Local distinction, quadratic base change and automorphic induction for $\protect \mathrm{GL}_n$

Nadir Matringe

doi:10.5802/jtnb.1233

Local distinction, quadratic base change and automorphic induction for

{GL}_{n}

Nadir Matringe ¹

¹ Université Paris Cité, France

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 903-916.

Abstract
Résumé

Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual or conjugate-dual representations. When applied to semi-simple representations of the Weil–Deligne group $W_{F}^{'}$ of a non Archimedean local field $F$ , and further translated in terms of representations of ${GL}_{n} (F)$ via the local Langlands correspondence when $F$ has characteristic zero, it yields various statements concerning the behaviour of different types of distinction under quadratic base change and automorphic induction. When $F$ has residual characteristic different from $2$ , combining of one of the simple results that we obtain with the tiviality of conjugate-orthogonal root numbers ([8]), we recover without using the LLC a result of Serre on the parity of the Artin conductor of orthogonal representations of $W_{F}^{'}$ ([23]). On the other hand we discuss its parity for symplectic representations using the LLC and the Prasad and Takloo-Bighash conjecture.

Ce titre sophistiqué dissimule un exercice élémentaire sur la théorie de Clifford pour les sous-groupes d’indice deux et les représentations auto-duales ou conjuguées-duales. Appliqué aux représentations du groupe Weil–Deligne $W_{F}^{'}$ d’un corps local non archimédien $F$ , puis interprété en termes de représentations de ${GL}_{n} (F)$ via correspondance de Langlands locale lorsque $F$ est de caractéristique nulle, l’exercice en question établit divers énoncés concernant le comportememnt de différents types de distinction sous changement de base et induction automorphe quadratiques. Lorsque $F$ est de caractéristique résiduelle non $2$ , en combinant un des résultats simples obtenus ici avec la trivialité des valeurs centrales de facteurs epsilon des représentations de $W_{F}^{'}$ conjuguées-orthogonales ([8]), nous retrouvons sans faire appel à la correspondance de Langlands locale un résultat de Serre sur la parité du conducteur d’Artin de ces représentations ([23]). D’autre part, nous discutons cette parité pour les représentations symplectiques à l’aide de la correspondance de Langlands locale et de la conjecture dite de Prasad et Takloo-Bighash.

Received: 2021-08-18
Revised: 2022-07-16
Accepted: 2022-09-23
Published online: 2023-01-27

DOI: 10.5802/jtnb.1233

Classification: 1170, 22E50
Keywords: Représentations galoisiennes locales, représentations distinguées, conducteur d’Artin

Author's affiliations:

Nadir Matringe ¹

¹ Université Paris Cité, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Nadir Matringe},
     title = {Local distinction, quadratic base change and automorphic induction for $\protect \mathrm{GL}_n$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {903--916},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Nadir Matringe. Local distinction, quadratic base change and automorphic induction for $\protect \mathrm{GL}_n$. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 903-916. doi : 10.5802/jtnb.1233. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1233/

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