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REPRESENTATION THEORY of REAL GROUPS Contents 1 REPRESENTATION THEORY OF REAL GROUPS ZIJIAN YAO Contents 1. Introduction1 2. Discrete series representations6 3. Geometric realizations9 4. Limits of discrete series 14 Appendix A. Various cohomology theories 15 References 18 1. Introduction 1.1. Introduction. The goal of these notes is to survey some basic concepts and constructions in representation theory of real groups, in other words, \at the infinite places". In particular, some of the definitions reviewed here should be helpful towards the formulation of the Langlands program. To be more precise, we first fix some notation for the introduction and the rest of the survey. Let G be a connected reductive group over R and G = G(R). Let K ⊂ G be an R-subgroup such that K := K(R) ⊂ G is a maximal compact subgroup. Let G0 ⊂ GC be the 1 unique compact real form containing K. By a representation of G we mean a complete locally convex topological C-vector space V equipped with a continuous action G × V ! V . We denote a representation by (π; V ) with π : G ! GL(V ): 2 Following foundations laid out by Harish-Chandra, we focus our attention on irreducible admissible representations of G (Definition 1.2), which are more general than (irreducible) unitary representations and much easier to classify, essentially because they are \algebraic". More precisely, by restricting to the K-finite subspaces V fin we arrive at (g;K)- modules. By a theorem of Harish-Chandra (recalled in Subsection 1.2), there is a natural bijection between Irreducible admissible G- Irreducible admissible representations (g;K)-modules ∼! infinitesimal equivalence isomorphism In particular, this allows us to associate an infinitesimal character χπ to each irreducible ad- missible representation π (Subsection 1.3), which carries useful information for us later in the survey. The strategy to understand unitary representations (up to infinitesimal equivalence) is there- fore clear: first classify all irreducible admissible (g;K)-modules, then distinguish unitary ones 1 For example, take G = GLn, K = O(n) and G0 = U(n). The corresponding Cartan involutions of the split real form G and compact real form G0 are respectively the trivial one and the Chevalley involution, up to GLn(C)-conjugation. 2In the majority of the notes we will pretend that (π; V ) is a Hilbert space representation for simplicity. 1 2 from the rest. The first step is more or less fully understood by work of Harish-Chandra, Lang- lands and others. In particular, Langlands proves the \Langlands classification”, as well as the archimedean Langlands correspondence, which states that there is a natural bijection between L-packets of irreducible admissible (Relevant) Langlands pa- representations of G rameters of G ∼! infinitesimal equivalence conjugation by G_ For quasi-split groups G over R, every Langlands parameter is relevant. For G = GLn, each L-packet consists of a single (infinitesimal equivalence class of) admissible representations. The second step, namely to classify all unitary representations, seems more difficult and has not been completed. One of the simplest classes of irreducible admissible representations are discrete series { they unitary representations that are square integrable in a suitable sense. Such representations form a discrete subset in the unitary dual Gb of G (under the Fell topology), hence the name. They are in a certain sense infinite dimensional analogs of highest weight representations. We discuss discrete series in Section2, where examples of SL 2 = GL2 are considered Using the theory of global characters, Harish-Chandra showed that discrete series exist precisely when rank(G) = rank(K). Moreover, one can describe discrete series representations of G via the Harich-Chandra parameters, as stated in Theorem 2.9. We then discuss explicit realizations of discrete representations. The classical story of realizing highest weight representations of compact groups are given by the Borel{Weil{Bott's theorem, which states that if G is a compact real Lie group with flag variety X = GC=B for some complex Borel subgroup B ⊂ GC, then the coherent cohomology of holomorphic line bundles Lλ is non-trivial precisely when λ is regular after a ρ-shift, in which case is appears in one degree and is the irreducible representation of highest weight λ (for the precise statement and its proof see Theorem 3.1). Moreover, all highest weight representations of G appear this way. When G is non-compact, Langlands conjectured that a similar story should hold when X is replaced by the open subdomain D which is an orbit under the G-action 3 and when the coherent cohomology is replaced by the L2-cohomology. Langlands predicted that the corresponding irreducible representation of G should be a discrete series representation and this procedure exhausts all discrete series. Langlands's conjecture is now a theorem of Schmid, which we explain in Subsection 3.2. Finally, we end this survey by a brief discussion on limits of discrete series and some properties of (non-generate) limits of discrete series. Acknowledgement. Most of the results stated in the survey are due to Harish-Chandra and Schmid. Due to the depth of the theorems stated in the survey, only brief outlines are provided instead of detailed proofs. I would like to thank professor Schmid for patiently explaining many of his results to me, which helps me immensely in understanding his proofs and organizing this survey article. 1.2. Admissible representations. Let G be a reductive group over R as in the introduction, let G = G(R) and g its Lie algebra. 1.2.1. Smooth vectors. Let V = (V; π) be a representation of G. Recall that a vector v 2 V is 1 d C if for each X 2 g, its derivative Xv = dt exp(tX)v t=0 exists. This allows us to inductively define Ck. The smooth vectors in V form a subspace V sm = fv 2 V v is Ck for all k ≥ 1g ⊂ V: 3D = D(∆+) depends on a choice of positive roots, or equivalent a Borel subalgebra b ⊂ g 3 In other words, these are vectors v such that g 7! π(g)v defines a smooth map G ! V . It is not difficult to see that V sm is a subrepresentation of G. Moreover, the representation of g on V sm by taking derivatives is a Lie algebra representation. 1 Next recall the action of the Hecke algebra H = Cc (G) on V given by Z π(f)v := f(g)π(g)v dg G for any f 2 H and v 2 V . First we observe that this \averaging" process makes the vectors sufficiently smooth. More precisely, we have π(f)v 2 V sm for each f 2 H; v 2 V . To see this, rewrite d Z Xπ(f)v = f(g) π exp(tX)g v dg dt G t=0 d Z = f exp(−tX)g π(g)v dg dt G t=0 Z = fX (g) π(g)v dg = π(fX )v G d 1 sm where fX (g) := f(exp(−tX)g) , which lies in Cc (G). Therefore π(f)v 2 V by induction. dt t=0 From this it allows that Corollary 1.1. The subrepresentation V sm ⊂ V is dense. 1.2.2. K-finite vectors. Restricting a representation to its compact subgroups is a valuable tool in representation theory, and in our setup leads to the notion of (g;K)-modules. First let us recall that a representation (V; π) of G is unitary if it is a Hilbert space represen- tation such that the action of G is unitary. Let V = (V; π; h · i) be a Hilbert space representation, then V jK is unitarizable: we define a Hermitian form ( · ) by Z (v1; v2) := hπ(g)v1; π(g)v2idg; K which is K-invariant by construction. It induces the same topology on V as the original Her- mitian form h · i. Let σ be an equivalent class of irreducible representation of K, then the σ-isotypic component of V is ∼ V (σ) = fv 2 V : Chπ(K)vi = σg: A vector v 2 V is K-finite if it is contained in V (σ) for some irreducible representation σ of K. The subspace of K-finite vectors in V is V fin := ⊕ V (σ) σ2Kb where the (algebraic) direct sum is taken over equivalent classes of irreducible representations of K 4. Definition 1.2. A representation (π; V ) of G is admissible if • πjK is unitary; • each V (σ) appearing in V fin has finite multiplicity/dimension. Remark 1.3. From the discussion above, for a Hilbert space representation, to check admissi- bility it suffices to check the second condition. 4By our discussion on unitarizability above, the equivalent classes of irreducible representations can be identified with the unitary dual Kb 4 Theorem 1.4 (Harish-Chandra). Suppose that G is connected reductive as above, then every irreducible unitary representation of G is admissible. 1.2.3. (g;K)-modules. The K-finite vectors V fin ⊂ V in an admissible representation (V; π) (which is usually much smaller) is a lot easier to analyze. Note that the G-action does not in general preserve K-finiteness, so we consider its Lie algebra action and are led to the following definition Definition 1.5. A(g;K)-module is a vector space V with a g (as Lie algebra) action and K action satisfying the following conditions (1) V is a countable algebraic direct sum V = ⊕iVi with each Vi a finite dimensional K- invariant sub-representation. 5 (2) For any X 2 k = Lie(K) and v 2 V , we have d X · v = π exp(tX)v dt t=0 (3) For any k 2 K; X 2 g and v 2 V , we have π(k) X · π(k−1)v = (Ad(k)X) · v We similarly define admissible (g;K)-modules { namely V is admissible if the Vi's in part (1) of the definition can be chosen to be distinct K-representation.
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