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CHARACTERIZING THE MOD- LOCAL LANGLANDS CORRESPONDENCE BY NILPOTENT GAMMA FACTORS
Nagoya Mathematical Journal ( IF 0.8 ) Pub Date : 2020-05-12 , DOI: 10.1017/nmj.2020.8 GILBERT MOSS
Nagoya Mathematical Journal ( IF 0.8 ) Pub Date : 2020-05-12 , DOI: 10.1017/nmj.2020.8 GILBERT MOSS
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:
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。
更新日期:2020-05-12
中文翻译:
用 NILpotent Gamma 因子表征模态 LANGLANDS 对应
让