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

He, Xuhua

doi:10.24033/asens.2305

He, Xuhua

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

Abstract (Original language)
Résumé

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.

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.

MR Zbl

DOI: 10.24033/asens.2305

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

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     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/}
}

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

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