Representations of locally compact groups { Fall 2013 Fiona Murnaghan Contents Chapter 1. Topological groups 5 1. Definitions and basic properties 5 2. Locally compact, totally disconnected groups 10 3. Matrix groups 14 4. Haar measure on locally compact Hausdorff groups 16 5. Coset spaces and quasi-invariant measures 21 Chapter 2. Continuous representations of locally compact groups 29 1. Definition of continuous representation 29 2. Representations of compact groups are unitary 32 3. Irreducibility, subrepresentations, etc. 32 4. The regular representation on L2(G=H) 34 5. Intertwining operators and Schur's Lemma 35 Chapter 3. Representations of compact groups: the general theory 39 1. The Peter-Weyl Theorem 39 2. Faithful representations 43 3. Finite-dimensional representations of locally profinite groups 43 4. Finite-dimensional subrepresentations of the regular representation 44 5. Orthogonality relations for matrix coefficients 47 6. Decomposition of the regular representation 48 7. Decomposition of unitary representations of compact groups 49 8. Characters of finite-dimensional representations 50 9. Convolution, characters and orthogonal projections 53 10. Induced representations of compact groups 56 Chapter 4. Representations of compact Lie groups 59 1. Maximal tori and Weyl's Integral Formula 60 2. Weights of finite-dimensional representations 62 3. Relations between representations of compact Lie groups and complex Lie algebras 63 4. Highest weight theorem 65 5. The Weyl character formula 68 Chapter 5. Representations of reductive groups 69 3 4 CONTENTS 1. The Iwasawa decomposition 69 2. Principal series representations of SL2(F ) 71 3. Parabolic subgroups of Sp4(F ) 73 4. Parabolic subgroups and induction 77 5. Discrete series representations{general properties 78 6. Maximal Tori 80 7. Discrete series representations of reductive Lie groups 82 8. Discrete series representations of reductive p-adic groups 85 9. Maximal tori and representations of finite groups of Lie type 88 10. Harish-Chandra's Philosophy of Cusp Forms 89 Bibliography 93 CHAPTER 1 Topological groups 1. Definitions and basic properties Definition 1.1. A topological group G is a group which is a topological space and has the property that the map (x; y) 7! xy−1 from G × G to G is continuous. (Equivalently, the maps (x; y) 7! xy and x 7! x−1 are continuous.) Here, G × G is given the product topology. Lemma 1.2. Let G be a topological group. Then (1) The map g 7! g−1 is a homeomorphism of G onto itself. −1 (2) Fix g0 2 G. The maps g 7! g0g, g 7! gg0, and g 7! g0gg0 are homeomorphisms of G onto itself. Example 1.3. Let F be a field that is also a topological group under addition and the nonzero elements F × in F is a toplogical group under multiplication. (Some standard examples are R, C and finite extensions of the field Qp of p-adic numbers). The set Mn(F ) n2 of n × n matrices with entries in F is homoemorphic to F . Let GLn(F ) (also denoted by GL(n; F )) be the group of invertible matrices in Mn(F ). Since GL(n; F ) is a subset of Mn(F ), we can make GLn(F ) into a topological space using the the subspace topology. The determinant map det : Mn(F ) ! F is continuous (it's a polynomial in the matrix entries). It is easy to see that the multiplication map (A; B) 7! AB from Mn(F ) × Mn(F ) ! Mn(F ) is continuous, so the restriction to GLn(F )×GLn(F ) is also continuous. Given A 2 GLn(F ), let adj A be the adjoint of A. Using the fact that A−1 = adj A= det A, we can see that inversion is continuous. Hence GLn(F ) is a topological group under matrix multiplication. If F is Hausdorff and locally compact, then GLn(F ) is Hausdorff and locally compact. (Recall that a topological space X is locally compact if given a point x 2 X there exists a compact subset U of X such that U contains an open neighbourhood of x.) More generally, if R is a commutative ring with identity, R is a topological space, and multiplication and addition are continuous maps from R × R to R, let GLn(R) = f A 2 5 6 1. TOPOLOGICAL GROUPS × × Mn(R) j det A 2 R g. Here, R is the group of units in R. Then GLn(R) is a topological group under matrix multiplication. Example 1.4. Let B(H) be the set of bounded linear operators on a Hilbert space H. (Recall that a linear operator T on H is bounded if there exists a positive constant C such that kT (v)k ≤ Ckvk for all v 2 H.) If T 2 B(H) is bijective, then T −1 2 B(H). The strong operator topology on B(H) is the weakest locally convex topology such that the evaluation map sending T to kT (v)k is continuous for each vector v 2 H. If H is infinite-dimensional, the group of bijective operators in B(H) is not a topological group with respect to the subspace topology induced by the strong operator topology on B(H). Neither the multiplication map nor the inversion map is continuous. If we take the subgroup U(H) of unitary operators on H, this is a topological group with respect to the subspace topology. To see this, fix T1, T2 2 U(H). Fix > 0 and v 2 H. Then the set V := f S 2 U(H) j k(S − T1T2(v)k < g is an open neighbourhood of T1T2 in the strong operator topology. Let 0 V = f (S1;S2) 2 U(H) j k(S1 − T1)(T2(v))k < /2; and k(S2 − T2)(v)k < /2 g: 0 0 Then V is an open neighbourhood of (T1;T2) in U(H) × U(H). Let (S1;S2) 2 V . Then k(S1S2 − T1T2)(v)k = kS1(S2 − T2)(v) + (S1 − T1)(T2(v))k ≤ kS1(S2 − T2)(v)k + k(S1 − T1)(T2(v))k = k(S2 − T2)(v)k + k(S1 − T1)(T2(v))k < /2 + /2 To see that inversion is continuous, fix T 2 U(H). Then, given v 2 H and > 0, let V = f S 2 U(H) j k(S − T −1)(v)k < g. Set w = T −1(v). Show that the image of the open neighbourhood f S0 2 U(H) j k(S0 − T )(w)k < g of T under the inversion map is a subset of V. A subgroup H of a topological group G is a topological group in the subspace topology. Example 1.5. Let (I; ≤) be a directed partially ordered set. Suppose that fGigi2I is a family of finite groups. Assume that whenever i ≤ j 2 I, there exists a homomorphism fij : Gj ! Gi such that (1) fii is the identity map. (2) fik = fij ◦ fjk for i ≤ j ≤ k 2 I. 1. DEFINITIONS AND BASIC PROPERTIES 7 Then ((Gi)i2I ; (fij)i≤j2I ) is called an inverse (or projective) system of finite groups and ho- momorphisms. Let lim G be the subset of the direct product Q G consisting of elements − i i2I i (g ) such that g = f (g ) for all i ≤ j 2 I. It is easy to see that lim G is a subgroup of i i2I i ij j − i Q i2I Gi. We give each Gi the discrete topology and then we take the product topology on Q G . Then lim G is closed in Q G . Furthermore, lim G is compact and totally dis- i2I i − i i2I i − i connected. The details are left as an exercise. Here is a simple concrete example: fix a prime i j i p, take I to be the natural numbers, Gi = Z=p Z, and, for i ≤ j, set fij(n + p Z) = n + p Z, n 2 . Then := lim =pi is known as the group of p-adic integers. (In fact, is a Z Zp − Z Z Zp ring.) Lemma 1.6. Any open subgroup of a topological group is closed. Proof. Let H be an open subgroup of a topological subgroup G. If g 2 G, then gH is S open. Hence S := g2GnH gH is open. Since H = G n S, it follows that H is closed. Lemma 1.7. If H is a subgroup of a topological group G, the closure H is a subgroup of G. If H is normal in G, then H is normal in G. Proof. Let x, y 2 H. Let U be an open neighbourhood of xy. Let SU = f (u; v) 2 G×G j uv 2 U g. Then SU is open and contains (x; y). Let U1 and U2 be open neighbourhoods of x and y, respectively, such that U1 × U2 ⊂ SU . Because x, y 2 H, Uj \ H is nonempty, j = 1, 2. Fix u 2 U1 \ H and v 2 U2 \ H. Note that uv 2 U \ H. We have shown that U \ H is nonempty for any open neighbourhood U of xy. Hence xy 2 H. Let V be an open neighbourhood of y−1. Let V 0 = f v−1 j v 2 V g. Since V 0 is open and contains y and y 2 H, the intersection V 0 \ H is nonempty. The inverse of any point in V 0 \ H belongs to V \ H. Hence V \ H is nonempty. As before, this implies that y−1 2 H. The assertion about normality is left as an exercise. Let H be a subgroup of a topological group G, and let q : G ! G=H be the canonical mapping of G onto G=H.