Exercise 15, § 9

Let G be a locally compact topological group operating properly on a topological space E; let $\rho$ be a representation of G on a finite dimensional real vector space V. Denote by $dg$ a right Haar measure on G.

a) Let $\mathscr{P}$ be the set of subsets A of E having the following property: there exists an open covering $(U_{\alpha})_{\alpha\in I}$ of E such that, for all $\alpha \in I$, the set of $g\in G$ such that $gA\cap U_{\alpha}\not=\emptyset$ is relatively compact in G.

Prove that, for every continuous function $f: E\rightarrow V$ whose support belongs to $\mathscr{P}$, the map $x \rightarrow \int_G\rho (g)^{-1}f(gx)dg$ is a continuous map from E to V, compatible with the operations of G.

b) Make one of the following assumptions:

(i) the space $E/G$ is regular;

(ii) there exists an open subset U of E and a compact subset K of G such that E = GU and $gU\cap U =\emptyset$ for $g \notin K$.

Show that every point $x$ in E has a neighbourhood belonging to $\mathscr{P}$ (in case (i), take $A = V\cap W$, where V is a neighbourhood of $x$ such that $gV\cap V =\emptyset$ for $g$ outside a compact subset of G, and W to be a closed neighbourhood of $Gx$ stable under G and contained in GV).

Every point in E has an open neighbourhood stable under G and satisfying (ii). c) Assume in addition that E is completely regular. Let $x\in E, v\in V$ be such that the fixer of $x$ is contained in that of $v$. Prove that there exists a continuous map $F : E\rightarrow V$, compatible with the operations of G, such that $F(x) =v$. (Let $\mathscr{F}$ be the space of continuous numerical functions on E whose support belongs to $\mathscr{P}$, and let $u:\mathscr{F}\rightarrow V$ be the map $\alpha \rightarrow \int_G\alpha (gx)\rho (g^{-1}).v dg$. Let C be a convex neighbourhood of $v$ in V; construct a neighbourhood A of $x$ belonging to $\mathscr{P}$ such that $gx\in A$ implies that $\rho (g^{-1}).v\in C$, and a function $\alpha$ on E, with support in A, such that $\alpha (x)\not= 0$ and $\int_G\alpha (gx)dg= 1$. Show that $u(\alpha )$ belongs to C and deduce that $v\in$ Im $u.$)