A note on finiteness of Tate cohomology groups

Abstract: Let $G$ be a reductive algebraic group defined over a non-Archimedean local field $F$ of residue characteristic $p$. Let $\sigma$ be an automorphism of $G$ of order $\ell$ -- a prime number -- with $\ell\neq p$. Let $\Pi$ be a finite length $\overline{\mathbb{F}}_\ell$-representation of $G(F)\rtimes \langle\sigma\rangle$. We show that the Tate cohomology $\widehat{H}^i(\langle\sigma\rangle, \Pi)$ is a finite length representation of $G^\sigma(F)$. We give an application to genericity of these Tate cohomology spaces.