 Peter–Weyl theorem

For a compact group, the matrix coefficients of its finite-dimensional irreducible unitary representations span a dense subspace of $C(G)$ and form a complete orthogonal decomposition of $L^2(G)$ with Haar measure. If $K$ is a closed subgroup and $W$ is a finite-dimensional unitary $K$-representation, the space of L2 sections of the associated homogeneous bundle has multiplicity $\dim {\mathrm{Hom}}_K(E{|}_K,W)$ for each irreducible $G$-representation $E$. This is the compact-group form of Frobenius reciprocity and determines homogeneous-bundle multiplicities from isotropy weights.

The Peter–Weyl theorem is a fundamental result in the representation theory of compact topological groups. It describes how the regular representation of a compact group can be decomposed into irreducible representations. Here's a brief overview of the main points of the theorem: 1. **Compact Groups**: The theorem applies specifically to compact groups, which are groups that are also compact topological spaces. Examples include \(SU(n)\), \(SO(n)\), and \(U(n)\).