Centre de Bernstein dual pour les groupes classiques

Moussaoui, Ahmed

doi:10.1090/ert/503

Centre de Bernstein dual pour les groupes classiques
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by Ahmed Moussaoui

Represent. Theory 21 (2017), 172-246

DOI: https://doi.org/10.1090/ert/503

Published electronically: September 8, 2017

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Abstract:

Dans cet article, nous nous intéressons aux liens entre l’induction parabolique et la correspondance de Langlands locale. Nous énonçons une conjecture concernant les paramètres de Langlands (enrichis) des représentations supercuspidales pour les groupes réductifs $p$-adiques déployés. Nous vérifions la validité de cette conjecture grâce aux résultats connus pour le groupe linéaire et les groupes classiques. À la suite de cela, en définissant la notion de support cuspidal d’un paramètre de Langlands enrichi, nous obtenons une décomposition à la Bernstein de l’ensemble des paramètres de Langlands enrichis pour les groupes classiques. Nous vérifions que ces constructions se correspondent par la correspondance de Langlands et en conséquence, nous obtenons la compatibilité de la correspondance de Langlands avec l’induction parabolique.

ABSTRACT. In this article, we consider the links between parabolic induction and the local Langlands correspondence. We state a conjecture about the (enhanced) Langlands parameters of supercuspidal representation of split reductives $p$-adics groups. We are able to verify it thanks to the known cases of the local Langlands correspondence for linear groups and classical groups. Furthermore, in the case of classical groups, we can construct the "cuspidal support" of an enhanced Langlands parameter and get a decomposition of the set of enhanced Langlands parameters à la Bernstein. We check that these constructions match under the Langlands correspondence and as consequence, we obtain the compatibility of the Langlands correspondence with parabolic induction.

References

Dan Barbasch and Dan Ciubotaru, Unitary equivalences for reductive $p$-adic groups, Amer. J. Math. 135 (2013), no. 6, 1633–1674. MR 3145006, DOI 10.1353/ajm.2013.0048
Dan Barbasch and Allen Moy, Reduction to real infinitesimal character in affine Hecke algebras, J. Amer. Math. Soc. 6 (1993), no. 3, 611–635. MR 1186959, DOI 10.1090/S0894-0347-1993-1186959-0
J. N. Bernstein, Le “centre” de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32 (French). Edited by P. Deligne. MR 771671
A. Borel, Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR 546608
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2010. Reprint of the 1997 edition. MR 2838836, DOI 10.1007/978-0-8176-4938-8
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 501–597 (French). MR 0349635
David Goldberg, On dual $R$-groups for classical groups, On certain $L$-functions, Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 159–185. MR 2767516
Benedict H. Gross and Mark Reeder, Arithmetic invariants of discrete Langlands parameters, Duke Math. J. 154 (2010), no. 3, 431–508. MR 2730575, DOI 10.1215/00127094-2010-043
Thomas J. Haines, The stable Bernstein center and test functions for Shimura varieties, Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 118–186. MR 3444233
Volker Heiermann, Orbites unipotentes et pôles d’ordre maximal de la fonction $\mu$ de Harish-Chandra, Canad. J. Math. 58 (2006), no. 6, 1203–1228 (French, with French summary). MR 2270924, DOI 10.4153/CJM-2006-043-8
Volker Heiermann, Paramètres de Langlands et algèbres d’entrelacement, Int. Math. Res. Not. IMRN 9 (2010), 1607–1623 (French). MR 2643577, DOI 10.1093/imrn/rnp191
Volker Heiermann, Opérateurs d’entrelacement et algèbres de Hecke avec paramètres d’un groupe réductif $p$-adique: le cas des groupes classiques, Selecta Math. (N.S.) 17 (2011), no. 3, 713–756 (French, with English and French summaries). MR 2827179, DOI 10.1007/s00029-011-0056-0
Volker Heiermann, Algèbres de Hecke avec paramètres et représentations d’un groupe $p$-adique classique: préservation du spectre tempéré, J. Algebra 371 (2012), 596–608 (French, with English and French summaries). MR 2975415, DOI 10.1016/j.jalgebra.2012.09.003
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
Jean-Philippe Anker and Bent Orsted (eds.), Lie theory, Progress in Mathematics, vol. 228, Birkhäuser Boston, Inc., Boston, MA, 2004. Lie algebras and representations. MR 2042688, DOI 10.1007/978-0-8176-8192-0
Tasho Kaletha, Epipelagic $L$-packets and rectifying characters, Invent. Math. 202 (2015), no. 1, 1–89. MR 3402796, DOI 10.1007/s00222-014-0566-4
Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, DOI 10.1215/S0012-7094-84-05129-9
Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, DOI 10.1023/A:1000102604688
G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, DOI 10.1007/BF01388564
George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145–202. MR 972345, DOI 10.1007/BF02699129
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
George Lusztig, Classification of unipotent representations of simple $p$-adic groups, Internat. Math. Res. Notices 11 (1995), 517–589. MR 1369407, DOI 10.1155/S1073792895000353
George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145–202. MR 972345, DOI 10.1007/BF02699129
G. Lusztig, Cuspidal local systems and graded Hecke algebras. III, Represent. Theory 6 (2002), 202–242. MR 1927954, DOI 10.1090/S1088-4165-02-00172-3
George Lusztig, Families and Springer’s correspondence, Pacific J. Math. 267 (2014), no. 2, 431–450. MR 3207592, DOI 10.2140/pjm.2014.267.431
G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), no. 1, 41–52. MR 527733, DOI 10.1112/jlms/s2-19.1.41
C. Mœglin, Sur la classification des séries discrètes des groupes classiques $p$-adiques: paramètres de Langlands et exhaustivité, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 2, 143–200 (French, with English summary). MR 1913095, DOI 10.1007/s100970100033
C. Mœglin, Multiplicité 1 dans les paquets d’Arthur aux places $p$-adiques, On certain $L$-functions, Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 333–374 (French, with English summary). MR 2767522, DOI 10.24033/bsmf.1947
Colette Mœglin, 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, Contemp. Math., vol. 614, Amer. Math. Soc., Providence, RI, 2014, pp. 295–336 (French, with English summary). MR 3220932, DOI 10.1090/conm/614/12254
Colette Mœglin and Marko Tadić, Construction of discrete series for classical $p$-adic groups, J. Amer. Math. Soc. 15 (2002), no. 3, 715–786. MR 1896238, DOI 10.1090/S0894-0347-02-00389-2
David Renard, Représentations des groupes réductifs $p$-adiques, Cours Spécialisés [Specialized Courses], vol. 17, Société Mathématique de France, Paris, 2010 (French). MR 2567785
Peter Scholze and Sug Woo Shin, On the cohomology of compact unitary group Shimura varieties at ramified split places, J. Amer. Math. Soc. 26 (2013), no. 1, 261–294. MR 2983012, DOI 10.1090/S0894-0347-2012-00752-8
David A. Vogan Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379. MR 1216197, DOI 10.1090/conm/145/1216197
Jean-Loup Waldspurger, Représentations de réduction unipotente pour $\textrm {SO}(2n+1)$: quelques conséquences d’un article de Lusztig, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 803–910 (French). MR 2058629
Jean-Loup Waldspurger, Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2e partie: extension aux représentations tempérées, Astérisque 346 (2012), 171–312 (French, with English and French summaries). Sur les conjectures de Gross et Prasad. I. MR 3202558
Bin Xu, On the cuspidal support of discrete series for p-adic quasisplit $\textrm {Sp}(N)$ and $\textrm {SO}(N)$, Manuscripta mathematica , posted on (Mar 17, 2017)., DOI 10.1007/s00229-017-0923-x