On twisted exterior and symmetric square $\gamma $-factors

[Sur les facteurs $\gamma $ des carres exterieurs et symetriques tordus]

Ganapathy, Radhika ; Lomelí, Luis

Annales de l'Institut Fourier, Tome 65 (2015), p. 1105-1132 / Harvested from Numdam

Résumé

On établit l’existence et l’unicité des facteurs $\gamma $ des carrés extérieurs et symétriques tordus en caractéristique positive en étudiant le sous groupe de Siegel Lévi d’un groupe spinoriel généralisé. La théorie en caractéristique zéro est due à Shahidi. En caractéristique $p$, on prouve que les facteurs tordus sont compatibles avec la correspondance de Langlands. Comme conséquence, on prouve une propriété de stabilité des facteurs $\gamma $ tordus par un caractère assez ramifié. De plus, on utilise les résultats de compatibilité des coefficients locaux de Langlands-Shahidi avec la philosophie de Deligne-Kazhdan sur les corps locaux proches et on prouve que les facteurs $\gamma $, fonctions $L$ et facteurs $\varepsilon $ des carrés extérieur et symétrique tordus sont préservés. Finalement, on conclut avec une formule en termes de facteurs $\gamma $ pour les mesures de Plancherel et on prouve qu’elles sont préservées sur les corps locaux proches.

Abstract

We establish the existence and uniqueness of twisted exterior and symmetric square $\gamma $-factors in positive characteristic by studying the Siegel Levi case of generalized spinor groups. The corresponding theory in characteristic zero is due to Shahidi. In addition, in characteristic $p$ we prove that these twisted local factors are compatible with the local Langlands correspondence. As a consequence, still in characteristic $p$, we obtain a proof of the stability property of $\gamma $-factors under twists by highly ramified characters. Next we use the results on the compatibility of the Langlands-Shahidi local coefficients with the Deligne-Kazhdan theory over close local fields to show that the twisted symmetric and exterior square $\gamma $-factors, $L$-functions and $\varepsilon $-factors are preserved. Furthermore, we obtain a formula for Plancherel measures in terms of local factors and we also show that they are preserved over close local fields.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2952
Classification:  11F70,  11M38,  22E50,  22E55
Mots clés: Fonctions L, correspondance de Langlands locale, corps locaux proches

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