Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 5, 2016

On descent and the generic packet conjecture

Sandeep Varma
From the journal Forum Mathematicum


Suppose F is a p-adic field, and H1 is (a z-extension of) a group that is twisted endoscopic to a connected reductive quasi-split group G over F. Suppose G satisfies the strong form of the generic packet conjecture (also called tempered packet conjecture in literature). Under certain assumptions, we show that the twisted endoscopic character identities associated to this situation imply the strong form of the generic packet conjecture for H1. This generalizes a result of T. Konno, and lets us deduce, under the assumption that appropriate character identities are satisfied, the generic packet conjecture for general spin (GSpin) groups.

MSC 2010: 22E50

Communicated by Freydoon Shahidi


The contents of this paper are partly based on the author’s thesis, [32], done a few years ago under the guidance of Professor Freydoon Shahidi. I am very grateful to Professor Shahidi for suggesting the problem of proving the generic packet conjecture in its stronger as well as more general form (under suitable assumptions), and for patient guidance and encouragement. This work and I have benefited a lot from the guidance and encouragement of Professors D. Goldberg and J.-K. Yu. I also gratefully acknowledge useful communication and encouragement from Professors D. Prasad and R. Kottwitz. I am thankful to Dr. R. Ganapathy, for it was joint work with her that motivated me to write this paper up. I thank Professor J.-L. Waldspurger for kindly clarifying some points over an email. I thank the referee for a careful and thorough reading of this article and for pointing out several omissions.


[1] Arthur J., On local character relations, Selecta Math. (N.S.) 2 (1996), no. 4, 501–579. 10.1007/BF02433450Search in Google Scholar

[2] Arthur J., The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups, Amer. Math. Soc. Colloq. Publ. 61, American Mathematical Society, Providence, 2013. Search in Google Scholar

[3] Beuzart-Plessis R., La conjecture locale de gross-prasad pour les représentations tempérées des groupes unitaires, preprint 2012, 10.24033/msmf.457Search in Google Scholar

[4] Borel A., Automorphic L-functions, Proc. Sympos. Pure Math. 33 (1979), no. 2, 27–61. 10.1090/pspum/033.2/546608Search in Google Scholar

[5] Clozel L., Characters of nonconnected, reductive p-adic groups, Canad. J. Math. 39 (1987), no. 1, 149–167. 10.4153/CJM-1987-008-3Search in Google Scholar

[6] Ferrari A., Théorème de l’indice et formule des traces, Manuscripta Math. 124 (2007), no. 3, 363–390. 10.1007/s00229-007-0130-2Search in Google Scholar

[7] Harish-Chandra , Harmonic Analysis on Reductive p-Adic Groups, Lecture Notes in Math. 162, Springer, Berlin, 1970. 10.1007/BFb0061269Search in Google Scholar

[8] Harish-Chandra , Admissible Invariant Distributions on Reductive p-Adic Groups. Notes by Stephen DeBacker and Paul J. Sally, Jr., Univ. Lecture Ser. 16, American Mathematical Society, Providence, 1999. 10.1090/ulect/016Search in Google Scholar

[9] Jiang D. and Soudry D., The local converse theorem for SO(2n+ 1) and applications, Ann. of Math. (2) 157 (2003), no. 3, 743–806. 10.4007/annals.2003.157.743Search in Google Scholar

[10] Kazhdan D. and Varshavsky Y., On endoscopic transfer of Deligne–Lusztig functions, Duke Math. J. 161 (2012), no. 4, 675–732. 10.1215/00127094-1548371Search in Google Scholar

[11] Konno T., Twisted endoscopy and the generic packet conjecture, Israel J. Math. 129 (2002), 253–289. 10.1007/BF02773167Search in Google Scholar

[12] Kostant B., Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. 10.2307/2373130Search in Google Scholar

[13] Kottwitz R. E., Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), no. 4, 785–806. 10.1215/S0012-7094-82-04939-0Search in Google Scholar

[14] Kottwitz R. E., Harmonic analysis on reductive p-adic groups and Lie algebras, Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc. 4, American Mathematical Society, Providence (2005), 393–522. Search in Google Scholar

[15] Kottwitz R. E. and Shelstad D., Foundations of Twisted Endoscopy, Astérisque 255, Société Mathématique de France, Paris, 1999. Search in Google Scholar

[16] Langlands R. P. and Shelstad D., On the definition of transfer factors, Math. Ann. 278 (1987), no. 1–4, 219–271. 10.1007/BF01458070Search in Google Scholar

[17] Langlands R. P. and Shelstad D., Descent for transfer factors, The Grothendieck Festschrift. Vol. II, Progr. Math. 87, Birkhäuser, Boston (1990), 485–563. 10.1007/978-0-8176-4575-5_12Search in Google Scholar

[18] Lemaire B., Caractéres tordus des représentations admissibles, preprint 2010, Search in Google Scholar

[19] Mœglin C., Représentations elliptiques; caractérisation et formule de transfert de caracteres, preprint 2013. 10.1007/978-3-319-30058-0_6Search in Google Scholar

[20] Mœglin C., Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands, Automorphic Forms and Related Geometry: Assessing the Legacy of I. I. Piatetski-Shapiro (New Haven 2012), Contemp. Math. 614, American Mathematical Society, Providence (2014), 295–336. 10.1090/conm/614/12254Search in Google Scholar

[21] Mœglin C. and Waldspurger J.-L., Modèles de Whittaker dégénérés pour des groupes p-adiques, Math. Z. 196 (1987), no. 3, 427–452. 10.1007/BF01200363Search in Google Scholar

[22] Mœglin C. and Waldspurger J.-L., La conjecture locale de Gross–Prasad pour les groupes spéciaux orthogonaux: Le cas général, Sur les Conjectures de Gross et Prasad. II, Astérisque 347, Société Mathématique de France, Paris (2012), 167–216. Search in Google Scholar

[23] Mok C. P., Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015), Paper No. 1108. 10.1090/memo/1108Search in Google Scholar

[24] Ngô B. C., Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169. 10.1007/s10240-010-0026-7Search in Google Scholar

[25] Prasad G. and Raghunathan M. S., On the Kneser–Tits problem, Comment. Math. Helv. 60 (1985), no. 1, 107–121. 10.1007/BF02567402Search in Google Scholar

[26] Ranga Rao R., Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505–510. 10.2307/1970822Search in Google Scholar

[27] Rodier F., Whittaker models for admissible representations of reductive p-adic split groups, Harmonic Analysis on Homogeneous Spaces, Proc. Sympos. Pure Math. 26, American Mathematical Society, Providence (1973), 425–430. 10.1090/pspum/026/0354942Search in Google Scholar

[28] Shahidi F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. 10.2307/1971524Search in Google Scholar

[29] Shelstad D., A formula for regular unipotent germs, Orbites Unipotentes et Représentations. II: Groupes p-Adiques et Réels, Astérisque 171–172, Société Mathématique de France, Paris (1989), 275–277. Search in Google Scholar

[30] Springer T. A., Linear Algebraic Groups, 2nd ed., Progr. Math. 9, Birkhäuser, Boston, 1998. 10.1007/978-0-8176-4840-4Search in Google Scholar

[31] van Dijk G., Computation of certain induced characters of p-adic groups, Math. Ann. 199 (1972), 229–240. 10.1007/BF01429876Search in Google Scholar

[32] Varma S., Descent and the generic packet conjecture, PhD thesis, Purdue University, West Lafayette, 2009. Search in Google Scholar

[33] Varma S., On a result of Moeglin and Waldspurger in residual characteristic 2, Math. Z. 277 (2014), no. 3–4, 1027–1048. 10.1007/s00209-014-1292-8Search in Google Scholar

[34] Vogan, Jr. D. A., Gel’fand–Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75–98. 10.1007/BF01390063Search in Google Scholar

[35] Waldspurger J.-L., Une formule des traces locale pour les algèbres de Lie p-adiques, J. Reine Angew. Math. 465 (1995), 41–99. Search in Google Scholar

[36] Waldspurger J.-L., Le lemme fondamental implique le transfert, Compos. Math. 105 (1997), no. 2, 153–236. 10.1023/A:1000103112268Search in Google Scholar

[37] Waldspurger J.-L., Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5 (2006), 423–525. 10.1017/S1474748006000041Search in Google Scholar

[38] Waldspurger J.-L., L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 194 (2008), Paper No. 908. 10.1090/memo/0908Search in Google Scholar

[39] Waldspurger J.-L., Les facteurs de transfert pour les groupes classiques: Un formulaire, Manuscripta Math. 133 (2010), no. 1–2, 41–82. 10.1007/s00229-010-0363-3Search in Google Scholar

[40] Waldspurger J.-L., La conjecture locale de Gross–Prasad pour les représentations tempérées des groupes spéciaux orthogonaux, Sur les Conjectures de Gross et Prasad. II, Astérisque 347, Société Mathématique de France, Paris (2012), 103–165. Search in Google Scholar

Received: 2015-6-11
Revised: 2015-10-30
Published Online: 2016-5-5
Published in Print: 2017-1-1

© 2017 by De Gruyter

Scroll Up Arrow