Families over the integral Bernstein Center and Tate cohomology of local Base change lifts for GL(n, F)

Let $p$ and $l$ be distinct odd primes, and let $F$ be a $p$-adic field. Let $\pi$ be a generic smooth integral representation of ${\mathrm{G}\mathrm{L}}_n(F)$ over an ${\bar{\mathbb{Q}}}_l$-vector space. Let $E$ be a finite Galois extension of $F$ with $[E:F]=l$. Let $\mathrm{\Pi }$ be the base change lift of $\pi$ to the group ${\mathrm{G}\mathrm{L}}_n(E)$. Let ${\mathbb{W}}^0(\mathrm{\Pi },{\psi }_E)$ be the lattice of ${\bar{\mathbb{Z}}}_l$-valued functions in the Whittaker model of $\mathrm{\Pi }$, with respect to a standard $\mathrm{G}\mathrm{a}\mathrm{l}(E\mathrm{/}F)$-equivaraint additive character ${\psi }_E:E\to {\bar{\mathbb{Q}}}_l^{\times }$. We show that the unique generic sub-quotient of the zero-th Tate cohomology group of ${\mathbb{W}}^0(\mathrm{\Pi },{\psi }_E)$ is isomorphic to the Frobenius twist of the unique generic sub-quotient of the mod-$l$ reduction of $\pi$. We first prove a version of this result for a family of smooth generic representations of ${\mathrm{G}\mathrm{L}}_n(E)$ over the integral Bernstein center of ${\mathrm{G}\mathrm{L}}_n(F)$. Our methods use the theory of Rankin-selberg convolutions and simple identities of local $\gamma$-factors. The results of this article remove the hypothesis that $l$ does not divide the pro-order of ${\mathrm{G}\mathrm{L}}_{n-1}(F)$ in our previous work.