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On the connected components of affine Deligne-Lusztig varieties

Xuhua He Rong Zhou

Algebraic Geometry mathscidoc:1912.43208

arXiv preprint arXiv:1610.06879, 2016.10
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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@inproceedings{xuhua2016on,
  title={On the connected components of affine Deligne-Lusztig varieties},
  author={Xuhua He, and Rong Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112943590766768},
  booktitle={arXiv preprint arXiv:1610.06879},
  year={2016},
}
Xuhua He, and Rong Zhou. On the connected components of affine Deligne-Lusztig varieties. 2016. In arXiv preprint arXiv:1610.06879. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112943590766768.
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