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 of a non Archimedean local field , and further translated in terms of representations of via the local Langlands correspondence when has characteristic zero, it yields various statements concerning the behaviour of different types of distinction under quadratic base change and automorphic induction. When has residual characteristic different from , 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 ([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 d’un corps local non archimédien , puis interprété en termes de représentations de via correspondance de Langlands locale lorsque 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 est de caractéristique résiduelle non , en combinant un des résultats simples obtenus ici avec la trivialité des valeurs centrales de facteurs epsilon des représentations de 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.
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Accepted:
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Keywords: Représentations galoisiennes locales, représentations distinguées, conducteur d’Artin

@article{JTNB_2022__34_3_903_0, 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}, volume = {34}, number = {3}, year = {2022}, doi = {10.5802/jtnb.1233}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1233/} }
TY - JOUR AU - Nadir Matringe TI - Local distinction, quadratic base change and automorphic induction for $\protect \mathrm{GL}_n$ JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 903 EP - 916 VL - 34 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1233/ DO - 10.5802/jtnb.1233 LA - en ID - JTNB_2022__34_3_903_0 ER -
%0 Journal Article %A Nadir Matringe %T Local distinction, quadratic base change and automorphic induction for $\protect \mathrm{GL}_n$ %J Journal de théorie des nombres de Bordeaux %D 2022 %P 903-916 %V 34 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1233/ %R 10.5802/jtnb.1233 %G en %F JTNB_2022__34_3_903_0
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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