Twisted eigenvarieties and self-dual representations
[Variétés propres tordues et représentations autoduales]
Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2381-2444.

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=Gl 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=Gl 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 :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3212
Classification : 11F33, 11F55, 11F75, 11F85
Keywords: eigenvariety, p-adic automorphic form, self-dual representation
Mot clés : variété propre, forme automorphe p-adique, représentation autoduale
Xiang, Zhengyu 1

1 SCMS and Fudan University East Guanghua Main Tower, Room 2214 220 Handan Road, Shanghai 200433 (P.R.C.)
@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{\textquoteright}institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3212},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3212/}
}
TY  - JOUR
AU  - Xiang, Zhengyu
TI  - Twisted eigenvarieties and self-dual representations
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2381
EP  - 2444
VL  - 68
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3212/
DO  - 10.5802/aif.3212
LA  - en
ID  - AIF_2018__68_6_2381_0
ER  - 
%0 Journal Article
%A Xiang, Zhengyu
%T Twisted eigenvarieties and self-dual representations
%J Annales de l'Institut Fourier
%D 2018
%P 2381-2444
%V 68
%N 6
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3212/
%R 10.5802/aif.3212
%G en
%F 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/articles/10.5802/aif.3212/

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

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

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

[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

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

[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

[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

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

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

[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

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

[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

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

[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

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

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

[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

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

Cité par Sources :