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arXiv · 1610.06879

On the connected components of affine Deligne-Lusztig varieties

Xuhua He·Rong Zhou
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Abstract

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 $\s$-conjugacy classes. As an application, we verify the Axioms in \cite{HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s$-conjugacy class in a residually split group, the set of connected components is "controlled" by the set of straight elements associated to the $\s$-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with \cite{Zhou}, this allows one to verify in the residually split case, the description of the mod-$p$ isogeny classes on Shimura varieties conjectured by Langland and Rapoport. Along the way, we determine the Picard group of the Witt vector affine Grassmannian of \cite{BS} and \cite{Zhu} which is of independent interest.

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Xuhua He, Rong Zhou. 2016-10-21. On the connected components of affine Deligne-Lusztig varieties. https://doi.org/10.1215/00127094-2020-0020

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