Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \mathfrak a\_{M,\mathbb C}^{\ast}$, we have the induced representation


$$I(\nu,\pi) = \operatorname{Ind}\_P^G \pi q^{\langle \nu+\rho,H\_M(-)\rangle}$$
of $G$. For $w$ in the Weyl group sending $\theta$ to $\theta' \subset \Delta$, and $P' = M'N'$ corresponding to $\theta'$, we have the intertwining operator $A = A(\nu,\sigma,w): I(\nu,\pi) \rightarrow I(w(\nu),w(\pi))$ defined by 


$$A(f)(g) = \int\limits\_{N\_w} f(w^{-1}ng)dn$$


where $N\_w$ is generated by the root subgroups of positive roots made negative by $w^{-1}$. The given integration takes place in the vector space $V$, and I am trying to understand:


* What is the meaning of this vector valued integral?
* Why does the integral converge (whatever that means, depending on the answer to my first question) for $\nu$ in a suitable cone?


I had asked a [question](https://mathoverflow.net/questions/293826/making-sense-out-of-intertwining-operators-defined-by-a-vector-valued-integral/293833?noredirect=1#comment780429_293833) about the meaning of the integral before, but I am sorry to say that after all this time I still do not understand what is going on. Paul Garrett provided an answer in which he suggested that we should not think of $V$ as having the discrete topology, but having a locally convex, quasi-complete topological vector space structure (coming as a colimit of its f.d. subspaces) in which one could make sense of the integral as a [Pettis](https://en.wikipedia.org/wiki/Pettis_integral) integral. That is, we should show that there exists a vector $v = A(f)(g)$ in $V$ with the property that for all $v^{\ast}$ in the algebraic dual of $V$,


$$\langle v^{\ast},v \rangle \rangle = \int\limits\_N \langle v^{\ast}, f(w^{-1}ng)\rangle dn$$


He also suggested that taking a good maximal compact subgroup $K$ of $G$, so that we have $G = PK = P'K$, we could use the fact that elements of the induced representation are determined by their effect on $K$ to reduce to the case where the vector valued integrals are just finite sums. I still have not figured out how to do this, and wanted to ask math overflow again for help.


These intertwining operators are unfortunately still very much a mystery to me, and I have not seen any reference explain them rigorously.



Convergence of the intertwining operator as a vector valued integral
A reference for this material is Waldspurger's article "La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra," [(pdf)](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1474748003000082). See section IV.1.


Here I will make a few remarks only about the definition. All serious mathematical arguments (e.g. convergence) are contained in the above reference (in particular Theorem IV.1.1).


Assume V is admissible. The definition of
$$
\int\_N f(w^{-1}ng)\,dn=v
$$
where v∈V is that
$$
\int\_N \langle f(w^{-1}ng),\check{v}\rangle\,dn=\langle v,\check{v}\rangle
$$
for all $\check{v}$ in the contragredient of V. If we ask this for all $\check{v}$ in the algebraic dual of V, then I believe that condition is too strong.


To check the integral converges, it suffices to check the integral of $\langle f(w^{-1}ng),\check{v}\rangle$ converges for all $\check{v}$. For once you have this convergence, it defines v in the algebraic dual of the contragredient of V. It's not hard to see that v is a smooth vector, so lies in the double contragredient, which is the same as V since V is admissible.


