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

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 $GL_n(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $π$ of $GL_n(F)$ is uniquely determined by its collection of Rankin--Selberg gamma factors $γ(π\times \widetildeτ,X,ψ)$ as $\widetildeτ$ varies over nilpotent lifts of irreducible generic $k$-representations $τ$ of $GL_t(F)$ 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.

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