CHARACTERIZING THE MOD- LOCAL LANGLANDS CORRESPONDENCE BY NILPOTENT GAMMA FACTORS

2020 ◽  
pp. 1-17
Author(s):  
GILBERT MOSS
Keyword(s):  
Algebraic Closure ◽  
Modular Representation ◽  
Langlands Correspondence ◽  
Local Langlands Correspondence

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.

scholarly journals On the Notion of Conductor in the Local Geometric Langlands Correspondence

2017 ◽  
Vol 69 (1) ◽  
pp. 107-129
Author(s):  
Masoud Kamgarpour
Keyword(s):  
Irreducible Representation ◽  
Galois Representation ◽  
Loop Group ◽  
Langlands Correspondence ◽  
Categorical Representation ◽  
Swan Conductor ◽  
Geometric Langlands ◽  
Meromorphic Connection ◽  
Local Langlands Correspondence ◽  
Geometric Analogue

AbstractUnder the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

scholarly journals Sur les -blocs de niveau zéro des groupes -adiques

2018 ◽  
Vol 154 (7) ◽  
pp. 1473-1507
Author(s):  
Thomas Lanard
Keyword(s):  
Abelian Category ◽  
Langlands Correspondence ◽  
Parabolic Induction ◽  
Langlands Parameters ◽  
Tits Building ◽  
Local Langlands Correspondence

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

scholarly journals Local Langlands correspondence and ramification for Carayol representations

2019 ◽  
Vol 155 (10) ◽  
pp. 1959-2038
Author(s):  
Colin J. Bushnell ◽  
Guy Henniart
Keyword(s):  
Langlands Correspondence ◽  
Smooth Complex ◽  
Compact Field ◽  
Weil Group ◽  
Detailed Interpretation ◽  
Simple Character ◽  
Local Langlands Correspondence ◽  
Class Of Functions ◽  
Residual Characteristic

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group ${\mathcal{W}}_{F}$. Let $\unicode[STIX]{x1D70E}$ be an irreducible smooth complex representation of ${\mathcal{W}}_{F}$, realized as the Langlands parameter of an irreducible cuspidal representation $\unicode[STIX]{x1D70B}$ of a general linear group over $F$. In an earlier paper we showed that the ramification structure of $\unicode[STIX]{x1D70E}$ is determined by the fine structure of the endo-class $\unicode[STIX]{x1D6E9}$ of the simple character contained in $\unicode[STIX]{x1D70B}$, in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}$. In this paper we concentrate on the fundamental Carayol case in which $\unicode[STIX]{x1D70E}$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $\unicode[STIX]{x1D70E}$, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$, as $\unicode[STIX]{x1D6EF}$ varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of $\unicode[STIX]{x1D70E}$ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.

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