Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T15:24:55.122Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE LOCAL LANGLANDS CORRESPONDENCE FOR SPLIT CLASSICAL GROUPS OVER LOCAL FUNCTION FIELDS

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  30 September 2015

Radhika Ganapathy  and
Sandeep Varma
Radhika Ganapathy
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, Canada (radhika@math.tifr.res.in)
Sandeep Varma
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai, India (sandeepvarmav@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

Keywords

local Langlands correspondenceclose local fieldsendoscopy

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 5 , November 2017 , pp. 987 - 1074
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, J. D., Refined anisotropic K-types and supercuspidal representations, Pacific J. Math. 185(1) (1998), 1–32.CrossRefGoogle Scholar
Adler, J. D. and DeBacker, S., Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group, Michigan Math. J. 50(2) (2002), 263–286.Google Scholar
Adler, J. D. and DeBacker, S., Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, J. Reine Angew. Math. 575 (2004), 1–35.Google Scholar
Adler, J. D. and Korman, J., The local character expansion near a tame, semisimple element, Amer. J. Math. 129(2) (2007), 381–403.Google Scholar
Adler, J. D. and Roche, A., An intertwining result for p-adic groups, Canad. J. Math. 52(3) (2000), 449–467.Google Scholar
Arthur, J., On local character relations, Selecta Math. (N.S.) 2(4) (1996), 501–579.Google Scholar
Arthur, J., Orthogonal and symplectic groups, in The Endoscopic Classification of Representations, American Mathematical Society Colloquium Publications, Volume 61 (American Mathematical Society, Providence, RI, 2013).Google Scholar
Aubert, A.-M., Baum, P., Plymen, R. and Solleveld, M., The local Langlands correspondence for inner forms of $SL_{n}$ . arXiv:1305.2638, March 2014.Google Scholar
Badulescu, A. I., Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle, Ann. Sci. Éc. Norm. Supér. (4) 35(5) (2002), 695–747.Google Scholar
Borel, A., Automorphic L-functions, in Automorphic Forms, Representations and L-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, OR, 1977, Part 2, Proceedings of Symposia in Pure Mathematics, Volume XXXIII, pp. 27–61 (American Mathematical Society, Providence, RI, 1979).Google Scholar
Broussous, P. and Lemaire, B., Building of GL(m,D) and centralizers, Transform. Groups 7(1) (2002), 15–50.CrossRefGoogle Scholar
Bushnell, C. J. and Henniart, G., An upper bound on conductors for pairs, J. Number Theory 65(2) (1997), 183–196.Google Scholar
Chaput, P.-E. and Romagny, M., On the adjoint quotient of Chevalley groups over arbitrary base schemes, J. Inst. Math. Jussieu 9(4) (2010), 673–704.CrossRefGoogle Scholar
Cluckers, R., Cunningham, C., Gordon, J. and Spice, L., On the computability of some positive-depth supercuspidal characters near the identity, Represent. Theory 15 (2011), 531–567.Google Scholar
Cluckers, R., Hales, T. and Loeser, F., Transfer principle for the fundamental lemma, in On the Stabilization of the Trace Formula, Stab. Trace Formula Shimura Var. Arith. Appl., Volume 1, pp. 309–347 (International Press, Somerville, MA, 2011).Google Scholar
Cogdell, J. W., Kim, H. H., Piatetski-Shapiro, I. I. and Shahidi, F., Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163–233.Google Scholar
Cogdell, J. W., Shahidi, F. and Tsai, T.-L., Local Langlands correspondence for $GL_{n}$ and the exterior and symmetric square $\unicode[STIX]{x1D700}$ -factors. ArXiv e-prints, December 2014.Google Scholar
Conrad, B., Reductive group schemes (SGA3 summer school). notes.Google Scholar
Conrad, K., The different ideal. Expository papers/Lecture notes. Available at: http://www.math.uconn. edu/∼kconrad/blurbs/gradnumthy/different.pdf, 2009.Google Scholar
Debacker, S., Homogeneity results for invariant distributions of a reductive p-adic group, Ann. Sci. Éc. Norm. Supér. (4) 35(3) (2002), 391–422.Google Scholar
DeBacker, S., Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. (2) 156(1) (2002), 295–332.CrossRefGoogle Scholar
Deligne, P., Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0, in Representations of Reductive Groups over a Local Field, Travaux en Cours, pp. 119–157 (Hermann, Paris, 1984).Google Scholar
Durfee, W. H., Congruence of quadratic forms over valuation rings, Duke Math. J. 11 (1944), 687–697.CrossRefGoogle Scholar
Ferrari, A., Théorème de l’indice et formule des traces, Manuscripta Math. 124(3) (2007), 363–390.Google Scholar
Gan, W. T., Gross, B. H. and Prasad, D., Symplectic local root numbers, central critical l-values, and restriction problems in the representation theory of classical groups, Astérisque, pp. 1–109. (2012).Google Scholar
Gan, W. T. and Lomelí, L., Globalization of supercuspidal representations over function fields and applications. Preprint available at http://www.math.nus.edu.sg/∼matgwt/globalization-2015.pdf, 2015.Google Scholar
Gan, W. T. and Takeda, S., The local Langlands conjecture for GSp(4), Ann. of Math. (2) 173(3) (2011), 1841–1882.Google Scholar
Gan, W. T. and Yu, J.-K., Group schemes and local densities, Duke Math. J. 105(3) (2000), 497–524.CrossRefGoogle Scholar
Ganapathy, R., The local Langlands correspondence for GSp4 over local function fields, Amer. J. Math. (2013), (to appear), arXiv:1305.6088.Google Scholar
Ganapathy, R. and Lomelí, L., On twisted exterior and symmetric square 𝛾-factors, Ann. Inst. Fourier 65(3) (2015), 1105–1132. doi:10.5802/aif.2952.CrossRefGoogle Scholar
Ginzburg, D., Rallis, S. and Soudry, D., Generic automorphic forms on SO(2n + 1): functorial lift to GL(2n), endoscopy, and base change, Int. Math. Res. Not. IMRN 14 (2001), 729–764.Google Scholar
Hales, T. C., On the fundamental lemma for standard endoscopy: reduction to unit elements, Canad. J. Math. 47(5) (1995), 974–994.Google Scholar
Harish-Chandra, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr., in Admissible Invariant Distributions on Reductive p-Adic Groups, University Lecture Series, Volume 16(American Mathematical Society, Providence, RI, 1999).Google Scholar
Henniart, G., Une caractérisation de la correspondance de Langlands locale pour GL(n), Bull. Soc. Math. France 130(4) (2002), 587–602.Google Scholar
Henniart, G., Correspondance de Langlands et fonctions L des carrés extérieur et symétrique, Int. Math. Res. Not. IMRN (4) (2010), 633–673.Google Scholar
Henniart, G. and Lomelí, L., Local-to-global extensions for GL n in non-zero characteristic: a characterization of 𝛾 F (s, 𝜋, Symsp2, 𝜓) and 𝛾 F (s, 𝜋, ∧sp2, 𝜓), Amer. J. Math. 133(1) (2011), 187–196.Google Scholar
Howe, R., Harish-Chandra homomorphisms for p-adic groups, CBMS Regional Conference Series in Mathematics, Volume 59 (Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1985). With the collaboration of Allen Moy.Google Scholar
Iwahori, N. and Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 5–48.Google Scholar
Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103(4) (1981), 777–815.Google Scholar
Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103(3) (1981), 499–558.CrossRefGoogle Scholar
Kazhdan, D., Representations of groups over close local fields, J. Anal. Math. 47 (1986), 175–179.CrossRefGoogle Scholar
Kazhdan, D. and Varshavsky, Y., On endoscopic transfer of Deligne-Lusztig functions, Duke Math. J. 161(4) (2012), 675–732.Google Scholar
Kottwitz, R. and Viehmann, E., Generalized affine Springer fibres, J. Inst. Math. Jussieu 11(3) (2012), 569–609.Google Scholar
Kottwitz, R. E., Stable trace formula: elliptic singular terms, Math. Ann. 275(3) (1986), 365–399.Google Scholar
Kottwitz, R. E., Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings, Volume 4, pp. 393–522 (American Mathematical Society, Providence, RI, 2005).Google Scholar
Kottwitz, R. E. and Shelstad, D., Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190.Google Scholar
Langlands, R. P., On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Mathematical Surveys and Monographs, Volume 31, pp. 101–170 (American Mathematical Society, Providence, RI, 1989).Google Scholar
Lemaire, B., Mœglin, C. and Waldspurger, J.-L., Le lemme fondamental pour l’endoscopie tordue: réduction aux éléments unités. Available at arXiv:1506.03383, June 2015.Google Scholar
Lemaire, B., Représentations génériques de GL N et corps locaux proches, J. Algebra 236(2) (2001), 549–574.Google Scholar
Lemaire, B., Comparison of lattice filtrations and Moy-Prasad filtrations for classical groups, J. Lie Theory 19(1) (2009), 29–54.Google Scholar
Lemaire, B., Caractéres tordus des représentations admissibles. Preprint (2010), available at arXiv:1003.2135.Google Scholar
Li, W.-W., On a pairing of Goldberg-Shahidi for even orthogonal groups, Represent. Theory 17 (2013), 337–381.Google Scholar
Lomelí, L. A., The LS method for the classical groups in positive characteristic and the Riemann Hypothesis. ArXiv e-prints, June 2012.Google Scholar
Lomelí, L. A., Functoriality for the classical groups over function fields, Int. Math. Res. Not. IMRN 22 (2009), 4271–4335.Google Scholar
Mœglin, C., Reprèsentations elliptiques ; caractèrisation et formule de transfert de caractères. Preprint.Google Scholar
Mœglin, C., Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité, J. Eur. Math. Soc. (JEMS) 4(2) (2002), 143–200.Google Scholar
Mœglin, C. and Waldspurger, J.-L., Stabilisation de la formule des traces tordue X: stabilisation spectrale. arXiv:1412.2981, December 2014.Google Scholar
Mœglin, C., Caractérisation des paramètres d’arthur, une remarque (Forthcoming).Google Scholar
Mœglin, C., Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands, in Automorphic Forms and Related Geometry: Assessing the Legacy of I. I. Piatetski–Shapiro, Contemparary Mathematics, Volume 614, pp. 295–336 (American Mathematical Society, Providence, RI, 2014).Google Scholar
Mœglin, C. and Tadić, M., Construction of discrete series for classical p-adic groups, J. Amer. Math. Soc. 15(3) (2002), 715–786 (electronic).Google Scholar
Mœglin, C. and Waldspurger, J.-L., La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général, Astérisque 347 (2012), 167–216. Sur les conjectures de Gross et Prasad. II.Google Scholar
Moy, A. and Prasad, G., Unrefined minimal K-types for p-adic groups, Invent. Math. 116(1–3) (1994), 393–408.Google Scholar
Moy, A. and Prasad, G., Jacquet functors and unrefined minimal K-types, Comment. Math. Helv. 71(1) (1996), 98–121.Google Scholar
Muić, G., A geometric construction of intertwining operators for reductive p-adic groups, Manuscripta Math. 125(2) (2008), 241–272.Google Scholar
Ngô, B. C., Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169.Google Scholar
O’Meara, O. T., Introduction to Quadratic Forms, Classics in Mathematics (Springer, Berlin, 2000). Reprint of the 1973 edition.Google Scholar
Pan, S.-Y., Depth preservation in local theta correspondence, Duke Math. J. 113(3) (2002), 531–592.Google Scholar
Ranga Rao, R., Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505–510.Google Scholar
Roche, A., The Bernstein decomposition and the Bernstein centre, in Ottawa Lectures on Admissible Representations of Reductive p-Adic Groups, Fields Institute Monographs, Volume 26, pp. 3–52 (American Mathematical Society, Providence, RI, 2009).Google Scholar
Satake, I., Theory of spherical functions on reductive algebraic groups over p-adic fields, Publ. Math. Inst. Hautes Études Sci. 18 (1963), 5–69.Google Scholar
Sauvageot, F., Principe de densité pour les groupes réductifs, Compositio Math. 108(2) (1997), 151–184.Google Scholar
Serre, J.-P., Local Fields, Graduate Texts in Mathematics, Volume 67 (Springer, New York–Berlin, 1979). Translated from the French by Marvin Jay Greenberg.Google Scholar
Shahidi, F., On certain L-functions, Amer. J. Math. 103(2) (1981), 297–355.Google Scholar
Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132(2) (1990), 273–330.Google Scholar
Shahidi, F., Poles of intertwining operators via endoscopy: the connection with prehomogeneous vector spaces, Compositio Math. 120(3) (2000), 291–325. With an appendix by Diana Shelstad.Google Scholar
Soudry, D., On Langlands functoriality from classical groups to GL n , Astérisque 298 (2005), 335–390. Automorphic forms. I.Google Scholar
Springer, T. A., Linear Algebraic Groups, Second ed., Modern Birkhäuser Classics (Birkhäuser Boston Inc., Boston, MA, 2009).Google Scholar
Tits, J., Reductive groups over local fields, in Automorphic Forms, Representations and L-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Part 1), Proceedings of Symposia in Pure Mathematic, XXXIII, pp. 29–69 (American Mathematical Society, Providence, RI, 1979).Google Scholar
Tsai, P.-Y., On Newforms for Split Special Odd Orthogonal Groups (ProQuest LLC, Ann Arbor, MI, 2013). Thesis (Ph.D.)–Harvard University.Google Scholar
Waldspurger, J.-L., Une formule des traces locale pour les algèbres de Lie p-adiques, J. Reine Angew. Math. 465 (1995), 41–99.Google Scholar
Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu 2(2) (2003), 235–333.Google Scholar
Waldspurger, J.-L., L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 194(908) (2008), x+261.Google Scholar
Waldspurger, J.-L., Les facteurs de transfert pour les groupes classiques: un formulaire, Manuscripta Math. 133(1–2) (2010), 41–82.Google Scholar
Waldspurger, J.-L., Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque (269) (2001), vi+449.Google Scholar
Waldspurger, J.-L., Endoscopie et changement de caractéristique: intégrales orbitales pondérées, Ann. Inst. Fourier (Grenoble) 59(5) (2009), 1753–1818.Google Scholar
Yu, J.-K., Bruhat-Tits theory and buildings, in Ottawa Lectures on Admissible Representations of Reductive p-Adic Groups, Fields Institute Monographs, Volume 26, pp. 53–77 (American Mathematical Society, Providence, RI, 2009).Google Scholar
Yu, J.-K., On the local Langlands correspondence for tori, in Ottawa Lectures on Admissible Representations of Reductive p-Adic Groups, Fields Institute Monographs, Volume 26, pp. 177–183 (American Mathematical Society, Providence, RI, 2009).Google Scholar