Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi_1$ and $\pi_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows?
$$E_2^{p,q} = \operatorname H^p(G/H,\operatorname{Ext}_H^q(\pi_1,\pi_2)) \implies \operatorname{Ext}_G^{p+q}(\pi_1,\pi_2)$$
Here $\operatorname H^{\bullet}(G/H,\cdot) \simeq \operatorname{Ext}_{G/H}^{\bullet}(\mathbb C,\cdot)$ denotes the group cohomology in the context of smooth representations.
I did find a paper On the Continuous (Co) Homology of Locally Profinite Groups and the Künneth Theorem by Hitta proving the existence of a Hochschild–Serre spectral sequence in the case $\pi_1 \simeq \mathbb C$ (Theorem 2 loc. cit.). But what about arbitrary smooth $\pi_1$?
Best Answer
I found a way to prove the existence of such a spectral sequence, at least when $H$ is open in $G$. In order not to leave this question unanswered, let me write things down.
If $G$ is a locally profinite group, let $\mathcal R(G)$ denote the category of smooth representations of $G$. If $V$ is an abstract representation of $G$, I will write $V^{G-\mathrm{sm}}$ for the subspace of smooth vectors. Thus $V^{G-\mathrm{sm}}\in \mathcal R(G)$.
Let $(\pi_1,V) \in \mathcal R(G)$. The functor $\mathcal R(G) \to \mathcal R(G)$ mapping $(\pi_2,W)$ to $\mathrm{Hom}(\pi_1,\pi_2)^{G-\mathrm{sm}}$ is exact and preserves injectives, see for instance the theorem in the section III.1.15 of the book by Renard and Schwartz. Here, the $\mathrm{Hom}$ space consists of $\mathbb C$-linear morphisms $V\rightarrow W$, and the $G$-action is given by $$gf: v \mapsto g\cdot f(g^{-1}\cdot v).$$ If we compose this functor with taking the $G$-invariants, we obtain the functor $\mathcal R(G) \to \mathrm{Ab}$ given by $(\pi_2,W) \mapsto \mathrm{Hom}_G(\pi_1,\pi_2)$. Thus, we deduce that $$\mathrm{Ext}^i_G(\pi_1,\pi_2) = \mathrm H^i(G,\mathrm{Hom}(\pi_1,\pi_2)^{G-\mathrm{sm}}).$$ Now, by the paper of Hitta which I refer to in my question, we have a Hochschild-Serre spectral sequence computing the right-hand side. $$E_2^{i,j} = \mathrm H^i(G/H,\mathrm H^j(H,\mathrm{Hom}(\pi_1,\pi_2)^{G-\mathrm{sm}})) \implies \mathrm H^i(G,\mathrm{Hom}(\pi_1,\pi_2)^{G-\mathrm{sm}})$$
Inside the $E_2^{i,j}$ term, one must be a litte bit careful. Namely, we want to replace $\mathrm H^j(H,\mathrm{Hom}(\pi_1,\pi_2)^{G-\mathrm{sm}})$, seen as a representation of $H$ by restriction, with $\mathrm{Ext}_H^j(\pi_{1|H},\pi_{2|H}) \simeq \mathrm H^j(H,\mathrm{Hom}(\pi_{1|H},\pi_{2|H})^{H-\mathrm{sm}})$.
But $\mathrm{Hom}(\pi_1,\pi_2)^{G-\mathrm{sm}}_{|H}$ consists of all linear morphisms $f:V\to W$ such that there exists $K \subset G$ an open compact subgroup with $f(k\cdot) = k\cdot f(\cdot)$ for all $k \in K$, and equipped with the action of $H$ restricted from $G$.
And $\mathrm{Hom}(\pi_{1|H},\pi_{2|H})^{H-\mathrm{sm}}$ consists of all linear morphisms $f:V\to W$ such that there exists $K' \subset H$ an open compact subgroup with $f(k'\cdot) = k'\cdot f(\cdot)$ for all $k' \in K'$, and equipped with the natural action of $H$.
Clearly the former is a subspace of the latter. If we assume that $H$ is open, then any $K'$ is also open compact in $G$, so that the two spaces are equal. Then, the Hochschild-Serre spectral sequence above is the desired result.