Let $G$ be a unimodular topological group of totally disconnected type, and $H$ a closed subgroup of $G$. Let $\delta_H$ be the modular character of $H$, and let $dh$ be a right Haar measure on $H$. Let $C_c^{\infty}(G)$ be the space of locally constant complex valued functions of compact support, and let $\textrm{c-Ind}_H^G(\delta_H)$ be the space of functions $F: G \rightarrow \mathbb{C}$ with the following properties:

1 . There exists an open compact subgroup $K$ (depending on $F$) such that $F(gk) = F(g)$ for all $g \in G, k \in K$.

2 . For all $g \in G, h \in H$, we have $F(hg) = \delta_H(h)F(g)$.

3 . There exists a compact set $\Omega \subseteq G$ (depending on $F$) such that $F(g) \neq 0$ for all $g$ outside the product set $H.\Omega$.

I'm reading Casselman's notes on representation theory and trying to understand the proof of why normalized parabolic induction commutes with the contragredient. My lack of understanding comes down to the following proposition:

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It is easy to see that for $f \in C_c^{\infty}(G)$, $\mathcal P_{\delta}(f)$ satisfies the first two properties. For the third, if $\Omega$ is the support of $f$, and $\mathcal P_{\delta}(f)(g) \neq 0$, then there must exist an $h \in H$ such that $f(hg) \neq 0$, so $hg \in \Omega$, hence $g \in h^{-1}\Omega \subseteq H.\Omega$.

What I'm having trouble with is seeing why $\mathcal P_{\delta}$ is surjective. I do not see how I would go about constructing a suitable function $f \in C_c^{\infty}(G)$ which maps to a given $F \in \textrm{c-Ind}_H^G(\delta_H)$. In general, I have a lot of trouble verifying that functions can be expressed as integrals of other functions. I would appreciate any hint for how to do this.


For each $K$, you can show it for the $K$-fixed vectors in $\textrm{c-Ind}_H^G(\delta_H)$. If $f$ is $K$-fixed and has support $H\Omega K$ write $\Omega = \coprod g_iK$. Modulo $H$,$f$ takes finitely many values $f(g_{i}K)$, so for each $i$ define $r_i(g_{i}K)=f(g_{i})$, and $r_i(x)=0$ for $x \not\in g_iK$. Applying $\mathcal P_{\delta}$ to the constant function $r_i$ just returns the "same" function except it now transforms according to $\delta_H$ on the left, and also is scaled by the measure of $\{h: hg_iK \in g_iK\}$. After normalizing the $r_i$ to account for that, the sum of the $\mathcal P_{\delta}(r_i)$ should give $f$.


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