1 Matrix Coefficients and Schur Orthogonality

Math 222 notes for Feb. 29

Alison Miller

1 Matrix coefﬁcients and Schur Orthogonality

Last time, we stated the Peter-Weyl theorem:

Theorem 1.1 (Peter-Weyl). For any compact Lie group G,

2 ∼ M ∗ L (G) = Vρ Vρ. (1) ρ∈G^

where G^ denotes the set of (isomorphism classes of) irreducible ﬁnite-dimensional representations ρ : G GL(Vρ) of G.

We’ll be spending this week proving it. Today we’ll find the copies of V∗ V inside → ρ ρ L2(G). Let V be a finite-dimensional representation of G. We know that V is unitarizable; let (·, ·)V be a G-invariant hermitian inner product on V. (Exercise: use Schur’s lemma to prove that (·, ·) is well-defined up to scaling.) Let v1, ... , vn be an orthonormal basis for V. Our representation gives us a map ∼ ρ : G GL(V) = GLn(C), where the latter isomorphism uses the basis {vi} of V. Then for any i, j between 1 and n, we can define a function ρij : G C by: ρij(g) is the ijth 2 entry→ of the matrix ρ(g). Clearly ρij ∈ C(G) ⊂ L (G). We can also define the function ρij without picking the→ entire basis by ρij(g) = (gvj, vi).

Deﬁnition. For any v1, v2 ∈ V (not necessarily part of an orthonormal basis), the function g 7 (gv1, v2) is called an matrix coefﬁcient of G.

There’s an alternate way defining matrix coefficients without the inner product. Let → V∗ be the dual space of V, and denote the canonical bilinear pairing V∗ × V C by v∗, v 7 hv∗, vi. Then the matrix coefficients are precisely the functions of the form ∗ ∗ ∗ g 7 hv , gvi for v ∈ V and v inV . Note that the function thus defined depends→ bilinearly→ on v and v∗. Hence we can define → ∗ 2 ∗ ∗ Definition. We define a map ΦV : V ⊗ V L (G) by ΦV (v ⊗ v)(g) = hv , gvi.

→1 Taking the direct sum of all the ΦV gives us the map

M M ∗ 2 ΦV : V ⊗ V L (G) V Virred rep → We will prove that this map is an isometry, that is, it preserves the inner product (and is hence also injective). In order to do this, we first need to give an inner product on the left hand side. Let V be any irreducible representation of G. As noted above, we have a G- invariant inner product (·, ·)V on V (unique up to scaling). We can use this to obtain a G-invariant ∗ ∗ ∗ ∗ inner product on V as follows: for any v1, v2 ∈ V , there exist unique v1, v2 ∈ V such ∗ ∗ ∗ that (v, vi)V = hvi , vi for all v ∈ V and i = 1, 2. Then define (v1, v2)V∗ = (v1, v2)V . We will then define an inner product (·, ·)V∗⊗V by: ∗ ∗ ∗ ∗ (v1, v2)V∗ (v1, v2)V (v ⊗ v1, v ⊗ v2)V∗⊗V = . 1 2 dim V L ∗ This then gives us an inner product on V ⊗ V which restricts to (·, ·)V∗⊗V on each V∗ ⊗ V, and such that V∗ ⊗ V is orthogonal to (V 0)∗ ⊗ V∗ if V 0 6= V. L Showing that V ΦV is an isometry ultimately boils down to the following: Theorem 1.2 (Schur Orthogonality). Let V, W be irreducible representations of G with invariant inner products (·, ·)V and (·, ·)W respectively. Let v1, v2 ∈ V, w1, w2 ∈ W, then (gv v ) (gw w ) dg 0 V ∼ W (v1,w1)V (v2,w2)V V = W g∈G 1, 2 V 1, 2 W is if 6= and dim V if . R To prove this theorem, we will need the following version of Schur’s lemma:

Lemma 1.3. Let V, W be irreducible representations of G, and let L : V W be any linear map. L˜ = gLg−1dg = 0 V ∼ W = tr L 1 Then g∈G if 6= and dim V V otherwise. → of lemma. RThe map L˜ is a morphism of representations, since for any h ∈ G

h−1Lh˜ = (gh)−1L(gh) = L˜ Zg∈G by right-invariance of Haar measure. We now apply Schur’s lemma; this gives the result for V 6= W, and tells us that L˜ = λ1V for some λ ∈ C. To determine λ, note that L˜ = Ldg = L λ = tr L tr g∈G tr tr , so dim V .. NowR we show the orthogonality relations: 0 0 Proof of Schur orthogonality. Apply the lemma with L : W V given by L(w ) = (w , w1)W · v1. L˜ = 0 V ∼ W L˜ = tr L 1 = (v1,w1)V 1 V = W Schur’s lemma tells us that if 6= , and dim→ V V dim V V if . (Lw˜ v ) 0 V ∼ W (v1,w1)V (v2,w2)V Consider the quantity 2, 2 W. By the above this is if 6= , and is dim V .

2 On the other hand,

−1 (Lw˜ 2, v2)V = (g((g w2, w1)W · v1)w2, v2)V dg Zg∈G −1 = (g w2, w1)V (gv1, v2)Wdg Zg∈G = (v1, gv2)V (w1, gw2)Wdg. Zg∈G

As an immediate corollary we get the following.

Corollary 1.4 (Schur Orthogonality, equivalent nform). Let v ∈ V, w ∈ W, v∗ ∈ V∗, w∗ ∈ W 0. Then

∗ ∗ (ΦV (v ⊗ v), ΦW(w ⊗ w))L2 ∗ 0 V ∼ W (v,w)V (v ,w∗)V∗ is if 6= and is dim V otherwise.

This shows that ⊕V ΦV is an isometry.