LECTURE 23-24: PETER-WEYL THEOREM AND ITS APPLICATIONS 1. Some Functional Analysis Let H be a (complex) Hilbert space, i.e. a (finite or infinite dimensional) vector space with an inner product, such that H is complete with respect to the induced metric jvj = hv; vi1=2. A linear operator T : H ! H is said to be bounded if there exists C > 0 such that jT vj ≤ Cjvj; 8v 2 H: Definition 1.1. Let H be a Hilbert space, and T : H ! H a bounded operator. (1) T is self-adjoint if for any v; w 2 H, hT v; wi = hv; T wi. (2) T is compact if for any bounded sequence v1; v2; ··· in H, the sequence T v1; T v2; ··· has a convergent subsequence. We say that λ 2 C is an eigenvalue of an operator T on H, with eigenvector 0 6= v 2 H, if T v = λv. We will denote the set of all eigenvalues of T by Spec(T ), and the eigenspace for λ 2 Spec(T ) by Hλ. Self-adjoint operators on Hilbert spaces are the (infinite dimensional) generalization of n × n Hermitian matrices acting on Cn. For example, we have Lemma 1.2. If T is self-adjoint, then (1) Spec(T ) ⊂ R, (2) for any λ 6= µ 2 Spec(T ), Hλ ? Hµ. Proof. If λ 2 Spec(T ) with eigenvector v, and T is self-adjoint, then λhv; vi = hT v; vi = hv; T vi = hv; λvi = λ¯hv; vi: So λ = λ¯, i.e. λ 2 R. Similarly if λ 6= µ 2 Spec(T ), 0 6= v 2 Hλ and 0 6= w 2 Hµ, then λhv; wi = hT v; wi = hv; T wi = hv; µwi = µhv; wi: So hv; wi = 0, i.e. v ? w. The following spectral theorem for compact self-adjoint operators in Hilbert space will play a crucial rule in the proof of Peter-Weyl theorem. Theorem 1.3 (The Spectral Theorem). Let T be a compact self-adjoint operator on a Hilbert space H, and denote by N(T ) = Ker(T ) the null space of T . Then (1) The set of nonzero eigenvalues is a countable set (and thus discrete). 1 2 LECTURE 23-24: PETER-WEYL THEOREM AND ITS APPLICATIONS (2) If the set Spec(T ) = fλ1; λ2; · · · g is not a finite set, then limk!1 λk = 0. (3) For each eigenvalue 0 6= λ 2 Spec(T ), dim Hλ < 1. (k) (k) (4) Denote by v1 ; ··· ; vn(k) an orthonormal basis of Hλk , where n(k) = dim Hλk . Then (k) fvj j 1 ≤ j ≤ n(k); k = 1; 2; · · · g form an orthonormal basis of N(T )?. 2. Convolution on Compact Lie Groups Let G be a compact Lie group with the normalized Haar measure dg, and C(G) the ring of continuous functions on G. Since G has total volume 1, we have inequalities kfk1 ≤ kfk2 ≤ kfk1 p for any f 2 C(G), where k · kp is the is the L norm, i.e. Z 1=p p kfkp = jf(g)j dg ; G and kfk1 = sup jf(g)j: g2G It follows that C(G) ⊂ L1(G) ⊂ L2(G) ⊂ L1(G). Recall that L2(G) is a Hilbert space, with inner product Z hf1; f2i = f1(g)f2(g)dg: G 1 Definition 2.1. For any f1 2 C(G) and f2 2 L (G), we can define the convolution by Z −1 (f1 ∗ f2)(g) := f1(gh )f2(h)dh: G Remark. Use the change of variable h 7! h−1g one can Z −1 (f1 ∗ f2)(g) = f1(h)f2(h g)dh: G Now for any φ 2 C(G) and any f 2 L1(G), we define Tφ(f) = φ ∗ f: 1 Proposition 2.2. For any φ 2 C(G) and f 2 L (G), Tφ(f) 2 C(G), and kTφ(f)k1 ≤ kφk1 · kfk1: 2 In particular, the operator Tφ is a bounded operator on L (G). LECTURE 23-24: PETER-WEYL THEOREM AND ITS APPLICATIONS 3 Proof. Since φ is uniformly continuous, there exits a neighborhood U of e such that jφ(g) − φ(kg)j < " for all g 2 G and k 2 U. It follows that for any g; g0 2 G with g0g−1 2 U, Z 0 −1 0 −1 jTφf(g) − Tφf(g )j = (φ(gh ) − φ(g h ))f(h)dh G Z −1 0 −1 −1 ≤ φ(gh ) − φ(g g gh ) jf(h)jdhj ≤ "kfk1: G So Tφf is a continuous function. Moreover, Z Z −1 kTφ(f)k1 = sup φ(gh )f(h)dh ≤ kφk1 jf(h)jdh = kφk1kfk1: g2G G G The main result in this section is Proposition 2.3. Let φ 2 C(G). Then 2 (1) The operator Tφ is a compact operator on L (G). −1 2 (2) If φ(g ) = φ(g), then Tφ is self-adjoint on L (G). Before proving this let's first remind you the classical Ascoli-Arzela theorem. Recall that a subset U ⊂ C(X) is equicontinuous if for any x 2 X and " > 0, there exists a neighborhood N of x such that jf(x) − f(y)j < " holds for all y 2 N and all f 2 U. Theorem 2.4 (Ascolli-Arzela). Let X be compact, U ⊂ C(X) a bounded and equicon- tinuous subset. Then every sequence in U has a uniformly convergent subsequence. Proof of Proposition 2.3. (1) It suffices to show that any sequence in 2 B = fTφ(f) j f 2 L (G); kfk2 ≤ 1g has a convergent subsequence. According to the fact kfk2 ≤ kfk1 and the Arzela- Ascoli theorem, it suffices to show that B is bounded and equicontinuous in (C(G); k · k1). The boundedness follows from the previous proposition. Now let's show the equicontinuity: Since G is compact, φ is uniformly continuous, i.e. 8" > 0, there exists a neighborhood U of e 2 G such that jφ(kg) − φ(g)j < " for all g 2 G and k 2 U. So if kfk2 ≤ 1, Z −1 −1 j(φ ∗ f)(kg) − (φ ∗ f)(g)j = (φ(kgh ) − φ(gh ))f(h)dh G Z −1 −1 ≤ φ(kgh ) − φ(gh ) jf(h)jdh G ≤ "kfk1 ≤ " 2 holds for all k 2 N and all f 2 L (G) with kfk2 ≤ 1. 4 LECTURE 23-24: PETER-WEYL THEOREM AND ITS APPLICATIONS (2) We have by definition Z Z −1 hTφ(f1); f2i = φ(gh )f1(h)f2(g)dgdh G G while Z Z −1 hf1;Tφ(f2)i = φ(hg )f1(h)f2(g)dgdh: G G −1 So if φ(g ) = φ(g), Tφ is self-adjoint. 3. The Peter-Weyl Theorem The right translation R(g) of G induces a linear G-action on C(G), given by (R(g)f)(x) = f(xg): Lemma 3.1. Suppose φ 2 C(G), and λ 2 Spec(Tφ). Then the λ-eigenspace Hλ is G-invariant. Proof. For any f 2 Hλ and any g 2 G, Z Z −1 −1 (TφR(g)f)(x) = φ(xh )f(hg)dh = φ(xgh )f(h)dh = R(g)(Tφf)(x) = λR(g)f(x): G G Recall that a matrix coefficient of G associated to a representation (V; π) is a function on G of the form L(π(g)v) for some v 2 V and L 2 V ∗. Lemma 3.2. A function f 2 C(G) is a matrix coefficient on G if and only if the functions R(g)f span a finite dimensional vector space. Proof. It is easy to check that if f 2 C(G)π for some finite dimensional representation π, then R(g)f 2 C(G)π for any g. It follows that R(g)f's span a vector space of dimension no more than n2, where n = dim π. Conversely, suppose the functions R(g)f span a finite dimensional vector space V , then (R; V ) is a finite dimensional representation of G, and if we define a functional L : V ! C by L(φ) = φ(e), it is clear that f 2 V and L(R(g)f) = f(g); so as a function f is a matrix coefficient of G associated to this representation. Definition 3.3. Given representation (V; π), we will denote by R(π) the linear span of of all matrix coefficients in C(G)π. Now we are ready to prove Theorem 3.4 (Peter-Weyl Theorem). Let G be a compact Lie group. Then the matrix coefficients of G are dense in C(G). LECTURE 23-24: PETER-WEYL THEOREM AND ITS APPLICATIONS 5 Proof. Let f 2 C(G) be a continuous function on G. Since G is compact, f is uniformly continuous, i.e. there exists a neighborhood U of e in G such that for any g 2 U and h 2 G, jf(g−1h) − f(h)j < "=2: R Now let φ be a nonnegative (real-valued) function supported in U such that G φ(g)dg = −1 2 1 and φ(g) = φ(g ). It follows that Tφ is a self-adjoint compact operator on L (G), and for any h 2 G, Z Z −1 −1 " j(Tφf)(h)−f(h)j = (φ(g)f(g h) − φ(g)f(h))dg ≤ φ(g)jf(g h)−f(h)jdg ≤ : G U 2 It follows by the spectral theorem that for all λ 2 Spec(Tφ), Hλ are finite dimensional 2 P for λ 6= 0, mutually orthogonal, and span L (G).