Twisted eigenvarieties and self-dual representations

Xiang, Zhengyu

doi:10.5802/aif.3212

Twisted eigenvarieties and self-dual representations [ Variétés propres tordues et représentations autoduales ]

Xiang, Zhengyu

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2381-2444.

Résumé
Abstract

Pour un groupe réductif $G$ et un automorphisme d’ordre fini $ι$ de type Cartan de $G$ nous construisons une variété propre paramétrant les systèmes propres de Hecke automorphes cuspidaux $ι$ -invariants de $G$ . En particulier, pour $G = G l_{n}$ , on prouve que chaque système propre de Hecke cuspidale autoduale de pente finie peut être déformé dans une famille $p$ -adique de sytèmes propres de Hecke cuspidaux autoduaux contenant un sous-ensemble Zariski-dense de points classiques.

For a reductive group $G$ and a finite order Cartan-type automorphism $ι$ of $G$ , we construct an eigenvariety parameterizing $ι$ -invariant cuspidal Hecke eigensystems of $G$ . In particular, for $G = G l_{n}$ , we prove, any self-dual cuspidal Hecke eigensystem can be deformed in a p-adic family of self-dual cuspidal Hecke eigensystems containing a Zariski dense subset of classical points.

Reçu le : 2016-03-08
Révisé le : 2017-09-07
Accepté le : 2017-12-12
Publié le : 2018-11-22

DOI : https://doi.org/10.5802/aif.3212
Classification : 11F33, 11F55, 11F75, 11F85
Mots clés : variété propre, forme automorphe p-adique, représentation autoduale

@article{AIF_2018__68_6_2381_0,
     author = {Xiang, Zhengyu},
     title = {Twisted eigenvarieties and self-dual representations},
     journal = {Annales de l'Institut Fourier},
     pages = {2381--2444},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3212},
     language = {en},
     url = {archive.numdam.org/item/AIF_2018__68_6_2381_0/}
}

Xiang, Zhengyu. Twisted eigenvarieties and self-dual representations. Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2381-2444. doi : 10.5802/aif.3212. http://archive.numdam.org/item/AIF_2018__68_6_2381_0/

Bibliographie

[1] Arthur, James The $L^{2}$ -Lefschetz numbers of Hecke operators, Invent. Math., Volume 97 (2003) no. 2, pp. 257-290 | Zbl 0692.22004

[2] Arthur, James; Clozel, Laurent Simple algebras, Base change and the advanced theory of the trace formula, Annals of Mathematics Studies, Volume 120, Princeton University Press, 1989, xiii+230 pages | Zbl 0682.10022

[3] Ash, Avner; Pollack, David; Stevens, Glenn Rigidity of $p$ -adic cohomology classes of congruence subgroups of $GL (n, ℤ)$ , Proc. Lond. Math. Soc., Volume 96 (2008) no. 2, pp. 367-388 | Zbl 1206.11067

[4] Ash, Avner; Stevens, Glenn $p$ -adic deformation of arithmetic cohomology (2008) (Preprint)

[5] Barbasch, Dan; Speh, Birgit Cuspidal representations of reductive groups (2008) (https://arxiv.org/abs/0810.0787)

[6] Borel, Armand; Labesse, Jean-Pierre; Schwermer, Joachim On the cuspidal cohomology of S-arithmetic subgroups of reductive groups over number fields, Compos. Math., Volume 102 (1996) no. 1, pp. 1-40 | Zbl 0853.11044

[7] Borel, Armand; Wallach, Nolan Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical Surveys and Monographs, Volume 67, American Mathematical Society, 1999, xvii+260 pages | Zbl 0980.22015

[8] Buzzard, Kevin Eigenvarieties, $L$ -functions and Galois representations (London Mathematical Society Lecture Note Series) Volume 320, Cambridge University Press, 2007, pp. 59-120 | Zbl 1230.11054

[9] Clozel, Laurent Motifs et formes automorphes: Applications du principe de fonctorialité, Automorphic Forms, Shimura Varieties, and $L$ -functions. Volume I (Perspectives in Mathematics) Volume 10, Academic Press, 1990, pp. 77-159 | Zbl 0705.11029

[10] Coleman, Robert $p$ -adic Banach spaces and families of modular forms, Invent. Math., Volume 127 (1997) no. 3, pp. 417-479 | Zbl 0918.11026

[11] Franke, Jens Harmonic analysis in weighted $L_{2}$ -spaces, Ann. Sci. Éc. Norm. Supér., Volume 31 (1998) no. 2, pp. 181-279 | Zbl 0938.11026

[12] Franke, Jens; Schwermer, Joachim A decomposition of spaces of automorphic forms and the Eisenstein cohomology of arithmetic groups, Math. Ann., Volume 331 (1998) no. 4, pp. 765-790 | Zbl 0924.11042

[13] Jantzen, Jens Representations of algebraic groups, Mathematical Surveys and Monographs, Volume 107, American Mathematical Society, 2003 | Zbl 1034.20041

[14] Milne, James Stuart Algebraic groups, Lie Groups, and their arithmetic subgroups (2010) (available at www.jmilne.org/math/)

[15] Serre, Jean-Pierre Endomorphismes completement continus des espaces de Banach $p$ -adiques, Publ. Math., Inst. Hautes Étud. Sci., Volume 12 (1962), pp. 62-85 | Zbl 0104.33601

[16] Speh, Birgit Unitary representations of $GL (n, ℝ)$ with non-trivial $(𝔤, K)$ -cohomology, Invent. Math., Volume 71 (1983) no. 3, pp. 443-465 | Zbl 0505.22015

[17] Springer, Tonny A. Reductive groups, Automorphic forms, representations and $L$ -functions (Proceedings of Symposia in Pure Mathematics) Volume 33, American Mathematical Society, 1979, pp. 3-29 | Zbl 0416.20034

[18] Urban, Eric Eigenvarieties for reductive groups, Ann. Math., Volume 174 (2011) no. 3, pp. 1685-1784 | Zbl 1285.11081

[19] Vogan, David A. Jr.; Zuckerman, Gregg J. Unitary representations with nonzero cohomology, Compos. Math., Volume 53 (1984) no. 1, pp. 51-90 | Zbl 0692.22008

[20] Xiang, Zhengyu A construction of the full eigenvariety of a reductive group, J. Number Theory, Volume 132 (2012) no. 5, pp. 938-952 | Zbl 1272.11073

[21] Xiang, Zhengyu Twisted Lefschetz number formula and p-adic trace formula (2016) (to appear in Trans. Am. Math. Soc.)