Tate Cohomology of Whittaker Lattices and Base Change of Generic Representations of GLn

Let $p$ and $l$ be two distinct odd primes, and let $n\geq 2$ 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 $\pi _{F}$ be an integral $l$-adic generic representation of $\mathrm{GL}_{n}(F)$, and let $\pi _{E}$ be the base change of $\pi _{F}$. Let $J_{l}(\pi _{F})$ (resp. $J_{l}(\pi _{E})$) be the unique generic component of the mod-$l$ reduction $r_{l}(\pi _{F})$ (resp. $r_{l}(\pi _{E})$). Assuming that $l$ does not divide $|\mathrm{GL}_{n-1}(\mathbb{F}_{q})|$, we prove that the Frobenius twist of $J_{l}(\pi _{F})$ is the unique generic subquotient of the Tate cohomology group $\widehat{H}^{0}(\mathrm{Gal}(E/F), J_{l}(\pi _{E}))$—considered as a representation of $\mathrm{GL}_{n}(F)$.