DOCSLIB.ORG

Localization and Standard Modules for Real Semisimple Lie Groups Ii: Irreducibility, Vanishing Theorems and Classification

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Localization and Standard Modules for Real Semisimple Lie Groups Ii: Irreducibility, Vanishing Theorems and Classification LOCALIZATION AND STANDARD MODULES FOR REAL SEMISIMPLE LIE GROUPS II: IRREDUCIBILITY, VANISHING THEOREMS AND CLASSIFICATION HENRYK HECHT, DRAGAN MILICIˇ C,´ WILFRIED SCHMID, AND JOSEPH A. WOLF Contents 1. Introduction 1 2. Generalities on intertwining functors 1 3. Supports and n-homology 13 4. Calculations for sl(2; C) 20 5. Some results on root systems with involution 32 6. K-orbits in the flag variety 38 7. Intertwining functors and standard Harish-Chandra sheaves 48 8. Irreducibility of standard Harish-Chandra sheaves 57 9. Geometric classification of irreducible Harish-Chandra modules 65 10. Decomposition of global sections of standard Harish-Chandra sheaves 71 11. n-homology of Harish-Chandra modules 75 12. Tempered Harish-Chandra modules 79 13. Quasispherical Harish-Chandra modules 81 References 84 1. Introduction 2. Generalities on intertwining functors ∗ Let θ be a Weyl group orbit in h . We consider the category M(Uθ) of Uθ- modules. For each λ 2 θ we also consider the category Mqc(Dλ) of (quasicoherent) Dλ-modules. Assigning to a Dλ-module V its global sections Γ(X; V) defines a functor Γ : Mqc(Dλ) −! M(Uθ). Its left adjoint is the localization functor ∆λ : M(Uθ) −! Mqc(Dλ) given by ∆λ(V ) = Dλ ⊗Uθ V . Let Σλ be the set of roots integral with respect to λ, i.e., Σλ = fα 2 Σ j αˇ(λ) 2 Zg: Then the subgroup Wλ of the Weyl group W generated by the reflections with respect to the roots from Σλ is equal to Wλ = fw 2 W j wλ − λ 2 Q(Σ)g; ∗ where Q(Σ) is the root lattice of Σ in h ([7], Ch. VI, §2, Ex. 2). Let Πλ be the set of simple roots in the root system Σλ attached to the set of positive roots + + Σλ = Σλ \ Σ . Denote by `λ : Wλ −! Z+ the corresponding length function. 2020 Mathematics Subject Classification. Primary 22E46. 1 2 H. HECHT, D. MILICIˇ C,´ W. SCHMID, AND J. A. WOLF We say that λ is antidominant if αˇ(λ) is not a strictly positive integer for any α 2 Σ+. For arbitrary λ we define n(λ) = minf`λ(w) j wλ is antidominant, w 2 Wλg: The following result was established in [4] and [11]. Theorem 2.1. Let λ 2 h∗ and θ = W · λ. Then (i) The right cohomological dimension of Γ: Mqc(Dλ) −! M(Uθ) is ≤ n(λ). (ii) The left cohomological dimension of ∆λ : M(Uθ) −! Mqc(Dλ) is finite if and only if λ is regular. (iii) If λ is regular, the left cohomological dimension of ∆λ is ≤ n(λ). Consider the derived category D(Uθ) of complexes of Uθ-modules and the de- rived category D(Dλ) of complexes of Dλ-modules. By (i), there exists the de- rived functor RΓ: D(Dλ) −! D(Uθ). This functor also induces functors be- tween the corresponding full subcategories of bounded complexes. On the other hand, for arbitrary λ, there exists also the derived functor of localization functor − − L∆λ : D (Uθ) −! D (Dλ) between derived categories of complexes bounded from above. If λ is regular, the left cohomological dimension of ∆λ is finite by (ii), and L∆λ extends to the derived functor between D(Dλ) and D(Uθ). Moreover, it maps bounded complexes into bounded complexes. We have the following result [4]. ∗ b b Theorem 2.2. Let λ 2 h be regular and θ = W ·λ. Then RΓ: D (Dλ) −! D (Uθ) b b and L∆λ : D (Uθ) −! D (Dλ) are mutually quasiinverse equivalences of categories. b This implies, in particular, that for any two λ, µ 2 θ, the categories D (Dλ) and b D (Dµ) are equivalent. This equivalence is given by the functor L∆µ ◦ RΓ from b b D (Dλ) into D (Dµ). In this section we describe a functor, defined in geometric terms, which is (under certain conditions) isomorphic to this functor. This is the intertwining functor of Beilinson and Bernstein [4]. Most of the following results on the intertwining functors are due to Beilinson and Bernstein and were announced in [4], [2]. Complete details can be found in [14]. We start with some geometric remarks. Define the action of G = Int(g) on X ×X by g · (x; x0) = (g · x; g · x0) for g 2 G and (x; x0) 2 X × X. The G-orbits in X × X can be parametrized in the following way. First we introduce a relation between Borel subalgebras in g. Let b and b0 be two Borel subalgebras in g, n and n0 their nilpotent radicals and N and N 0 the corresponding subgroups of G. Let c be a Cartan subalgebra of g contained in b \ b0. Denote by R the root system of (g; c) in c∗ and by R+ the set of positive roots determined by b. This determines a specialization of the Cartan triple (h∗; Σ; Σ+) into (c∗; R; R+) [12]. On the other hand, b0 determines another set of positive roots in R, which corresponds via this specialization to w(Σ+) for some uniquely determined w 2 W . The element w 2 W does not depend on the choice of c, and we say that b0 is in relative position w with respect to b. Let 0 Zw = f(x; x ) 2 X × X j bx0 is in the relative position w with respect to bxg IRREDUCIBILITY, VANISHING THEOREMS AND CLASSIFICATION 3 for w 2 W . Then the map w −! Zw is a bijection of W onto the set of G-orbits in X × X, hence the sets Zw, w 2 W , are smooth subvarieties of X × X. Denote by p1 and p2 the projections of Zw onto the first and second factor in the product X × X, respectively. The fibrations pi : Zw −! X, i = 1; 2, are locally trivial with fibres isomorphic to `(w)-dimensional affine spaces. Hence, they are are affine morphisms. Let ΩZwjX be the invertible OZw -module of top degree relative differential forms for the projection p1 : Zw −! X. Let Tw be its inverse. Since the tangent space at 0 (x; x ) 2 Zw to the fibre of p1 can be identified with nx=(nx \ nx0 ), and ρ − wρ is the sum of roots in Σ+ \ (−w(Σ+)), we see that ∗ Tw = p1(O(ρ − wρ)): It is easy to check that p1 p2 Tw (Dwλ) = (Dλ ) ([12], Appendix A). Since the morphism p2 : Zw −! X is a surjective submersion, + p2 p2 is an exact functor from Mqc(Dλ) into Mqc((Dλ) ). Twisting by Tw defines an exact functor V −! T ⊗ p+(V) from M(D ) into M ((D )p1 ) ([12], A.3.3.1). w OZw 2 λ qc wλ Therefore, we have a functor V· −! T ⊗ p+(V·) from Db(D ) into Db((D )p1 ). w OZw 2 λ wλ b p1 b Composing it with the direct image functor Rp1+ : D ((Dwλ) ) −! D (Dwλ), we b b get the functor Jw : D (Dλ) −! D (Dwλ) by the formula J (V·) = Rp (T ⊗ p+(V·)) w 1+ w OZw 2 · b for any V 2 D (Dλ). Let V 2 Mqc(Dλ). Since p1 is an affine morphism with i `(w)-dimensional fibres, it follows that H (Jw(D(V))) vanishes for i < −`(w) and i > 0. Moreover, the functor I (V) = R0p (T ⊗ p+(V)) w 1+ w OZw 2 from Mqc(Dλ) into Mqc(Dwλ) is right exact. This is the intertwining functor (attached to w 2 W ) between Mqc(Dλ) and Mqc(Dwλ). One knows that Jw is actually the left derived functor LIw of Iw ([4], [14]); moreover, ∗ b b Proposition 2.3. Let w 2 W and λ 2 h . Then LIw = Jw : D (Dλ) −! D (Dwλ) is an equivalence of categories. For a weight ν 2 P (Σ) we denote by O(ν) the corresponding homogeneous in- vertible OX -module. From the construction of the intertwining functors one can easily check that they behave nicely with respect to twists by homogeneous invert- ible OX -modules: Lemma 2.4. Let w 2 W , λ 2 h∗ and ν 2 P (Σ). Then · · LIw(V (ν)) = LIw(V )(wν) · b for any V 2 D (Dλ). Intertwining functors satisfy a natural \product formula". To formulate it we need some additional geometric information on G-orbits in X × X. Let w; w0 2 0 0 W . Denote by p1 and p2 the projections of Zw into X, and by p1 and p2 the corresponding projections of Zw0 into X. Let Zw0 ×X Zw be the fibre product of Zw0 0 0 and Zw with respect to the morphisms p2 and p1. Denote by q : Zw0 ×X Zw −! Zw0 and q : Zw0 ×X Zw −! Zw the corresponding projections to the first, resp. second 0 0 factor. Finally, the morphisms p1 ◦q : Zw0 ×X Zw −! X and p2 ◦q : Zw0 ×X Zw −! 4 H. HECHT, D. MILICIˇ C,´ W. SCHMID, AND J. A. WOLF X determine a morphism r : Zw0 ×X Zw −! X×X. Therefore, we have the following commutative diagram. X × X O r Zw0 ×X Zw q0 q { # Zw0 Zw 0 p 1 p2 0 p1 p 2 ~ XXX$ { All morphisms in the diagram are G-equivariant. From the construction it follows that the image of r is contained in Zw0w, and by the G-equivariance of r it is a 0 surjection of Zw0 ×X Zw onto Zw0w. Assume in addition that w; w 2 W are such 0 0 that `(w w) = `(w ) + `(w). Then r : Zw0 ×X Zw −! Zw0w is an isomorphism.
Load more

Recommended publications
  • Hyperpolar Homogeneous Foliations on Symmetric Spaces of Noncompact Type
    HYPERPOLAR HOMOGENEOUS FOLIATIONS ON SYMMETRIC SPACES OF NONCOMPACT TYPE JURGEN¨ BERNDT Abstract. A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jos´eCarlos D´ıaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in [1], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SLr+1(R)/SOr+1. 1. Isometric actions on the real hyperbolic plane The special linear group a b G = SL2(R) = a, b, c, d ∈ R, ad − bc = 1 c d acts on the upper half plane H = {z ∈ C | =(z) > 0} by linear fractional transformations of the form a b az + b · z = . c d cz + d By equipping H with the Riemannian metric dx2 + dy2 dzdz¯ ds2 = = (z = x + iy) y2 =(z)2 we obtain the Poincar´ehalf-plane model of the real hyperbolic plane RH2. The above action of SL2(R) is isometric with respect to this metric. The action is transitive and the stabilizer at i is cos(s) sin(s) K = s ∈ R = SO2.
    [Show full text]
  • Invariant Differential Operators for Quantum Symmetric Spaces, I
    Invariant Differential Operators for Quantum Symmetric Spaces, I Gail Letzter∗ Mathematics Department Virginia Tech Blacksburg, VA 24061 [email protected] January 1, 2018 Abstract This is the first paper in a series of two which proves a version of a theorem of Harish-Chandra for quantum symmetric spaces in the maximally split case: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential op- erators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Here, we establish this result for all quantum symmetric spaces defined using irreducible sym- metric pairs not of type EIII, EIV, EVII, or EIX. A quantum version of a related theorem due to Helgason is also obtained: The image of the center under this Harish-Chandra map is the entire invariant ring arXiv:math/0406193v1 [math.QA] 9 Jun 2004 if and only if the underlying irreducible symmetric pair is not one of these four types. Introduction In [HC2, Section 4], Harish-Chandra proved the following fundamental result for semisimple Lie groups: the Harish-Chandra map induces an isomorphism ∗supported by NSA grant no. MDA904-03-1-0033. AMS subject classification 17B37 1 between the ring of invariant differential operators on a symmetric space and invariants of an appropriate polynomial ring under the restricted Weyl group. When the symmetric space is simply a complex semisimple Lie group, this result is Harish-Chandra’s famous realization of the center of the enveloping algebra of a semisimple Lie algebra as Weyl group invariants of the Cartan subalgebra ([HC1]).
    [Show full text]
  • The Absolute Continuity of Convolution Products of Orbital Measures In
    The Absolute continuity of convolution products of orbital measures in exceptional symmetric spaces K. E. Hare and Jimmy He Abstract. Let G be a non-compact group, let K be the compact subgroup fixed by a Cartan involution and assume G/K is an exceptional, symmetric space, one of Cartan type E,F or G. We find the minimal integer, L(G), such that any convolution product of L(G) continuous, K-bi-invariant measures on G is absolutely continuous with respect to Haar measure. Further, any product of L(G) double cosets has non-empty interior. The number L(G) is either 2 or 3, depending on the Cartan type, and in most cases is strictly less than the rank of G. 1. Introduction In [18] Ragozin showed that if G is a connected simple Lie group and K a compact, connected subgroup fixed by a Cartan involution of G, then the convo- lution of any dim G/K, continuous, K-bi-invariant measures on G is absolutely continuous with respect to the Haar measure on G. This was improved by Graczyk and Sawyer in [8], who showed that when G is non-compact and n = rank G/K, then the convolution product of any n + 1 such measures is absolutely continuous. They conjectured that n + 1 was sharp with this property. This conjecture was shown to be false, in general, for the symmetric spaces of classical Cartan type in [11] (where it was shown that rank G/K suffices except when the restricted root system is type An).
    [Show full text]
  • The Absolute Continuity of Convolutions of Orbital Measures in Symmetric Spaces
    THE ABSOLUTE CONTINUITY OF CONVOLUTIONS OF ORBITAL MEASURES IN SYMMETRIC SPACES SANJIV KUMAR GUPTA AND KATHRYN E. HARE Abstract. We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characteri- zation can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rankG/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rankG/K +1 is needed. 1. Introduction In this paper, we study the smoothness properties of K-bi-invariant measures on the irreducible Riemannian symmetric spaces G/K, where G is a connected Lie group and K is a compact, connected subgroup fixed by a Cartan involution of G. Inspired by earlier work of Dunkl [6] on zonal measures on spheres, Ragozin in [27] and [28] showed that the convolution of any dim G/K, continuous, K-bi-invariant measures on G was absolutely continuous with respect to the Haar measure on G. This was improved by Graczyk and Sawyer, who in [10] showed that if G was non-compact and n = rankG/K, then any n + 1 convolutions of such measures is absolutely continuous and that this is sharp for the symmetric spaces whose restricted root system was type An.
    [Show full text]
  • View Technical Report
    Available at: http://www.ictp.it/~pub− off IC/2007/015 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS INVARIANT DIFFERENTIAL OPERATORS FOR NON-COMPACT LIE GROUPS: PARABOLIC SUBALGEBRAS V.K. Dobrev Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract In the present paper we start the systematic explicit construction of invariant differen- tial operators by giving explicit description of one of the main ingredients - the cuspidal parabolic subalgebras. We explicate also the maximal parabolic subalgebras, since these are also important even when they are not cuspidal. Our approach is easily generalised to the supersymmetric and quantum group settings and is necessary for applications to string theory and integrable models. MIRAMARE – TRIESTE March 2007 1 1. Introduction Invariant differential operators play very important role in the description of physical sym- metries - starting from the early occurrences in the Maxwell, d’Allembert, Dirac, equations, (for more examples cf., e.g., [1]), to the latest applications of (super-)differential operators in conformal field theory, supergravity and string theory, (for a recent review, cf. e.g., [2]). Thus, it is important for the applications in physics to study systematically such operators. In the present paper we start with the classical situation, with the representation theory of semisimple Lie groups, where there are lots of results by both mathematicians and physi- cists, cf., e.g. [3-38]. We shall follow a procedure in representation theory in which such operators appear canonically [22] and which has been generalized to the supersymmetry setting [39] and to quantum groups [40].
    [Show full text]
  • Dynamical R-Matrices for Calogero Models
    Nuclear Physics B 621 [PM] (2002) 523–570 www.elsevier.com/locate/npe Dynamical R-matrices for Calogero models Michael Forger 1, Axel Winterhalder 2 Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, BR-05315-970 São Paulo, SP, Brazil Received 17 July 2001; accepted 18 September 2001 Abstract We present a systematic study of the integrability of the Calogero models, degenerate as well as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras, where “integrability” is understood to encompass not only the existence of a Lax representation for the equations of motion but also the—more far-reaching—existence of a (dynamical) R-matrix. Using the standard group-theoretical machinery available in this context, we show that integrability of these models, in this sense, can be reduced to the existence of a certain function, denoted here by F , defined on the relevant root system and taking values in the respective Cartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, all given by explicit formulas. We also show that among the simple Lie algebras, only those belonging to the A-series admit a solution of these constraints, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p, q)/S(U(p) × U(q)), allows non-trivial solutions when |p − q| 1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our method provides new ones, namely for the degenerate models when |p − q|=1 and for the elliptic models when |p − q|=1orp = q.
    [Show full text]
  • On the Classification of Anosov Flows
    Topology Vol. 14. pp 179-189 Pergamon Press. 197! Prmkd in Great Brltam ON THE CLASSIFICATION OF ANOSOV FLOWS PER TOMTER (Received 13 May 1974; reuised 10 September 1974) 50. INTRODUCTION IN HIS survey article[20], Smale pointed out the importance of classifying Anosov flows on compact manifolds. We refer to [20] for the role played by Anosov flows in the general theory of differentiable dynamical systems. In this paper some progress is obtained by determining all Anosov flows which satisfy an additional symmetry condition (symmetric Anosov flows). The first known examples of Anosov flows were the geodesic flows on the unit tangent bundles of compact Riemannian manifolds of negative curvature and the suspensions of Anosov diff eomorphisms on infranilmanifolds defined by hyperbolic automorphisms. In [24] a new example was constructed on a compact 7-dimensional manifold. This is generalized here to a new class of Anosov flows (symmetric Anosov flows of mixed type), and these are described in terms of invariants from the theory of algebraic groups. This raises the following question: Is a general Anosov flow on a compact manifold topologically conjugate to a symmetric Anosov flow? This generalizes the well-known question: Is an arbitrary Anosov ditfeomorphism on a compact manifold topologically conjugate to a hyperbolic infranilmanifold diffeomorphism? It has been proved that the answer to the latter question is positive if one of the invariant foliations associated with the diffeomorphism has codimension one (see Newhouse [ IS]). Section 1 contains some preliminary remarks on symmetric Anosov flows, and some of the results from [24] are summarized. In the second section the structure of the lifted flow to a covering manifold is studied.
    [Show full text]
  • Invariant Differential Operators for Non-Compact Lie Groups: Parabolic
    Invariant Differential Operators for Non-Compact Lie Groups: Parabolic Subalgebras V.K. Dobrev Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria (permanent address) and The Abdus Salam International Center for Theoretical Physics P.O. Box 586, Strada Costiera 11 34014 Trieste, Italy Abstract In the present paper we start the systematic explicit construction of invariant differen- tial operators by giving explicit description of one of the main ingredients - the cuspidal parabolic subalgebras. We explicate also the maximal parabolic subalgebras, since these are also important even when they are not cuspidal. Our approach is easily generalised to the supersymmetric and quantum group settings and is necessary for applications to string arXiv:hep-th/0702152v13 7 Oct 2020 theory and integrable models. 1. Introduction Invariant differential operators play very important role in the description of physical sym- metries - starting from the early occurrences in the Maxwell, d’Allembert, Dirac, equations, (for more examples cf., e.g., [1]), to the latest applications of (super-)differential operators in conformal field theory, supergravity and string theory, (for a recent review, cf. e.g., [2]). Thus, it is important for the applications in physics to study systematically such operators. 1 In the present paper we start with the classical situation, with the representation theory of semisimple Lie groups, where there are lots of results by both mathematicians and physi- cists, cf., e.g. [3-41]. We shall follow a procedure in representation theory in which such operators appear canonically [24] and which has been generalized to the supersymmetry setting [42] and to quantum groups [43].
    [Show full text]
  • Classifying the Fine Structures of Involutions Acting on Root Systems
    Abstract IVY, SAMUEL JAMAL. Classifying the Fine Structures of Involutions Acting on Root Systems. (Under the direction of Aloysius Helminck.) Symbolic Computation is a growing and exciting intersection of Mathematics and Com- puter Science that provides a vehicle for illustrations and algorithms for otherwise difficult to describe mathematical objects. In particular, symbolic computation lends an invaluable hand to the area of symmetric spaces. Symmetric spaces, as the name suggests, offers the study of symmetries. Indeed, it can be realized as spaces acted upon by a group of symmetries or motions (a Lie Group). Though the presence of symmetric spaces reaches to several other areas of Mathematics and Physics, the point of interest reside in the realm of Lie Theory. More specifically, much can be determine and described about the Lie algebra/group from the root system. This dissertation focuses on the algebraic and combinatorial structures of symmetric spaces including the action of involutions on the underline root systems. The characterization of the orbits of parabolic subgroups acting on these symmetric spaces involves the action of both the symmetric space involution q on the maximal k-split tori and their root system and its opposite q. While the action of q is often known, the action of q − − is not well understood. This thesis focuses on building results and algorithms that enable one to derive the root system structure related to the action of q from the root system − structure related to q. This work involves algebraic group theory,
    [Show full text]
  • Notes on Symmetric Spaces
    Notes on symmetric spaces Jonathan Holland Bogdan Ion arXiv:1211.4159v1 [math.DG] 17 Nov 2012 Contents Preface 5 Chapter1. Affine connections and transformations 7 1. Preliminaries 7 2. Affine connections 10 3. Affine diffeomorphisms 13 4. Connections invariant under parallelism 17 Chapter 2. Symmetric spaces 21 1. Locally symmetric spaces 21 2. Riemannian locally symmetric spaces 22 3. Completeness 23 4. Isometry groups 25 Chapter 3. Orthogonal symmetric Lie algebras 27 1. Structure of orthogonal symmetric Lie algebras 28 2. Rough classification of orthogonal symmetric Lie algebras 30 3. Cartan involutions 31 4. Flat subspaces 33 5. Symmetric spaces from orthogonal symmetric Lie algebras 35 6. Cartan immersion 38 Chapter 4. Examples 41 1. Flat examples 41 2. Spheres 41 3. Projective spaces 44 4. Negatively curved spaces 48 Chapter 5. Noncompact symmetric spaces 53 Chapter 6. Compact semisimple Lie groups 59 1. Introduction 59 2. Compact groups 59 3. Weyl’s theorem 61 4. Weight and root lattices 62 5. The Weyl group and affine Weyl group 63 6. Determining the center of G 65 7. Symmetric spaces of type II 67 8. Relative root systems 68 9. Fixed points of involutions 72 10. Symmetric spaces of type I 74 3 4 CONTENTS Chapter 7. Hermitian symmetric spaces 75 1. Hermitian symmetric Lie algebras 75 2. Hermitian symmetric spaces 78 3. Bounded symmetric domains 81 4. Structure of Hermitian symmetric Lie algebras 84 5. Embedding theorems 89 Chapter 8. Classification of real simple Lie algebras 93 1. Classical structures 93 2. Vogan diagrams 94 Bibliography 107 Index 109 Preface These are notes prepared from a graduate lecture course by the second author on symmetric spaces at the University of Pittsburgh during the Fall of 2010.
    [Show full text]
  • Complex Geometry and Representations of Lie Groups
    http://dx.doi.org/10.1090/conm/288/04826 Contemporary Mathematics Volume 288, 2001 Complex Geometry and Representations of Lie Groups Joseph A. Wolf Dedicated to the memory of my friend and colleague Alfred Gray ABSTRACT. Let Z = G/Q be a complex flag manifold and let Go be a real form of G. Then the representation theory of the real reductive Lie group Go is intimately connected with the geometry of Go-orbits on Z. The open orbits correspond to the discrete series representations and their analytic con- tinuations, closed orbits correspond to the principal series, and certain other orbits give the other series of tempered representations. Here I try to indicate some of that interplay between geometry and analysis, concentrating on the complex geometric aspects of the open orbits and the related representations. Contents: §0. Introduction Part I. Geometry of Flag Manifolds §1. Parabolic Subalgebras and Complex Flags §2. Real Group Orbits on Complex Flags §3. The Closed Orbit §4. Open Orbits §5. Cycle Spaces §6. The Double Fibration Transform Part II. Representations of Reductive Lie Groups §7. The Principal Series §8. The Discrete Series §9. The Various Tempered Series §10. The Plancherel Formula Part III. Geometric Realizations of Representations §11. L 2 Realizations of the Principal Series §12. £ 2 Realizations of the Discrete Series §13. L 2 Realizations of the Various Tempered Series §14. Approaches to Non-Tempered Representations 2000 Mathematics Subject Classification. Primary 22E15, 22E46, 43A80; secondary 22E10, 32E10, 32M10, 32T15. Key words and phrases. reductive Lie group, reductive Lie algebra, semisimple Lie group, semisimple Lie algebra, representation, flag manifold, flag domain.
    [Show full text]
  • Hermitian Symmetric Spaces and Kähler Rigidity
    HERMITIAN SYMMETRIC SPACES AND KAHLER¨ RIGIDITY MARC BURGER, ALESSANDRA IOZZI, AND ANNA WIENHARD Abstract. We characterize irreducible Hermitian symmetric spaces of non- tube type both in terms of the topology of the space of triples of pairwise transverse points in the Shilov boundary, and of two invariants which we in- troduce, the Hermitian triple product and its complexification. We apply these results and the techniques introduced in [6] to characterize conjugacy classes of Zariski dense representations of a locally compact group into the connected component G of the isometry group of an irreducible Her- mitian symmetric space which is not of tube type, in terms of the pullback of the bounded K¨ahler class via the representation. We conclude also that if the second bounded cohomology of a finitely generated group Γ is finite dimen- sional, then there are only finitely many conjugacy classes of representations of Γ into G with Zariski dense image. This generalizes results of [6]. Contents 1. Introduction 1 2. Hermitian Triple Product 6 3. Characterization of NonTube Type Domains 14 4. The K¨ahler Class, the Bounded K¨ahler Class and Boundaries 19 5. Proofs 23 References 26 1. Introduction A symmetric space X is Hermitian if it carries a complex structure which is Isom(X )◦-invariant and hence, together with the Riemannian metric, defines a closed Isom(X )◦-invariant differential two-form on X , which thus turns X into a K¨ahler manifold. The K¨ahler form gives rise to invariants for isometric actions of finitely generated groups Γ on X itself and on geometric objects related to X .
    [Show full text]