= Peter-Weyl theorem
{c}
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 $\dim\operatorname{Hom}_K(E|_K,W)$ for each irreducible $G$-representation $E$. This is the compact-group form of Frobenius reciprocity and determines homogeneous-bundle multiplicities from isotropy weights.
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