MAT 449 : Representation theory Sophie Morel December 17, 2018 Contents I Representations of topological groups5 I.1 Topological groups . .5 I.2 Haar measures . .9 I.3 Representations . 17 I.3.1 Continuous representations . 17 I.3.2 Unitary representations . 20 I.3.3 Cyclic representations . 24 I.3.4 Schur’s lemma . 25 I.3.5 Finite-dimensional representations . 27 I.4 The convolution product and the group algebra . 28 I.4.1 Convolution on L1(G) and the group algebra of G ............ 28 I.4.2 Representations of G vs representations of L1(G) ............. 33 I.4.3 Convolution on other Lp spaces . 39 II Some Gelfand theory 43 II.1 Banach algebras . 43 II.1.1 Spectrum of an element . 43 II.1.2 The Gelfand-Mazur theorem . 47 II.2 Spectrum of a Banach algebra . 48 II.3 C∗-algebras and the Gelfand-Naimark theorem . 52 II.4 The spectral theorem . 55 III The Gelfand-Raikov theorem 59 III.1 L1(G) ....................................... 59 III.2 Functions of positive type . 59 III.3 Functions of positive type and irreducible representations . 65 III.4 The convex set P1 ................................. 68 III.5 The Gelfand-Raikov theorem . 73 IV The Peter-Weyl theorem 75 IV.1 Compact operators . 75 IV.2 Semisimplicity of unitary representations of compact groups . 77 IV.3 Matrix coefficients . 80 IV.4 The Peter-Weyl theorem . 86 IV.5 Characters . 87 3 Contents IV.6 The Fourier transform . 90 IV.7 Characters and Fourier transforms . 93 V Gelfand pairs 97 V.1 Invariant and bi-invariant functions . 97 V.2 Definition of a Gelfand pair . 100 V.3 Gelfand pairs and representations . 102 V.3.1 Gelfand pairs and vectors fixed by K ................... 103 V.3.2 Gelfand pairs and multiplicity-free representations . 104 V.4 Spherical functions . 106 V.5 Spherical functions of positive type . 110 V.6 The dual space and the spherical Fourier transform . 112 V.7 The case of compact groups . 114 VI Application of Fourier analysis to random walks on groups 119 VI.1 Finite Markov chains . 119 VI.2 The Perron-Frobenius theorem and convergence of Markov chains . 123 VI.3 A criterion for ergodicity . 127 VI.4 Random walks on homogeneous spaces . 130 VI.5 Application to the Bernoulli-Laplace diffusion model . 133 VI.6 Random walks on locally compact groups . 135 VI.6.1 Setup . 135 VI.6.2 Random walks . 136 VI.6.3 Compact groups . 136 VI.6.4 Convergence of random walks with Fourier analysis . 139 VI.6.5 Random walks on noncompact groups . 141 A Urysohn’s lemma and some consequences 143 A.1 Urysohn’s lemma . 143 A.2 The Tietze extension theorem . 143 A.3 Applications . 143 B Useful things about normed vector spaces 145 B.1 The quotient norm . 145 B.2 The open mapping theorem . 146 B.3 The Hahn-Banach theorem . 146 B.4 The Banach-Alaoglu theorem . 152 B.5 The Krein-Milman theorem . 152 B.6 The Stone-Weierstrass theorem . 153 4 I Representations of topological groups I.1 Topological groups Definition I.1.1. A topological group is a topological set G with the structure of a group such that the multiplication map G×G ! G, (x; y) 7−! xy and the inversion map G ! G, x 7−! x−1 are continuous. We usually will denote the unit of G by 1 or e. Example I.1.2. - Any group with the discrete topology is a topology group. Frequently used examples include finite groups, free groups (both commutative and noncommutative) and “arithmetic” matrix groups such as GLn(Z) and SLn(Z). - The additive groups of R and C are topological groups. - The group GLn(C), with the topology given by any norm on the C-vector space Mn(C), is a topological group,1 hence so are all its subgroups if we put the induced topology on 1 them. For example S := fz 2 Cjjzj = 1g, GLn(R), SU(n), SO(n) etc. - (See problem set 1.) The additive group of Qp and the group GLn(Qp) are topological groups. Definition I.1.3. We say that a topological space X is locally compact if every point of X has a compact neighborhood. Remark I.1.4. If X is Hausdorff, this is equivalent to the fact that every point of X has a basis of compact neighborhoods. Note that we do not assume that neighborhoods of points in topological spaces are open. Notation I.1.5. Let G be a group, and let A; B ⊂ G, x 2 G and n ≥ 1. We use the following notation : xA = fxy; y 2 Ag and Ax = fyx; y 2 Ag AB = fyz; y 2 A; z 2 Bg 1See problem set 1 for a proof. 5 I Representations of topological groups An = AA : : : A (n factors) A−1 = fy−1; y 2 Ag Definition I.1.6. We say that a subset A of G is symmetric if A = A−1. Proposition I.1.7. Let G be a topological group. 1. If U is an open subset of G and A is any subset of G, then the sets UA, AU and U −1 are open. 2. If U is a neighborhood of 1 in G, then there is an open symmetric neighborhood V of 1 such that V 2 ⊂ U. 3. If H is a subgroup of G, then its closure H is also a subgroup of G. 4. If H is an open subgroup of G, then it is also closed. 5. If A and B are compact subsets of G, then the set AB is also compact. 6. Let H be a subgroup of G. Then the quotient G=H (with the quotient topology) is : a) Hausdorff if H is closed; b) Locally compact if G is locally compact; c) A topological group if H is normal. Proof. 1. For x 2 G, we denote by lx : G ! G (resp. rx : G ! G) left (resp. right) multiplication by x. We also denote by ι : G ! G the map x 7−! x−1. By the axioms for topological groups, all these maps are continuous. −1 −1 S −1 S −1 −1 Now note that U = ι (U), AU = x2A lx−1 (U) and UA = x2A rx−1 (U). So U , AU and UA are open. 2. We may assume that U is open. Let m : G×G ! G, (x; y) 7−! xy. Then m is continuous, so W := s−1(U) is open. We have (1; 1) 2 W because 12 = 1 2 U. By definition of the product topology on G × G, there exists an open subset Ω 3 1 of G such that Ω × Ω ⊂ W . We have Ω2 ⊂ U by definition of W . Let V = Ω \ Ω−1. We know that Ω−1 is open by (a), so V is open, and it is symmetric by definition. We clearly have 1 2 V and V 2 ⊂ Ω2 ⊂ U. 3. Consider the map u : G × G ! G, (x; y) 7−! xy−1; then a nonempty subset A of G is a subgroup if and only if u(A×A) ⊂ A. Alos, by the axioms of topological groups, the map u is continuous. Hence, for every Z ⊂ G × G, u(Z) ⊂ u(Z)). Applying this to H × H (whose closure is H × H), we see that H is a subgroup of G. 4. We have G = H t ((G − H)H). If H is open, then (G − H)H is also open by (a), hence H is closed. 5. The multiplication map m : G×G ! G is continuous by hypothesis. As AB = m(A×B) and A × B is compact, the set AB is also compact. 6 I.1 Topological groups 6. a) Let x; y 2 G be such that xH 6= yH. By question (a), x(G − H)y−1 is open, so its complement xHy−1 is closed. Also, by the assumption that xH 6= yH, the unit 1 is not in xHy−1. By (b), there exists a symmetric open set 1 2 U such that U 2 ⊂ G − xHy−1. Let’s show that UxH \ UyH = ?, which will prove the result because UxH (resp. UyH) is an open neighborhood of xH (resp. yH) in G=H. If UxH \ UyH 6= ?, then we can find u1; u2 2 U and h1; h2 2 H such that −1 −1 −1 −1 2 u1xh1 = u2yh2. But then xh1h2 y = u1 u2 2 xHy \U , which is not possible. b) Let xH 2 G=H. If K is a compact neighborhood of x in G, then its image in G=H is a compact neighborhood of xH in G=H. c) If H is normal, then G=H is a group. Let’s show that its multiplication is continuous. Let x; y 2 G. Any open neighborhood of xyH in G=H is of the form UxyH, with U an open neighborhood of xy in G. By the continuity of multiplication on G, there exists open neighborhoods V and W of x and y in G such that VW ⊂ U. Then VH and WH are open neighborhoods of xH and yH in G=H, and we have (VH)(WH) ⊂ UH. (Remember that, as H is normal, AH = HA for every subset A of G.) Let’s show that inversion is continuous on G=H. Let x 2 G. Any open neighborhood of x−1H in G=H is of the form UH, with U an open neighborhood of x−1 in G.