Proposition 1

Let f be a continuous real-valued function on X whose support S has compact intersection with the saturation of every compact subset of X.

a) For every $x \in X$, the function $\xi \mapsto f(x\xi)$ on H belongs to $\mathscr{K}(H)$; one sets

b) The function $f^\chi$ is continuous, is zero outside SH, and satisfies $f^\chi(x\xi) = \chi(\xi)f^\chi(x)$.

c) If g is a continuous real-valued function on X and satisfies $g(x\xi) = \chi(\xi)g(x)$, then $(fg)^\chi = f^1g$ ($f^1$ being given by the formula (1) with $\chi$ replaced by the representation $\xi \mapsto 1$ of H in $\mathbf{R}_+^*$).

d) If $\eta \in \mathbf{H}$, then $(\delta(\eta)f)^{\chi} = \chi(\eta)\Delta_{\mathbf{H}}(\eta)^{-1}f^{\chi}$.

Let $x_0 \in X$ and let $V$ be a compact neighborhood of $x_0$ in $X$. The set of $\xi \in \mathbf{H}$ such that $V\xi$ intersects $S$ is also the set of $\xi \in \mathbf{H}$ such that $V\xi$ intersects $S \cap VH$, hence is compact in $\mathbf{H}$ since $S \cap VH$ is compact and $\mathbf{H}$ operates properly in $X$ (GT, III, §4, No. 5, Th. 1); then Lemma 1 of §1, No. 1 proves a) and the continuity of $f^{\chi}$. The rest of b) is obvious. Finally, c) and d) result from the following calculations: