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

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, Volume 65 (2015) no. 3, pp. 1105-1132.

Abstract
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

Author's affiliations:

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)

@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
AU  - Ganapathy, Radhika
AU  - Lomelí, Luis
TI  - On twisted exterior and symmetric square $\gamma $-factors
JO  - Annales de l'Institut Fourier
PY  - 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/
DO  - 10.5802/aif.2952
LA  - en
ID  - AIF_2015__65_3_1105_0
ER  -

%0 Journal Article
%A Ganapathy, Radhika
%A Lomelí, Luis
%T On twisted exterior and symmetric square $\gamma $-factors
%J Annales de l'Institut Fourier
%D 2015
%P 1105-1132
%V 65
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2952/
%R 10.5802/aif.2952
%G en
%F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2952/

References
Cited by

[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 | Zbl

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

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

[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, Volume 65 (1997) no. 2, pp. 183-196 | DOI | MR | Zbl

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

[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 (Travaux en Cours), Hermann, Paris, 1984, pp. 119-157 | MR | Zbl

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

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

[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

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

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

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

[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 | DOI | MR | Zbl

[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 $γ_{F} (s, π, \land^{2}, ψ)$ , Amer. J. Math., Volume 133 (2011) no. 1, pp. 187-196 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[27] Lemaire, Bertrand Représentations génériques de ${GL}_{N}$ et corps locaux proches, J. Algebra, Volume 236 (2001) no. 2, pp. 549-574 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

[33] Shahidi, Freydoon On certain $L$ -functions, Amer. J. Math., Volume 103 (1981) no. 2, pp. 297-355 | DOI | MR | Zbl

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

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

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

[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) | DOI | MR | Zbl

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

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

Cited by Sources:

ANNALES DE L'INSTITUT FOURIER

ARTICLES TO APPEAR

BROWSE ISSUES

ABOUT THE JOURNAL

EDITORIAL BOARD

SUBMIT A PAPER

SUBSCRIPTION