Frobenius reciprocity for compact groups (${\mathrm{Hom}}_G(E,{\mathrm{Ind}}_K^{G}W)\cong {\mathrm{Hom}}_K(E{|}_K,W)$)

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 $\varphi :E\to W$ produces the $G$-intertwiner $v\mapsto (g\mapsto \varphi (g^{-1}v))$. 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 $\dim {\mathrm{Hom}}_K(E{|}_K,W)$.