Let $G$ be the rational points of a connected, reductive group over a $p$-adic field $F$. Let $S$ be a maximal split torus of $G$ with $\Delta$ a set of simple roots corresponding to a minimal parabolic. Let $(\pi,V)$ be an irreducible, admissible representation of a Levi subgroup $M$ of $G$. For $P = MN$ corresponding to some set of simple roots $\theta \subseteq \Delta$, we have the induced representation

$$I(\nu,\pi) = \operatorname{Ind}_{MN}^G \pi \otimes q^{\langle \nu + \rho, H_M(-) \rangle}$$

where $\nu$ is in the complexified real Lie algebra of $M$, and $\rho$ is half the sum of the roots of $S$ in $N$ counting multiplicity. For $w$ in the Weyl group of $S$, sending $\theta$ to $\theta' \subseteq \Delta$, we get an intertwining operator $A: I(\nu,\pi) \rightarrow I(w(\nu),w(\pi))$ defined by an integral

$$Af(g) = \int\limits_{N_w} f(\dot w^{-1}ng)dn$$

where $N_w$ is generated by the root subgroups of those roots which are made negative by $w^{-1}$, and $\dot w$ is a nice choice of representative for $w$.

So, I understand formally why this integral intertwines the action of $G$ on each space. What I don't understand and haven't yet found a reference for is how to make sense of the integral itself. It is a vector valued integral, over a function with values in the underlying space of $\pi$. Since we are dealing with smooth representations of $p$-adic groups, this space need not have any topology associated with it.

Usually, vector valued integrals for $p$-adic representations are finite sums, taken over compact sets. I don't believe this should be the case here. If one is stating everything precisely, how should we make sense out of this intertwining operator?