A note on finiteness of Tate cohomology groups

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 $\mathrm{\ell }$ – a prime number – with $\mathrm{\ell }\mathrm{\ne }p$. Let $\mathrm{\Pi }$ be a finite length ${\bar{\mathbb{F}}}_{\mathrm{\ell }}$-representation of $G(F)⋊⟨\sigma ⟩$. We show that the Tate cohomology ${\widehat{H}}^i(⟨\sigma ⟩,\mathrm{\Pi })$ is a finite length representation of $G^{\sigma }(F)$. We give an application to genericity of these Tate cohomology spaces.