On twisted exterior and symmetric square γ-factors
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1105-1132.

We establish the existence and uniqueness of twisted exterior and symmetric square γ-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 γ-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 γ-factors, L-functions and ε-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.

On établit l’existence et l’unicité des facteurs γ 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 γ 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 γ, fonctions L et facteurs ε des carrés extérieur et symétrique tordus sont préservés. Finalement, on conclut avec une formule en termes de facteurs γ pour les mesures de Plancherel et on prouve qu’elles sont préservées sur les corps locaux proches.

DOI: 10.5802/aif.2952
Classification: 11F70, 11M38, 22E50, 22E55
Keywords: L-functions, local Langlands correspondence, close local fields
Ganapathy, Radhika 1; Lomelí, Luis 2

1 The University of British Columbia Department of Mathematics 1984 Mathematics Road Vancouver, BC V6T 1Z2 (Canada)
2 Department of Mathematics University of Oklahoma Norman, OK 73019-3103 (USA)
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Ganapathy, Radhika; Lomelí, Luis. On twisted exterior and symmetric square $\gamma $-factors. Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1105-1132. doi : 10.5802/aif.2952. https://aif.centre-mersenne.org/articles/10.5802/aif.2952/

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