Jacquet modules of Tate cohomology and base change lifting

Let $G$ be a connected reductive group defined over a non-Archimedean local field $F$ of residue characteristic $p$. Let $\ell$ be a prime number distinct from $p$. Let $E$ be a cyclic Galois extension of $F$ with $[E:F]=\ell$. Let $\Pi$ be a finite length $\overline{\mathbb{F}}_\ell$-representation (or an $\ell$-modular representation) of $G(E)\rtimes {\rm Gal}(E/F)$. In this context, we prove a conjecture of Treumann and Venkatesh which predicts that the Tate cohomology groups $\widehat{H}^i({\rm Gal}(E/F), \Pi)$ are finite length representations of $G(F)$. We discuss the explicit computation of these Tate cohomology groups when $G$ is ${\rm GL}_n$ and $\Pi$ is obtained as a base change lifting of a depth-zero cuspidal representation of ${\rm GL}_n(F)$. The primary novelty from our previous work is that we treat the case where $\Pi$ is possibly non-cuspidal. We also study the ${\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$-Tate cohomology groups of the mod-$\ell$ reduction of the unipotent cuspidal representation of ${\rm Sp}_4(\mathbb{F}_{q^\ell})$.