We establish the existence and uniqueness of twisted exterior and symmetric square -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 we prove that these twisted local factors are compatible with the local Langlands correspondence. As a consequence, still in characteristic , we obtain a proof of the stability property of -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 -factors, -functions and -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 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 , on prouve que les facteurs tordus sont compatibles avec la correspondance de Langlands. Comme conséquence, on prouve une propriété de stabilité des facteurs 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 , fonctions et facteurs des carrés extérieur et symétrique tordus sont préservés. Finalement, on conclut avec une formule en termes de facteurs pour les mesures de Plancherel et on prouve qu’elles sont préservées sur les corps locaux proches.
Revised:
Accepted:
Published online:
Classification: 11F70, 11M38, 22E50, 22E55
Keywords: L-functions, local Langlands correspondence, close local fields
@article{AIF_2015__65_3_1105_0, author = {Ganapathy, Radhika and Lomel{\'\i}, Luis}, 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}, number = {3}, year = {2015}, doi = {10.5802/aif.2952}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2952/} }
TY - JOUR TI - On twisted exterior and symmetric square $\gamma $-factors JO - Annales de l'Institut Fourier PY - 2015 DA - 2015/// SP - 1105 EP - 1132 VL - 65 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2952/ UR - https://doi.org/10.5802/aif.2952 DO - 10.5802/aif.2952 LA - en ID - AIF_2015__65_3_1105_0 ER -
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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