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The Peter-Weyl theorem


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From Tao’s Blog: The Peter-Weyl theorem, and non-abelian Fourier analysis on compact groups.

Let G be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then G has a number of unitary representations, i.e. continuous homomorphisms ρ:GU(H) to the group U(H) of unitary operators on a Hilbert space H, equipped with the strong operator topology. In particular, one has the left-regular representation τ:GU(L2(G)), where we equip G with its normalised Haar measure μ (and the Borel σ-algebra) to form the Hilbert space L2(G), and τ is the translation operation
τ(g)f(x):=f(g1x).
We call two unitary representations ρ:GU(H) and ρ:GU(H) isomorphic if one has ρ(g)=Uρ(g)U1 for some unitary transformation U:HH, in which case we write ρρ.
Given two unitary representations ρ:GU(H) and ρ:GU(H), one can form their direct sum ρρ:GU(HH) in the obvious manner: ρρ(g)(v):=(ρ(g)v,ρ(g)v). Conversely, if a unitary representation ρ:GU(H) has a closed invariant subspace VH of H (thus ρ(g)VV for all gG), then the orthogonal complement V is also invariant, leading to a decomposition ρρVρV of ρ into the subrepresentations ρV:GU(V), ρV:GU(V). Accordingly, we will call a unitary representation ρ:GU(H) irreducible if H is nontrivial (i.e. H{0}) and there are no nontrivial invariant subspaces (i.e. no invariant subspaces other than {0} and H); the irreducible representations play a role in the subject analogous to those of prime numbers in multiplicative number theory. By the principle of infinite descent, every finite-dimensional unitary representation is then expressible (perhaps non-uniquely) as the direct sum of irreducible representations.
The Peter-Weyl theorem asserts, among other things, that the same claim is true for the regular representation:

Theorem 1 (Peter-Weyl theorem). Let G be a compact group. Then the regular representation τ:GU(L2(G)) is isomorphic to the direct sum of irreducible representations. In fact, one has τξˆGρdim(Vξ)ξ, where (ρξ)ξˆG is an enumeration of the irreducible finite-dimensional unitary representations ρξ:GU(Vξ) of G (up to isomorphism). (It is not difficult to see that such an enumeration exists.)

In the case when G is abelian, the Peter-Weyl theorem is a consequence of the Plancherel theorem; in that case, the irreducible representations are all one dimensional, and are thus indexed by the space ˆG of characters ξ:GR/Z (i.e. continuous homomorphisms into the unit circle R/Z), known as the Pontryagin dual of G. (See for instance my lecture notes on the Fourier transform.) Conversely, the Peter-Weyl theorem can be used to deduce the Plancherel theorem for compact groups, as well as other basic results in Fourier analysis on these groups, such as the Fourier inversion formula.
Because the regular representation is faithful (i.e. injective), a corollary of the Peter-Weyl theorem (and a classical theorem of Cartan) is that every compact group can be expressed as the inverse limit of Lie groups, leading to a solution to Hilbert’s fifth problem in the compact case. Furthermore, the compact case is then an important building block in the more general theory surrounding Hilbert’s fifth problem, and in particular a result of Yamabe that any locally compact group contains an open subgroup that is the inverse limit of Lie groups.
I’ve recently become interested in the theory around Hilbert’s fifth problem, due to the existence of a correspondence principle between locally compact groups and approximate groups, which play a fundamental role in arithmetic combinatorics. I hope to elaborate upon this correspondence in a subsequent post, but I will mention that versions of this principle play a crucial role in Gromov’s proof of his theorem on groups of polynomial growth (discussed previously on this blog), and in a more recent paper of Hrushovski on approximate groups (also discussed previously). It is also analogous in many ways to the more well-known Furstenberg correspondence principle between ergodic theory and combinatorics (also discussed previously).
Because of the above motivation, I have decided to write some notes on how the Peter-Weyl theorem is proven. This is utterly standard stuff in abstract harmonic analysis; these notes are primarily for my own benefit, but perhaps they may be of interest to some readers also.
0.1. Proof of the Peter-Weyl theorem Throughout these notes, G is a fixed compact group.
Let ρ:GU(H) and ρ:GU(H) be unitary representations. An (linear) equivariant map T:HH is defined to be a continuous linear transformation such that Tρ(g)=ρ(g)T for all gG.
A fundamental fact in representation theory, known as Schur’s lemma, asserts (roughly speaking) that equivariant maps cannot mix irreducible representations together unless they are isomorphic. More precisely:
Lemma 2 (Schur’s lemma for unitary representations). Suppose that ρ:GU(H) and ρ:GU(H) are irreducible unitary representations, and let T:HH be an equivariant map. Then T is either the zero transformation, or a constant multiple of an isomorphism. In particular, if ρρ, then there are no non-trivial equivariant maps between H and H.

Proof . The adjoint map T:HH of the equivariant map T is also equivariant, and thus so is TT:HH. As TT is also a bounded self-adjoint operator, we can apply the spectral theorem to it. Observe that any closed invariant subspace of TT is G-invariant, and is thus either {0} or H. By the spectral theorem, this forces TT to be a constant multiple of the identity. Similarly for TT. This forces T to either be zero or a constant multiple of a unitary map, and the claim follows. (Thanks to Frederick Goodman for this proof.)

Schur’s lemma has many foundational applications in the subject. For instance, we have the following generalisation of the well-known fact that eigenvectors of a unitary operator with distinct eigenvalues are necessarily orthogonal:
Corollary 3. Let ρV:GU(V) and ρW:GU(W) be two irreducible subrepresentations of a unitary representation ρ:GU(H). Then one either has ρVρW or VW.

Proof . Apply Schur’s lemma to the orthogonal projection from W to V.

Another application shows that finite-dimensional linear representations can be canonically identified (up to constants) with finite-dimensional unitary representations:
Corollary 4. Let ρ:GGL(V) be a linear representation on a finite-dimensional space V. Then there exists a Hermitian inner product , on V that makes this representation unitary. Furthermore, if V is irreducible, then this inner product is unique up to constants.

Proof . To show existence of the Hermitian inner product that unitarises ρ, take an arbitrary Hermitian inner product ,0 and then form the average
v,w:=Gρ(g)v,ρ(g)w0 dμ(g).
(this is the “Weyl averaging trick”, which crucially exploits compactness of G). Then one easily checks (using the fact that V is finite dimensional and thus locally compact) that , is also Hermitian, and that ρ is unitary with respect to this inner product, as desired. (This part of the argument does not use finite dimensionality.)
To show uniqueness up to constants, assume that one has two such inner products ,, , on V, and apply Schur’s lemma to the identity map between the two Hilbert spaces (V,,) and (V,,). (Here, finite dimensionality is used to establish

A third application of Schur’s lemma allows us to express the trace of a linear operator as an average:
Corollary 5. Let ρ:GGL(H) be an irreducible unitary representation on a non-trivial finite-dimensional space H, and let T:HH be a linear transformation. Then
1dim(H)trH(T)IH=Gρ(g)Tρ(g) dμ(g),
where IH:HH is the identity operator.

Proof . The right-hand side is equivariant, and hence by Schur’s lemma is a multiple of the identity. Taking traces, we see that the right-hand side also has the same trace as T. The claim follows.

Let us now consider the irreducible subrepresentations ρV:GU(V) of the left-regular representation ρ:GU(L2(G)). From Corollary 3, we know that those subrepresentations coming from different isomorphism classes in ˆG are orthogonal, so we now focus attention on those subrepresentations coming from a single class ξˆG. Define the ξ-isotypic component L2(G)ξ of the regular representation to be the finite-dimensional subspace of L2(G) spanned by the functions of the form
fξ,v,w:gv,ρξ(g)wVξ
where v,w are arbitrary vectors in Vξ. This is clearly a left-invariant subspace of L2(G) (in fact, it is bi-invariant, a point which we will return to later), and thus induces a subrepresentation of the left-regular representation. In fact, it captures precisely all the subrepresentations of the left-regular representation that are isomorphic to ρξ:
Proposition 6. Let ξˆG. Then every irreducible subrepresentation τV:GU(V) of the left-regular representation τ:GU(L2(G)) that is isomorphic to ρξ is a subrepresentation of L2(G)ξ. Conversely, L2(G)ξ is isomorphic to the direct sum ρdim(Vξ)ξ of dim(Vξ) copies of ρξ:GU(Vξ). (In particular, L2(G)ξ has dimension dim(Vξ)2).

Proof . Let τV:GU(V) be a subrepresentation of the left-regular representation that is isomorphic to ρξ. Thus, we have an equivariant isometry ι:VξL2(G) whose image is V; it has an adjoint ι:L2(G)Vξ.
Let vVξ and KL2(G). The convolution
ι(v)K(g):=Gι(v)(gh)K(h1) dμ(h)
can be re-arranged as
Gτ(g1)(ι(v))(h)¯˜K(h)dμ(h)=τ(g1)(ι(v)),˜KL2(G)=ι(ρξ(g1)v),˜KL2(G)=ρξ(g1)v,ι˜KVξ=v,ρξ(g)ι˜KVξ
where
˜K(g):=¯K(g1).
In particular, we see that ι(v)KL2(G)ξ for every K. Letting K be a sequence (or net) of approximations to the identity, we conclude that ι(v)L2(G)ξ as well, and so VL2(G)ξ, which is the first claim.
To prove the converse claim, write n:=dim(Vξ), and let e1,,en be an orthonormal basis for Vξ. Observe that we may then decompose L2(G)ξ as the direct sum of the spaces
L2(G)ξ,ei:=fξ,v,ei:vVξ
for i=1,,n. The claim follows.

From cor:3, the ξ-isotypic components L2(G)ξ for ξˆG are pairwise orthogonal, and so we can form the direct sum ξˆGL2(G)ξξˆGρdim(G)ξ, which is an invariant subspace of L2(G) that contains all the finite-dimensional irreducible subrepresentations (and hence also all the finite-dimensional representations, period). The essence of the Peter-Weyl theorem is then the assertion that this direct sum in fact occupies all of L2(G):
Proposition 7. We have L2(G)=ξˆGL2(G)ξ.

Proof . Suppose this is not the case. Taking orthogonal complements, we conclude that there exists a non-trivial fL2(G) which is orthogonal to all L2(G)ξ, and is in particular orthogonal to all finite-dimensional subrepresentations of L2(G).
Now let KL2(G) be an arbitrary self-adjoint kernel, thus ¯K(g1)=K(g) for all gG. The convolution operator T:ffK is then a self-adjoint Hilbert-Schmidt operator and is thus compact. (Here, we have crucially used the compactness of G.) By the spectral theorem, the cokernel ker(T) of this operator then splits as the direct sum of finite-dimensional eigenspaces. As T is equivariant, all these eigenspaces are invariant, and thus orthogonal to f; thus f must lie in the kernel of T, and thus fK vanishes for all self-adjoint KL2(G). Using a sequence (or net) of approximations to the identity, we conclude that f vanishes also, a contradiction.

thm:1 follows by combining this proposition with 6.

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