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1 October 2020 On the connected components of affine Deligne–Lusztig varieties
Xuhua He, Rong Zhou
Duke Math. J. 169(14): 2697-2765 (1 October 2020). DOI: 10.1215/00127094-2020-0020


We study the set of connected components of certain unions of affine Deligne–Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic σ -conjugacy classes. As an application, we verify the Axioms in recent work by the first author and Rapoport for certain PEL-type Shimura varieties. We also show that, for any nonbasic σ -conjugacy class in a residually split group, the set of connected components is “controlled” by the set of straight elements associated to the σ -conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with the second author’s earlier article, this allows one to verify, in the residually split case, the description of the mod- p isogeny classes on Shimura varieties conjectured by Langlands and Rapoport. Along the way, we determine the Picard group of the Witt vector affine Grassmannian first proposed by Bhatt, Scholze, and Zhu, which is of independent interest.


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Xuhua He. Rong Zhou. "On the connected components of affine Deligne–Lusztig varieties." Duke Math. J. 169 (14) 2697 - 2765, 1 October 2020.


Received: 21 October 2016; Revised: 20 February 2020; Published: 1 October 2020
First available in Project Euclid: 11 August 2020

MathSciNet: MR4149507
Digital Object Identifier: 10.1215/00127094-2020-0020

Primary: 14G35
Secondary: 20G25

Keywords: affine Deligne–Lusztig varieties , Shimura varieties

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 14 • 1 October 2020
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