Paquets d’Arthur des groupes classiques et unitaires

Nicolás Arancibia; Colette Mœglin; David Renard

doi:10.5802/afst.1590

Nicolás Arancibia ¹ ; Colette Mœglin ² ; David Renard ³

¹ Institut Mathématique de Jussieu
² CNRS, Institut Mathématique de Jussieu
³ Centre de Mathématiques Laurent Schwartz, École Polytechnique

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 5, pp. 1023-1105.

Résumé
Abstract

Soit $G = G (ℝ)$ le groupe des points réels d’un groupe algébrique connexe réductif quasi-déployé défini sur $ℝ$ . Supposons de plus que $G$ soit un groupe classique (symplectique, spécial orthogonal ou unitaire). Nous montrons que les paquets de représentations irréductibles unitaires et cohomologiques définies par Adams et Johnson en 1987 coïncident avec ceux definis plus récemment par J. Arthur dans son travail sur la classification du spectre automorphe discret des groupes classiques (C.-P. Mok pour les groupes unitaires). Pour cela, nous calculons le transfert endoscopique des distributions stables sur $G$ supportées par ces paquets vers le groupe ${GL}_{N}$ tordu en termes de modules standard et nous montrons qu’il est égal à la trace tordue prescrite par Arthur.

Let $G = G (ℝ)$ be the group of real points of a quasi-split connected reductive algebraic group defined over $ℝ$ . Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on $G$ supported by these packets to twisted ${GL}_{N}$ in terms of standard modules and show that it coincides with the twisted trace prescribed by Arthur.

Reçu le : 2016-11-29
Accepté le : 2017-02-06
Publié le : 2019-01-21

MR Zbl

DOI : 10.5802/afst.1590

Affiliations des auteurs :

Nicolás Arancibia ¹ ; Colette Mœglin ² ; David Renard ³

¹ Institut Mathématique de Jussieu
² CNRS, Institut Mathématique de Jussieu
³ Centre de Mathématiques Laurent Schwartz, École Polytechnique

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AFST_2018_6_27_5_1023_0,
     author = {Nicol\'as Arancibia and Colette M{\oe}glin and David Renard},
     title = {Paquets {d{\textquoteright}Arthur} des groupes classiques et unitaires},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1023--1105},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 27},
     number = {5},
     year = {2018},
     doi = {10.5802/afst.1590},
     zbl = {1420.22018},
     mrnumber = {3919547},
     language = {fr},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1590/}
}

TY  - JOUR
AU  - Nicolás Arancibia
AU  - Colette Mœglin
AU  - David Renard
TI  - Paquets d’Arthur des groupes classiques et unitaires
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
SP  - 1023
EP  - 1105
VL  - 27
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1590/
DO  - 10.5802/afst.1590
LA  - fr
ID  - AFST_2018_6_27_5_1023_0
ER  -

%0 Journal Article
%A Nicolás Arancibia
%A Colette Mœglin
%A David Renard
%T Paquets d’Arthur des groupes classiques et unitaires
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2018
%P 1023-1105
%V 27
%N 5
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1590/
%R 10.5802/afst.1590
%G fr
%F AFST_2018_6_27_5_1023_0

Nicolás Arancibia; Colette Mœglin; David Renard. Paquets d’Arthur des groupes classiques et unitaires. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 5, pp. 1023-1105. doi : 10.5802/afst.1590. https://afst.centre-mersenne.org/articles/10.5802/afst.1590/

Bibliographie
Cité par

[1] Jeffrey Adams; Dan Barbasch; David A. Vogan The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, 104, Birkhäuser, 1992, xii+318 pages | DOI | MR | Zbl

[2] Jeffrey Adams; Fokko du Cloux Algorithms for representation theory of real reductive groups, J. Inst. Math. Jussieu, Volume 8 (2009) no. 2, pp. 209-259 | DOI | MR | Zbl

[3] Jeffrey Adams; Joseph F. Johnson Endoscopic groups and packets of nontempered representations, Compos. Math., Volume 64 (1987) no. 3, pp. 271-309 | MR | Zbl

[4] Nicolás-José Arancibia-Robert Paquets d’Arthur des représentations cohomologiques, Institut de Mathématiques de Jussieu (France) (2015) (Ph. D. Thesis)

[5] James Arthur On some problems suggested by the trace formula, Lie group representations, II (College Park, Md., 1982/1983) (Lecture Notes in Mathematics), Volume 1041, Springer, 1984, pp. 1-49 | DOI | MR | Zbl

[6] James Arthur Intertwining operators and residues. I. Weighted characters, J. Funct. Anal., Volume 84 (1989) no. 1, pp. 19-84 | DOI | MR | Zbl

[7] James Arthur The $L^{2}$ -Lefschetz numbers of Hecke operators, Invent. Math., Volume 97 (1989) no. 2, pp. 257-290 | DOI | MR | Zbl

[8] James Arthur Unipotent automorphic representations : conjectures, Orbites unipotentes et représentations, II Groupes $p$ -adiques et réels (Astérisque), Volume 171-172, Société Mathématique de France, 1989, pp. 13-71 | Numdam | MR | Zbl

[9] James Arthur A stable trace formula III. Proof of the main theorems, Ann. Math., Volume 158 (2003) no. 2, pp. 769-873 | MR | Zbl

[10] James Arthur The endoscopic classification of representations. Orthogonal and symplectic groups, Colloquium Publications, 61, American Mathematical Society, 2013, xviii+590 pages | MR | Zbl

[11] Dan Barbasch; David A. Vogan Unipotent representations of complex semisimple groups, Ann. Math., Volume 121 (1985) no. 1, pp. 41-110 | DOI | MR | Zbl

[12] Ehud Moshe Baruch A proof of Kirillov’s conjecture, Ann. Math., Volume 158 (2003) no. 1, pp. 207-252 | DOI | MR | Zbl

[13] Alexandre Beilinson; Joseph Bernstein Localisation de $𝔤$ -modules, C. R. Math. Acad. Sci. Paris, Volume 292 (1981) no. 1, pp. 15-18 | MR | Zbl

[14] Nicolas Bergeron; John Millson; Colette Mœglin The Hodge conjecture and arithmetic quotients of complex balls, Acta Math., Volume 216 (2016) no. 1, pp. 1-125 | DOI | MR | Zbl

[15] Nicolas Bergeron; John Millson; Colette Mœglin Hodge type theorems for arithmetic manifolds associated to orthogonal groups, Int. Math. Res. Not. (2017) no. 15, pp. 4495-4624 | DOI | MR | Zbl

[16] Armand Borel Automorphic $L$ -functions, Automorphic forms, representations and $L$ -functions (Corvallis, 1977), Part 2 (Proceedings of Symposia in Pure Mathematics), Volume 33, American Mathematical Society, 1979, pp. 27-61 | MR | Zbl

[17] Abderrazak Bouaziz Quelques remarques sur les distributions invariantes dans les algèbres de Lie réductives, Noncommutative harmonic analysis (Progress in Mathematics), Volume 220, Birkhäuser, 2004, pp. 119-130 | MR | Zbl

[18] Gaëtan Chenevier; Laurent Clozel Corps de nombres peu ramifiés et formes automorphes autoduales, J. Am. Math. Soc., Volume 22 (2009) no. 2, pp. 467-519 | DOI | MR | Zbl

[19] Gaëtan Chenevier; Jean Lannes Formes automorphes et voisins de Kneser des réseaux de Niemaier (http://gaetan.chenevier.perso.math.cnrs.fr/pub.html) | Zbl

[20] Gaëtan Chenevier; David Renard On the vanishing of some non-semisimple orbital integrals, Expo. Math., Volume 28 (2010) no. 3, pp. 276-289 | DOI | MR | Zbl

[21] Gaëtan Chenevier; David Renard Level One algebraic cusp forms of classical groups of small rank, Mem. Am. Math. Soc., Volume 237 (2015) no. 1121 | MR | Zbl

[22] Laurent Clozel Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982) no. 1, pp. 45-115 | Numdam | MR | Zbl

[23] Brent Fraser; Paul Mezo Twisted endoscopy in miniature (http://people.math.carleton.ca/~mezo/research.html)

[24] Federico Incitti The Bruhat order on the involutions of the symmetric group, J. Algebr. Comb., Volume 20 (2004) no. 3, pp. 243-261 | DOI | MR | Zbl

[25] Joseph F. Johnson Lie algebra cohomology and the resolution of certain Harish-Chandra modules, Math. Ann., Volume 267 (1984) no. 3, pp. 377-393 | DOI | MR | Zbl

[26] Joseph F. Johnson Stable base change $C / R$ of certain derived functor modules, Math. Ann., Volume 287 (1990) no. 3, pp. 467-493 | DOI | MR | Zbl

[27] Anthony W. Knapp; Elias M. Stein Intertwining operators for semisimple groups, Ann. Math., Volume 93 (1971), pp. 489-578 | DOI | MR | Zbl

[28] Anthony W. Knapp; Elias M. Stein Intertwining operators for semisimple groups. II, Invent. Math., Volume 60 (1980) no. 1, pp. 9-84 | DOI | MR | Zbl

[29] Anthony W. Knapp; David A. Vogan Cohomological induction and unitary representations, Princeton Mathematical Series, 45, Princeton University Press, 1995, xx+948 pages | MR | Zbl

[30] Robert E. Kottwitz Shimura varieties and $λ$ -adic representations, Automorphic forms, Shimura varieties, and $L$ -functions, Vol. I (Ann Arbor, 1988) (Perspectives in Mathematics), Volume 10, Academic Press, 1990, pp. 161-209 | MR | Zbl

[31] Robert E. Kottwitz; Diana Shelstad Foundations of twisted endoscopy, Astérisque, 255, Société Mathématique de France, 1999, vi+190 pages | Numdam | MR | Zbl

[32] Jean-Pierre Labesse; Robert P. Langlands $L$ -indistinguishability for $SL (2)$ , Can. J. Math., Volume 31 (1979) no. 4, pp. 726-785 | DOI | MR | Zbl

[33] Jean-Pierre Labesse; Jean-Loup Waldspurger La formule des traces tordue d’après le Friday Morning Seminar, CRM Monograph Series, 31, American Mathematical Society, 2013, xxvi+234 pages (With a foreword by Robert Langlands) | MR | Zbl

[34] Robert P. Langlands On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups (Mathematical Surveys and Monographs), Volume 31, American Mathematical Society, 1989, pp. 101-170 | DOI | MR | Zbl

[35] Hisayosi Matumoto On the representations of $S p (p, q)$ and ${SO}^{*} (2 n)$ unitarily induced from derived functor modules, Compos. Math., Volume 140 (2004) no. 4, pp. 1059-1096 | DOI | MR | Zbl

[36] William M. McGovern; Peter E. Trapa Pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety, J. Algebra, Volume 322 (2009) no. 8, pp. 2713-2730 | DOI | MR | Zbl

[37] Paul Mezo Tempered spectral transfer in the twisted endoscopy of real groups, J. Inst. Math. Jussieu, Volume 15 (2016) no. 3, pp. 569-612 | DOI | MR | Zbl

[38] Dragan Miličić Localization and representation theory of reductive Lie groups (http://www.math.utah.edu/~milicic/Eprints/book)

[39] Colette Mœglin Multiplicité 1 dans les paquets d’Arthur aux places $p$ -adiques, On certain $L$ -functions (Clay Mathematics Proceedings), Volume 13, American Mathematical Society, 2011, pp. 333-374 | MR | Zbl

[40] 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 (Contemporary Mathematics), Volume 614, American Mathematical Society, 2014, pp. 295-336 | DOI | MR | Zbl

[41] Colette Mœglin; Jean-Loup Waldspurger Endoscopie tordue sur un corps local, Stabilisation de la formule des traces tordue I (Progress in Mathematics), Volume 316, Birkhäuser, 2016, pp. 1-180 | DOI | Zbl

[42] Colette Mœglin; Jean-Loup Waldspurger Stabilisation spectrale, Stabilisation de la formule des traces tordue II (Progress in Mathematics), Volume 317, Birkhäuser, 2016, pp. 1145-1255 | DOI

[43] Colette Mœglin; Jean-Loup Waldspurger La formule des traces locale tordue, Mem. Am. Math. Soc., Volume 251 (2018) no. 1198, v+183 pages | DOI | MR | Zbl

[44] Chung Pang Mok Endoscopic classification of representations of quasi-split unitary groups, Mem. Am. Math. Soc., Volume 235 (2015) no. 1108, vi+248 pages | DOI | MR | Zbl

[45] Sophie Morel; Junecue Sue The standard sign conjecture on algebraic cycles : the case of Shimura varieties (http://arxiv.org/abs/1408.0461, à paraître dans J. Reine Angew. Math.) | DOI | Zbl

[46] Susana A. Salamanca-Riba On the unitary dual of real reductive Lie groups and the $A_{g} (λ)$ modules : the strongly regular case, Duke Math. J., Volume 96 (1999) no. 3, pp. 521-546 | DOI | MR | Zbl

[47] Gérard Schiffmann Intégrales d’entrelacement et fonctions de Whittaker, Bull. Soc. Math. Fr., Volume 99 (1971), pp. 3-72 | DOI | MR | Zbl

[48] Freydoon Shahidi On certain $L$ -functions, Am. J. Math., Volume 103 (1981) no. 2, pp. 297-355 | DOI | MR | Zbl

[49] Freydoon Shahidi Eisenstein series and automorphic $L$ -functions, Colloquium Publications, 58, American Mathematical Society, 2010, vi+210 pages | MR | Zbl

[50] Joseph A. Shalika The multiplicity one theorem for ${GL}_{n}$ , Ann. Math., Volume 100 (1974), pp. 171-193 | DOI | MR

[51] Diana Shelstad Characters and inner forms of a quasi-split group over $R$ , Compos. Math., Volume 39 (1979) no. 1, pp. 11-45 | MR | Zbl

[52] Diana Shelstad On geometric transfer in real twisted endoscopy, Ann. Math., Volume 176 (2012) no. 3, pp. 1919-1985 | DOI | MR | Zbl

[53] Birgit Speh The unitary dual of $G l (3, R)$ and $G l (4, R)$ , Math. Ann., Volume 258 (1981) no. 2, pp. 113-133 | DOI | MR | Zbl

[54] Marko Tadić $GL (n, ℂ) \hat{}$ and $GL (n, ℝ) \hat{}$ , Automorphic forms and $L$ -functions II. Local aspects (Contemporary Mathematics), American Mathematical Society, 2009, pp. 285-313 | MR | Zbl

[55] Olivier Taïbi Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula, Ann. Sci. Éc. Norm. Supér., Volume 50 (2017) no. 2, pp. 269-344 | DOI | MR | Zbl

[56] David A. Vogan Representations of real reductive Lie groups, Progress in Mathematics, 15, Birkhäuser, 1981, xvii+754 pages | MR | Zbl

[57] David A. Vogan Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality, Duke Math. J., Volume 49 (1982) no. 4, pp. 943-1073 http://projecteuclid.org/euclid.dmj/1077315538 | MR | Zbl

[58] David A. Vogan Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case, Invent. Math., Volume 71 (1983) no. 2, pp. 381-417 | DOI | MR | Zbl

[59] David A. Vogan The unitary dual of $GL (n)$ over an Archimedean field, Invent. Math., Volume 83 (1986) no. 3, pp. 449-505 | DOI | MR | Zbl

[60] David A. Vogan; Gregg J. Zuckerman Unitary representations with nonzero cohomology, Compos. Math., Volume 53 (1984) no. 1, pp. 51-90 | MR | Zbl

[61] Atsuko Yamamoto Orbits in the flag variety and images of the moment map for classical groups. I, Represent. Theory, Volume 1 (1997), pp. 329-404 | DOI | MR | Zbl

Cité par Sources :