On Tate cohomology and base change of representations of $D^{\times }$

Abstract

Let $F$ be a non-Archimedean local field with residue characteristic $p$, and let $D$ be a central $F$-division algebra of degree $d$. Let $E/F$ be a finite Galois extension of prime degree $\ell$, where $\ell \ne p$ and $\ell$ does not divide $d$. Set $D_E=D{\otimes }_{F}E$. Let $\mathcal{K}$ be the maximal unramified extension of ${\mathbb{Q}}_{\ell }$ in ${\overline{\mathbb{Q}}}_{\ell }$. In this article, we explicitly compute the Tate cohomology groups of absolutely irreducible, integral, depth-zero, $\mathcal{K}$-representations of $D_E^{\times }$ in the context of local base change lifting, which verifies a conjecture of Treumann--Venkatesh on mod-$\ell$ functoriality.

9 pages, Comments are welcome