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There is a claim in a proof in Casselman's notes on representation theory which I have not been able to verify. I have asked several people and nobody knows why it is true.

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The thing I can't figure out is why if $f \in \operatorname{c-Ind}_Q^{P} \sigma$, then the restriction $R_pf$ to $N$ has compact support modulo $Q \cap N$. Clearly we may assume $p = 1$.

Our hypothesis on $f$ is that there exists a compact set $\Omega \subset P$ such that $\{ p \in P : f(p) \neq 0\}$ is contained in the product set $Q. \Omega$. And we want to find a compact set $\Omega_0 \subset N$ such that $\{n \in N : f(n) \neq 0\}$ is contained in $(Q \cap N).\Omega_0$.

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  • $\begingroup$ Take $\Omega_0$ to be $\Omega \cap N$ - this is compact in $N$. Let $S$ be the support of $f$ in $P$. Assume that there exists a compact set $\Omega \subset P$ such that $S \subset Q.\Omega$. The support of the restriction of $f$ to $N$ is just $S \cap N$.Then $S \cap N \subset (Q.\Omega) \cap N = (Q \cap N).(\Omega \cap N)$. $\endgroup$
    – Not a grad student
    Commented Nov 18, 2018 at 1:15
  • $\begingroup$ How do we know that $(Q.\Omega) \cap N \subseteq (Q \cap N).(\Omega \cap N)$? $\endgroup$
    – D_S
    Commented Nov 18, 2018 at 8:28

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