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

Ganapathy, Radhika; Lomelí, Luis

doi:10.5802/aif.2952

On twisted exterior and symmetric square

γ

-factors
[Sur les facteurs

γ

des carres exterieurs et symetriques tordus]

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)

Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1105-1132.

Abstract (VO)
Résumé

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
Mots-clés : Fonctions L, correspondance de Langlands locale, corps locaux proches

Affiliations des auteurs :

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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     title = {On twisted exterior and symmetric square $\gamma $-factors},
     journal = {Annales de l'Institut Fourier},
     pages = {1105--1132},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
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Ganapathy, Radhika; Lomelí, Luis. On twisted exterior and symmetric square $\gamma $-factors. Annales de l'Institut Fourier, Tome 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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