Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $ B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. Denote by $i_P^G=\mathrm{Ind}_P^G(\mathbf{1})$ the smooth parabolic induction of the trivial representation. In Borel & Wallachs's "Continous Cohomology, Discrete subgroups, and Representations of Reductive Groups" it is stated in Lemma 4.5 of chapter $X$ that for $V_0,\ldots, V_m$ $G$-submodules of $V:=i_P^G$ $$ V_0 \cap (V_1+ \ldots V_m)=(V_0 \cap V_1) + \ldots + (V_0 \cap V_m)$$ holds (in the book it is in a litte more general version, but for understanding the proof it should be enough).
They state first that the equality holds obviously for $V_N$-modules (Jaquet module of V). Why??
Because the RHS is trivially contained in the LHS, the equation follows by some observations of section 3.2 in the same chapter, which are the following:
For an unramified character $\chi$ of $G$ they denote by $PS(\chi)$ the space $\mathrm{Ind}_P^G(\mathbb{C}_\phi)$ where $P$ acts $\mathbb{C}_\phi$ by $\phi=\chi\delta^{1/2}$ and $\delta$ being the modular function on $P$. Then
- The semi-simplification of $PS(\chi)_N$ as an $T$-module is isomorphic to $\bigoplus_{s \in W}\mathbb{C}_{\delta^{1/2}\cdot s\chi}$.
- $PS(\chi)$ has a finite Jordan-Höder series.
- If $\chi' \notin W(\chi)$ then $PS(\chi)$ and $PS(\chi')$ have no constituent in common
- If $s\chi \neq \chi$ for $s \neq 1, s \in W$, then $PS(\chi)$ has a unique non-zero irreducible sub representation, which is denoted by $W_\chi$.
Then by 4.
- If $s\chi \neq \chi$ for $s \neq 1, s \in W$,then $PS(\chi)$ has a multiplicity free Jordan-Holder series and each constituent is equivalent to some $W_{s\chi}$.
- An irreducible, admissible module $V$ is a constituent (resp. submodule) of some $PS(\chi)$ if and only if $V^B \neq (0)$.
But at the moment I am more or less unable to make a link from the observations to the equality above.
