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 $\pi$ of $GL_n(F)$ is uniquely determined by its collection of Rankin--Selberg gamma factors $\gamma(\pi\times \widetilde{\tau},X,\psi)$ as $\widetilde{\tau}$ varies over nilpotent lifts of irreducible generic $k$-representations $\tau$ 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.