On twisted exterior and symmetric square γ-factors
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, p. 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 : https://doi.org/10.5802/aif.2952
Classification:  11F70,  11M38,  22E50,  22E55
Keywords: L-functions, local Langlands correspondence, close local fields
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     author = {Ganapathy, Radhika and Lomel\'\i , Luis},
     title = {On twisted exterior and symmetric square $\gamma $-factors},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     pages = {1105-1132},
     doi = {10.5802/aif.2952},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_3_1105_0}
}
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. http://www.numdam.org/item/AIF_2015__65_3_1105_0/

[1] Artin, Emil; Tate, John Class field theory, AMS Chelsea Publishing, Providence, RI (2009), pp. viii+194 (Reprinted with corrections from the 1967 original) | MR 2467155 | Zbl 1179.11040

[2] Asgari, Mahdi Local L-functions for split spinor groups, Canad. J. Math., Tome 54 (2002) no. 4, pp. 673-693 | Article | MR 1913914 | Zbl 1011.11034

[3] Asgari, Mahdi; Shahidi, Freydoon Generic transfer for general spin groups, Duke Math. J., Tome 132 (2006) no. 1, pp. 137-190 | Article | MR 2219256 | Zbl 1099.11028

[4] Aubert, A.-M.; Baum, P.; Plymen, R.; Solleveld, M. The local Langlands correspondence for inner forms of S L n (http://arxiv.org/abs/1305.2638)

[5] Bushnell, C. J.; Henniart, G. An upper bound on conductors for pairs, J. Number Theory, Tome 65 (1997) no. 2, pp. 183-196 | Article | MR 1462836 | Zbl 0884.11049

[6] Cogdell, J. W.; Piatetski-Shapiro, I. I.; Shahidi, F. Stability of γ-factors for quasi-split groups, J. Inst. Math. Jussieu, Tome 7 (2008) no. 1, pp. 27-66 | Article | MR 2398146 | Zbl 1175.11024

[7] Cogdell, J. W.; Shahidi, F.; Tsai, T. L. Local Langlands correspondence for GL ( n ) and the exterior and symmetric square ϵ -factors (http://arxiv.org/abs/1412.1448)

[8] Conrad, B. Reductive group schemes (SGA3 summer school)

[9] Deligne, P. Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0, Representations of reductive groups over a local field, Hermann, Paris (Travaux en Cours) (1984), pp. 119-157 | MR 771673 | Zbl 0578.12014

[10] Deligne, P.; Henniart, G. Sur la variation, par torsion, des constantes locales d’équations fonctionnelles de fonctions L, Invent. Math., Tome 64 (1981) no. 1, pp. 89-118 | Article | MR 621771 | Zbl 0442.12012

[11] Gan, Wee Teck; Takeda, Shuichiro The local Langlands conjecture for GSp (4), Ann. of Math. (2), Tome 173 (2011) no. 3, pp. 1841-1882 | Article | MR 2800725 | Zbl 1230.11063

[12] Ganapathy, Radhika The local Langlands correspondence for GSp 4 over local function fields (http://arxiv.org/abs/1305.6088)

[13] Ganapathy, Radhika The Deligne-Kazhdan philosophy and the Langlands conjectures in positive characteristic, ProQuest LLC, Ann Arbor, MI (2012), pp. 89 (Thesis (Ph.D.)–Purdue University) | MR 3103737

[14] Harris, Michael; Taylor, Richard The geometry and cohomology of some simple Shimura varieties, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 151 (2001), pp. viii+276 (With an appendix by Vladimir G. Berkovich) | MR 1876802 | Zbl 1036.11027

[15] Henniart, Guy Une preuve simple des conjectures de Langlands pour GL (n) sur un corps p-adique, Invent. Math., Tome 139 (2000) no. 2, pp. 439-455 | Article | MR 1738446 | Zbl 1048.11092

[16] Henniart, Guy Une caractérisation de la correspondance de Langlands locale pour GL (n), Bull. Soc. Math. France, Tome 130 (2002) no. 4, pp. 587-602 | Numdam | MR 1947454 | Zbl 1029.22023

[17] Henniart, Guy Correspondance de Langlands et fonctions L des carrés extérieur et symétrique, Int. Math. Res. Not. IMRN (2010) no. 4, pp. 633-673 | Article | MR 2595008 | Zbl 1184.22009

[18] Henniart, Guy; Lomelí, Luis Local-to-global extensions for GL n in non-zero characteristic: a characterization of γ F (s,π, Sym 2 ,ψ) and γ F (s,π, 2 ,ψ), Amer. J. Math., Tome 133 (2011) no. 1, pp. 187-196 | Article | MR 2752938 | Zbl 1219.11076

[19] Henniart, Guy; Lomelí, Luis Characterization of γ-factors: the Asai case, Int. Math. Res. Not. IMRN (2013) no. 17, pp. 4085-4099 | MR 3096920

[20] Henniart, Guy; Lomelí, Luis Uniqueness of Rankin-Selberg products, J. Number Theory, Tome 133 (2013) no. 12, pp. 4024-4035 | Article | MR 3165629

[21] Howe, Roger Harish-Chandra homomorphisms for p -adic groups, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, CBMS Regional Conference Series in Mathematics, Tome 59 (1985), pp. xi+76 (With the collaboration of Allen Moy) | MR 821216 | Zbl 0957.22021

[22] Kazhdan, D. Representations of groups over close local fields, J. Analyse Math., Tome 47 (1986), pp. 175-179 | Article | MR 874049 | Zbl 0634.22010

[23] Knus, Max-Albert Quadratic and Hermitian forms over rings, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 294 (1991), pp. xii+524 (With a foreword by I. Bertuccioni) | Article | MR 1096299 | Zbl 0756.11008

[24] Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre The book of involutions, American Mathematical Society, Providence, RI, American Mathematical Society Colloquium Publications, Tome 44 (1998), pp. xxii+593 (With a preface in French by J. Tits) | MR 1632779 | Zbl 0955.16001

[25] Lafforgue, Laurent Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math., Tome 147 (2002) no. 1, pp. 1-241 | Article | MR 1875184 | Zbl 1038.11075

[26] Laumon, G.; Rapoport, M.; Stuhler, U. 𝒟-elliptic sheaves and the Langlands correspondence, Invent. Math., Tome 113 (1993) no. 2, pp. 217-338 | Article | MR 1228127 | Zbl 0809.11032

[27] Lemaire, Bertrand Représentations génériques de GL N et corps locaux proches, J. Algebra, Tome 236 (2001) no. 2, pp. 549-574 | Article | MR 1813491 | Zbl 0970.22014

[28] Lomelí, Luis Alberto On automorphic L -functions in positive characteristic (http://arxiv.org/abs/1201.1585)

[29] Lomelí, Luis Alberto Functoriality for the classical groups over function fields, Int. Math. Res. Not. IMRN (2009) no. 22, pp. 4271-4335 | Article | MR 2552304 | Zbl 1210.22014

[30] Moy, Allen; Prasad, Gopal Unrefined minimal K-types for p-adic groups, Invent. Math., Tome 116 (1994) no. 1-3, pp. 393-408 | Article | MR 1253198 | Zbl 0804.22008

[31] Moy, Allen; Prasad, Gopal Jacquet functors and unrefined minimal K-types, Comment. Math. Helv., Tome 71 (1996) no. 1, pp. 98-121 | Article | MR 1371680 | Zbl 0860.22006

[32] Serre, Jean-Pierre Local fields, Springer-Verlag, New York-Berlin, Graduate Texts in Mathematics, Tome 67 (1979), pp. viii+241 (Translated from the French by Marvin Jay Greenberg) | MR 554237 | Zbl 0423.12016

[33] Shahidi, Freydoon On certain L-functions, Amer. J. Math., Tome 103 (1981) no. 2, pp. 297-355 | Article | MR 610479 | Zbl 0467.12013

[34] Shahidi, Freydoon Local coefficients and normalization of intertwining operators for GL (n), Compositio Math., Tome 48 (1983) no. 3, pp. 271-295 | Numdam | MR 700741 | Zbl 0506.22020

[35] Shahidi, Freydoon On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2), Tome 127 (1988) no. 3, pp. 547-584 | Article | MR 942520 | Zbl 0654.10029

[36] Shahidi, Freydoon A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2), Tome 132 (1990) no. 2, pp. 273-330 | Article | MR 1070599 | Zbl 0780.22005

[37] Shahidi, Freydoon On non-vanishing of twisted symmetric and exterior square L-functions for GL (n), Pacific J. Math. (1997) no. Special Issue, pp. 311-322 (Olga Taussky-Todd: in memoriam) | Article | MR 1610812 | Zbl 1001.11021

[38] Tate, J. Number theoretic background, Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Amer. Math. Soc., Providence, R.I. (Proc. Sympos. Pure Math., XXXIII) (1979), pp. 3-26 | MR 546607 | Zbl 0422.12007

[39] Yu, Jiu-Kang Bruhat-Tits theory and buildings, Ottawa lectures on admissible representations of reductive p -adic groups, Amer. Math. Soc., Providence, RI (Fields Inst. Monogr.) Tome 26 (2009), pp. 53-77 | MR 2508720 | Zbl 1179.22020