Characterizing the mod-$\ell$ local Langlands correspondence by nilpotent gamma factors

Abstract

Let $F$ be a $p$-adic field and choose $k$ an algebraic closure of ${\mathbb{F}}_{\ell }$, with $\ell$ different from $p$. We define ``nilpotent lifts'' of irreducible generic $k$-representations of $G{L}_n(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $\pi$ of $G{L}_n(F)$ is uniquely determined by its collection of Rankin--Selberg gamma factors $\gamma (\pi \times \tilde{\tau },X,\psi )$ as $\tilde{\tau }$ varies over nilpotent lifts of irreducible generic $k$-representations $\tau$ of $G{L}_t(F)$ for $t=1,\dots ,\lfloor \frac{n}{2}\rfloor$. This gives a characterization of the mod-$\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

Definition of "nilpotent lift" revised to include all Artin local k-algebras. Proof of main theorem simplified