Exercise 20, § 2

Let $G$ be a locally compact group countable at infinity and let $H$ be a closed subgroup of $G$. Let $\pi$ be a unitary representation of $H$ in a Hilbert space $E$ and $\varrho$ the unitary representation of $G$ induced by $\pi$. Let $F$ denote the space of $\varrho$.

Fix Haar measures $\mu$ on $G$ and $\beta$ on $H$, and let $\kappa : G\rightarrow \mathbf{R}^*_+$ be a continuous function such that $\kappa (xh) = \Delta_H(h)\Delta_G(h)^{-1}\kappa (x)$ for $(x, h)\in G\times H$ and such that $\kappa (e) = 1$.

a) If $G/H$ admits a bounded $G$-invariant measure, and if there exists an $H$-invariant nonzero vector $x\in E$, then there exists a nonzero vector $y\in F$ which is $G$-invariant.

b) Conversely, suppose that $y\in F$ is a nonzero $G$-invariant vector of $\varrho$. Now prove that there exists a nonzero function $f\in \mathscr{L}_{\overline{\pi}}^2$ such that

for all $(g, x)\in G^2$. Deduce that $\kappa = 1$, then that the measure $\mu/\beta$ on $G/H$ is defined and bounded, and finally that $f(e)\in E$ is a nonzero $H$-invariant vector.