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Simple wild -packets

Published online by Cambridge University Press:  16 May 2012

Tasho Kaletha
Tasho Kaletha*
Affiliation:
School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA([email protected])
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Abstract

In a recent paper, Gross and Reeder study the arithmetic properties of discrete Langlands parameters for semi-simple -adic groups, and they conjecture that a special class of these – the simple wild parameters – should correspond to -packets consisting of simple supercuspidal representations. We provide a construction of this correspondence, and show that the simple wild -packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group, and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

Keywords

local Langlands correspondenceL-packetsp-adic groupssupercuspidal representations22E5020G2511S3711F70
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 1 , January 2013 , pp. 43 - 75
Copyright
Copyright © Cambridge University Press 2013

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References

1.Adler, J. D. , Refined anisotropic -types and supercuspidal representations, Pacific J. Math. 185 (1) (1998), 1–32.CrossRefGoogle Scholar
2.Adler, J. D. and DeBacker, S. , Murnaghan–Kirillov theory for supercuspidal representations of general linear groups, J. Reine Angew. Math. 575 (2004), 1–35.Google Scholar
3.Adler, J. D. and Roche, A. , An intertwining result for -adic groups, Canad. J. Math. 52 (3) (2000), 449–467.Google Scholar
4.Adler, J. D. and Spice, L. , Supercuspidal characters of reductive -adic groups, Amer. J. Math. 131 (4) (2009), 1137–1210.CrossRefGoogle Scholar
5.Bosch, S., Lütkebohmert, W. and Raynaud, M. , Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 21. (Springer-Verlag, Berlin, 1990), x+325 pp. ISBN: 3-540-50587-3.Google Scholar
6.Bourbaki, N. , Lie groups and Lie algebras, in Elements of mathematics. (Springer-Verlag, Berlin, 2002), xii+300 pp. (Chapters 4–6).Google Scholar
7.Chai, C. L. and Yu, J. K. , Congruences of Néron models for tori and the Artin conductor, Ann. of Math. (2) 154 (2) (2001), 347–382. With an appendix by Ehud de Shalit.Google Scholar
8.DeBacker, S. and Reeder, M. , Depth-zero supercuspidal -packets and their stability, Ann. of Math. (2) 169 (3) (2009), 795–901.Google Scholar
9.DeBacker, S. and Reeder, M. , Some very cuspidal generic representations, Compos. Math. 146 (2010), 1029–1055.Google Scholar
10.Demazure, M. and Grothendieck, A. , Schémas en groupes I, II, III, Lecture Notes in Math., Volume 151–153. (Springer-Verlag, New York, 1970).Google Scholar
11.Deligne, P. , Cohomologie étale, in Séminaire de Géométrie Algébrique du Bois-Marie SGA ½, Lecture Notes in Mathematics, Volume 569. (Springer-Verlag, Berlin, New York, 1977), iv+312pp.Google Scholar
12.Dieudonné, J. and Grothendieck, A. , Éléments de géométrie algébrique IV. Publ. Math. IHÉS, (1964–1967), 20, 24, 28, 32, pp.CrossRefGoogle Scholar
13.Gross, B. , Irreducible cuspidal representations with prescribed local behaviour, Amer. J. Math. 133 (2011), 1231–1258.Google Scholar
14.Gross, B., Levy, P., Reeder, M. and Yu, J. K. , Gradings of positive rank on simple Lie algebras, preprint.Google Scholar
15.Gross, B. and Reeder, M. , Arithmetic invariants of discrete Langlands parameters, Duke Math. J. 154 (3) (2010), 431–508.CrossRefGoogle Scholar
16.Hales, T. C. , A simple definition of transfer factors for unramified groups, Contemp. Math. 145 (1993), 109–134.CrossRefGoogle Scholar
17.Admissible Invariant Distributions on Reductive p-adic Groups. Preface and notes by Stephen DeBacker and Paul J. Sally, Jr. University Lecture Series, Volume 16. (American Mathematical Society, Providence, RI, 1999).Google Scholar
18.Hiraga, K., Ichino, A. and Ikeda, T. , Formal degrees and adjoint gamma-factors, J. Amer. Math. Soc. 21 (2008), 283–304.CrossRefGoogle Scholar
19.Heinloth, J., Ngo, B. C. and Yun, Z. , Kloosterman sheaves for reductive groups (arXiv:1005.2765).Google Scholar
20.Kostant, B. , The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032.CrossRefGoogle Scholar
21.Kottwitz, R. E. , Stable trace formula: Elliptic singular terms, Math. Ann. 275 (1986), 365–399.CrossRefGoogle Scholar
22.Kottwitz, R. E. , Isocrystals with additional structure. II, Compositio Math. 109 (3) (1997), 255–339.CrossRefGoogle Scholar
23.Kutzko, P. , Mackey’s theorem for nonunitary representations, Proc. Amer. Math. Soc. 64 (1) (1977), 173–175.Google Scholar
24.Kunyavskii, B. and Sansuc, J. , Réduction des groupes algébriques commutatifs, J. Math. Soc. Japan 53 (2) (2001), 457–483.CrossRefGoogle Scholar
25.Labesse, J. P. , Cohomologie, -groupes et fonctorialité, Compositio Math. 55 (2) (1985), 163–184.Google Scholar
26.Langlands, R. P. and Shelstad, D. , On the definition of transfer factors, Math. Ann. 278 (1987), 219–271.Google Scholar
27.Moeglin, C. and Waldspurger, J.-L. , Modèles de Whittaker dégénérées pour des groupes p-adiques, Math. Zeit. 196 (1987), 427–452.Google Scholar
28.Moy, A. and Prasad, G. , Jacquet functors and unrefined minimal -types, Comment. Math. Helv. 71 (1) (1996), 98–121.CrossRefGoogle Scholar
29.Nart, E. and Xarles, X. , Additive reduction of algebraic tori, Arch. Math. (Basel) 57 (5) (1991), 460–466.CrossRefGoogle Scholar
30.Prasad, G. , Galois-fixed points in the Bruhat–Tits building of a reductive group, Bull. Soc. Math. France 129 (2) (2001), 169–174.CrossRefGoogle Scholar
31.Rapoport, M. , A guide to the reduction modulo p of Shimura varieties, Automorphic forms. I, Astérisque 298 (2005), 271–318.Google Scholar
32.Reeder, M. , Supercuspidal -packets of positive depth and twisted Coxeter elements, J. Reine Angew. Math. 620 (2008), 1–33.CrossRefGoogle Scholar
33.Serre, J. P. , Local fields, Graduate Texts in Mathematics, Volume 67. (Springer-Verlag, New York, Berlin, 1979), viii+241 pp. ISBN: 0-387-90424-7.Google Scholar
34.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.CrossRefGoogle Scholar
35.Shelstad, D. , A formula for regular unipotent germs, Orbites unipotentes et representations, II, Astérisque 171-2 (1989), 275–277.Google Scholar
36.Springer, T. A. and Steinberg, R. , Conjugacy classes, in Seminar on algebraic groups and related finite groups (the institute for advanced study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, Volume 131. pp. 167–266. (Springer, Berlin, 1970).CrossRefGoogle Scholar
37.Steinberg, R. , Regular elements of semisimple algebraic groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 281–312.CrossRefGoogle Scholar
38.Waldspurger, J.-L. , Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), 153–236.CrossRefGoogle Scholar
39.Xarles, X. , The scheme of connected components of the Neron model of an algebraic torus, J. Reine Angew. Math. 437 (1993), 167–179.Google Scholar
40.Yu, J. K. , Smooth models associated to concave functions in Bruhat–Tits theory, preprint.Google Scholar