The split case of the Prasad–Takloo-Bighash conjecture for cuspidal representations of level zero

Chommaux, Marion; Matringe, Nadir

doi:10.5802/aif.3456

The split case of the Prasad–Takloo-Bighash conjecture for cuspidal representations of level zero
[Le cas déployé de la conjecture de Prasad–Takloo-Bighash pour les représentations cuspidales de niveau zéro]

Chommaux, Marion ¹ ; Matringe, Nadir ¹

¹ LMA, Site du Futuroscope - Téléport 2 11 Boulevard Marie et Pierre Curie Bâtiment H3- TSA 61125 86073 Poitiers Cedex 9, (France)

Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 123-153.

Résumé
Abstract

Soit $E / F$ une extension quadratique de corps locaux nonarchimédiens de caractéristique résiduelle impaire. On prouve une conjecture de Prasad et Takloo-Bighash dans le cas des représentations cuspidales de niveau zéro de ${GL}_{2 m} (F)$ . Cette conjecture caractérise la distinction pour la paire $({GL}_{2 m} (F), {GL}_{m} (E))$ selon un caractère $μ \circ det$ de ${GL}_{m} (E)$ , en termes de certaines conditions sur le paramètre de Langlands incluant une valeur spéciale de facteur epsilon. On montre aussi que l’espace des formes linéaires équivariantes vaut un lorsque $E / F$ est non ramifiée, et aussi lorsque $μ$ est modéré.

Let $E / F$ be a quadratic extension of non archimedean local fields of odd residual characteristic. We prove a conjecture of Prasad and Takloo-Bighash, in the case of cuspidal representations of depth zero of ${GL}_{2 m} (F)$ . This conjecture characterizes distinction for the pair $({GL}_{2 m} (F), {GL}_{m} (E))$ with respect to a character $μ \circ det$ of ${GL}_{m} (E)$ , in terms of certain conditions on Langlands paremeters, including an epsilon value. We also compute the multiplicity of the involved equivariant linear forms when $E / F$ is unramified, and also when $μ$ is tame. In both cases this multiplicity is at most one.

Reçu le : 2020-04-25
Révisé le : 2020-11-02
Accepté le : 2020-11-19
Publié le : 2022-07-01

DOI : 10.5802/aif.3456

Classification : 22E50, 11F70
Keywords: Cuspidal representations of level zero, Distinction, The Prasad–Takloo-Bighash conjecture
Mot clés : Représentations cuspidales de niveau zéro, Distinction, Conjecture de Prasad–Takloo-Bighash

Affiliations des auteurs :

Chommaux, Marion ¹ ; Matringe, Nadir ¹

¹ LMA, Site du Futuroscope - Téléport 2 11 Boulevard Marie et Pierre Curie Bâtiment H3- TSA 61125 86073 Poitiers Cedex 9, (France)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The split case of the {Prasad{\textendash}Takloo-Bighash} conjecture for cuspidal representations of level zero},
     journal = {Annales de l'Institut Fourier},
     pages = {123--153},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {1},
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     doi = {10.5802/aif.3456},
     language = {en},
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Chommaux, Marion; Matringe, Nadir. The split case of the Prasad–Takloo-Bighash conjecture for cuspidal representations of level zero. Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 123-153. doi : 10.5802/aif.3456. https://aif.centre-mersenne.org/articles/10.5802/aif.3456/

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