Tate cohomology and local base change of generic representations of ${\mathrm{G}\mathrm{L}}_3$ -- non-banal case

Let $F$ be a finite extension of ${\mathbb{Q}}_p$, and let $E$ be a finite Galois extension of $F$ with degree of extension $l$, where $l$ and $p$ are distinct odd primes. Let ${\pi }_F$ be an integral, $l$-adic generic representation of ${\mathrm{G}\mathrm{L}}_3(F)$, and let ${\pi }_E$ be the base change lifting of ${\pi }_F$ to ${\mathrm{G}\mathrm{L}}_3(E)$. Let $J_l({\pi }_F)$ (resp. $J_l({\pi }_E)$) be the unique generic sub-quotient of the mod-$l$ reduction of ${\pi }_F$ (resp. ${\pi }_E$). In this article, using the local converse theorem over local Artinian ${\bar{\mathbb{F}}}_l$-algebras, we prove that the Frobenius twist of $J_l({\pi }_F)$ is isomorphic to the Tate cohomology group ${\widehat{H}}^0(\mathrm{G}\mathrm{a}\mathrm{l}(E\mathrm{/}F),J_l({\pi }_E))$. The result of this article removes the hypothesis that the prime $l$ does not divide the pro-order of ${\mathrm{G}\mathrm{L}}_2(F)$.