4 - Finite-dimensional representations of compact groups (Peter-Weyl theorem)

Summary

In sec. 2, we have shown that all compact groups have faithful representations. For that purpose, the regular representations were constructed and examined. However, these last representations are infinite dimensional in general (i.e. when G is infinite). In particular, this does not prove the existence of irreducible representations (different from the identity in dimension 1) for compact groups.

This section is devoted mainly to the proof of the following basic result (and its consequences).

Theorem (Peter-Weyl). Let G be a compact group. For any s ≠ e in G, there exists a finite dimensional, irreducible representation π of G such that π(S) ≠ id.

Since certain compact groups have no faithful finite dimensional representations (groups with arbitrarily small subgroups are in this class when infinite), this result is the best possible. This theorem is sometimes stated in the following terms: all compact groups haveenoughfinite dimensional representations, or: all compact groups have acomplete systemof (irreducible) finite dimensional representations. As we have already seen that all finite dimensional representations of compact groups are completely reducible, the theorem will already be proved if we show that for s ≠ e in G, there exists a finite dimensional representation π with π(s) ≠ id.

The proof of the preceding theorem is based on the spectral properties of compact hermitian operators in Hilbert spaces. Let us review the main points needed.