Exercise 3, § 3

Let X be a locally compact space in which a locally compact group H operates on the right, continuously and properly, by $(x, \xi) \mapsto x\xi$ ($x \in X, \xi \in H$). Let $\pi$ be the canonical mapping $X \to X/H$. Let $\rho$ be a continuous representation of H in $\mathbf{GL}(n, \mathbf{C})$. Show that for every $b \in X/H$, there exist a neighborhood U of b in X/H and a continuous mapping r of $\overline{\pi}(U)$ into $\mathbf{GL}(n, \mathbf{C})$ such that $r(x\xi) = r(x)\rho(\xi)$ for all $x \in \overline{\pi}(U)$ and $\xi \in H$. (One may assume X/H to be compact. Let f be a continuous function $\geqslant 0$ on X, not identically zero on any orbit, and with compact support. Set

where $\beta$ denotes a left Haar measure on H. Show that, for f and U suitably chosen, $r(x) \in \mathbf{GL}(n, \mathbf{C})$ for every $x \in \overline{\pi}(U)$.