Peter-Weyl theorem

From Academic Kids

The Peter-Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Peter, in the setting of a compact Lie group, G. It generalises the significant facts about the decomposition of the regular representation of any finite group, as discovered by F. G. Frobenius and Issai Schur.

To state the theorem we need first the idea of the Hilbert space over G, L2(G); this makes sense because Haar measure exists on G. Calling it H, the group G has a unitary representation on H by acting on the left, or on the right. This implies a representation of G×G (via ρ((h,k))[f](g)=f(h-1gk)).

This representation decomposes into the sum of <math>\bar{r}\otimes r<math> for each finite irreducible unitary representation of G where <math>\bar{r}<math> is the dual representation. That is, there is a direct sum description of H with the indexation by all the classes (up to isomorphism) of irreducible unitary representations of G.

This implies immediately the structure of H for the left or right representations of G, which comes out as a direct sum of each ρ as many times as its dimension (always finite).

Structure of compact topological groups

From the theorem, one can deduce a significant general structure theorem. Let G be a compact topological group, which we assume Hausdorff. For any finite-dimensional G-invariant subspace V in L2(G), where G acts on the left, we consider the image of G in GL(V). It is closed, since G is compact, and a subgroup of the Lie group GL(V). It follows by a basic theorem (of Élie Cartan) that the image of G is a Lie group also.

If we now take the limit (in the sense of category theory) over all such spaces V, we get a result about G - because G acts faithfully on L2(G). We can say that G is an inverse limit of Lie groups. It may of course not itself be a Lie group: it may for example be a profinite group.

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