Peter–Weyl theorem

Let $G$ be a compact Lie group .

Theorem (Peter–Weyl)

Consider the left-regular representation of $G$ on $L^2(G)$ (with respect to Haar measure). Then:

1. (Density of matrix coefficients) The complex vector space spanned by matrix coefficients of finite-dimensional continuous unitary representations of $G$ is dense in $C(G)$ in the uniform norm, and dense in $L^2(G)$ in the $L^2$-norm.

2. (Hilbert space decomposition) As a unitary representation,

   $$L^2(G)\ \cong\ \widehat{\bigoplus}_{\pi\in \widehat G}\ (\dim \pi)\, \pi,$$

   a Hilbert direct sum over the set $\widehat G$ of isomorphism classes of irreducible representations , where each irreducible $\pi$ occurs with multiplicity $\dim \pi$.

3. (Orthogonality) Matrix coefficients of inequivalent irreducibles are orthogonal in $L^2(G)$, refining the Schur orthogonality relations .

Consequences

• Every finite-dimensional continuous representation of a compact Lie group is completely reducible (compare complete reducibility ).
• Harmonic analysis on $G$ reduces to the study of its irreducibles; for connected compact $G$, these are classified by highest weights (see the highest weight theorem ).

Context

Peter–Weyl is the nonabelian analogue of Fourier series on a torus: irreducible representations play the role of characters, and their matrix coefficients play the role of exponentials.