On the connected components of affine Deligne–Lusztig varieties

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 $\sigma$-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 $\sigma$-conjugacy class in a residually split group, the set of connected components is “controlled” by the set of straight elements associated to the $\sigma$-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.