Supplement to the Peter-Weyl Theorem

Max Hallgren

November 2, 2015

Let G be a compact group, and let be the bi-invariant Haar measure on G.

We first want to extend the notion of a representation of G to any Hilbert space V (possiblyinfinite-dimensional). Let V be a complex Hilbert space. Then GLcont(V ) is the set of contin-uous C-linear isomorphisms V V . By the Banach isomorphism theorem, this implies thateach member of GLcont has is in fact a homeomorphism, so GLcont(V ) is a group with composi-tion. A representation of G on V is then a continuous group homomorphism pi : G GLcont(V ).

Lemma 1. Let V be a Banach space, and let pi : G GLcont(V ) be a group homomorphism.pi is continuous (and thus a representation of G on V ) if and only if the following both hold:i. The map G V, g 7 pi(g)v is continuous at 1 for any fixed v V ,ii. The map G R+, g 7 ||pi(g)|| is bounded in a neighborhood of 1 G.Proof.

In particular, we may conclude that any map pi : G U(V ) with g 7 pi(g)v continuous for anyfixed v V is a (unitary) representation.

DefineL : G EndC(L2(G)) by (L(g)f)(x) := f(g1x)

for f L2(G) and x, g G. By the invariance of the Haar measure and the continuity of themap g 7 L(g)f for each f L2(G) (Lemma 4.17 in Knapp), we know that L is a unitaryrepresentation of G on L2(G). Similarly, the map

R : G U(L2(G)) with (R(g)f))(x) := f(xg)for f L2(G) and x, g G is a unitary representation. L,R are called the left, right regularrepresentations of G on L2(G).

Suppose we are given a unitary representation pi of a compact group G on the finite-dimensionalcomplex inner product space (V, , ). We may define an inner product on End(V ) by setting

T, S := tr(T S)for T, S End(V ), where T End(V ) is the Hilbert space adjoint of T given by

Tv,w = v, T w for v, w V.Also define a representation pi of GG on End(V ) by

pi(g, h)T := pi(g) T pi(h1)for g, h G and T End(V ).

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Lemma 2. pi is a unitary representation of GG on End(V ).Proof. Let T, S End(V ) and g, h G. Because pi is a unitary representation, we havepi(g) = pi1 and pi(h1) = pi(h), so

pi(g, h)T, pi(g, h)S = tr((pi(h1) T pi(g)) (pi(g) S pi(h1)))

= tr(pi(h) T S pi(h)1) = tr(T S) = T, S.

Let ( , V) be a unitary irreducible representation of G. Define

: L2(G) End(V), f 7

Gf(g)(g)dg.

Also define : End(V) L2(G) by (T )(g) := tr(T ).

Suppose ( , V) is a finite-dimensional unitary representation for in some indexed set .Let V := V be the Hilbert sum of the V . Because the set {|| ||2 ; } is bounded(by 1), elementary Hilbert space theory tells us that, for each g G, there is a unique map(g) : V V such that | V = for each . By Plancherels theorem, we have||(g)||2 = 1 for each g G, so is a unitary representation of G on V , and each projectionmap V V is a continuous G-equivariant map.

Theorem 1. Peter-Weyl Theorem (first version) Let be a mutually inequivalent set contain-ing every finite-dimensional irreducible unitary representation of G. Then the map

: L2(G)

End(V), f 7(

Gf(g)(g)dg

)

(1)

is a G-equivariant isometric isomorphism, where G acts on L2(G) by the left regular represen-tation and G acts on each

Proof. Choose a representative (V , pi) for each , and let {v1, ..., vn()} be an orthonormalbasis for V . Let {v1, ..., v(n)} be the dual basis for V , and set pii,j(x) := pi(x)vi , vj for eachx G and 1 i, j n(). We know that , 7 is a bijection, so (dim()pikl)k,l isan orthonormal basis for End(V). For any f L2(G), we compute

(f)vi, vj =Gf(x)pii,j(x)dx = f, pii,j.

Letting f = ij = ij for some , we obtain (by the Schur orthogonality relations), we seethat (piij) = 0 for all i, j when pi, and

pi(piij)vk, vl ={

dim(Vpi), i = k and j = l0 otherwise

.

We may conclude that (piij) = (dimVpi)1Epiij , where Eij End(Vpi) takes vi to vj and all

other basis elements to 0.

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Suppose (pi, V ) is a unitary representation of G on V . For each g G we may define pi(g) End(V ) by

(pi(g))(x) := (pi(g1)x) = (pi(g)1x) for all V , x G.

Moreover, by the Riesz representation theorem, we have a C-linear isomorphism : V V , v 7 , v, so we may define an inner product on V by

, := 1,1 for , V .

(pi, V ) is a unitary representation of G with respect to this inner product.

Moreover, for any unitary representations (pi, V ), (,W ) of G, we obtain a unitary repre-sentation (pi , V W ) given on simple tensors by (pi )(g)v w := pi(g)v (g)w, whereV W is a Hilbert space with inner product extended sesquilinearly from

v w, v w := v, v w,w, v, v V,w,w W.

Lemma 3. For any irreducible representation (pi, V ) of G, there is a canonical G-intertwiningisometric isomorphism

: V V End(V ) which gives pi pi = pi.

Proof.

Theorem 2. Let be a mutually inequivalent set containing every finite-dimensional irre-ducible unitary representation of G. Then the map

: L2(G)

V V (2)

is a G-equivariant isometric isomorphism, where G acts on L2(G) by the left regular represen-tation and G acts on each V V by the above description.Proof. This follows by extending the isomorphisms from the previous lemma to all of V .

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