Peter-Weyl theorem

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(g))[f])(h)=f(ghg^-1)).

This representation decomposes into the sum of for each finite irreducible unitary representation of G where 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 Elie 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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