NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS MATH 224, SPRING 2017 DENNIS GAITSGORY 1. Week 1, Day 1 (Tue, Jan 24) 1.1. Action of distributions. 1.1.1. Let K be a compact topological group. Let C(K) denote the space of continuous complex- valued functions on K. Unless specified otherwise, integration over K will be with respect to the Haar measure µHaar on K, normalized so that the volume of K equals 1. Let Meas(K) be the topological dual of C(K) (it is known that Meas(K) can be described as Radon measures). Unless specified otherwise, we will regard it as equipped with the weak topology. For k 2 K we let δk 2 Meas(K) denote the corresponding δ-function, i.e., hδk; fi = f(k). Note that we have a canonical embedding (1.1) L1(K) ,! Meas(K); f 7! f · µHaar with dense image (with respect to the weak topology on Meas(K)). Remark 1.1.2. Note that, being the topological dual of a Banach space, Meas(K) has a natural norm, isn which it is a Banach space. The corresponding topology (called the norm topology) is stronger than the weak topology. The map (1.1) is an isometric embedding with respect to the corresponding norms (in particular, the image of L1(K) in Meas(K) is not dense in the norm topology). The above discussion is applicable to (K; µHaar) replaced by an arbitrary compact topological set equipped with a Borel measure. 1.1.3. Pushforward of measures along the multiplication map K × K ! K (i.e., the operator dual to that of pullback of functions) makes Meas(K) into an associative algebra. We denote the corresponding operation by f1; f2 7! f1 ? f2 and refer to it as the convolution product. The unit for this algebra is δ1. We have δk1 · δk2 = δk1·k2 : This algebra structure is compatible with the embedding (1.1), i.e., it induces a structure of associative algebra on L1(K). We have µHaar ? µHaar = µHaar: Date: September 22, 2021. 1 2 DENNIS GAITSGORY 1.1.4. Let V be a finite-dimensional continuous representation of K. For k 2 K we let Tk the corresponding element of End(V ). Since for every v 2 V; v∗ 2 V ∗, the matrix coefficient function ∗ k 7! hv ;Tk(v)i is continuous, we obtain that the assignment k 7! Tk uniquely extends to a continuous linear map (1.2) Meas(K) ! End(V ); f 7! Tf ; determined by the condition that Tδk = Tk. The map (1.2) is a homomorphism of associative algebras. 1.2. Matrix coefficients. 1.2.1. We view C(K) as a representation of K × K by −1 ((k1; k2) · f)(k) = f(k1 · k · k2): By adjunction, we obtain action of K × K on Meas(K). We have (k1; k2) · δk = δ −1 : k1·k·k2 1.2.2. For a finite-dimensional continuous representation V of K, we view End(V ) as a repre- sentation of K × K via −1 (k1; k2) · S = Tk1 ◦ S ◦ T : k2 The action map (1.3) ActV : Meas(K) ! End(V ); f 7! Tf and the matrix coefficient map (1.4) MCV : End(V ) ! C(K); MCS;V (k) = Tr(S ◦ Tk−1 ;V ) are maps of K × K-representations. 1.2.3. Note that we can canonically identify End(V ) with the dual of End(V ∗) as K × K- representations by setting ∗ ∗ hS1;S2i = Tr(S1 ◦ S2 ;V );S1 2 End(V );S2 2 End(V ); where S 7! S∗ denotes the operation of taking the dual linear operator. In terms of this identification, the maps (1.3) and (1.4) are each other's duals. Remark 1.2.4. Note that the map S 7! S∗ defines an isomorphism of vector spaces End(V ) ' End(V ∗): This isomorphism is compatible with the K × K-actions up to the swap of factors in K × K. 1.3. Orthogonality formulas. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 3 1.3.1. Let U be a finite-dimensional continuousrepresentation of K. Note that the operator inv K PU = TµHaar is an idempotent projection onto U , the subspace of K-invariant vectors. For S 2 End(U) we have Z inv inv inv K (1.5) MCS;U (k) = Tr(S ◦ PU ;U) = Tr(PU ◦ S ◦ PU ;U ): k2K Proposition 1.3.2. Let V1 and V2 be finite-dimensional continuous irreducible representations. Then we have: Z MCS1;V1 (k) · MCS2;V2 (k) = 0 k2K ∗ ∗ unless V2 ' V1 (both reps are irreducible), and for V1 = V and V2 = V , we have Z 1 ∗ MC (k) · MC ∗ (k) = · Tr(S ◦ S ;V ): S1;V S2;V dim(V ) 1 2 k2K ∗ Proof. Consider U = V1 ⊗ V2, and apply (1.5) to S = S1 ⊗ S2. If V2 is non-isomorphic to V1 , then U K = 0, and the assertion follows. ∗ If V1 = V and V2 = V , we identify V ⊗ V ∗ ' End(V ); ∗ and under this identification, for S1 2 End(V ) and S2 2 End(V ), the corresponding endomor- ∗ phism S1 ⊗ S2 of V ⊗ V corresponds to the endomorphism of End(V ) given by 0 0 ∗ (1.6) S 7! S1 ◦ S ◦ S2 : By Schur's lemma, the map K C ! End(V ) ; 1 7! IdV is an isomorphism. The corresponding projector inv K PEnd(V ) : End(V ) ! End(V ) identifies with S0 7! Tr(S0;V ): The operator 0 inv 0 inv S 7! PEnd(V ) ◦ S ◦ PEnd(V ); End(End(V )) ! C sends the endomorphism of End(V ) given by (1.6) to 1 · Tr(S ◦ S∗;V ): dim(V ) 1 2 This establishes the desired identity. 4 DENNIS GAITSGORY 1.3.3. Note that −1 MCS∗;V ∗ (k) = MCS;V (k ): Hence, the orthogonality relation can also be interpreted as a formula for Z −1 (1.7) MCS1;V1 (k) · MCS2;V2 (k ): k2K Namely, (1.7) vanishes unless V1 ' V2, and for V1 = V = V2, it equals 1 · Tr(S ◦ S ;V ): dim(V ) 1 2 1.3.4. As a corollary of Proposition 1.3.2, we obtain: Corollary 1.3.5. For V1 and V2 irreducible, the composite ActV1 MCV2 End(V1) −! C(K) ,! Meas(K) −! End(V2) 1 is zero unless V1 ' V2 and equals dim(V ) · IdEnd(V ) for V1 = V = V2. 1.3.6. Using an invariant Hermitian form, we also have −1 MCS;V (k ) = MCSy;V (k): Thus, we obtain that the images of MCV for distinct V 's are pair-wise orthogonal in L2(K), and that for an individual V , the map MCV is an isometric embedding of End(V ) into L2(K), where we endow End(V ) with a Hermitian structure by the formula 1 (1.8) (S ;S ) = · Tr(S ◦ Sy): 1 2 dim(V ) 1 2 1.4. Characters. 1.4.1. Let V be a finite-dimensional continuous representation of K. Let χV 2 C(K) be equal MCV (IdV ). From Proposition 1.3.2 we obtain: Corollary 1.4.2. Let V and W be irreducible. Then Z Z Z −1 χV (k) · χW (k ) = χV (k) · χW ∗ (k) = χV (k) · χW (k) k2K k2K k2K equals 1 if V ' W , and vanishes otherwise. 1.4.3. For V irreducible, set ξV := dim(V ) · χV . From Corollary 1.3.5, we obtain: Corollary 1.4.4. For W irreducible, the element TξV 2 End(W ) equals IdW for W ' V and zero otherwise. From Corollary 1.7, we obtain: Corollary 1.4.5. ξV ? ξW = 0 unless W ' V and ξV ? ξV = ξV . 2. Week 1, Day 2 (Tue, Jan 26) 2.1. Continuous representations. 2.1.1. Let G be a topological group and V a topological vector space. We shall say that a representation of G on V is continuous if the action map G × V ! V is continuous. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 5 2.1.2. In this course we will restrict our attention to the following class of topological vector spaces: we will assume that they are Hausdorff, 2nd countable, locally convex and complete. Locally convex is equivalent to saying that there is a fundamental system of neighborhoods of 0 2 V of the form fv 2 V ; ρ(v) < 1g; where ρ : V ! R is a continuous map satisfying ρ(c · v) = jcj · ρ(v); ρ(v1 + v2) ≤ ρ(v1) + ρ(v2): Such maps are called semi-norms. 2.1.3. Suppose that G is locally compact. Here is a (tautological) way to reformulate the condition of continuity of action. It amounts to the combination of the following two: (i) For every v 2 V , a semi-norm ρ and , there exists a neighborhood 1 2 U ⊂ G such that ρ(Tg(v) − v) < for g 2 U: (ii) For every compact Ω ⊂ G and every semi-norm ρ, there exists a semi-norm ρ0 so that 0 ρ(Tg(v)) ≤ ρ (v); for all v 2 V and g 2 Ω: 2.1.4. Here are some examples of continuous representations. Let G act continuously on a topological space X. (a) Let C(X) be the space of continuous functions on G. We topologize by convergence on compact subsets. I.e., the fundamental system of neighborhoods of 0 is given by semi-norms max jf(x)j; x2Ω where Ω is a compact subset of X. We consider the action of G on V = C(X) by −1 (Tg · f)(x) = f(g · x): Conditions (i) and (ii) are evident in this case.