Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties

He, Xuhua

doi:10.24033/asens.2305

Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties
[La conjecture de Kottwitz-Rapoport sur les unions de variétés de Deligne-Lusztig affines]

He, Xuhua

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1125-1141.

Résumé
Abstract

Dans cet article nous prouvons une conjecture de Kottwitz et Rapoport sur l'union de variétés de Deligne-Lusztig affines (généralisées) $X {(μ, b)}_{J}$ pour $G$ un groupe $p$ -adique et $P_{J}$ son sous-groupe parahorique. Nous montrons que $X {(μ, b)}_{J}$ est non vide si et seulement si la version de l'inégalité de Mazur pour les groupes est satisfaite. Au cours de la preuve, nous obtenons une généralisation de la conjecture de Grothendieck sur les inclusions des adhérences de classes de $σ$ -conjugaison d'un groupe de lacets tordu.

In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X {(μ, b)}_{J}$ for a $p$ -adic group $G$ and its parahoric subgroup $P_{J}$ . We show that $X {(μ, b)}_{J} \neq \emptyset$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of $σ$ -conjugacy classes of a twisted loop group.

MR Zbl

DOI : 10.24033/asens.2305

Classification : 14M15, 14G35, 20G25
Keywords: Shimura varieties, affine Deligne-Lusztig varieties, Newton strata.
Mot clés : Cariétés de Shimura, variétés de Deligne-Lusztig affines, strates de Newton.

@article{ASENS_2016__49_5_1125_0,
     author = {He, Xuhua},
     title = {Kottwitz-Rapoport conjecture on unions of affine {Deligne-Lusztig} varieties},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1125--1141},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {5},
     year = {2016},
     doi = {10.24033/asens.2305},
     mrnumber = {3581812},
     zbl = {1375.14166},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2305/}
}

TY  - JOUR
AU  - He, Xuhua
TI  - Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 1125
EP  - 1141
VL  - 49
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - https://www.numdam.org/articles/10.24033/asens.2305/
DO  - 10.24033/asens.2305
LA  - en
ID  - ASENS_2016__49_5_1125_0
ER  -

%0 Journal Article
%A He, Xuhua
%T Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 1125-1141
%V 49
%N 5
%I Société Mathématique de France. Tous droits réservés
%U https://www.numdam.org/articles/10.24033/asens.2305/
%R 10.24033/asens.2305
%G en
%F ASENS_2016__49_5_1125_0

He, Xuhua. Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1125-1141. doi : 10.24033/asens.2305. https://www.numdam.org/articles/10.24033/asens.2305/

Bibliographie
Cité par

Gashi, Q. R. On a conjecture of Kottwitz and Rapoport, Ann. Sci. Éc. Norm. Supér., Volume 43 (2010), pp. 1017-1038 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Görtz, U.; He, X. Basic loci of Coxeter type in Shimura varieties, Camb. J. Math., Volume 3 (2015), pp. 323-353 (ISSN: 2168-0930) | DOI | MR | Zbl

Görtz, U.; Haines, T. J.; Kottwitz, R. E.; Reuman, D. C. Affine Deligne-Lusztig varieties in affine flag varieties, Compos. Math., Volume 146 (2010), pp. 1339-1382 (ISSN: 0010-437X) | DOI | MR | Zbl

Görtz, U.; He, X.; Nie, S. $𝐏$ -alcoves and nonemptiness of affine Deligne-Lusztig varieties, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015), pp. 647-665 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Hamacher, P. The Dimension of Affine Deligne–Lusztig Varieties in the Affine Grassmannian, Int. Math. Res. Not., Volume 2015 (2015), pp. 12804-12839 (ISSN: 1073-7928) | DOI | MR | Zbl

Hamacher, P. The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type, Duke Math. J., Volume 164 (2015), pp. 2809-2895 (ISSN: 0012-7094) | DOI | MR | Zbl

He, X. Minimal length elements in some double cosets of Coxeter groups, Adv. Math., Volume 215 (2007), pp. 469-503 (ISSN: 0001-8708) | DOI | MR | Zbl

He, X. Closure of Steinberg fibers and affine Deligne-Lusztig varieties, Int. Math. Res. Not., Volume 2011 (2011), pp. 3237-3260 (ISSN: 1073-7928) | DOI | MR | Zbl

He, X. Geometric and homological properties of affine Deligne-Lusztig varieties, Ann. of Math., Volume 179 (2014), pp. 367-404 (ISSN: 0003-486X) | DOI | MR | Zbl

Haines, T. J.; He, X. Vertexwise criteria for admissibility of alcoves (preprint arXiv:1411.5450 ) | MR

He, X.; Nie, S. On the acceptable sets (preprint arXiv:1408.5836 )

He, X.; Nie, S. Minimal length elements of extended affine Weyl groups, Compos. Math., Volume 150 (2014), pp. 1903-1927 (ISSN: 0010-437X) | DOI | MR | Zbl

Haines, T. J.; Rapoport, M. On parahoric subgroups, Adv. Math., Volume 219 (2008), pp. 188-198 (appendix to [22] ) | DOI | MR

Hartl, U.; Viehmann, E. The Newton stratification on deformations of local $G$ -shtukas, J. reine angew. Math., Volume 656 (2011), pp. 87-129 (ISSN: 0075-4102) | DOI | MR | Zbl

Kottwitz, R. E. On the Hodge-Newton decomposition for split groups, Int. Math. Res. Not., Volume 2003 (2003), pp. 1433-1447 (ISSN: 1073-7928) | DOI | MR | Zbl

Kottwitz, R. E. Isocrystals with additional structure, Compositio Math., Volume 56 (1985), pp. 201-220 (ISSN: 0010-437X) | Numdam | MR | Zbl

Kottwitz, R. E. Isocrystals with additional structure. II, Compositio Math., Volume 109 (1997), pp. 255-339 (ISSN: 0010-437X) | DOI | MR | Zbl

Kottwitz, R. E.; Rapoport, M. On the existence of $F$ -crystals, Comment. Math. Helv., Volume 78 (2003), pp. 153-184 (ISSN: 0010-2571) | DOI | MR | Zbl

Lucarelli, C. A converse to Mazur's inequality for split classical groups, J. Inst. Math. Jussieu, Volume 3 (2004), pp. 165-183 (ISSN: 1474-7480) | DOI | MR | Zbl

Lusztig, G., CRM Monograph Series, 18, Amer. Math. Soc., Providence, RI, 2003, 136 pages (ISBN: 0-8218-3356-1) | MR | Zbl

Moy, A.; Prasad, G. Unrefined minimal $K$ -types for $p$ -adic groups, Invent. math., Volume 116 (1994), pp. 393-408 (ISSN: 0020-9910) | DOI | MR | Zbl

Pappas, G.; Rapoport, M. Twisted loop groups and their affine flag varieties, Adv. Math., Volume 219 (2008), pp. 118-198 (ISSN: 0001-8708) | DOI | MR | Zbl

Pappas, G.; Zhu, X. Local models of Shimura varieties and a conjecture of Kottwitz, Invent. math., Volume 194 (2013), pp. 147-254 (ISSN: 0020-9910) | DOI | MR | Zbl

Rapoport, M. A guide to the reduction modulo $p$ of Shimura varieties, Astérisque, Volume 298 (2005), pp. 271-318 (ISSN: 0303-1179) | Numdam | MR | Zbl

Rapoport, M.; Richartz, M. On the classification and specialization of $F$ -isocrystals with additional structure, Compositio Math., Volume 103 (1996), pp. 153-181 (ISSN: 0010-437X) | Numdam | MR | Zbl

Rapoport, M.; Zink, T. A finiteness theorem in the Bruhat-Tits building: an application of Landvogt's embedding theorem, Indag. Math. (N.S.), Volume 10 (1999), pp. 449-458 (ISSN: 0019-3577) | DOI | MR | Zbl

Viehmann, E. Newton strata in the loop group of a reductive group, Amer. J. Math., Volume 135 (2013), pp. 499-518 (ISSN: 0002-9327) | DOI | MR | Zbl

Wintenberger, J.-P. Existence de $F$ -cristaux avec structures supplémentaires, Adv. Math., Volume 190 (2005), pp. 196-224 (ISSN: 0001-8708) | DOI | MR | Zbl

Zhu, X. private communication

Zhu, X. On the coherence conjecture of Pappas and Rapoport, Ann. of Math., Volume 180 (2014), pp. 1-85 (ISSN: 0003-486X) | DOI | MR | Zbl

Cité par Sources :