Exercise 20, § 4

Let H be a locally compact topological group endowed with a left Haar measure $\nu$, and G a compact subgroup of H. Let $\varrho$ be an irreducible unitary representation of H in a Hilbert space E.

Assume that the multiplicity of $\pi$ in the restriction of $\varrho$ to G is finite for every irreducible representation $\pi \in \widehat{G}$. We identify a function on G with a measure with compact support on H.

a) For every $\pi \in \widehat{G}$, the endomorphism $\varrho (f*\chi_{\pi})$ of E is of finite rank.

b) For every $f\in L^1$(H), the endomorphism $\varrho (f)$ is compact.