= Frobenius reciprocity for compact groups
{c}
{title2=$\operatorname{Hom}_G(E,\operatorname{Ind}_K^G W)\cong\operatorname{Hom}_K(E|_K,W)$}
For a <compact group> $G$, closed subgroup $K$ and finite-dimensional unitary $K$-representation $W$, induced sections are functions satisfying $f(gk)=k^{-1}f(g)$. A $K$-intertwiner $\phi:E\to W$ produces the $G$-intertwiner $v\mapsto(g\mapsto\phi(g^{-1}v))$. Evaluation at the identity reverses this construction on each finite-dimensional continuous representation summand. Thus the multiplicity of an irreducible $E$ in the induced section space is $\dim\operatorname{Hom}_K(E|_K,W)$.
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