Families over the integral Bernstein Center and Tate cohomology of local Base change lifts for GL(n, F)
Sabyasachi Dhar, Santosh Nadimpalli
TL;DR
The paper investigates how local base change for GL$_n$ over a degree-$l$ Galois extension $E/F$ interacts with Tate cohomology in the mod-$l$ setting. It develops a framework in smooth representations in families, using the integral Bernstein center and co-Whittaker modules to interpolate base change and Langlands data, and establishes universal Rankin–Selberg gamma factors that control functional equations in families. The main result identifies the zeroth Tate cohomology subquotient of the base-changed representation with the Frobenius twist of the mod-$l$ unique generic subquotient, i.e. $J_l(\pi_F)^{(l)}$, thus removing prior hypotheses on $l$ and providing a robust local Langlands-in-families picture. The work advances understanding of how functorial lifts behave under Tate cohomology and base change, with potential implications for mod-$l$ automorphic forms and local-global compatibility in Langlands theory.
Abstract
Let $p$ and $l$ be distinct odd primes, and let $F$ be a $p$-adic field. Let $π$ be a generic smooth integral representation of ${\rm GL}_n(F)$ over an $\overline{\mathbb{Q}}_l$-vector space. Let $E$ be a finite Galois extension of $F$ with $[E:F]=l$. Let $Π$ be the base change lift of $π$ to the group ${\rm GL}_n(E)$. Let $\mathbb{W}^0(Π, ψ_E)$ be the lattice of $\overline{\mathbb{Z}}_l$-valued functions in the Whittaker model of $Π$, with respect to a standard ${\rm Gal}(E/F)$-equivaraint additive character $ψ_E:E\rightarrow \overline{\mathbb{Q}}_l^\times$. We show that the unique generic sub-quotient of the zero-th Tate cohomology group of $\mathbb{W}^0(Π, ψ_E)$ is isomorphic to the Frobenius twist of the unique generic sub-quotient of the mod-$l$ reduction of $π$. We first prove a version of this result for a family of smooth generic representations of ${\rm GL}_n(E)$ over the integral Bernstein center of ${\rm GL}_n(F)$. Our methods use the theory of Rankin-selberg convolutions and simple identities of local $γ$-factors. The results of this article remove the hypothesis that $l$ does not divide the pro-order of ${\rm GL}_{n-1}(F)$ in our previous work.