Exponents in Archimedean Arthur packets
Annales de l'Institut Fourier, Volume 63 (2013) no. 1, p. 113-154
Generalizing the proof – by Hecht and Schmid – of Osborne’s conjecture we prove an Archimedean (and weaker) version of a theorem of Colette Moeglin. The result we obtain is a precise Archimedean version of the general principle – stated by the second author – according to which a local Arthur packet contains the corresponding local L-packet and representations which are more tempered.
En généralisant la démonstration de Hecht et Schmid de la conjecture d’Osborne, nous démontrons une version archimédienne – et plus faible – d’un théorème de Colette Moeglin. Cela donne un sens archimédien précis au principe énoncé par le second auteur selon lequel on trouve dans un paquet d’Arthur des représentations qui appartiennent au paquet de Langlands associé et des représentations plus tempérées.
DOI : https://doi.org/10.5802/aif.2757
Classification:  22E45,  22E46
Keywords: Représentations unitaires, exposants, conjecture d’Osborne, paquets d’Arthur
@article{AIF_2013__63_1_113_0,
     author = {Bergeron, Nicolas and Clozel, Laurent},
     title = {Exponents in Archimedean Arthur packets},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {1},
     year = {2013},
     pages = {113-154},
     doi = {10.5802/aif.2757},
     zbl = {1276.22002},
     mrnumber = {3097944},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2013__63_1_113_0}
}
Exponents in Archimedean Arthur packets. Annales de l'Institut Fourier, Volume 63 (2013) no. 1, pp. 113-154. doi : 10.5802/aif.2757. https://aif.centre-mersenne.org/item/AIF_2013__63_1_113_0/

[1] Arthur, James An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Amer. Math. Soc., Providence, RI (Clay Math. Proc.) Tome 4 (2005), pp. 1-263 | MR 2192011 | Zbl 1152.11021

[2] Baruch, Ehud Moshe A proof of Kirillov’s conjecture, Ann. of Math. (2), Tome 158 (2003) no. 1, pp. 207-252 | Article | MR 1999922 | Zbl 1034.22010

[3] Bernstein, Joseph N. P-invariant distributions on GL (N) and the classification of unitary representations of GL (N) (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983), Springer, Berlin (Lecture Notes in Math.) Tome 1041 (1984), pp. 50-102 | MR 748505 | Zbl 0541.22009

[4] Bouaziz, Abderrazak Sur les caractères des groupes de Lie réductifs non connexes, J. Funct. Anal., Tome 70 (1987) no. 1, pp. 1-79 | Article | MR 870753 | Zbl 0622.22009

[5] Casselman, William; Osborne, M. Scott The 𝔫-cohomology of representations with an infinitesimal character, Compositio Math., Tome 31 (1975) no. 2, pp. 219-227 | Numdam | MR 396704 | Zbl 0343.17006

[6] Chenevier, G.; Clozel, L. Corps de nombres peu ramifiés et formes automorphes autoduales, J. Amer. Math. Soc., Tome 22 (2009) no. 2, pp. 467-519 | Article | MR 2476781 | Zbl 1206.11066

[7] Clozel, Laurent The ABS principle: consequences for L 2 (G/H), On certain L -functions, Amer. Math. Soc., Providence, RI (Clay Math. Proc.) Tome 13 (2011), pp. 99-115 | MR 2767512 | Zbl 1243.22016

[8] Duflo, Michel Représentations irréductibles des groupes semi-simples complexes, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973–75), Springer, Berlin (1975), p. 26-88. Lecture Notes in Math., Vol. 497 | MR 399353 | Zbl 0315.22008

[9] Fomin, A. I.; Šapovalov, N. N. A certain property of the characters of irreducible representations of real semisimple Lie groups, Funkcional. Anal. i Priložen., Tome 8 (1974) no. 3, p. 87-88 | MR 367115 | Zbl 0302.22017

[10] Harish-Chandra Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc., Tome 119 (1965), pp. 457-508 | Article | MR 180631 | Zbl 0199.46402

[11] Hecht, Henryk; Schmid, Wilfried Characters, asymptotics and 𝔫-homology of Harish-Chandra modules, Acta Math. (1983) no. 1-2, pp. 49-151 | Article | MR 716371 | Zbl 0523.22013

[12] Hirai, Takeshi The characters of some induced representations of semi-simple Lie groups, J. Math. Kyoto Univ., Tome 8 (1968), pp. 313-363 | MR 238994 | Zbl 0185.21503

[13] Jacquet, H.; Langlands, R. P. Automorphic forms on GL ( 2 ) , Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 114 (1970) | MR 401654 | Zbl 0236.12010

[14] Knapp, Anthony W. Representation theory of semisimple groups, Princeton University Press, Princeton, NJ, Princeton Landmarks in Mathematics (2001) (An overview based on examples, Reprint of the 1986 original) | MR 1880691 | Zbl 0993.22001

[15] Knapp, Anthony W.; Vogan, David A. Jr. Cohomological induction and unitary representations, Princeton University Press, Princeton, NJ, Princeton Mathematical Series, Tome 45 (1995) | MR 1330919 | Zbl 0863.22011

[16] Kottwitz, Robert E.; Shelstad, Diana Foundations of twisted endoscopy, Astérisque (1999) no. 255, pp. vi+190 | MR 1687096 | Zbl 0958.22013

[17] Labesse, J.-P. Stable twisted trace formula: elliptic terms, J. Inst. Math. Jussieu, Tome 3 (2004) no. 4, pp. 473-530 | Article | MR 2094449 | Zbl 1061.11025

[18] Langlands, R. P.; Shelstad, D. On the definition of transfer factors, Math. Ann., Tome 278 (1987) no. 1-4, pp. 219-271 | Article | MR 909227 | Zbl 0644.22005

[19] Mœglin, Colette Comparaison des paramètres de Langlands et des exposants à l’intérieur d’un paquet d’Arthur, J. Lie Theory, Tome 19 (2009) no. 4, pp. 797-840 | MR 2599005 | Zbl 1189.22010

[20] Vogan, David A. Jr. The unitary dual of GL (n) over an Archimedean field, Invent. Math., Tome 83 (1986) no. 3, pp. 449-505 | Article | MR 827363 | Zbl 0598.22008

[21] Waldspurger, J.-L. Le groupe GL N tordu, sur un corps p-adique. I, Duke Math. J., Tome 137 (2007) no. 2, pp. 185-234 | Article | MR 2309147 | Zbl 1113.22013

[22] Waldspurger, J.-L. Les facteurs de transfert pour les groupes classiques: un formulaire, Manuscripta Math., Tome 133 (2010) no. 1-2, pp. 41-82 | Article | MR 2672539 | Zbl 1207.22011