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 $\unicode[STIX]{x1D70B}$ of $GL_{n}(F)$ is uniquely determined by its collection of Rankin–Selberg gamma factors $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$ as $\widetilde{\unicode[STIX]{x1D70F}}$ varies over nilpotent lifts of irreducible generic $k$-representations $\unicode[STIX]{x1D70F}$ of $GL_{t}(F)$ for $t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$. 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.

中文翻译：

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用 NILpotent Gamma 因子表征模态 LANGLANDS 对应

让$F$做一个$p$-adic 领域并选择$k$的代数闭包$\mathbb{F}_{\ell }$， 和$\ell$不同于$p$. 我们定义了不可约泛型的“幂零提升”$k$- 的表示$GL_{n}(F)$，它采用 Artin local 中的系数$k$-代数。我们证明了一个不可约泛型$\ell$- 模块化表示$\unicode[STIX]{x1D70B}$的$GL_{n}(F)$由其 Rankin–Selberg 伽马因子的集合唯一确定$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$作为$\widetilde{\unicode[STIX]{x1D70F}}$在不可约泛型的幂零提升上变化$k$- 陈述$\unicode[STIX]{x1D70F}$的$GL_{t}(F)$为了$t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$. 这给出了 mod- 的表征$\ell$就伽马因子而言的局部朗兰兹对应，假设它可以扩展到幂零提升上的满射局部朗兰兹对应。