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

Ganapathy, Radhika; Lomelí, Luis

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

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 $\epsilon$ 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.

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 $\epsilon$-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.

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