Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Construction of discrete series for classical $p$-adic groups

Authors: Colette Moeglin and Marko Tadic
Journal: J. Amer. Math. Soc. 15 (2002), 715-786
MSC (1991): Primary 22E50, 22E35; Secondary 11F70, 11S37
Published electronically: April 5, 2002
MathSciNet review: 1896238
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The classification of irreducible square integrable representations of classical $p$-adic groups is completed in this paper, under a natural local assumption. Further, this classification gives a parameterization of irreducible tempered representations of these groups. Therefore, it implies a classification of the non-unitary duals of these groups (modulo cuspidal data). The classification of irreducible square integrable representations is directly related to the parameterization of irreducible square integrable representations in terms of dual objects, which is predicted by Langlands program.

References [Enhancements On Off] (What's this?)

  • [Ad] Adams, J., L-functoriality for dual pairs, Astérisque 171-172 (1989), 85-129. MR 91e:22020
  • [Ar] Arthur, J., Unipotent automorphic representations: global motivations, Automorphic forms, Shimura varieties and L-functions, Progress in Mathematics 10, Birkhäuser, 1990, pp. 1-75. MR 92a:11059
  • [B] Ban, D., Parabolic induction and Jacquet modules of representations of $O(2n,F)$, Glasnik Mat. 34(54) (3) (1999), 147-185. MR 2001m:22033
  • [B-Z] Bernstein I.N., Zelevinsky A.V., Induced Representations of Reductive $p$-adic groups I, Ann. Sci. École Norm. Sup 10 (1977), 441-472. MR 58:28310
  • [G] Goldberg, D., Reducibility of induced representations for ${Sp}(2n)$ and ${SO}(n)$, Amer. J. Math. 116 (5) (1994), 1101-1151. MR 95g:22016
  • [JPSS] Jacquet, H.; Piateteski-Shapiro I. and Shalika J., Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), 367-463. MR 85g:11044
  • [Jn1] Jantzen, C., On supports of induced representations for symplectic and odd-orthogonal groups, Amer. J. Math. 119 (1997), 1213-1262. MR 99b:22028
  • [Jn2] -, On square integrable representations of classical p-adic groups II, Representation Theory 4 (2000), 127-180. MR 2001j:22024
  • [KaL] Kazhdan, D. and Lusztig, G., Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153-215. MR 88d:11121
  • [KuR] Kudla, S. and Rallis, S., A regularized Siegel-Weil formula: The first term identity, Ann. of Math. 140 (1994), 1-80. MR 95f:11036
  • [M1] M\oeglin C., Normalisation des opérateurs d'entrelacement et réductibilité des induites de cuspidales; le cas des groupes classiques $p$-adiques, Ann. of Math. 151 (2) (2000), 817-847. MR 2002b:22032
  • [M2] -, Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité, in Journal of the European Mathematical Society (to appear).
  • [M3] -, Représentations quadratiques unipotentes des groupes classiques p-adiques, Duke Math. J. 84 (1996), 267-332. MR 97h:22013
  • [M4] -, Non nullité de certains relèvements par série théta, Journal of Lie Theory 7 (1997), 201-229. MR 98m:11044a
  • [M5] -, Points de réductibilité pour les induites de cuspidales, Prépublication, Institut de mathématiques de Jussieu (2001).
  • [MViW] M\oeglin, C.; Vignéras, M.-F. and Waldspurger, J.-L., Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Math. 1291, Springer-Verlag, Berlin, 1987. MR 91f:11040
  • [MW] M\oeglin, C. and Waldspurger J.-L., Le spectre discret de GL(n), Ann. Sci. École Norm. Sup. 22 (1989), 605-674. MR 91b:22028
  • [Mu] Muic, G., On generic irreducible representations of $Sp(n,F)$ and $SO(2n+1,F)$, Glasnik Mat. 33(53) (1998), 19-31. MR 2000j:22020
  • [Sh1] Shahidi, F., A proof of Langlands conjecture on Plancherel measures: complementary series for p-adic groups, Ann. of Math. 132 (1990), 273-330. MR 91m:11095
  • [Sh2] -, On certain L-functions, Amer. J. Math. 103 (1981), 297-356. MR 82i:10030
  • [Sh3] -, Local coefficients and intertwining operators for GL(n), Compositio Math. 48 (1983), 271-295. MR 85a:22027
  • [Sh4] -, Twisted endoscopy and reducibility of induced representations for $p$-adic groups, Duke Math. J. 66 (1992), 1-41. MR 93b:22034
  • [Sh5] -, Fourier transforms of intertwining operators and Plancherel measures for GL(n), Amer. J. Math. 106 (1984), 67-111. MR 86b:22031
  • [Si] Silberger, A., Special representations of reductive p-adic groups are not integrable, Ann. of Math. 111 (1980), 571-587. MR 82k:22015
  • [T1] Tadic, M., On regular square integrable representations of $p$-adic groups, Amer. J. Math. 120 (1) (1998), 159-210. MR 99h:22026
  • [T2] -, On reducibility of parabolic induction, Israel J. Math. 107 (1998), 29-91. MR 2001d:22012
  • [T3] -, Square integrable representations of classical $p$-adic groups corresponding to segments, Representation Theory 3 (1999), 58-89. MR 2000d:22020
  • [T4] -, A family of square integrable representations of classical $p$-adic groups, preprint (1998).
  • [T5] -, Structure arising from induction and Jacquet modules of representations of classical $p$-adic groups, Journal of Algebra 177 (1) (1995), 1-33. MR 97b:22023
  • [T6] -, Representations of $p$-adic symplectic groups, Compositio Math. 90 (1994), 123-181. MR 95a:22025
  • [Vo] Vogan, D.A., The local Langlands conjecture, Contemp. Math. 145 (1993), 305-379. MR 94e:22031
  • [W1] Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques, d'après Harish-Chandra, Journal de l'Institut de Mathématiques de Jussieu (to appear).
  • [W2] -, Un exercice sur $GSp(4,F)$ et les représentations de Weil, Bull. Soc. Math. France 115 (1987), 35-69. MR 89a:22033
  • [Z] Zelevinsky, A. V., Induced representations of reductive p-adic groups II. On irreducible representations of GL(n), Ann. Sci. École Norm. Sup. 13 (1980), 165-210. MR 83g:22012

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 22E50, 22E35, 11F70, 11S37

Retrieve articles in all journals with MSC (1991): 22E50, 22E35, 11F70, 11S37

Additional Information

Colette Moeglin
Affiliation: Institut de Mathématiques de Jussieu, CNRS, F-75251 Paris Cedex 05, France

Marko Tadic
Affiliation: Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Keywords: Classical groups, $p$-adic fields, irreducible square integrable representations, irreducible tempered representations, non-unitary dual, local Langlands correspondences
Received by editor(s): December 1, 2000
Received by editor(s) in revised form: January 2, 2002
Published electronically: April 5, 2002
Additional Notes: The second author was partly supported by Croatian Ministry of Science and Technology grant # 37001.
Article copyright: © Copyright 2002 American Mathematical Society