From Casselman's notes on representation theory.
If $p \in P$, it's not clear to me at all that $R_pf|_N$ must have compact support modulo $Q \cap N$.
To prove this, we may assume $p = 1_P$. Since $f \in \operatorname{c-Ind}_Q^P \sigma$, we have $f(qp) = \sigma(q)f(p)$ for all $q \in Q, p \in P$, and there exists a compact set $K \subset P$ such that if $f(p) \neq 0$, then $qp \in K$ for some $q \in Q$.
Now considering the restriction of $f$ to $N$, if we suppose that $f(n) \neq 0$, then again $qn \in K$ for some $q \in Q$. What we want to find is a compact set $K_0 \subseteq N$ such that whenever $f(n) \neq 0$, we have $qn \in K_0$ for some $q \in Q$ (then automatically $q \in Q \cap N$).