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

This is a direct extension to our paper arXiv:2507.09773; comments are welcome