Tate cohomology of Whittaker lattices and base change of generic representations of ${\mathrm{GL}}_n$

Abstract

Let $p$ and $l$ be distinct odd primes and let $n\ge 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let ${\overline{\pi }}_F$ be a generic mod-$l$ representation of ${\mathrm{GL}}_n(F)$ and let ${\pi }_F$ be an $l$-adic lift of ${\overline{\pi }}_F$. Let ${\mathbb{W}}^0({\pi }_E,{\psi }_E)$ be the integral Whittaker model of ${\pi }_E$, i.e., the lattice of ${\overline{\mathbb{Z}}}_l$-valued functions in the Whittaker model of ${\pi }_E$. Assuming that $l$ does not divide $|{\mathrm{GL}}_{n-1}({\mathbb{F}}_q)|$, we prove that the Frobenius twist of ${\overline{\pi }}_F$ is a $G_n(F)$ sub-quotient of the Tate cohomology group ${\hat{H}}^0(\mathrm{Gal}(E/F),{\mathbb{W}}^0({\pi }_E,{\psi }_E))$.

A revised version, final version to appear in IMRN