REPRESENTATION THEORY of REAL GROUPS Contents 1
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
A NEW APPROACH to RANK ONE LINEAR ALGEBRAIC GROUPS An
A NEW APPROACH TO RANK ONE LINEAR ALGEBRAIC GROUPS DANIEL ALLCOCK Abstract. One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material before one can even characterize PGL2. Our goal here is to show how to develop the root system, etc., using only the completeness of the flag variety, its immediate consequences, and some facts about solvable groups. In particular, one can skip over the usual analysis of Cartan subgroups, the fact that G is the union of its Borel subgroups, the connectedness of torus centralizers, and the normalizer theorem (that Borel subgroups are self-normalizing). The main idea is a new approach to the structure of rank 1 groups; the key step is lemma 5. All algebraic geometry is over a fixed algebraically closed field. G always denotes a connected linear algebraic group with Lie algebra g, T a maximal torus, and B a Borel subgroup containing it. We assume the structure theory for connected solvable groups, and the completeness of the flag variety G/B and some of its consequences. Namely: that all Borel subgroups (resp. maximal tori) are conjugate; that G is nilpotent if one of its Borel subgroups is; that CG(T )0 lies in every Borel subgroup containing T ; and that NG(B) contains B of finite index and (therefore) is self-normalizing. -
Lecture Notes on Compact Lie Groups and Their Representations
Lecture Notes on Compact Lie Groups and Their Representations CLAUDIO GORODSKI Prelimimary version: use with extreme caution! September , ii Contents Contents iii 1 Compact topological groups 1 1.1 Topological groups and continuous actions . 1 1.2 Representations .......................... 4 1.3 Adjointaction ........................... 7 1.4 Averaging method and Haar integral on compact groups . 9 1.5 The character theory of Frobenius-Schur . 13 1.6 Problems.............................. 20 2 Review of Lie groups 23 2.1 Basicdefinition .......................... 23 2.2 Liealgebras ............................ 25 2.3 Theexponentialmap ....................... 28 2.4 LiehomomorphismsandLiesubgroups . 29 2.5 Theadjointrepresentation . 32 2.6 QuotientsandcoveringsofLiegroups . 34 2.7 Problems.............................. 36 3 StructureofcompactLiegroups 39 3.1 InvariantinnerproductontheLiealgebra . 39 3.2 CompactLiealgebras. 40 3.3 ComplexsemisimpleLiealgebras. 48 3.4 Problems.............................. 51 3.A Existenceofcompactrealforms . 53 4 Roottheory 55 4.1 Maximaltori............................ 55 4.2 Cartansubalgebras ........................ 57 4.3 Case study: representations of SU(2) .............. 58 4.4 Rootspacedecomposition . 61 4.5 Rootsystems............................ 63 4.6 Classificationofrootsystems . 69 iii iv CONTENTS 4.7 Problems.............................. 73 CHAPTER 1 Compact topological groups In this introductory chapter, we essentially introduce our very basic objects of study, as well as some fundamental examples. We also establish some preliminary results that do not depend on the smooth structure, using as little as possible machinery. The idea is to paint a picture and plant the seeds for the later development of the heavier theory. 1.1 Topological groups and continuous actions A topological group is a group G endowed with a topology such that the group operations are continuous; namely, we require that the multiplica- tion map and the inversion map µ : G G G, ι : G G × → → be continuous maps. -
A Taxonomy of Irreducible Harish-Chandra Modules of Regular Integral Infinitesimal Character
A taxonomy of irreducible Harish-Chandra modules of regular integral infinitesimal character B. Binegar OSU Representation Theory Seminar September 5, 2007 1. Introduction Let G be a reductive Lie group, K a maximal compact subgroup of G. In fact, we shall assume that G is a set of real points of a linear algebraic group G defined over C. Let g be the complexified Lie algebra of G, and g = k + p a corresponding Cartan decomposition of g Definition 1.1. A (g,K)-module is a complex vector space V carrying both a Lie algebra representation of g and a group representation of K such that • The representation of K on V is locally finite and smooth. • The differential of the group representation of K coincides with the restriction of the Lie algebra representation to k. • The group representation and the Lie algebra representations are compatible in the sense that πK (k) πk (X) = πk (Ad (k) X) πK (k) . Definition 1.2. A (Hilbert space) representation π of a reductive group G with maximal compact subgroup K is called admissible if π|K is a direct sum of finite-dimensional representations of K such that each K-type (i.e each distinct equivalence class of irreducible representations of K) occurs with finite multiplicity. Definition 1.3 (Theorem). Suppose (π, V ) is a smooth representation of a reductive Lie group G, and K is a compact subgroup of G. Then the space of K-finite vectors can be endowed with the structure of a (g,K)-module. We call this (g,K)-module the Harish-Chandra module of (π, V ). -
Algebraic D-Modules and Representation Theory Of
Contemporary Mathematics Volume 154, 1993 Algebraic -modules and Representation TheoryDof Semisimple Lie Groups Dragan Miliˇci´c Abstract. This expository paper represents an introduction to some aspects of the current research in representation theory of semisimple Lie groups. In particular, we discuss the theory of “localization” of modules over the envelop- ing algebra of a semisimple Lie algebra due to Alexander Beilinson and Joseph Bernstein [1], [2], and the work of Henryk Hecht, Wilfried Schmid, Joseph A. Wolf and the author on the localization of Harish-Chandra modules [7], [8], [13], [17], [18]. These results can be viewed as a vast generalization of the classical theorem of Armand Borel and Andr´e Weil on geometric realiza- tion of irreducible finite-dimensional representations of compact semisimple Lie groups [3]. 1. Introduction Let G0 be a connected semisimple Lie group with finite center. Fix a maximal compact subgroup K0 of G0. Let g be the complexified Lie algebra of G0 and k its subalgebra which is the complexified Lie algebra of K0. Denote by σ the corresponding Cartan involution, i.e., σ is the involution of g such that k is the set of its fixed points. Let K be the complexification of K0. The group K has a natural structure of a complex reductive algebraic group. Let π be an admissible representation of G0 of finite length. Then, the submod- ule V of all K0-finite vectors in this representation is a finitely generated module over the enveloping algebra (g) of g, and also a direct sum of finite-dimensional U irreducible representations of K0. -
LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0. -
Orbits of Real Forms in Complex Flag Manifolds 71
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 69-109 Orbits of real forms in complex flag manifolds ANDREA ALTOMANI,COSTANTINO MEDORI AND MAURO NACINOVICH Abstract. We investigate the CR geometry of the orbits M of a real form G0 of a complex semisimple Lie group G in a complex flag manifold X = G/Q.Weare mainly concerned with finite type and holomorphic nondegeneracy conditions, canonical G0-equivariant and Mostow fibrations, and topological properties of the orbits. Mathematics Subject Classification (2010): 53C30 (primary); 14M15, 17B20, 32V05, 32V35, 32V40, 57T20 (secondary). Introduction In this paper we study a large class of homogeneous CR manifolds, that come up as orbits of real forms in complex flag manifolds. These objects, that we call here parabolic CR manifolds, were first considered by J. A. Wolf (see [28] and also [12] for a comprehensive introduction to this topic). He studied the action of a real form G0 of a complex semisimple Lie group G on a flag manifold X of G.Heshowed that G0 has finitely many orbits in X,sothat the union of the open orbits is dense in X, and that there is just one that is closed. He systematically investigated their properties, especially for the open orbits and for the holomorphic arc components of general orbits, outlining a framework that includes, as special cases, the bounded symmetric domains. We aim here to give a contribution to the study of the general G0-orbits in X, by considering and utilizing their natural CR structure. In [2] we began this program by investigating the closed orbits. -
The Recursive Nature of Cominuscule Schubert Calculus
Manuscript, 31 August 2007. math.AG/0607669 THE RECURSIVE NATURE OF COMINUSCULE SCHUBERT CALCULUS KEVIN PURBHOO AND FRANK SOTTILE Abstract. The necessary and sufficient Horn inequalities which determine the non- vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson co- efficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities. Introduction We investigate the following general problem: Given Schubert subvarieties X, X′,...,X′′ of a flag variety, when is the intersection of their general translates (1) gX ∩ g′X′ ∩···∩ g′′X′′ non-empty? When the flag variety is a Grassmannian, it is known that such an intersection is non-empty if and only if the indices of the Schubert varieties, expressed as partitions, satisfy the linear Horn inequalities. The Horn inequalities are themselves indexed by lists of partitions corresponding to such non-empty intersections on smaller Grassmannians. This recursive answer to our original question is a consequence of work of Klyachko [15] who linked eigenvalues of sums of hermitian matrices, highest weight modules of sln, and the Schubert calculus, and of Knutson and Tao’s proof [16] of Zelevinsky’s Saturation Conjecture [28]. -
Ind-Varieties of Generalized Flags: a Survey of Results
Ind-varieties of generalized flags: a survey of results∗ Mikhail V. Ignatyev Ivan Penkov Abstract. This is a review of results on the structure of the homogeneous ind-varieties G/P of the ind-groups G = GL∞(C), SL∞(C), SO∞(C), Sp∞(C), subject to the condition that G/P is a inductive limit of compact homogeneous spaces Gn/Pn. In this case the subgroup P ⊂ G is a splitting parabolic subgroup of G, and the ind-variety G/P admits a “flag realization”. Instead of ordinary flags, one considers generalized flags which are, generally infinite, chains C of subspaces in the natural representation V of G which satisfy a certain condition: roughly speaking, for each nonzero vector v of V there must be a largest space in C which does not contain v, and a smallest space in C which contains v. We start with a review of the construction of the ind-varieties of generalized flags, and then show that these ind-varieties are homogeneous ind-spaces of the form G/P for splitting parabolic ind-subgroups P ⊂ G. We also briefly review the characterization of more general, i.e. non-splitting, parabolic ind-subgroups in terms of generalized flags. In the special case of an ind-grassmannian X, we give a purely algebraic-geometric construction of X. Further topics discussed are the Bott– Borel–Weil Theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of G/P for arbitrary splitting parabolic ind-subgroups P ⊂ G, as well as the orbits of real forms on G/P for G = SL∞(C). -
Linear Algebraic Groups
Clay Mathematics Proceedings Volume 4, 2005 Linear Algebraic Groups Fiona Murnaghan Abstract. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 1. Algebraic groups Let K be an algebraically closed field. An algebraic K-group G is an algebraic variety over K, and a group, such that the maps µ : G × G → G, µ(x, y) = xy, and ι : G → G, ι(x)= x−1, are morphisms of algebraic varieties. For convenience, in these notes, we will fix K and refer to an algebraic K-group as an algebraic group. If the variety G is affine, that is, G is an algebraic set (a Zariski-closed set) in Kn for some natural number n, we say that G is a linear algebraic group. If G and G′ are algebraic groups, a map ϕ : G → G′ is a homomorphism of algebraic groups if ϕ is a morphism of varieties and a group homomorphism. Similarly, ϕ is an isomorphism of algebraic groups if ϕ is an isomorphism of varieties and a group isomorphism. A closed subgroup of an algebraic group is an algebraic group. If H is a closed subgroup of a linear algebraic group G, then G/H can be made into a quasi- projective variety (a variety which is a locally closed subset of some projective space). If H is normal in G, then G/H (with the usual group structure) is a linear algebraic group. Let ϕ : G → G′ be a homomorphism of algebraic groups. Then the kernel of ϕ is a closed subgroup of G and the image of ϕ is a closed subgroup of G. -
Math 222 Notes for Apr. 1
Math 222 notes for Apr. 1 Alison Miller 1 Real lie algebras and compact Lie groups, continued Last time we showed that if G is a compact Lie group with Z(G) finite and g = Lie(G), then the Killing form of g is negative definite, hence g is semisimple. Now we’ll do a converse. Proposition 1.1. Let g be a real lie algebra such that Bg is negative definite. Then there is a compact Lie group G with Lie algebra Lie(G) =∼ g. Proof. Pick any connected Lie group G0 with Lie algebra g. Since g is semisimple, the center Z(g) = 0, and so the center Z(G0) of G0 must be discrete in G0. Let G = G0/Z(G)0; then Lie(G) =∼ Lie(G0) =∼ g. We’ll show that G is compact. Now G0 has an adjoint action Ad : G0 GL(g), with kernel ker Ad = Z(G). Hence 0 ∼ G = Im(Ad). However, the Killing form Bg on g is Ad-invariant, so Im(Ad) is contained in the subgroup of GL(g) preserving the Killing! form. Since Bg is negative definite, this subgroup is isomorphic to O(n) (where n = dim(g). Since O(n) is compact, its closed subgroup Im(Ad) is also compact, and so G is compact. Note: I found a gap in this argument while writing it up; it assumes that Im(Ad) is closed in GL(g). This is in fact true, because Im(Ad) = Aut(g) is the group of automorphisms of the Lie algebra g, which is closed in GL(g); but proving this fact takes a bit of work. -
Lie Algebras from Lie Groups
Preprint typeset in JHEP style - HYPER VERSION Lecture 8: Lie Algebras from Lie Groups Gregory W. Moore Abstract: Not updated since November 2009. March 27, 2018 -TOC- Contents 1. Introduction 2 2. Geometrical approach to the Lie algebra associated to a Lie group 2 2.1 Lie's approach 2 2.2 Left-invariant vector fields and the Lie algebra 4 2.2.1 Review of some definitions from differential geometry 4 2.2.2 The geometrical definition of a Lie algebra 5 3. The exponential map 8 4. Baker-Campbell-Hausdorff formula 11 4.1 Statement and derivation 11 4.2 Two Important Special Cases 17 4.2.1 The Heisenberg algebra 17 4.2.2 All orders in B, first order in A 18 4.3 Region of convergence 19 5. Abstract Lie Algebras 19 5.1 Basic Definitions 19 5.2 Examples: Lie algebras of dimensions 1; 2; 3 23 5.3 Structure constants 25 5.4 Representations of Lie algebras and Ado's Theorem 26 6. Lie's theorem 28 7. Lie Algebras for the Classical Groups 34 7.1 A useful identity 35 7.2 GL(n; k) and SL(n; k) 35 7.3 O(n; k) 38 7.4 More general orthogonal groups 38 7.4.1 Lie algebra of SO∗(2n) 39 7.5 U(n) 39 7.5.1 U(p; q) 42 7.5.2 Lie algebra of SU ∗(2n) 42 7.6 Sp(2n) 42 8. Central extensions of Lie algebras and Lie algebra cohomology 46 8.1 Example: The Heisenberg Lie algebra and the Lie group associated to a symplectic vector space 47 8.2 Lie algebra cohomology 48 { 1 { 9. -
Geometry of Compact Complex Manifolds Associated to Generalized Quasi-Fuchsian Representations
Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations David Dumas and Andrew Sanders 1 Introduction This paper is concerned with the following general question: which aspects of the complex- analytic study of discrete subgroups of PSL2C can be generalized to discrete subgroups of other semisimple complex Lie groups? To make this more precise, we recall the classical situation that motivates our discussion. A torsion-free cocompact Fuchsian group Γ < PSL2R acts freely, properly discontinuously, 2 and cocompactly by isometries on the symmetric space PSL2R=PSO(2) ' H . The quotient 2 S = ΓnH is a closed surface of genus g > 2. When considering Γ as a subgroup of 3 PSL2C, it is natural to consider either its isometric action on the symmetric space H ' PSL =PSU(2) or its holomorphic action on the visual boundary 1 ' PSL =B . 2C PC 2C PSL2C The latter action has a limit set Λ = 1 and a disconnected domain of discontinuity PR Ω = H t H. The quotient ΓnΩ is a compact K¨ahler manifold|more concretely, it is the union of two complex conjugate Riemann surfaces. Quasiconformal deformations of such groups Γ give quasi-Fuchsian groups in PSL2C. Each such group acts on 1 in topological conjugacy with a Fuchsian group, hence the limit PC set Λ is a Jordan curve, the domain of discontinuity has two contractible components, and the quotient manifold is a union of two Riemann surfaces (which are not necessarily complex conjugates of one another). If G is a complex simple Lie group of adjoint type (such as PSLnC, n > 2), there is a distinguished homomorphism ιG : PSL2C ! G introduced by Kostant [Kos59] and called the principal three-dimensional embedding.