (Σ\Θ)C = exp(h(Σ\Θ)), Nα = exp(gα ),

it is worthwhile to point out that g(Θ)C denotes the complex semisimple Lie algebra associ-ated to the subset Θ ∈ Σ and h(Σ\Θ) denotes the abelian subalgebra generated by hα ∈ h withα ∈ Σ\Θ.

Generalities and notations

204

By keeping the previous data, consider for every α ∈ Π+ the following isomorphism

ϕα : sl (2,C) → sl (α ) ⊂ gC, (D.1.3)

where sl (α ) = SpanC

xα ,yα , [xα ,yα ]

, such that0 1

0 0

→ xα , and

0 0

1 0

→ yα .

from this we get SL(2,C) SL(α ) = 〈exp(sl (α ))〉. rough of the last homomorphisms wecan consider the compact real form G of GC as being the simply connected subgroup

G =⟨ϕα (SU(2))

∣∣ α ∈ Π+⟩,here we also denote byϕα the induced homomorphisms on SU(2) ⊂ SL(2,C). Now we considerthe following characterization for the Cartan subalgebra h ⊂ gC

h = SpanZ

h∨α =

2κ (α ,α )hα

∣∣∣ α ∈ Σ ⊗ C.

By taking the dual basis of h∨α , α ∈ Σ, we have the laice generated by fundamental weights

Λ∗Z =⊕α∈Σ

Zωα ,

where ωα (h∨α ) = δαβ ,∀α , β ∈ Σ. We will denote by

ρα : gC → gl (V (ωα )),

each irreducible fundamental representation with highest weight ωα . If we also denote by ραthe induced representation on GC, we obatin by restriction a representation

ρα |G : G → GL(V (ωα )),

here it is worth to point out that each irreducible representation V (ωα ) can be wrien as

V (ωα ) = U (gC)v+ωα ,

where v+ωα ∈ V (ωα ) denotes the highest weight vector associated to ωα ∈ h∗.

Now once eachωα , α ∈ Σ, is an integral dominant weight we can associate to it a holomorphiccharacter χωα : TC → C×, such that (dχωα )e = ωα , see for instance [157, p. 466]. Given aparabolic subgroup P ⊂ GC, we can take the extension χωα : P → C× and dene a holomorphicline bundle

Lχωα = GC × C−ωα /P = G

C ×χωα C−ωα , (D.1.4)

which is an associated vector bundle to the principal bundle P → GC → GC/P . We call Lχωαfundamental line bundle associated to ωα , α ∈ Σ.

205

Remark D.1.1. We recall that in the above denition of Lχωα we consider C−ωα as a P-space withthe action

p · z = χωα (p−1)z,

for every z ∈ C and p ∈ P . erefore Lχωα is dened by the orbit space of GC × C−ωα with theaction

p · (д, z) = (дp, χωα (p)z) = (дp,p−1 · z),

for every (д, z) ∈ GC × C−ωα and p ∈ P .

It is worth to observe that our construction of line bundles by means of the characters χ ∈Hom(P ,C×) is dierent from [8], the main dierence is just the P-space structure dened on C.Actually, the holomorphic line bundle Lχ dened in [8] is obtained as the orbit space of GC × Cthrough the action

p · (д, z) = (дp, χ (p)−1z),

therefore our convention will provide dierent signs in some expressions, see [11, p. 281].

Remark D.1.2. We can also describe the previous construction in the following way. In terms ofCech cocycle, if we consider an open cover XP =

⋃i∈I Ui and GC = (Ui )i∈I ,ψij : Ui ∩ Uj → P ,

then we have

Lχωα =(Ui )i∈I , χ

−1ωα ψij : Ui ∩Uj → C×

,

thus Lχωα = дij ∈ H1(XP ,O∗XP

), with дij = χ−1ωα ψij , where i, j ∈ I .

Global potentials on GC and topological generalities

In order to show that every closed real (1,1)-form η ∈ Ω1,1(GC) has a global potential we willanalyse the following cohomological and homological facts

H 2(GC,Z) = 0, H 1(GC,OGC ) = 0, (D.1.5)

here OGC denotes the structure sheaf of holomorphic functions of GC.

In order to show that H 2(GC,Z) = 0, we consider the following results

• Consider the dieomorphism induced by the polar decomposition

GC G × ig.

• e above decomposition allows us to write

H •(GC,R) H •(G,R) H •L (G,R),

206

where H •L (G,R) denotes the cohomology group of le invariant forms, it follows that

H •(GC,R) H •L (G,R) H •(g),

• By Whitehead’s lemma [92], we obtain

H 1(GC,R) = H 2(GC,R) = 0. (D.1.6)

• Since GC is simply connected, by Hurewicz’s theorem [83, p. 366], we have

H1(GC,Z) π1(G

C)/[π1(G

C),π1(GC)]= 0.

By looking at the short exact sequence obtained from the universal coecients theoremfor cohomology [83, p. 195], we get

0→ Ext(H0(GC,Z),Z) → H 1(GC,Z) → Hom(H1(G

C,Z),Z) → 0,

thus we have H 1(GC,Z) = 0.

• Now applying the universal coecients theorem for homology [83, p. 264], we obtain

0→ H2(GC,Z) ⊗ R→ H2(G

C,R) → Tor(H1(GC,Z),R) → 0,

it follows that H2(GC,Z) ⊗ R H2(G

C,R), i.e. H2(GC,Z) is torsion-free.

From the universal coecients theorem for cohomology and D.1.6, it follows that

0 = H 2(GC,R) = Hom(H2(GC,Z),R),

since H2(GC,Z) is torsion-free, we have H2(G

C,Z) = 0. us we conclude that

H 2(GC,Z) = Free(H2(GC,Z)) ⊕ Torsion(H1(G

C,Z)) = 0.

e fact that H 1(GC,OGC ) = 0 follows from the following results

• SinceGC is a complex simple Lie group it can be embedded holomorphically as a closedcomplex matrix Lie subgroup of GL(N ,C) ⊂ M(N ,C) = CN×N , for a suciently largeN , see for instance [157, p. 453, eorem 15.8.4].

• e above fact shows us that GC is a Stein manifold, see [53, p. 47], therefore we haveH 1(GC,OGC ) = 0, see [53, p. 52, eorem 2.4.1] for more details.

Proposition D.1.1. Let η ∈ Ω(1,1) (GC) be a closed real (1, 1)-form, then

η = i∂∂φ,

for some φ ∈ C∞(GC,R).

207

Proof. Let η ∈ Ω(1,1) (GC) be a closed real (1,1)-form. By local i∂∂-lemma [126, p. 68], on anopen set Uk ⊂ GC, we can write

η = i∂∂uk ,

where uk ∈ C∞(Uk ,R). erefore, we can take a cover of GC given by a collection of patchesU = (Uk )k∈I such that the dierences

u =(ukj = uk − uj ,Uk ∩Uj ) | Uk ,Uj ∈ U

,

provides a Cech 1-cocycle u ∈ Z 1(U ,P ), where P denotes the sheaf of germs of pluri-harmonic functions onGC, we recall that a real valued smooth function f is pluri-harmonic if∂∂ f = 0, see [147, p. 15].

Since locally every pluri-harmonic function is the real part (or the imaginary part) of a holo-morphic function, see for instance [147, p. 16, eorem 2], we have the following short exactsequence of sheaves

0 R OGC P 0,Im

here R denotes the sheaf of locally constant smooth functions. Notice that the exactness atOGC follows from the fact that if a holomorphic function just assume real values then it needsto be locally constant.

By taking the long exact sequence of cohomology associated to the previous short exact se-quence of sheaves [147, p. 249], we obtain

· · · H 1(GC,OGC ) H 1(GC,P ) H 2(GC,R) · · · .δ

From the previous comments we have H 1(GC,P ) = 0, thus we obtain a cochain

f =( fj ,Uj )

∣∣Uj ∈ U∈ C0(U ,P ), (D.1.7)

such that

fj − fk = uj − uk ⇐⇒ uj − fj = uk − fk , in Uk ∩Uj .

hence we can take φk = uk − fk ∈ C∞(Uk ,R), for each k ∈ I . By gluing (φk ,Uk ) we obtainφ ∈ C∞(GC,R) satisfying φ |Uk = φk and

i∂∂(φ |Uk ) = i∂∂uk = η,

on eachUk ∈ U , i.e. η ∈ Ω(1,1) (GC) ∩ Ω2(GC) admits a global potential φ ∈ C∞(GC,R), there-fore we have the result stated.

Remark D.1.3. e statement of Proposition D.1.1 is also true for any complex manifoldX whichsatises H 1(X ,OX ) = H 2(X ,R) = 0, the proof for this fact is exactly the same as above.

Remark D.1.4. In the above calculations we have used the fact that the sheaf cohomology coin-cides with the Cech cohomology, i.e. for a sheaf of abelian groups F we have

208

H i (GC,F ) = H i (GC,F ) = lim→

H i (U ,F ),

for more details and related results see for instance [89, p. 287].

Remark D.1.5. It is worthwhile to point out that the cohomology group H 1(GC,P ) is obtainedfrom the coboundary homomorphisms given by

δ : C0(U ,P ) → C1(U ,P ),

sucht that for a =(aj ,Uj )

∣∣Uj ∈ U∈ C0(U ,P ), we have

δ (a) =(ak − aj ,Uk ∩Uj )

∣∣Uk ,Uj ∈ U∈ C1(U ,P ),

and δ : C1(U ,P ) → C2(U ,P ) is dened by the cocycle condition

δ (c ) =(cjk − cik + cij ,Ui ∩Uj ∩Uk )

∣∣Ui ,Uj ,Uk ∈ U∈ C2(U ,P ),

for every c =(ckj ,Uk ∩ Uj )

∣∣ Uk ,Uj ∈ U∈ C1(U ,P ), see for example [147, p. 242-247].

erefore, the result of the previous proposition can be summarized as follows. For every η ∈Ω(1,1) (GC) ∩ Ω2(GC) we can associate to a 1-cocycle u ∈ C1(U ,P ) dened by the dierences ofthe local potentials, since H 1(GC,P ) = 0 we have f ∈ C0(U ,P ) such that δ ( f ) = u.

Now we analyse the topological consequence of H2(GC,Z) = 0 over the complex homoge-

neous space GC/P . Consider the short exact sequence

P → GC → GC/P ,

by looking at the long exact sequence of homotopy [83, p. 376] associated the above shortexact sequence, we have

· · · → π2(GC) → π2(G

C/P ) → π1(P ) → π1(GC) → · · · .

Since GC is simply connected (1-connected) by Hurewicz’s theorem [83, p. 366] we have

π2(GC) H2(G

C,Z) = 0,

therefore from the previous long exact sequence we obtain

π2(GC/P ) π1(P ).

We observe now that from D.1.1 we have

P = PΘ = LΘRu (PΘ),

it implies that π1(PΘ) π1(LΘ), since Ru (PΘ) C#(Π+\〈Θ〉+) , here #(Π+\〈Θ〉+) denotes thecardinality of the set Π+\〈Θ〉+. Now once we have from D.1.2 that LΘ = G (Θ)T (Σ\Θ)C itfollows that

π2(GC/P ) π1(LΘ) π1(T (Σ\Θ)

C).

209

Since G (Θ) is the simply connected subgroup which integrates the complex semisimple Liealgebra dened by Θ ⊂ Σ.

From the above comments, and by considering that T (Σ\Θ)C (C×)#(Σ\Θ) , we conclude that

π2(GC/P ) π1(T (Σ\Θ)

C) Z#(Σ\Θ) . (D.1.8)

us we obtain for XP = GC/P the following fact

Pic(XP ) H 1(XP ,O∗XP) ⊆ Z#(Σ\Θ) .

Let us make the above statement more clear. Since XP is simply connected, by Hurewicz’stheorem [83, p. 366], we have

H2(XP ,Z) π2(XP ) Z#(Σ\Θ) . (D.1.9)

From this we get

H 2(XP ,Z) = Hom(H2(XP ,Z),Z) Hom(Z#(Σ\Θ),Z) Z#(Σ\Θ) ,

we point out that the last isomorphism on the right side above follows from the fact that Z is acommutative ring, see for instance [140, p. 56]. Now we remember that from the exponentialsequence of sheaves

0 Z OXP O∗XP0,i exp

we have the long exact sequence

· · · H 1(XP ,OXP ) H 1(XP ,O∗XP)

Pic(XP )

H 2(XP ,Z) · · · ,c1

where the map c1 : H 1(XP ,O∗XP) → H 2(XP ,Z) is given by

c1(L) =[

i2π F∇

]∈ H 2(XP ,Z),

where F∇ denotes the curvature associated the connection ∇ = d + ∂(H )H−1 for some Hermi-tian structure H on L → XP . In order to see that this map is injective we notice that since XP

is a compact complex manifold we have H 1(XP ,OXP ) = H 0,1(XP ) = 0 1, in fact once XP issimply connected it implies that

H 1(XP ,C) = H 1(XP ,R) ⊗ C = 0,1Here we notice that if X is a compact Kahler manifold we have H 0,q (X ) Hq (X ,Ω0

hol) = Hq (X ,OX ), whereΩphol denotes the space of holomorphic p-forms o n X, see [89, p. 125-129] or [163, p. 137-143].

210

thus 0 = H 1(XP ,C) = H 1,0(XP ) ⊕H0,1(XP ). From the last long exact sequence it follows that

the map c1 : H 1(XP ,O∗XP) → H 2(XP ,Z) is injective, i.e.

H 1(XP ,O∗XP) ⊆ H 2(XP ,Z) Z#(Σ\Θ) .

Our main task in the next subsections will be providing a suitable expression for the generatorsof Pic(XP ) and concluding that in fact Pic(XP ) Z#(Σ\Θ) .

Remark D.1.6. It is worth to point out that from the Lefschetz theorem on (1, 1)-classes and theabove comments we showed that

Pic(XP ) H 1,1(XP ,Z) = H 2(XP ,Z) ∩ H 1,1(XP )

see for instance [89, p. 133, Proposition 3.3.2] for more details.

D.1.2 Local potential and representation theory

In this subsection and the next one we will be concerned to provide a suitable expression tothe local potential for the Chern class of holomorphic line bundles over XP = G

C/P . We startby looking at some features of the local potential associated to the curvature form of holomor-phic line bundles over ag manifolds.

Given L ∈ Pic(XP ) as we have seen in the previous section we have c1(L) ∈ H1,1(XP ,Z), since

XP is a compact manifold and the compact real formG ofGC acts by le translations on XP , itfollows that

H •G (XP ,R) H •DR (XP ,R) = H •(XP ,R),

where H •G (XP ,R) denotes the cohomology of G-invariant forms. Once H 2(XP ,Z) is torsion-free we have H 1,1(XP ,Z) ⊂ H 2(XP ,R), see Remark D.1.6, therefore we can choose a Hermitianstructure H on L such that the connection ∇ = d + ∂(H )H−1 has its curvature given by aG-invariant real (1, 1)-form

ω = i2π F∇ ∈ Ω

(1,1) (XP ) ∩ Ω2(XP )G .

Now consider the natural projection π : GC → GC/P , from this we have a G-invariant, closedand real (1, 1)-form on GC given by

η = π ∗ω.

By applying Proposition D.1.1, we obtain

π ∗ω = i∂∂φ,

for some φ ∈ C∞(GC,R), we claim that we can choose φ ∈ C∞(GC,R)G . In fact, if we x abi-invariant measure dVд on G, such that

∫G dVд = 1, we can take the le invariant function

φ ∈ C∞(GC,R)G dened by

211

φ (x ) =

∫GL∗дφ (x )dVд.

Since π ∗ω = i∂∂φ is invariant by the le action of G, we have L∗дφ − φ = c (д, ·), ∀д ∈ G, fromthis we obtain

φ (x ) =

∫GL∗дφ (x )dVд = φ (x ) +

∫Gc (д,x )dVд = φ (x ) + c (x )

for every x ∈ GC, a straightforward calculation shows us that i∂∂c = 0. erefore we considerφ ∈ C∞(GC,R)G .

Given p ∈ P we can take the right translation Rp : GC → GC, since π Rp = π we have

R∗pη = R∗pπ∗ω = (π Rp )

∗ω = π ∗ω = η,

it follows that R∗pφ − φ = θ (p, ·), ∀p ∈ P . From the last comment, given д ∈ GC and p ∈ P wehave

φ (дp) = R∗pφ (д) = φ (д) + θ (p,д),

thus we have

θ (pp′,д) = φ (дpp′) − φ (д) = R∗p ′φ (дp) − φ (д).

Once R∗p ′φ = θ (p′, ·) + φ, we obtain

θ (pp′,д) = θ (p′,дp) + φ (дp) − φ (д) = θ (p,д) + θ (p′,дp), (D.1.10)

for every p,p′ ∈ P and every д ∈ GC. A straightforward calculation shows us that the functionθ (p, ·) : GC → R is a smooth le-invariant pluriharmonic function, if we suppose it is constant,namely θ (p, ·) = c (p), it follows from Equation D.1.10 that

c (pp′) = c (p) + c (p′),

i.e. the function c : P → R denes a real valued additive character. Based on the above ideaswe have the following proposition

Proposition D.1.2. [8] Let φ : GC → R be a G-invariant smooth function and c : P → R a realvalued smooth additive character on P , such that

φ (дp) = φ (д) + c (p), (D.1.11)

for every д ∈ GC and p ∈ P . en there exists a G-invariant real (1, 1)-form ω ∈ Ω2(XP ) suchthat i∂∂φ = π ∗ω.

212

Proof. e idea to construct the 2-form ω is by gluing patches (ωU ,U ), where U ⊂ XP is anopen set andωU ∈ Ω

2(U ). Let (U , sU ) be a pair given by an open setU ⊂ XP and a local sectionsU : U → GC, remember that P → GC → XP is a principal bundle. By means of (U , sU ) we candene

ωU = i∂∂(s∗Uφ),

therefore we obtain ωU ∈ Ω2(U ). Now we take a trivializing cover U of XP and consider

(ωU ,U )U ∈U .

For U ,V ∈ U , such that U ∩V , ∅, we have

x = π (sU (x )) = π (sV (x )),

∀x ∈ U ∩V , it follows that sU (x ) = sV (x )ψUV (x ), whereψUV : U ∩V → P is a smooth function.From these we have

φ (sU (x )) = φ (sV (x ) f (x )) = φ (sV (x )) + c (ψUV (x )),

thus on U ∩V we have

s∗Uφ = s∗Vφ +ψ

∗UVc ,

therefore in order to show that ωU = ωV on U ∩V is enough to verify that i∂∂c = 0.

Now we observe that since c : P → R is an additive character, we have c ≡ 0 on [P , P]. If wedenote q : P → P/[P , P], the last comment implies that c = q∗χ where

χ : P/[P , P]→ R, sucht that χ (q(p)) = c (p),

∀p ∈ P . Since P/[P , P] is an abelian group we can consider its universal cover

π : Ck → P/[P , P],

for some k . By pulling back χ we get an additive character π ∗χ : Ck → R. A straightforwardcalculation show us that

π ∗(i∂∂χ ) = i∂∂(π ∗χ ) = 0⇒ i∂∂χ = 0,

the implication above follows from the fact that π is a local dieomorphism, see for example[98, p. 90, Proposition 1.101]. Finally, we obtain

i∂∂c = i∂∂(q∗χ ) = q∗(i∂∂χ ) = 0,

thus we have ω ∈ Ω2(XP ) by patching together ω |U = ωU = i∂∂(s∗Uφ). In order to see thatπ ∗ω = i∂∂φ, we notice that for a local trivialization U ⊂ XP , we have

π−1(U ) = U P U × P , and π (x ,p) = x ,∀x ∈ U ,∀p ∈ P ,

for some subset U ⊂ GC. It follows from D.1.11 that φ (x ,p) = φ (x , e ) + c (p) on π−1(U ),therefore if we denote f : U → R such that f (x ) = φ (x , e ), since i∂∂c = 0, we have

213

i∂∂φ |π−1 (U ) = π∗(i∂∂ f ).

We notice thatφ (sU (x )) = f (x )+c (pr2(sU (x ))), where pr2 : π−1(U ) → P denotes the projectionon the second coordinate. From these we obtain i∂∂(s∗Uφ) = i∂∂ f , thus

π ∗(ω |U ) = π∗(i∂∂(s∗Uφ)) = π

∗(i∂∂ f ) = i∂∂φ |π−1 (U ) .

In order to see that ω ∈ Ω(1,1) (XP ) is G-invariant, we look at the following identity

π ∗l∗дω = L∗дπ∗ω = π ∗ω,

for every д ∈ G, where lд denotes the le translation on XP induced by Lд. Since π ∗ is aninjective map, the last equation implies that l∗дω = ω, ∀д ∈ G.

Now suppose that φ : GC → R is a smooth function which satises the hypothesis of Proposi-tion D.1.2, from the Iwasawa decomposition GC = GAN +, where

N + = exp ∑α∈Π+

zαxα

∣∣∣ zα ∈ C

,

and

A = exp∑

α∈Σ

aαhα

∣∣∣ aα ∈ R

,

we can write GC = GP . erefore, given x = дp ∈ GP = GC it follows that

φ (x ) = φ (дp) = φ (д) + c (p) = φ (e ) + c (p),

here we use theG-invariance to obtainφ (д) = φ (e ). Without loss of generality we can supposeφ (e ) = 0, it follows that

φ (x ) = φ (дp) = φ (д) + c (p) = c (p),

thus φ : GC → R is completely determined by its values on P ⊂ GC. From the decompositionsD.1.1 and D.1.2, we have the following fact associated to the Lie algebra of P = PΘ

[pΘ,pΘ] = g(Θ)C ⊕∑

α∈Π+\〈Θ〉+

gα , where Lie(Ru (PΘ)) =∑

α∈Π+\〈Θ〉+

gα ,

see also [2, p. 16] and [98, p. 325], therefore we can write

P = PΘ = G (Θ)CT (Σ\Θ)CRu (PΘ) = [PΘ, PΘ]T (Σ\Θ)C = [P , P]T (Σ\Θ)C.

us, once that φ : GC → R is completely determined by its values on P , and c vanishes on[P , P], we conclude that φ is completely determined by its values on T (Σ\Θ)C.

Now, by means of the homomorphism D.1.3 we have the following characterization forT (Σ\Θ)C

214

T (Σ\Θ)C =

∏α∈Σ\Θ

ϕα

tα 0

0 t−1α

∣∣∣∣∣ zα ∈ C×,∀α ∈ Σ\Θ

.

From the above characterization and the previous comments, given t ∈ T (Σ\Θ)C we have

t =∏

α∈Σ\Θ

ϕα

tα 0

0 t−1α

,

thus we obtain

φ (t ) = c (t ) =∑α∈Σ\Θ

c(ϕα

tα 0

0 t−1α

) = ∑α∈Σ\Θ

φ(ϕα

tα 0

0 t−1α

).

It follows that

φ (t ) =∑α∈Σ\Θ

(ϕ∗αφ)

tα 0

0 t−1α

, (D.1.12)

this last expression will be useful for our purpose.

Now we observe that there is a constructive way to obtain functions which satises the hy-potheses of Proposition D.1.2, the construction goes as follows. Given any integral dominantweight µ ∈ h∗, namely

µ ∈ Λ∗Z≥0=⊕α∈Σ

Z≥0ωα ,

letV (µ ) = U (gC)v+µ be the irreducible gC-module with high-weight µ, herev+µ ∈ V (µ ) denotesthe high-weight vector. Since V (µ ) is also an irreducible G-module, we can x a G-invariantinner product (·, ·) on V (µ ), from these we can dene a smooth function φµ : GC → R byseing

φµ (x ) = log | |xv+µ | |, (D.1.13)

here | |xv+µ | |2 = (xv+µ ,xv+µ ). Now we consider Pµ ⊂ GC as being the subgroup which stabilizes

the line generated by v+µ ∈ V (µ ). Since n+v+µ = 0 and hv+µ = µ (h)v+µ , ∀h ∈ h, we have

B = TCN + ⊂ Pµ ,

i.e. Pµ ⊂ GC is a parabolic subgroup [2, p. 16]. We claim that the data (φµ , Pµ ) satises thehypotheses of Proposition D.1.2. In order to see this we notice that φµ is certainlyG invariant,since | | · | | is obtained from a G-invariant inner product. Now, given д ∈ G and p ∈ Pµ , wehave

215

φµ (дp) = log | |дpv+µ | | = log | |l (p)дv+µ | | = log |l (p) | | |дv+µ | | = log | |дv+µ | | + log |l (p) |,

here l (p) ∈ C. Once φµ is G-invariant, we have

φµ (дp) = log | |v+µ | | + log |l (p) |, (D.1.14)

therefore if we take | |v+µ | | = 1, we obtain

φµ (дp) = log |l (p) |.

Since GC = GPµ , the function φµ is completely deterined by its values on Pµ . It remains toverify that the function cµ : p 7→ log |l (p) | denes an real valued character on Pµ , but it followsfrom the following identity

pp′v+µ = l (p)l (p′)v+µ = l (pp

′)v+µ ,

thus l (pp′) = l (p)l (p′), notice that l (p) , 0, ∀p ∈ Pµ . erefore we have that cµ : Pµ → Rdenes an additive real valued character.

Remark D.1.7. We notice that the function l : Pµ → C× described above denes an extension ofthe character χµ ∈ Hom(TC,C×), thus we can denote l = χµ .

Now we look more closely the expression of φµ . At rst we note that Pµ = PΘ, for

Θ =α ∈ Σ

∣∣∣ µ (hα ) = 0

,

see for instance [29, p. 300, Proposition 3.2.5], from this given t ∈ T (Σ\Θ)C ⊂ Pµ , we havetv+µ = χµ (t )v

+µ , where χµ denotes the character of TC = exp(h) associated to µ ∈ Λ∗Z≥0

[157, p.466, Proposition 16.2.2], namely

(dχµ )e = µ.

Now we use the expression D.1.11 to obtain

φµ (t ) =∑α∈Σ\Θ

(ϕ∗αφµ )

tα 0

0 t−1α

,

we notice that each element in the sum above yields

(ϕ∗αφµ )

tα 0

0 t−1α

= φµ(ϕαtα 0

0 z−1α

) = log∣∣∣χµ(ϕα

tα 0

0 t−1α

)∣∣∣,therefore we get


log∣∣∣(ϕ∗α χµ )

tα 0

0 t−1α

∣∣∣.

216

We observe that each element (ϕ∗α χµ ) denes a character of SL(2,C), we also denote by ϕα thehomomorphism from sl (2,C) to gC, see D.1.3, therefore we have the following expression


log∣∣∣(ϕ∗α χµ )

tα 0

0 t−1α

∣∣∣ = ∑α∈Σ\Θ

log |tα | (dχµ )e (h∨α ) =

∑α∈Σ\Θ

log |tα |µ (h∨α ) .

thus the function φµ : GC → R can be described on T (Σ\Θ)C as follows


〈µ,h∨α 〉 log |tα |,

for every t ∈ T (Σ\Θ)C. We have from the previous comments the following result

Proposition D.1.3 ([8]). Let GC be a complex, simply connected and simple Lie group. Givenµ ∈ Λ∗Z≥0

, there exists a function φµ : GC → R which satises the properties of Proposition D.1.2for P = Pµ . Furthermore this function is completely described by the expression


〈µ,h∨α 〉 log |tα |, (D.1.15)

for every t ∈ T (Σ\Θ)C.

Proof. e proof for this follows directly from the previous comments.

RemarkD.1.8. In the above calculation we used some properties of the representations of SL(2,C)and sl (2,C) which we explain now. Consider the canonical basis of sl (2,C) given by the elements

x =

0 1

0 0

, y =0 0

1 0

, h =1 0

0 −1

.

From the representation theory, for each r ∈ Z>0 we have an irreducible sl (2,C)-moduleVr givenby

Vr =P (u,v ) ∈ C[u,v]

∣∣∣ P (u,v ) is homogeneous and deg(P (u,v )) = r

,

where the basic elements act as the following dierential operators

x = u ∂∂u , y = v∂∂v , h = u

∂∂u −v

∂∂v .

From this the weight spaces of h ∈ sl (2,C) are the 1-dimensional spaces spanned by the elements

wk = ur−kvk , k = 0, 1, . . . , r ,

thus the action of the basic elements can be described as follows

xwk = k (r − 1 + 1)wk−1, ywk = wk+1, hwk = (r − 2k )wk .

217

Once xw0 = 0, we have w0 ∈ Vr as the high-weight vector with high-weight r , i.e. Vr =U (sl (2,C))w0. By meas of the exponential map we haveVr also as an irreducible SL(2,C)-module.

In order to obtain the expression for the character χr : C× → C× we identify C× = 〈exp(th)〉 andfor the above representation of SL(2,C) we proceed as follows. Given t ∈ C×, we can write

t =

t 0

0 t−1

=|t |eiarg(t ) 0

0 ( |t |eiarg(t ) )−1

=elog |t |+iarg(t ) 0

0 (elog |t |+iarg(t ) )−1

,

it follows that t = exp((log |t | + iarg(t ))h). Since

(log |t | + iarg(t ))hw0 = r ((log |t | + iarg(t )))w0,

we obtain

tw0 = er ((log |t |+iarg(t )))w0 = trw0 = χr (t )w0,

therefore we have χr (t ) = tr , i.e. Vr denes a SL(2,C)-irreducible module with associated char-acter χr .

From the above comments, on the expression D.1.15 we have used the fact that ϕ∗α χµ correspondsto the character of χr for r = µ (h∨α ). In fact we have an irreducible copy ofVr inside ofV (µ ) givenby

Vr = U (ϕα (sl (2,C)))v+µ ⊂ V (µ ),

we just need to identify v+µ ∼ ur ∈ Vr to obtain the above equality.

We nish this subsection by describing a basic example for the complex Lie group SL(2,C) inorder to illustrate all we have done so far. e example below can be seen as a building blockfor the general case.

Example D.1.1. Consider GC = SL(2,C), as we have seen on the previous remark we havethe following triangular decomposition for sl (2,C)

sl (2,C) =⟨x =

0 1

0 0

⟩C⊕

⟨h =

1 0

0 −1

⟩C⊕

⟨y =

0 0

1 0

⟩C

.

Once the Cartan subalgebra is 1-dimensional, the set of integral dominant weights is given by

Λ∗Z>0= Z>0ωα Z>0,

therefore given µ ∈ Λ∗Z>0we identify µ = rωα = r ∈ Z>0. us we have for the irreducible

module described in D.1.8 the following characterization

V (µ ) = Vr .

218

In this case we have the parabolic subgroup given by

Pµ = B = TCN +,

where TC = exp(h) = C× and N + = exp(n+). By applying Proposition D.1.2 for the functionφµ : SL(2,C) → R, from the expression D.1.15 we obtain

φµ (t ) = 〈µ,h〉 log |t |.

Now we will describe the formω ∈ Ω(1,1) (XB ) obtained from Proposition D.1.2. First we noticethat in this case we have

XB = SL(2,C)/B = CP1,

as we have seen in the proof of Proposition D.1.2 the local expression of ω ∈ Ω(1,1) (XB ) isgiven by

ω |U = i∂∂(s∗Uφµ ),

where sU : U → SL(2,C) denotes a local section of the principal B-bundle

π : SL(2,C) → SL(2,C)/B.

Consider N − = 〈exp(zy)〉 ⊂ SL(2,C) and denote by p0 = eB ∈ XB . From these we have thefollowing cell decomposition for XB = CP1

XB = CP1 = N −p0 ∪ π( 0 1

−1 0

).Now we take the open set U = N −p0 ⊂ XB and the smooth local section sU : U → SL(2,C)dened by

sU (np0) = n, ∀n ∈ N −.

From the Gram-Schmidt process we have for every n ∈ N − the following expression

n =

1 0

z 1

= д√1 + |z |2 0

0(√

1 + |z |2)−1

1 z1+|z |2

0 1

,

for some д ∈ SU(2). In fact the decomposition above in the right side comes from the Iwasawadecomposition of SL(2,C), we will denote by√1 + |z |2 0

0(√

1 + |z |2)−1

=t (z) 0

0 t (z)−1

,

from these we obtain the following expression

φµ (n) = 〈µ,h〉 log |t (z) | = 〈µ,h〉2 log(1 + |z |2

),

therefore we have

ω |U = i∂∂(s∗Uφµ ) =

〈µ,h〉2 i∂∂ log

(1 + |z |2

).

us the 2-form obtained by Proposition D.1.2 is exactly a multiple of the canonical Fubini–StudyKahler form of XB = CP1.

219

D.1.3 Chern class for holomorphic line bundles over ag manifolds

In this subsection we will compute the Chern class of holomorphic line bundles L → XP . Infact, we will provide a suitable expression for the generators of Pic(XP ), our approach will bebased on the content developed on the previous subsection.

Consider the complex manifold given by XP = GC/P as before. Since P = PΘ, for some Θ ⊂ Σ,

we have the following characterization for PΘ

PΘ =⋂

α∈Σ\Θ

Pωα . (D.1.16)

e above equality follows from the following facts. At rst we notice that Pωα = PΣ\α

Lie(Pωα ) = n+ ⊕ h ⊕ n(Σ\α )−, ∀α ∈ Σ\Θ,

thus α ∈ Σ\Θ ⇒ Θ ⊂ Σ\α , ∀α ∈ Σ\Θ, it follows that PΘ ⊆⋂

α∈Σ\Θ

Pωα . Now we consider the

following facts:

• For every µ ∈⊕λ∈Σ\Θ

Z>0ωα , we have PΘ = Pµ , see for instance [29, p. 300, Proposition

3.2.5];

• For every µ ∈⊕λ∈Σ\Θ

Z>0ωα , we have V (µ ) = U (gC)v+µ ⊂⊗α∈Σ\Θ

V (ωα )⊗nα , and

v+µ =⊗α∈Σ\Θ

(v+ωα )⊗nα ,

where µ =∑α∈Σ\Θ

nαωα , see for example [144, p. 305, Proposition 11.13].

From the above facts, once Pωα is the group which stabilizes the line generated byv+ωα ∈ V (ωα ),given p ∈

⋂α∈Σ\Θ Pωα , we have

pv+µ = p⊗α∈Σ\Θ

(v+ωα )⊗nα =

⊗α∈Σ\Θ

(pv+ωα )⊗nα = lµ (p)v

+µ ,

where lµ (p) ∈ C×, notice that lµ is in fact a extension of χµ ∈ Hom(TC,C×), thus we have theequality D.1.16.

Now for each fundamental weight ωα , with α ∈ Σ\Θ, by propositions D.1.2 and D.1.3, we canassociate a function φωα : GC → R, such that

φωα (д) = log | |дv+ωα | |.

From the equality D.1.16 for every p ∈ P = PΘ and д ∈ GC we have

220

φωα (дp) = φωα (д) + cωα (p),

where cωα : Pωα → R denes an additive character. erefore if we consider the restrictioncωα |P , we can apply again Proposition D.1.2 and obtain a closed real G-invariant (1, 1)-formηα ∈ Ω

(1,1) (XP ), such that

π ∗ηα = i∂∂φωα ,

where π : GC → GC/P . We have the following result from the last developments

eorem D.1.4 ([8]). Let GC be a complex, simply connected and simple Lie group and P ⊂ GC

be a parabolic subgroup. en the function φ : GC → R dened by

φ (д) =∑α∈Σ\Θ

aα log | |дv+ωα | |, (D.1.17)

where aα ∈ R, ∀α ∈ Σ\Θ, denes a closed realG-invariant (1, 1)-formωφ ∈ Ω(1,1) (XP ), such that

π ∗ωφ = i∂∂φ,

where π : GC → GC/P . Furthermore, we have that

1. ωφ is positive if and only if aα > 0, ∀α ∈ Σ\Θ;

2. ωφ is nondegenerate if and only if∑α∈Σ\Θ

aα 〈ωα ,h∨β 〉

2 , 0, ∀β ∈ Π+\〈Θ〉+.

erefore, if the rst condition above holds, then ωφ ∈ Ω(1,1) (XP ) denes a Kahler form.

Proof. e fact that φ : GC → R dened above induces a closed real G-invariant (1, 1)-formωφ ∈ Ω

(1,1) (XP ) follows from the previous comments together with Proposition D.1.2. Actually

φ =∑α∈Σ\Θ

aαφωα .

In order to verify the conditions 1 and 2 of the above theorem we observe that we can associateto ωφ ∈ Ω(1,1) (XP ) a Hermitian form Hφ , see for instance [163, p. 65], such that

ωφ = −Im(Hφ ).

We have Hφ (u,v ) = дφ (u,v ) − iωφ (u,v ), such that дφ (u,v ) = ωφ (u, Jv ), ∀u,v ∈ TXP , denotesa symmetric bilinear form, here J : TXP → TXP denotes the complex structure of XP .

From the integrability of the complex structure of XP we have the following identication ofholomorphic vector bundles

(TXP , J ) = T(1,0)XP ,

where the isomorphism is dened by v → 12 (v − i Jv ), ∀v ∈ TXP , this identication provides

that

221

Hφ (u,v ) = Hφ (12 (u − i Ju),

12 (v − i Jv )).

A straightforward computation shows us that

−iωφ (12 (u − i Ju),

12 (v + i Jv )) = Hφ (u,v ) = Hφ (

12 (u − i Ju),

12 (v − i Jv )),

thus we have Hφ (X ,Y ) = −iωφ (X ,Y ), ∀X ,Y ∈ T (1,0)XP . On the other hand, by denition ofHφ we have

Hφ (v,v ) = дφ (v,v ),

it follows that

Hφ (v,v ) = дφ (v,v ) ≥ 0 ⇐⇒ −iωφ (12 (v − i Jv ),

12 (v + i Jv )) ≥ 0,

∀v ∈ TXP . Since ωφ is positive if and only if дφ is positive, given a local basis∂∂zj

is enough

to analyse

Hφ

( ∂∂zi,∂

∂zj

)= −iωφ

( ∂∂zi,∂

∂z j

).

Moreover, once ωφ is G-invariant we just need to analyse the condition on the tangent spaceat x0 = eP ∈ XP .

Now we consider (U , sU ), such that U ⊂ XP is a coordinate open neighbourhood of x0 ∈ XP

and sU : U → GC is a holomorphic section. From Proposition D.1.2 we have

ωφ |U = i∂∂(s∗Uφ),

it follows that

Hφ

( ∂∂zi,∂

∂zj

)=∂2

∂zi∂z j(φ sU ), (D.1.18)

i.e. the Hermitian form Hφ is completely determined by the Levi form

Lev(φ su ) =∑i,j

∂2

∂zi∂z j(φ sU )dzi ⊗ dz j

see [54] for more details about the Levi form. It is worth to notice that

T (1,0)x0 XP =

∑α∈Π−\〈Θ〉−

gα ,

furthermore have a distinguished neighbourhood U ⊂ XP dened by the opposite “big cell”,see C.4.10 on page 199, such that

U = B−x0 = Ru (PΘ)−x0 ⊂ XP ,

where

222

Ru (PΘ)− =

∏α∈Π−\〈Θ〉−

N −α , with N −α = exp(gα ), ∀α ∈ Π−\〈Θ〉−.

From these we take the local section sU : U → GC dened on U = Ru (PΘ)−x0 as

sU (n−x0) = n

−, ∀n− ∈ Ru (PΘ)−,

it follows that

φ (sU (n−x0)) = φ (n

−),

∀n− ∈ Ru (PΘ)−. By gathering together the above data it follows that

φ sU = φ |Ru (PΘ)− . (D.1.19)

Now we observe that each N −α is biholomorphic to C [153, p. 132, Proposition 8.1. 1], in factfor every α ∈ Π−\〈Θ〉− we can write

N −α =ϕ−α

( 1 0

zα 1

) ∣∣∣ zα ∈ C

,

here ϕ−α : SL(2,C) → GC, ∀α ∈ Π−\〈Θ〉−, are the homomorphisms dened in D.1.3.

From Remark D.1.8, for each β ∈ Σ\Θ and each α ∈ Π−\〈Θ〉− we have a character of SL(2,C)associated to the positive integer ωβ (h∨−α ) ∈ Z>0 dened by

χωβ (h∨−α ) = ϕ∗−α χωβ .

us for every t ∈ C× we have

ϕ−α

(t 0

0 t−1

)v+ωβ = tωβ (h∨−α )v+ωβ .

Now consider n− ∈ Ru (PΘ)− such that

n− = ϕ−α

( 1 0

zα 1

),

i.e. n− ∈ N −α , for some α ∈ Π−\〈Θ〉−, as in Example D.1.1 we can write 1 0

zα 1

= д√1 + |zα |2 0

0(√

1 + |zα |2)−1

1 zα1+|zα |2

0 1

.

Since φωβ is G-invariant and n+v+ωβ = 0, by denoting t (zα ) =√

1 + |zα |2, we obtain

223

φωβ (n−) = log

∣∣∣∣∣∣ϕ−α(t (zα ) 0

0 t (zα )−1

)v+ωβ ∣∣∣∣∣∣ = ωβ (h∨−α ) log |t (zα ) |

for every n− ∈ N −α . From the last comments we have for all n− ∈ N −α the expression

φ (n−) =∑β∈Σ\Θ

aβφωβ (n−) =

∑β∈Σ\Θ

aβωβ (h∨−α ) log |t (zα ) |,

and by replacing t (zα ) =√

1 + |zα |2 in the last expression it follows that

φ (n−) =∑β∈Σ\Θ

aβωβ (h∨−α )

2 log(1 + |zα |2

). (D.1.20)

By taking an enumeration for the set of roots Π−\〈Θ〉− = α1, . . . ,αn, where n = dimC(XP ),we have a coordinate system on U = Ru (PΘ)

−x0 ⊂ XP given by

n∑l=1

zlel = (z1, . . . , zn ) =n∏l=1

ϕ−αl

(1 0

zl 1

)x0.

Furthermore, at the point x0 = eP we have

∂

∂zi= π∗yαi

∀i = 1, . . . ,n, where π : GC → GC/P . In order to calculate Hφ (∂∂zi, ∂∂zi ), from D.1.20, we need

to look at the derivatives of the function

φ (0, . . . , zi , . . . , 0) = φ (ziei ) = φ(ϕ−αi

(1 0

zi 1

)). (D.1.21)

From D.1.20 and D.1.21 we obtain

Hφ

( ∂∂zi,∂

∂zi

)=∑β∈Σ\Θ

aβωβ (h∨−αi )

2 ,

for every i = 1, . . . ,n.

Given 1 ≤ i , j ≤ n, in order to calculate the mixed termHφ (∂∂zi, ∂∂zj ) we observe the following.

From the comments at the beginning we have

Hφ (∂

∂zi,∂

∂zj) = Hφ (π∗yαi ,π∗yα j ) = π

∗(Hφ ) (yαi ,yα j ),

since the elements yαi , i = 1, . . . ,n, are weight vectors corresponding to dierent weightsfor the adjoint action of the compact maximal torus T = TC ∩ G. For every 1 ≤ i , j ≤ nthe vectors yαi and yα j are orthogonal with respect to any T -invariant Hermitian form. Inparticular π ∗(Hφ ) denes a T -invariant Hermitian form on

∑α∈Π−\〈Θ〉−

gα . erefore we obtain

224

Hφ (∂

∂zi,∂

∂zj) = π ∗(Hφ ) (yαi ,yα j ) = 0,

for every 1 ≤ i , j ≤ n.

From the above discussion we conclude that

Hφ

( ∂∂zi,∂

∂zj

)= 0 and Hφ

( ∂∂zi,∂

∂zi

)=∑β∈Σ\Θ

aβωβ (h∨−αi )

2 , (D.1.22)

for 1 ≤ i, j ≤ n. us, if Hφ is positive we have aβ > 0, in fact, just take −αi = β ∈ Σ\Θ onthe right side of the above expression. Conversely, if aβ > 0, ∀β ∈ Σ\Θ, we have Hφ positivefrom the fact that for every i = 1, . . . ,n, we have ωβ (h∨−αi ) ≥ 0, ∀β ∈ Σ\Θ, and ωβ (h∨−αi ) > 0,for some β ∈ Σ\Θ.

By analysing the above equation we see that Hφ is nondegenerate if and only if

∑β∈Σ\Θ

aβ〈ωβ ,h∨−α 〉

2 , 0,

∀α ∈ Π−\〈Θ〉−. erefore if the rst condition of proposition holds we have a Hermitian met-ric дφ with fundamental form ωφ . Once dωφ = 0, it implies that дφ is in fact a Kahler metric.

In what follows we will apply the ideas of eorem D.1.4 in order to describe the generatorsof the following groups

Pic(XP ) and π2(XP ).

Let XP be a ag manifold associated to some parabolic subgroup P = PΘ ⊂ GC. By takinga fundamental weight ωα ∈ Λ∗Z>0

, such that α ∈ Σ\Θ, as we have seen in Proposition D.1.3we can associate to this weight a closed real G-invariant (1, 1)-form ηα ∈ Ω(1,1) (XP ) whichsatises

π ∗ηα = i∂∂φωα ,

where π : GC → GC/P = XP and φωα (д) = log | |дv+ωα | |, ∀д ∈ GC.

Since [ηα ] ∈ H 2(XP ,R), ∀α ∈ Σ\Θ, we will verify that these elements are in fact algebraicallyindependent over Z. In order to do this, suppose we have aα ∈ Z, α ∈ Σ\Θ, such that∑

α∈Σ\Θ

aα [ηα ] = 0,

with aα0 , 0, for some α0 ∈ Σ\Θ. It follows that∫S

∑α∈Σ\Θ

aαηα = 0, (D.1.23)

225

for every closed 2-dimensional oriented submanifold S → XP2.

For α0 ∈ Σ\Θ, consider the homomorphism D.1.3, namely

ϕα0 : sl (2,C) → sl (α0) ⊂ gC,

and also consider the induced homomorphism ϕα0 : SL(2,C) → GC. ese two homomor-phisms allows us to dene a holomorphic action of SL(2,C) on XP given by

д · x = ϕα0 (д)x ,

for д ∈ SL(2,C) and x ∈ XP . erefore the orbit through the origin x0 = eP ∈ XP for thisaction provides a biholomorphism

SL(2,C)/B SL(2,C)x0 ⊂ XP .

It follows from the Example D.1.1 that we have an embedding

ια0 : CP1 → XP ,

where ια0 is the map obtained from the commutative diagram

SL(2,C) GC

CP1 XP

ϕα0

π π

ια0

notice that the above map ια0 denes in fact a homotopy class since CP1 = S2, we denote therepresentative of this class by

P1α0 = (CP1, ια0 ).

From Equation D.1.23 we obtain

∑α∈Σ\Θ

aα

∫CP1

ι∗α0ηα = 0,

once we have a cell decomposition given by

CP1 = N −x0 ∪ π( 0 1

−1 0

),

it follows that2Here we consider

∫S β =

∫S ι∗β , for evry β ∈ Ω2 (XP ), where ι : S → XP , see [141, p. 178].

226

0 =∑α∈Σ\Θ

aα

∫CP1

ι∗α0ηα =∑α∈Σ\Θ

aα

∫N−x0

ι∗α0ηα .

From the previous commutative diagram we obtain

ια0 (N−x0) = ια0 (π (N

−)) = π (ϕα0 (N−)),

thus we have for each term of the above sum of integrals the following expression∫N−x0

ι∗α0ηα =

∫ια0 (N

−x0)ηα =

∫ϕα0 (N

−)π ∗ηα .

We notice that π ∗ηα = i∂∂φωα , ∀α ∈ Σ\Θ, furthermore for n− ∈ ϕα0 (N−) = N −α0 we have

φωα (n−) =

ωα (h∨α0 )

2 log(1 + |zα0 |

2),see D.1.20, it follows that∫

ϕα0 (N−)π ∗ηα =

∫N−

i∂∂(φωα ϕα0 ) =ωα (h

∨α0 )

2

∫N−

i∂∂ log(1 + |zα0 |

2).

By considering the natural identication N − = C, we obtain∫N−

i∂∂ log(1 + |zα0 |

2) =

∫C

i

(1 + |zα0 |2)2

dzα0 ∧ dzα0 = 2π ,

thus we have

0 =∑α∈Σ\Θ

aα

∫CP1

ι∗α0ηα =∑α∈Σ\Θ

aαωα (h∨α0 )

2

∫N−

i∂∂ log(1 + |zα0 |

2) = πaα0 ,

however we suppose aα0 , 0. erefore we have [ηα ] ∈ H 2(XP ,R), α ∈ Σ\Θ, algebraicallyindependent over Z.

From the above calculation we showed that theG-invariant real (1, 1)-forms [ηα ] ∈ H 2(XP ,R),α ∈ Σ\Θ, are algebraically independent over Z. Our next step will be showing that these (1, 1)-forms are in fact integral, i.e. [ηα ] ∈ H 2(XP ,Z), ∀α ∈ Σ\Θ.

In order to do this we remember that

H2(XP ,Z) π2(XP ) Z#(Σ\Θ) ,

see D.1.9, thus the homotopy classes [P1α ], α ∈ Σ\Θ, also can be regarded as a homology

classes. Now we take the normalization

φωα (д) =1π

log | |дv+ωα | | =1

2π log | |дv+ωα | |2,

227

and consider henceforth ηα the G-invariant real (1, 1)-form associated to φωα dened above.From the previous computations we have

⟨[ηα ], [P1

β]⟩=

∫P1β

ηα = δαβ , (D.1.24)

thus we obtain the following results

Proposition D.1.5. Let XP be a ag manifold associated to some parabolic subgroup P = PΘ ⊂GC. en we have

H2(XP ,Z) = π2(XP ) =⊕α∈Σ\Θ

Z[P1α ],

for each α ∈ Σ\Θ the class [P1α ] ∈ π2(XP ) is called Dynkin line 3.

Proof. Suppose we have [P1α ] = [P1

β], for α , β , it follows that

P1α − P1

β = ∂Aαβ ,

for some 3-cycle Aαβ , thus we have

0 ,⟨[ηα ], [P1

α ]⟩=

∫P1α

ηα =

∫P1β+∂Aα β

ηα .

but from the Equation D.1.24 and the Stokes’ theorem we have∫P1β+∂Aα β

ηα =⟨[ηα ], [P1

β]⟩+

∫Aα β

dηα = 0,

therefore we have [P1α ] , [P1

β], for α , β , α , β ∈ Σ\Θ.

Proposition D.1.6. Let XP be a ag manifold associated to some parabolic subgroup P = PΘ ⊂GC, then we have

Pic(XP ) = H 1,1(XP ,Z) = H 2(XP ,Z) =⊕α∈Σ\Θ

Z[ηα ].

Proof. We have seen that

Pic(XP ) = H 1,1(XP ,Z) ⊂ H 2(XP ,Z),

see Remark D.1.6. In order to show that H 1,1(XP ,Z) = H 2(XP ,Z), we observe at rst that

[ηα ] ∈ H 1,1(XP ,R), ∀α ∈ Σ\Θ.

SinceH 1,1(XP ,Z) = H 1,1(XP ,R)∩H 2(XP ,Z) and the classes [ηα ] ∈ H 1,1(XP ,R) are algebraicallyindependent over Z it is enough to show that they are integral classes.

From the universal coecient theorem for cohomology [83, p. 195], we get3See [9, p. 623] for more details about the denition of Dynkin line, an alternative discussion about the second

homotopy group of complex ag manifolds also can be found in [25].

228

0→ Ext(H1(XP ,Z),Z) → H 2(XP ,Z) → Hom(H2(XP ,Z),Z) → 0.

Since XP is simply connected from Hurewicz’s theorem [83, p. 366] we have H1(XP ,Z) = 0,thus

H 2(XP ,Z) Hom(H2(XP ,Z),Z).

From Equation D.1.24 and the previous proposition we obtain⟨[ηα ],−

⟩∈ Hom(H2(XP ,Z),Z),

∀α ∈ Σ\Θ, thus we have H 2(XP ,Z) =⊕α∈Σ\Θ

Z[ηα ].

e above result provide the following characterization

Pic(XP ) =⊕α∈Σ\Θ

Z[ηα ],

it remains to show what is the holomorphic line bundle L → XP which satises c1(L) = [ηα ],α ∈ Σ\Θ.

Let us collect some basic facts about holomorphic sections. Given a fundamental weight ωα ∈Λ∗Z>0

, such that α ∈ Σ\Θ, we can consider the fundamental holomorpic line bundle

Lχωα = GC ×χωα C−ωα ,

see Remark D.1.1 on page 205. Let σ ∈ H 0(XP ,Lχωα ) be a holomorphic section, we have

σ (дP ) = [д, fσ (д)], where fσ : GC → C−ωα ,

since σ does not depend on of the representative for the classes дP ∈ XP , we obtain

σ (дP ) = σ (дpP ) = [дp, fσ (дp)] = [д, χ−1ωα (p) fσ (дp)].

It follows that fσ (д) = χ−1ωα (p) fσ (дp), i.e. fσ satises the equivariance condition

fσ (дp) = χωα (p) fσ (д),

notice that χωα (p) fσ (д) = p−1 · fσ (д), ∀p ∈ P , see Remark D.1.1.

Now consider a locally dened holomorphic section σU : U → Lχωα , in this case we have

σU (дP ) = [д, fσU (д)], where fσU : π−1(U ) → C−ωα ,

with π : GC → GC/P = XP . As before the function fσU also satises the equivariance condition,thus if we have a holomorphic section sU : U → GC, for дP ∈ U we can write

π (sU (дP )) = дP ⇐⇒ sU (дP ) = дp, for some p ∈ P ,

229

therefore

σU (дP ) = [д, fσU (д)] = [sU (дP )p−1,pp−1 fσU (д)] = [sU (дP ),p−1 fσU (д)].

Since p−1 · fσU (д) = fσU (дp) and sU (дP ) = дp, we obtain

σU (дP ) = [sU (дP ), fσU (sU (дP ))],

thus we can writeσU = [sU , fσU sU ]. (D.1.25)

Now we can show the following result

Proposition D.1.7. Let XP be a ag manifold associated to some parabolic subgroup P = PΘ ⊂GC. en for every fundamental weight ωα ∈ Λ∗Z≥0

, such that α ∈ Σ\Θ, we have

c1(Lχωα ) = [ηα ], (D.1.26)

where Lχωα = GC ×χωα C−ωα , and ηα ∈ Ω(1,1) (XP ) is the G-invariant real (1, 1)-form associatedto the function dened by

φωα (д) =1π

log | |дv+ωα | | =1

2π log | |дv+ωα | |2,

for all д ∈ GC.

Proof. e proof of this result goes as follows, under the above hypotheses we can dene aHermitian structure on Lχωα given by the expression

H ([д, z], [д,w]) = e−2πφωα (д)zw =zw

| |дv+ωα | |2 ,

notice that

H ([дp,p−1z], [дp,p−1w]) = H ([дp, χωα (p)z], [дp, χωα (p)w]) = H ([д, z], [д,w]),

actually it follows from the fact that

H ([дp, χωα (p)z], [дp, χωα (p)w]) = |χωα (p) |2zw

| |дpv+ωα | |2 =

|χωα (p) |2zw

|χωα (p) |2 | |дv+ωα | |

2 =zw

| |дv+ωα | |2 ,

thusH is well-dened, see Remark D.1.1 for more details about the action involved in the abovecalculations. e Hermitian connection ∇ associated to above structure is dened locally onU ⊂ XP by

∇ = d +AU , where AU = H−1∂H ,

the (1, 0)-form dened byAU = H−1∂H can be described by taking a locally dened non-vanishholomorphic section σU : U → Lχωα , and taking the ∂-derivative of the function logH (σU ,σU ),i.e.

230

AU = H (σU ,σU )−1∂H (σU ,σU ),

notice that the above description does not depend on of σU since the transition functions of(Lχωα ,H ) take values in U(1).

Now in order to calculate the local expression of the curvature form F∇ = dAU we rememberthat

∂∂ logH = −∂∂ logH = −∂(∂HH

)= ∂AU − F∇,

since

∂AU = ∂(H−1∂H

)=

(∂2H )H − ∂H ∧ ∂H

H 2 = 0,

we have F∇ = −∂∂ logH .

On the other hand, we can take a locally dened holomorphic section sU : U → GC and con-sider the expression obtained in D.1.25, from our denition of H follows that

H (σU ,σU ) = e−2π (φωα sU ) | fσU sU |2,

thus we obtain

logH = −2π (φωα sU ) + log | fσU sU |2,

now since ∂∂ log | fσU sU |2 = 04, it follows that

F∇ = −∂∂ logH = 2π∂∂(φωα sU ),

therefore we have

ηα |U = i∂∂(s∗Uφωα ) =

i

2π F∇ =⇒ c1(Lχωα ) = [ηα ],

from these we conclude the proof.

RemarkD.1.9. ere is an alternative way to show the last result which goes as follows. Consideran open cover XP =

⋃i∈I Ui which trivializes both P → GC → XP and Lχωα → XP and take a

collection of local sections (si )i∈I , such that si : Ui → GC. From these we dene qi : Ui → R+ by

qi = e−2πφωα si =1

| |siv+ωα | |2 , (D.1.27)

for every i ∈ I . ese functions (qi )i∈I satisfy qj = |χ−1ωα ψij |

2qi onUi ∩Uj , ∅, here we have usedthat sj = siψij on Ui ∩ Uj , ∅, and pv+ωα = χωα (p)v

+ωα for every p ∈ P and α ∈ Σ\Θ. From the

4Notice that for every x ∈ U there exists an open neighbourhood Vx ⊂ U such that log fσU (sU (z)) =log | fσU (sU (z)) | + iarg( fσU (sU (z))), ∀z ∈ Vx ⊂ U , i.e. log | fσU sU | = Re(log( fσU sU )) on Vx , see for ex-ample [161, p. 227], since log( fσU sU ) is holomorphic in Vx it follows that ∂∂ log | fσU sU |2 = 0 on Vx ⊂ U ,once x ∈ U is arbitrary we conclude that log | fσU sU |2 is pluriharmonic on U , see [147, p. 15].

231

above collection of smooth functions we can dene a Hermitian structure H on Lχωα by taking oneach trivialization ϕi : Lχωα → Ui × C a metric dened by

H ((x ,v ), (x ,w )) = qi (x )vw, (D.1.28)

for (x ,v ), (x ,w ) ∈ Lχωα |Ui Ui × C. e above Hermitian metric induces a Chern connection∇ = d + ∂ logH with curvature F∇ satisfying

i

2π F∇ = ηα ,

thus we have c1(Lχωα ) = [ηα ].

Remark D.1.10. Here it is worth to point out that on the above proposition we have restrictedour aention just to weights ωα ∈ Λ∗Z>0

for which α ∈ Σ\Θ. Actually, if we have a parabolicsubgroup P ⊂ GC, with P = PΘ, the decomposition

PΘ = G (Θ)CT (Σ\Θ)CRu (PΘ) = [PΘ, PΘ]T (Σ\Θ)C,

shows us that Hom(P ,C×) = Hom(T (Σ\Θ)C,C×). erefore, if we takeωα ∈ Λ∗Z, such that α ∈ Θ,we obtain Lχωα = XP × C, i.e. the associated holomorphic line bundle Lχωα is trivial.

In what follows we will describe some results related to the previous proposition. ConsiderL ∈ Pic(XP ) as being an arbitrary holomorphic line bundle over XP , since P = PΘ, from thelast proposition we have

c1(L) =∑α∈Σ\Θ

⟨c1(L), [P1

α ]⟩c1(Lχωα ).

erefore if we denote nα =⟨c1(L), [P1

α ]⟩, ∀α ∈ Σ\Θ, we obtain

L ⊗α∈Σ\Θ

L⊗nαχωα,

thus Proposition D.1.7 allows to describe, up to isomorphism, any holomorphic line bundleover a ag manifold XP .

Remark D.1.11. We notice that Proposition D.1.7 allows us to describe the Cox ring of XP in thefollowing way

Cox(XP ) =⊕

nα ∈Z,∀α∈Σ\Θ

H0(XP ,⊗α∈Σ\Θ

L⊗nαχωα

).

e above ring contains a lot of geometric information about the manifold XP . In fact, if L ∈Pic(XP ) is a very ample line bundle overXP we have a projective embeddingΨL : XP → P(H0(XP ,L)

∗),furthermore we have XP Proj(R (XP ,L))

5 where

R (XP ,L) :=⊕k≥0

H 0(XP ,L⊗k ),

5A brief description of the “Proj” construction can be found in [109, p. 17].

232

notice that since L ⊗α∈Σ\Θ

L⊗nαχωα, we have R (XP ,L) as a subring of Cox(XP ). is illustrative

situation shows us that the Cox ring has all the “coordinate rings” as subrings, see [134] for moredetails about Cox rings of projective varieties, for the Kodaira embedding of XP we recommendthe content of Appendix C.3.

Besides of the above description for holomorphic line bundles over XP , from eorem D.1.4we can choose in the class c1(L) ∈ H 1,1(XP ,Z) the G-invariant real (1, 1)-form ωφ , whereφ : GC → R is dened by

φ (д) =1π

∑α∈Σ\Θ

⟨c1(L), [P1

α ]⟩

log | |дv+ωα | |, (D.1.29)

∀д ∈ GC. A holomorphic line bundle L → XP is called positive if there exists a Hermitianmetric on L with curvature F∇ such that i

2π F∇ denes a positive (1, 1)-form on XP [63, p. 148],in this case we denote c1(L) > 0. We have the following result

Proposition D.1.8. e set of positive holomorphic line bundles over a ag manifold XP , withP = PΘ, is completely described by the set⊕

α∈Σ\Θ

Z>0c1(Lχωα ) ⊂ Pic(XP ).

Proof. Let L ∈ Pic(XP ) be a positive holomorphic line bundle, it follows that there exists i2πω ∈

c1(L) which denes a positive real (1, 1)-form. From the previous comments we also have

L ⊗α∈Σ\Θ

L⊗nαχωα,

where nα =⟨c1(L), [P1

α ]⟩, ∀α ∈ Σ\Θ. erefore, from D.1.29 we have ωφ ∈ c1(L) such that

i2πω − ωφ = dΞ,

for some Ξ ∈ Ω1(XP ), notice that ∀α ∈ Σ\Θ we have∫P1α

ωφ =i

2π

∫P1α

ω =⟨c1(L), [P1

α ]⟩∈ Z>0.

Hence, we have the following correspondence

(L,ω) −→∑α∈Σ\Θ

( i

2π

∫P1α

ω)c1(Lχωα ) = [ωφ].

Conversely, if we take

[ω] =∑α∈Σ\Θ

kαc1(Lχωα ),

with kα ∈ Z>0, ∀α ∈ Σ\Θ, we can construct a holomorphic line bundle L with c1(L) = [ω].Actually, we just need to take the integral weight

233

µ =∑α∈Σ\Θ

kαωα ; χµ =∏

α∈Σ\Θ

χkαωα ,

and set L = Lχµ , the positivity of Lχµ follows from Equation D.1.29 and from eorem D.1.20.

e content of the last proposition can be understood as a description of all ample and veryample line bundles over XP . Moreover, we can also understand the last result as a descriptionof the Kahler cone of XP . In fact, we have from eorem D.1.4 and the last proposition thefollowing fact

KXP =⊕α∈Σ\Θ

R>0c1(Lχωα ),

here KXP denotes the Kahler cone of XP .

Now let us use the previous ideas to calculate the Chern class of the anti-canonical bundle


),

here we suppose dimC(XP ) = n. At rst we consider the following characterization ofT (1,0)XP .If we denote

m =∑

α∈Π+\〈Θ〉+

g−α = T(1,0)x0 XP ,

where x0 = eP ∈ XP , we have the following identication 6

T (1,0)XP = GC ×P m,

here the twisted product on the right side above is obtained from Ad: P → GL(m) as anassociated holomorphic vector bundle. erefore we have


)= det

(GC ×P m

)= GC ×χ C, (D.1.30)

here we identify∧n (m) = C and χ : P → C× is given by

χ (p) = det(Ad(p)),

∀p ∈ P , i.e. χ ∈ Hom(P ,C×). Now we notice that for P = PΘ we have

P = [P , P]T (Σ\Θ)C,

it follows that every p ∈ P can be wrien as p = p′t , where p′ ∈ [P , P] and t ∈ T (Σ\Θ)C, thus6is identication can be obtained as follows, rst we consider the Maurer–Cartan form of GC, i.e. the map

θд : TдGC → gC, from this form we can dene a map Ψ : GC ×P m → T (1,0)XP such that Ψ([д,X ]) = π∗ (θ−1д (X )),

∀[д,X ] ∈ GC ×P m, here π : GC → XP denotes the canonical projection. A straightforward calculation show usthat Ψ denes an isomorphism of holomorphic vector bundles.

234

χ (p) = det(Ad(p)) = det(Ad(p′)) det(Ad(t )) = det(Ad(t )).

e last comment implies that χ can be seen as an extension of χ ′ ∈ Hom(T (Σ\Θ)C,C×),dened by

χ ′(t ) = det(Ad(t )),

∀t ∈ T (Σ\Θ)C.

Now, once we have T (Σ\Θ)C = exp(h(Σ\Θ)), it follows that

χ ′(exp(h)) = det(Ad(exp(h))) = det(ead(h) ) = eTr(ad(h)) ,

∀h ∈ h(Σ\Θ), thus we can write

(dχ ′)e (h) = Tr(ad(h)) = −∑

α∈Π+\〈Θ〉+

α (h). (D.1.31)

From these, since we have nontrivial integral weights

α |h(Σ\Θ) =∑β∈Σ\Θ

α (h∨β )ωβ ,

notice that α (h∨β ) ∈ Z, α ∈ Π+\〈Θ〉+, if we denote by

δP =∑

α∈Π+\〈Θ〉+

α , (D.1.32)

the Equation D.1.31 shows us that

(dχ ′)e = −δP . (D.1.33)

Hence, we get

χ ′(exp(h)) = e−δP (h), (D.1.34)

∀h ∈ h(Σ\Θ), therefore χ (p) = χδP (p−1), ∀p ∈ P . From these we have the following result

Proposition D.1.9. Let XP be a ag manifold associated to some parabolic subgroup P = PΘ ⊂GC. en we have


)= LχδP , (D.1.35)

therefore the Chern class of XP is given by

c1(−KXP ) =∑α∈Σ\Θ

⟨δP ,h

∨α

⟩c (Lχωα ).

235

Proof. From Equation D.1.30 we have

−KXP = GC ×χ C,

where χ ∈ Hom(P ,C×) is a character which denes the following action on C

p · z = χ (p)z = det(Ad(p))z.

However, from the Equation D.1.34 we have χ (p)z = χδP (p−1)z, ∀p ∈ P and ∀z ∈ C, it implies

that the P-space∧n (m) = C is exactly C−δP . erefore we have

−KXP = GC ×χδP

C−δP = LχδP ,

see Remark D.1.1 for more details about the P-space structure of C−δP .

Now in order to compute c1(−KXP ) we observe that

χδP =∏

α∈Σ\Θ

χ〈δP ,h

∨α 〉

ωα ,

thus we have

−KXP = LχδP =⊗α∈Σ\Θ

L⊗〈δP ,h

∨α 〉

χωα ,

from the last equality above and Proposition D.1.7 we have the expression

c1(−KXP ) =∑α∈Σ\Θ

⟨δP ,h

∨α

⟩c1(Lχωα ),

therefore we obtain the result stated.

Now since c1(det(T (1,0)XP )) = c1(T(1,0)XP ), see for instance [126, p. 117], from the identica-

tion

T (1,0)XP = TXP ,

we have

c1(XP ) = c1(TXP ) = LχδP .

From the above comments and by means of Proposition D.1.7, we obtain the following corol-lary

Corollary D.1.10. Let XP be a ag manifold associated to some parabolic subgroup P = PΘ ⊂GC. en we have c1(XP ) > 0, i.e. XP is a fano manifold.

236

Consider now the function φ : GC → R dened by

φ (д) =1π

∑α∈Σ\Θ

⟨δP ,h

∨α

⟩log | |дv+ωα | | =

12π log

( ∏α∈Σ\Θ

| |дv+ωα | |2〈δP ,h∨α 〉

). (D.1.36)

In the same way that we did in the proof of Proposition D.1.7 we can dene a hermitian struc-ture on −KXP given by the following expression

H ([д, z], [д,w]) = e−2πφ (д)zw =zw∏

α∈Σ\Θ

| |дv+ωα | |2〈δP ,h∨α 〉

, (D.1.37)

therefore we have

ωφ =i

2π F∇ with ∇ = d + ∂ logH (locally),

where ωφ is the G-invariant closed real (1, 1)-form associated to the function φ dened abovein D.1.36, other way to see that ωφ ∈ c1(XP ) is to observe that

ωφ =∑α∈Σ\Θ

⟨δP ,h

∨α

⟩ηα .

In what follows we will denote by ωφ = ωXP .

Since 〈δP ,h∨α 〉 ∈ Z>0, ∀α ∈ Σ\Θ, from eorem D.1.4 the form

ωXP =∑α∈Σ\Θ

⟨δP ,h

∨α

⟩ηα , (D.1.38)

denes a Kahler form on XP . We have the following result from the last comments

eoremD.1.11. LetXP be a ag manifold associated to some parabolic subgroup P = PΘ ⊂ GC.en the Kahler manifold (XP ,ωXP ) is a Kahler–Einstein manifold.

Proof. Let ∇ be the Chern connection on TXP induced by the metric associated to ωXP . If wedenote by R∇ its curvature tensor, we have the Ricci tensor [13, p. 55] dened by

Ric(X ,Y ) = TrZ → R∇(Z ,X )Y

.

From this tensor we have the associated Ricci form ρ (X ,Y ) = Ric(JX ,Y ) which satises[ ρ2π

]= c1(TXP , ∇) = c1(XP ),

see for instance [126, p. 120] or [13, p. 57]. Hence, once [ωXP ] = c1(XP ), it follows that

ρ

2π − ωXP = i∂∂ f .

for some f ∈ C∞(XP ). Now by looking at the local expression of the Ricci form

237

ρ = −i∂∂ log det(дφ ),

we see that ρ is a G-invariant form, here дφ denotes the matrix associated to the G-invariantHermitian metric dened by the G-invariant Kahler form ωXP .

From the last comment we can suppose that f is also G-invariant. In fact we can replace fby its averaging

∫G (Lд)

∗ f dVд. erefore we have f constant, since the action of G on XP istransitive. us we obtain

ρ

2π − ωXP = 0 ⇐⇒ Ric(ωXP ) − 2πωXP = 0,

it follows that ωXP is a Kahler–Einstein metric.

An important consequence of the above theorem is the following. If we replace the Kahlermetric ωXP by its Ricci form ρ = 2πωXP , then the Einstein equation becomes

Ric(ρ) − ρ = 0, (D.1.39)

it follows from the fact that Ric(cωXP ) = Ric(ωXP ), for all constant c > 0. us, up to rescale,for the anti-canonical class metric ωXP we have the Einstein’s constant equal 1.

Other important consequence of the above result and previous comments is that we have thefollowing local expression for the Ricci form of (XP ,ωXP )

ρ = i∂∂ log( ∏α∈Σ\Θ

| |дv+ωα | |2〈δP ,h∨α 〉

), (D.1.40)

see Equation D.1.36. e nice feature of this local expression is that it shows us how we cancompletely describe a geometric object associated to XP in terms of the Lie-theoretical objectsassociated to the underlying Lie algebra gC, namely root system and fundamental irreduciblegC-modules.

Example D.1.2. Let us apply all the previous results in the basic example introduced in D.1.1.At rst we need to collect some basic Lie-theoretical facts. We consider in this case GC =

SL(2,C) and we x a triangular decomposition for sl (2,C) given by

sl (2,C) =⟨x =

0 1

0 0

⟩C⊕

⟨h =

1 0

0 −1

⟩C⊕

⟨y =

0 0

1 0

⟩C

.

e information about the above decomposition is codied in Σ = α and Π = α ,−α , ourset of integral dominant weights in this case is given by

Λ∗Z≥0= Z≥0ωα .

We take P∅ = B ⊂ SL(2,C), and consider the Kahler manifold

238

XB = SL(2,C)/B = CP1.

In this low dimensional case we have

−KCP1 = det(T (1,0)CP1) = T (1,0)CP1,

from Proposition D.1.9 we obtain

−KCP1 = LχδB = L⊗〈δB ,h

∨α 〉

χωα ,

where δB = α . Since Π+ = α , it follows that

δB = 〈δB,h∨α 〉ωα = 2ωα .

From the expression D.1.38 the SU(2)-invariant Kahler metric in the class c1(CP1) is given by

ωCP1 = 〈δB,h∨α 〉ηα = 2ηα ,

where ηα ∈ Ω(1,1) (CP1)SU(2) is determined by the function φωα : SL(2,C) → R dened by

φωα (д) =1π

log | |дv+ωα | |,

notice that Pic(CP1) = Z[ηα ]. From the cellular decomposition

XB = CP1 = N −x0 ∪ π( 0 1

−1 0

),we take the open set dened by the opposite big cell U = N −x0 ⊂ XB and the local sectionsU : U → SL(2,C) dened by

sU (nx0) = n, ∀n ∈ N −.

Now from the Gram-Schmidt process we have for every n ∈ N − the following expression

n =

1 0

z 1

= д√1 + |z |2 0

0(√

1 + |z |2)−1

1 z1+|z |2

0 1

,

from these we have φωα (n) = 1π log | |nv+ωα | | =

12π log(1 + |z |2). It follows that the local expres-

sion of ωCP1 is given by

ωCP1 =i

π∂∂ log(1 + |z |2).

If we denote by c1(CP1) = ωCP1 and consider the identication CP1 = S2, a straightforwardcalculation shows us that

239

χ (S2) =

∫CP1

c1(CP1) = 2,

as expected, here χ (S2) denotes the Euler characteristic of S2 = CP1. Notice that if we denoteby L⊗kχωα = O (k ), ∀k ∈ Z, we obtain the standard notation KCP1 = O (−2).

As we have seen in the last example the results which we described in this section provide asuitable constructive method to recover geometric ideas by means of the Lie-theoretical ob-jects.

Let us show how the construction explored in the above example works in a more generalcase. We point out that every step which we will describe in the next example for SL(n + 1,C)can be done for any complex simple Lie group, once we have essentially the same abstractstructural elements, namely root system, fundamental weights and Cartan matrices.

Example D.1.3. Consider nowGC = SL(n+1,C), as before we need to collect some basic datarelate to sl (n + 1,C), all the results which we will describe bellow can be found in [144], [87]and [29]. We rst x the Cartan subalgebra h ⊂ sl (n + 1,C) given by the diagonal matriceswhose trace is zero, from this we can take the simple root system

Σ =αl = ϵl − ϵl+1

∣∣∣ l = 1, . . . ,n

,

where ϵl : diaga1, . . . ,an+1 7→ al , ∀l = 1, . . . ,n + 1. All the information about the Lie algebrastructure of sl (n + 1,C) are encoded on its Dynkin diagram

Figure D.1: Dynkin diagram associated to sl (n + 1,C).

From the above comments we can consider the triangular decomposition

sl (n + 1,C) = n+ ⊕ h ⊕ n−,

where

n+ =⟨Eij

∣∣∣ 1 ≤ i < j ≤ n + 1⟩

Cand n− =

⟨Eij

∣∣∣ 1 ≤ j < i ≤ n + 1⟩

C.

We have for this case the following description for the fundamental weights ωα j , j = 1, . . . ,n,

ωα j = ϵ1 + . . . + ϵj , j = 1, . . . ,n.

Furthermore, we also have a complete description of the fundamental irreducible sl (n + 1,C)-modules, namely

V (ωα j ) =∧j (Cn+1),

240

for j = 1, . . . ,n. ese basic Lie-theoretical data allow us to describe for all P ⊂ SL(n+1,C) therst Chern class of XP , its Picard group, projective embedding, Chern class and many othersgeometric objects. Let us illustrate our last statement with an important standard complexKahler manifold. Consider Θ = Σ\α1, in this case we have PΣ\α1 = Pωα1

, from this forP = Pωα1

we obtain

XP = SL(n + 1.C)/Pωα1= CPn.

e irreducible representation associated to CPn is given by

V (ωα1 ) =∧1(Cn+1) = Cn+1,

in this case we have v+ωα = e1 = (1, 0, . . . , 0) ∈ Cn+1. We notice that from Proposition D.1.7 wehave

Pic(CPn ) = Z[ηα1],

where ηα1 ∈ Ω(1,1) (CPn )SU(n+1) is the SU(n + 1)-invariant form associated to the functionφωα1

: SL(n + 1,C) → R dened by

φωα1(д) =

1π

log | |дv+ωα | | =1π

log | |дe1 | |,

here | | · | | is the norm induced by the standard inner product of Cn+1. e projective embeddingis quite simple in this case, we have

CPn = XPωα1→ P(V(ωα1 )) = P(Cn+1).

Now we take the open set dened by the opposite big cell U = B−x0 ⊂ CPn, we notice that inthis case we have the open set parameterized by matrices n ∈ B− of the form

n =

1 0 · · · 0

z1 1 · · · 0....... . .

...

zn 0 · · · 1

,

with (z1, . . . , zn ) ∈ Cn. From this we can take a local section sU : U → SL(n + 1,C), such thatsU (nx0) = n ∈ SL(n + 1,C), thus we obtain

φωα1(sU (nx0)) = φωα1

(n) =1π

log | |ne1 | | =1

2π log | |ne1 | |2.

Hence, we have

φωα1(sU (nx0)) =

12π log

(1 +

n∑l=1|zl |

2)

.

241

which is the standard Kahler potential for the Fubini-Study metric. From Proposition D.1.9 itfollows that

−KCPn = LχδP = L⊗〈δP ,h

∨α1 〉

χωα1,

thus locally on U ⊂ CPn we can write

ωCPn = 〈δP ,h∨α1〉ηα =

〈δP ,h∨α1〉

2π i∂∂ log(

1 +n∑l=1|zl |

2)

.

As we have seen in the proof of eorem D.1.11 we have

ρ

2π − ωCPn = 0 ⇐⇒ Ric(ωCPn ) − 2πωCPn = 0,

therefore the Ricci form of CPn is locally given by

ρ = 〈δP ,h∨α1〉i∂∂ log

(1 +

n∑l=1|zl |

2)

.

In order to calculate the constant 〈δP ,h∨α1〉 we notice that

δP =∑

α∈Π+\〈Θ〉+

α ,

since Θ = Σ\α1 we have

Π+\〈Θ〉+ = k∑

l=1αl ∈ Π

+∣∣∣ 1 ≤ k ≤ n

.

erefore a simple inspection in the Cartan matrix of sl (n+1,C), see [87, p. 59] for a descriptionof the Cartan matrix in this case, shows us that

⟨ k∑l=1

αl ,h∨α1

⟩= 2 − 1 = 1, for 1 < k ≤ n,

and⟨α1,h

∨α1

⟩= 2. Hence, as expected we obtain

ρ = (n + 1)i∂∂ log(

1 +n∑l=1|zl |

2)

.

Here it is worth to compare our calculations with [126, p. 97] in order to see how the stan-dard geometric approach and the Lie-theoretical approach are well related. As in the previousexample, if we denote by

L⊗kχωα1= O (k ),

242

∀k ∈ Z, we obtain the standard notation KCPn = O (−n − 1) for the canonical bundle of CPn.

e main feature of the approach that we have explored throughout this chapter is that wecan use these ideas to give new interpretation for well known results as well as to establishnew ideas.

Remark D.1.12. It is worth to point out that from Proposition D.1.35 we have that for a complexag manifold XP , associated to P = PΘ ⊂ GC, its Fano index is given by

I (XP ) = gcd(〈δP ,h

∨α 〉

∣∣∣ α ∈ Σ\Θ).

We nish this subsection providing some basic consequences which the last remark allows tocalculate

eorem D.1.12 (Atiyah). Let X be a compact complex manifold. en there exists a bijectivecorrespondence between spin structures on X and holomorphic line bundles L ∈ Pic(X ) whichsatises L⊗2 = KX .

en we have the following result

eorem D.1.13. Let XP = GC/P be a complex ag manifold, then XP admits a G-invariantspin structure i its Fano index satises

I (XP ) = 0 mod 2 ⇐⇒ 〈δP ,h∨α 〉 = 0 mod 2,∀α ∈ Σ\Θ.

Furthermore, if such a structure exists on XP it is unique.

Proof. Let J be a G-invariant complex structure induced by GC on XP , then we have

w2(XP ) = c1(XP ) mod 2,

where w2(XP ) ∈ H 2(XP ,Z2) denotes the second Stiefel-Whitney class of XP and c1(XP ) ∈H 2(XP ,Z) denotes its rst Chern class. Form Proposition D.1.35 we have

c1(X ) = I (XP )(c1(XP )

I (XP )

),

thus if I (XP ) = 2`, for some ` ∈ N, it follows that KXP = L⊗2, where

L = K⊗ `I (XP )

XP.

e converse is a direct consequence of eorem D.1.12. Similarly, we have

〈δP ,h∨α 〉 = 0 mod 2, ∀α ∈ Σ\Θ ⇐⇒ ∃ L ∈ Pic(XP ) such that L⊗2 = KXP .

e uniqueness follows from the fact that π1(XP ) = 0.

243

D.2 Calabi ansatz technique onKahler-Einstein Fanoman-ifolds

In this section we will provide a complete description of how to construct Ricci-at metrics oncanonical bundles of Kahler-Einstein Fano manifolds. e main tool which we will apply inthis context is Calabi’s technique [26], which provides a constructive method to obtain Kahler-Einstein metrics on the total space of holomorphic vector bundles over Kahler-Einstein man-ifolds. e basic idea is to use the Hermitian vector bundle structure over a Kahler-Einsteinmanifold to reduce the Kahler-Einstein condition, which is generally a Monge-Ampere equa-tion, to an ordinary dierential equation [100]. e main result which we will cover in thesection below can be found in [143, p. 108, eorem 8.1], see also [100], [60] and [24].

D.2.1 Ricci-at Kahler metrics on canonical bundles ofKahler-Einstein Fano manifolds

e main task of this subsection will be to provide a proof for the following theorem

eorem D.2.1 (Calabi, [26]). Let (X ,ωX ) be a compact Kahler-Einstein manifold such thatc1(X ) > 0, i.e. a Kahler-Einstein Fano manifold, then there exists a complete Ricci-at metric onits canonical bundle KX =

∧(n,0) (X ).

As we have mentioned previously this result is part of a more general method of to constructKahler-Einstein metrics on total space of holomorphic vector bundles over Kahler-Einsteinmanifolds, the general ideas can be found in [26]. Although our notations are the same usedin [26], the proof that we will cover here is mainly based on [143, p. 108, eorem 8.1]. Werestrict the construction covered in [143] to the Kahler-Einstein Fano case because it is exactlythe context of complex ag manifolds, see Corollary D.1.10.

We start by seing some basic results related to canonical bundles, aer that we will providea proof for the above theorem.

Generalities on canonical bundles

Let (X ,ωX ) be a compact Kahler-Einstein manifold such that c1(X ) > 0, i.e. (X ,ωX ) is a Fanomanifold with Ric(ωX ) = tωX , for some constant t > 0. Consider the holomorphic line bundledened by the canonical bundle of (X ,ωX ), namely

π : KX → (X ,ωX ).

From this we take an open cover Uα α∈I of X which trivializes KX , through this open coverwe have biholomorphic maps

ψα : KX |Uα → Uα × C,

where KX |Uα = π−1(Uα ), ∀α ∈ I . We notice that by means of the trivialization Uα ,ψα α∈I we

can dene for each α ∈ I a locally nonvanishing section such that 7

7Notice that on the overlaps Uα ∩Uβ we have σβ = дα βσα .

244

σα : Uα → KX , σα (p) = ψ−1α (p, 1).

Since the restriction of ψα over the ber (KX )p is a C-linear isomorphism between (KX )p andp × C, we have the following characterization for the local coordinates in KX |Uα Uα × C,

(p, ξ ) := ψ−1α (p, ξ ) = ξσα (p).

Conversely, if we have a family of nonvanishing local sections σα : Uα → KX , α ∈ I , we candene local trivializationsψα : KX |Uα → Uα × C by

ψα : ξpσα (p) → (p, ξp ),

∀p ∈ Uα . In what follows we will deal with these two equivalent local descriptions of KX , see[156, p. 36] or [63, p. 66] for more details about the above comments.

From the above comments we x holomorphic coordinates z (α ) = (z (α )1 , . . . , z(α )n ) on Uα ⊂ X

for all α ∈ I , here we suppose dimC(X ) = n, and we denote by ξ (α ) the standard coordinate inC ⊂ Uα × C. Notice that these local coordinates in fact can be regarded as functions denedin KX |Uα , namely

z (α ) = pr1 ψ−1α and ξ (α ) = pr2 ψ

−1α ,

here prj denotes the natural projection in the j-factor of Uα × C, j = 1, 2.

Remark D.2.1. It is worthwhile to point out that by means of the local trivialization KX |Uα Uα × C we have a natural inclusion

π ∗ : Ω•(Uα ) → Ω•(KX |Uα ).

Hence, locally we will use in most cases the identication of Ω•(Uα ) with its image inside ofΩ•(KX |Uα ) by the pullback π ∗.

Now if we x a Hermitian structure H on KX , by means of the basic data which we havedescribed above we can write locally the associated Hermitian connection ∇ as follows

∇(α ) = d + ∂ log(H (σα ,σα )

)= d +Aα ,

here ∇(α ) = ∇|Uα and Aα = ∂ log(H (σα ,σα )

)∈ Ω(1,0) (Uα ).

Remark D.2.2. Without loss of generality we can suppose σα : U → KX unitary, ∀α ∈ I , thuswe have

0 = d log(H (σα ,σα )

)= Aα +Aα . (D.2.1)

erefore, we can suppose Aα purely imaginary, ∀α ∈ I , this procedure is equivalent to considerthe reduction of the structural group C× of KX to U(1).

In general a Hermitian structure is locally dened by

245

H ((z (α ), ξ (α ) ), (z (α ), ξ (α ) )) = a(z (α ), z (α ) ) |ξ (α ) |2

where a : Uα → R>0, see for instance [26, p. 271]. We will x on KX the Hermitian structureinduced by the Kahler metric дX associated to the Kahler form ωX , namely

H ((z (α ), ξ (α ) ), (z (α ), ξ (α ) )) =1

det(дX )|ξ (α ) |2. (D.2.2)

From this the curvature associated to the above Hermitian structure satises

F∇ = −∂∂ logH = ∂∂ log(det(дX )) = itωX

here we use the fact that ρ = −i∂∂ log(det(дX )) = tωX . Hence, locally we have

iF∇ = idAα = −tωX . (D.2.3)

see for example [126, p. 120, Proposition 17.4]. In order to analyse the covariant derivative ofa local section σ ′ ∈ Γ(Uα ,KX ) we notice that

σ ′(z (α ) ) = (z (α ), fσ ′ (z(α ) )) = fσ ′ (z

(α ) )σα (zα ) = ξ (α ) (σ ′(z (α ) ))σα (z

α ),

here ξ (α ) σ ′ = fσ ′ : Uα → C is a smooth function. us we obtain

∇(σ ′) = ∇((ξ (α ) σ ′)σα ) =(d (ξ (α ) σ ′) + (ξ (α ) σ ′)Aα

)⊗ σα .

From the above calculations we notice that adapted to the local geometry of KX we have alocally dened (1, 0)-form, which we denote by ∇ξ (α ) ∈ Ω(1,0) (π−1(Uα )), given by 8

∇ξ (α ) := dξ (α ) + ξ (α )Aα . (D.2.4)

erefore, instead of considering the standard local dual basis(dz (α )1 , . . . ,dz

(α )n ;dξ (α )

)and

(∂z(α )1, . . . , ∂

z(α )n

; ∂ξ (α ))

,

which respectively (locally) trivializes (T (1,0)KX )∗ and T (1,0)KX , we will work with the local

dual basis (dz (α )1 , . . . ,dz

(α )n ;∇ξ (α )

)and

(∇zα1 , . . . ,∇zα1 ; ∂ξ (α )

), (D.2.5)

see for instance [26, p. 272], where

∇zαj := ∂z(α )j− ξ (α )

1H

∂H

∂z (α )j

∂ξ (α ) ,

8Here our notation are according to [26], dierent from [143] we take Aα instead of π ∗Aα , see Remark D.2.1for more details.

246

∀j = 1, . . . ,n. It is worth to observe that on the above expression we consider H = H (σα ,σα ).

We denote by (z (α ), ξ (α ) ) the local coordinates in KX |Uα described above, ∀α ∈ I , on the changeof coordinates Uα ∩Uβ , ∅ we have the following facts

• (ψα ψ−1β ) (z (β ), ξ (β ) ) = (z (α ),дαβξ

(β ) ) = (z (β ), ξ (α ) ) =⇒ ξ (α ) = дαβξ(β ) , where

дαβ : Uα ∩Uβ → U(1) ⊂ C×,

see for example [63, p. 132] or [166, p. 105]. Notice that here we consider the reductionof the structural group C× to U(1) provided by the Hermitian structure.

• Since σβ = дαβσα , the Hermitian connection ∇ on the overlaps satises

Aα = Aβ − д−1αβdдαβ .

Actually, it follows from the equation ∇( f σβ ) = ∇( f дαβσα ), see for instance [63, p. 141].

From the above two facts on the transitions Uα ∩Uβ we have

∇ξ (α ) = dξ (α ) + ξ (α )Aα = d (дαβξ(β ) ) + дαβξ

(β ) (Aβ − д−1αβdдαβ ).

Since d (дαβξ (β ) ) = ξ (β )dдαβ + дαβdξ (β ) , the above equation becomes

∇ξ (β ) = дαβ∇ξ(α ), (D.2.6)

thus if we consider the (0, 1)- forms ∇ξ (α ) , ∀α ∈ I . e above computation shows that we cangluing the locally dened (1, 1)-forms

∇ξ (α ) ∧ ∇ξ (α ) ,

in order to obtain a globally dened (1, 1)-form ∇ξ ∧ ∇ξ which satises

∇ξ ∧ ∇ξ = ∇ξ (α ) ∧ ∇ξ (α ) , on KX |Uα = π−1(Uα ).

In order to see that the above (1, 1)-form is well dened we notice that on the transitionsUα ∩Uβ we have

∇ξ ∧ ∇ξ = ∇ξ (α ) ∧ ∇ξ (α ) =(дαβ∇ξ

(β ))∧(дαβ∇ξ (β )

),

since |дαβ | = 1, we have

∇ξ ∧ ∇ξ = ∇ξ (α ) ∧ ∇ξ (α ) = ∇ξ (β ) ∧ ∇ξ (β ) .

247

Hence, we obtain a well dened (1, 1)-form ∇ξ ∧ ∇ξ ∈ Ω(1,1) (KX ).

e main idea to construct a Ricci-at metric on KX is to work with the horizontal and vertical(1, 1)-forms, respectively, given by

π ∗ωX and ∇ξ ∧ ∇ξ .

We need to introduce one more element which will be fundamental in your explanation. Con-sider the locally dened smooth function given by the correspondence 9

u : KX |Uα → R, (z (α ), ξ (α ) ) → |ξ (α ) |2,

this function is in fact globally dened. In order to see that there is no ambiguity on its de-nition, we notice that on the overlaps Uα ∩Uβ we have

u (z (α ), ξ (α ) ) = |ξ (α ) |2 = |дαβξ(β ) |2 = |дαβ |

2 |ξ (β ) |2 = |ξ (β ) |2 = u (z (β ), ξ (β ) ),

thus we have a globally dened function u : KX → R≥0.

Given a smooth function f : R→ R, our next step will be investigate how to obtain a Ricci-atmetric on KX given by a (1, 1)-form with the following characterization

ω0 = f (u)π ∗ωX −1tf ′(u)i∇ξ ∧ ∇ξ , (D.2.7)

here t > 0 denotes the constant which satises Ric(ωX ) = tωX , and f ′ denotes the derivativewith respect to the parameter u of the function f .

Ricci-at metrics on canonical bundles via Calabi ansatz

Consider ω0 ∈ Ω(1,1) (KX ) as before, the rst thing we notice is that this form satises thefollowing condition

ω0(∇z(α )i,∇

z(α )j

)= f (u)π ∗ωX

(∂z(α )i, ∂

z(α )j

), ω0

(∇z(α )i, ∂

ξ (α )

)= 0.

Furthermore, we have

ω0(∂ξ (α ) , ∂ξ (α )

)= −

1tf ′(u)idξ (α ) ∧ dξ (α )

(∂ξ (α ) , ∂ξ (α )

),

therefore from the above comment if we take f : R→ R such that

f > 0 and f ′ > 0,

we have thatω0 ∈ Ω(1,1) (KX ) denes a nondegenerate positive form. In fact we notice that the

matrix associated to ω0 with respect to the basis D.2.2 can be wrien as9is function is just the square of the radial parameter in the vertical direction, we can also dene u : KX →

R≥0, directly by u ([p, ξ ]) = |ξ |2, here we used that KX =⊔α

Uα × C/ ∼, see [166, p. 105].

248

ω0 =

f (u)ωX 0

0 −1t f′(u)i∇ξ ∧ ∇ξ

.

Now in order to show that the above form is actually a Kahler form we need to show that ω0is closed. Consider the locally dened 1-form

η = −i

t

(f Aα −

f dξ (α )

ξ (α )

),

see for instance, [60, p. 331], we have

dη = −i

t

[d f ∧Aα + f dAα − d

( f

ξ (α )

)∧ dξ (α )

].

We observe that

du = d (ξ (α )ξ (α ) ) = ξ (α )dξ (α ) + ξ (α )dξ (α ) = ξ (α )∇ξ (α ) + ξ (α )∇ξ (α ) ,

on the right side of the last equality above we used the following

dξ (α ) = ∇ξ (α ) − ξ (α )Aα and dξ (α ) = ∇ξ (α ) + ξ (α )Aα ,

here we suppose Aα = −Aα , see Remark D.2.2. From these a straightforward computationshows us that

d f ∧Aα =d f

dudu ∧Aα =

d f

du

(∇ξ (α ) ∧ dξ (α ) − ∇ξ (α ) ∧ ∇ξ (α ) +

ξ (α )

ξ (α )∇ξ (α ) ∧ dξ (α )

),

and we also have

d( f

ξ (α )

)∧ dξ (α ) =

1ξ (α )

d f

dudu ∧ dξ (α ) =

d f

du∇ξ (α ) ∧ dξ (α ) +

d f

du

ξ (α )

ξ (α )∇ξ (α ) ∧ dξ (α ) .

erefore the expression of dη becomes

dη = −i

t

[f dAα +

d f

du∇ξ (α ) ∧ ∇ξ (α )

],

since − itdAα = ωX , see Equation D.2.3, we obtain

dη = ω0.

Now we will analyse the conditions over the Kahler form

ω0 = f (u)π ∗ωX −1tf ′(u)i∇ξ ∧ ∇ξ ,

249

in order to obtain Ric(ω0) = 0.

Consider the (n, 0)-tautological form θ ∈ Ω(n,0) (KX ) dened by

θν (X1, . . . ,Xn ) = ν (π∗X1, . . . ,π∗Xn ),

for ν ∈ KX and X1, . . . ,Xn ∈ T(1,0)ν KX . From the above denition of θ since KX =

∧(n,0) (X ) wehave

θν (X1, . . . ,Xn ) = π∗(ν ) (X1, . . . ,Xn ),

from this locally we can write ν = ξ (α ) (ν )σα (π (ν )) which follows that

θν = ξ(α ) (ν )π ∗(σα (π (ν ))).

us on KX |Uα we have the expression θ = ξ (α )π ∗(σα ). Now since dimC(KX ) = n+1, by takingthe exterior derivative of θ we obtain

dθ = dξ (α ) ∧ π ∗(σα ) + ξ(α )dπ ∗(σα ).

Since d (π ∗(σα ) = π ∗(dσα ) we have the following facts:

• σα ∈ Γ(Uα ;∧(n,0) (X )) =⇒ ∂σα = 0;

• σα is holomorphic (∂ = ∇(0,1)) =⇒ ∂σα = 0 10 .

From the above comments we have dθ ∈ Ω(n+1,0) (KX ) which locally is given by

dθ = dξ (α ) ∧ π ∗(σα ).

We denote by Ω = dθ and notice that

dΩ = 0 and ∂Ω = 0,

thus we have a closed and holomorphic (n + 1, 0)-form dened on KX .

Now we analyse the conditions over ω0 in order to obtain a Ricci-at metric in KX . We rstconsider the norm | | · | |ω0 on the canonical bundle

∧(n+1,0) (KX ), induced by the Kahler metricд0 associated to ω0. From this we have

(n + 1)!Ω ∧ Ω = i (n+1)2 | |Ω | |2ω0ωn+10 , (D.2.8)

see for instance [143, p. 108] or [126, p. 126]. Now a straightforward calculation shows us that

ωn+10 = −

i

tf (u)n f ′(u)π ∗(ωn

X ) ∧ ∇ξ ∧ ∇ξ ,

10We can conclude the same fact if we take σα unitary instead holomorphic. We take σα holomorphic in orderto provide a more direct argument to get the desired expression.

250

locally the above expression becomes

ωn+10 = −

i

tf (u)n f ′(u)ωn

X ∧ ∇ξ(α ) ∧ ∇ξ (α ) ,

here we have used the inclusion Ω•(Uα ) ⊂ Ω•(KX |Uα ), see Remark D.2.1. From these we noticethat since ∇ξ (α ) = dξ (α ) + ξ (α )Aα , it follows that 11

ωnX ∧ ∇ξ

(α ) ∧ ∇ξ (α ) = ωnX ∧ dξ

(α ) ∧ dξ (α ) ,

thus we obtainωn+1

0 = −i

tf (u)n f ′(u)ωn

X ∧ dξ(α ) ∧ dξ (α ) . (D.2.9)

Now let us look at the expression of Ω ∧ Ω, locally we have

Ω ∧ Ω =(dξ (α ) ∧ σα

)∧(dξ (α ) ∧ σα

),

here again we have used the inclusion Ω(n,0) (Uα ) ⊂ Ω(n,0) (KX |Uα ) in order to denote π ∗σα justby σα . A straightforward calculation shows us that

(dξ (α ) ∧ σα

)∧(dξ (α ) ∧ σα

)= (−1)n

(σα ∧ σα

)∧(dξ (α ) ∧ dξ (α )

),

here we have used the identity a ∧b = (−1)deg(a)deg(b)b ∧a for dierential forms a and b. Nowwe notice that since σα ∈ Ω(n,0) (Uα ) we have

σα ∧ σα =in

2

n! | |σα | |2ωXωnX ,

where | | · | |ωX denotes the norm induced by the Kahler metric associated to ωX , see D.2.2.erefore, if we take σα unitary instead of holomorphic, namely | |σα | |ωX = 1, we obtain

Ω ∧ Ω =(−1)nin2

n! ωnX ∧ dξ

(α ) ∧ dξ (α ) . (D.2.10)

By means of the expression D.2.10, on the le side of the Equation D.2.8 we have

(n + 1)!Ω ∧ Ω = (n + 1) (−1)nin2ωnX ∧ dξ

(α ) ∧ dξ (α ) .

rough of the expression D.2.1 the right side of the Equation D.2.1 can be wrien as

i (n+1)2 | |Ω | |2ω0ωn+10 =

(−1)nin2

tf (u)n f ′(u) | |Ω | |2ω0ω

nX ∧ dξ

(α ) ∧ dξ (α ) ,

11Notice that − it dAα = ωX and ∇ξ (α ) ∧ ∇ξ (α ) = dξ (α ) ∧ dξ (α ) + Aα ∧ (ξ αdξ

α+ ξ

αdξ α ), here we consider

Aα = −Aα .

251

thus the Equation D.2.1 is equivalent to

f (u)n f ′(u) | |Ω | |2ω0 = (n + 1)t . (D.2.11)

e above equation tells us what are the conditions over the functions f > 0 and f ′ > 0 weneed to obatin | |Ω | |ω0 constant. Since | |Ω | |ω0 is constant if and only if Ω is parallel, see forinstance [16, p. 81, Proposition 2.90], we need to seek for such a function f which satises

f > 0, f ′ > 0, f (u)n f ′(u) = const..

We consider the family of functions of the form

f (u) = (tu +C )1

n+1 ,

where C is a positive constant (C > 0). We have the following expression for f ′(u)

f ′(u) =t

n + 1 (tu +C )− nn+1 ,

the above formula show us that

f (u)n f ′(u) =t

n + 1 =⇒ ||Ω | |2ω0 = (n + 1)2, (D.2.12)

when we consider the Kahler form dened by

ω0 = (tu +C )1

n+1π ∗ωX −1

n + 1 (tu +C )− nn+1 i∇ξ ∧ ∇ξ , (D.2.13)

with C > 0. erefore if we denote by ∇ω0 the Hermitian connection on TKX induced by theKahler metric associated to ω0, we have ∇ω0Ω = 0. Hence

Hol0(∇ω0 ) ⊆ SU(n + 1),

see [94, p. 123], where Hol0(∇ω0 ) denotes the component of the identity of the holonomy groupHol(∇ω0 ) 12. From the above comments we conclude that (KX ,ω0) is a noncompact Ricci-atmanifold with holomorphic volume form given by Ω = dθ .

e Ricci-at metric associated to (KX ,ω0,Ω) is given by

д0 = (tu +C )1

n+1π ∗дX +1

n + 1 (tu +C )− nn+1 Re

(∇ξ ⊗ ∇ξ

), (D.2.14)

since we suppose (X ,ωX ) compact we have that (X ,дX ) is a complete Riemannian manifold.It will be important for us in our next step which is to verify that (KX ,д0) is a complete Rie-mannian manifold.

In order to check that the metric д0 dened above is complete, we consider the followingcharacterization of complete Riemannian manifolds

12Notice that if KX is simply connected we have Hol0 (∇ω0 ) = Hol(∇ω0 ), see [14, p. 8, Proposition 2.3], it willbe exactly the case of the canonical bundle of complex ag manifolds.

252

Denition D.2.2. A divergent path on a Riemannian manifold (M,д) is a continuous curveγ : [0,a) → (M,д), 0 < a ≤ +∞, such that for any compact subset K ⊂ M , there is a numbers0(K ) ∈ [0,a) such that γ (s ) ∈ M\K for every s > s0(K ).

Proposition D.2.3. A Riemannian manifold (M,д) is complete if and only if every divergentC1-path γ : [0,a) → (M,д), 0 < a ≤ +∞, has innite length.

Proof. e proof can be found in [40, p. 179, Proposition 1].

From the above result, if we take a divergentC1-path γ : [0,a) → (KX ,д0), we have essentiallytwo possibilities:

1. π γ : [0,a) → (X ,дX ) is a divergent C1-path (horizontal divergence);

2. π γ : [0,a) → (X ,дX ) is not a divergent C1-path (vertical divergence).

In the rst case above the completeness of (X ,дX ) shows us that γ : [0,a) → (KX ,д0) hasinnite length, in the second case the divergence of the integral 13

∫ +∞0

1(tu +C )

n2(n+1)

d (u12 ),

tells us that the curve γ : [0,a) → (KX ,д0) has innite length on the vertical radial direction,i.e. ifγ is a divergent curve on the vertical radial direction it has innite length, thus the metricD.2.14 is complete. Hence, we have the result stated in eorem D.2.1.

From eorem D.2.1 we also have the following direct consequence. If we consider the iden-tication of X with the 0-section of KX , namely

X ν ∈ KX

∣∣∣ u (ν ) = 0⊂ KX ,

we can take the restriction of д0 over X ⊂ KX . From this we obtain

д0 |X = C1

n+1π ∗дX ,

see D.2.14. erefore we have the following result

Corollary D.2.4. Let (X ,ωX ) be a Kahler-Einstein fano manifold, then X can be embedded as atotally geodesic submanifold of the Ricci-at manifold (KX ,ω0).

13It is worthwhile to point out that if we consider polar coordinates on the vertical direction, namely ξ = reiϕ ,it follows that u : KX → R≥0 satises u ([p, reiϕ]) = r 2, moreover we can suppose that γ : [0,a) → (KX ,д0) isa divergent path “constant” on the horizontal direction, from this a straighforward calculation shows us that∫ a

0 |γ |ds =∫ a

0√

f ′(r (s )2) drds ds =∫ r (a)r (0) (tr

2 +C )−n

2(n+1)dr

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