On the connected components of affine Deligne-Lusztig varieties

TL;DR

This paper investigates the connected components of affine Deligne-Lusztig varieties related to Shimura varieties, providing classifications for basic and nonbasic cases, and verifying conjectures about mod-$p$ isogeny classes.

Contribution

It determines the connected components for basic classes, relates nonbasic cases to straight elements and Levi obstructions, and verifies key axioms for certain Shimura varieties.

Findings

Connected components classified for basic $\s$-conjugacy classes.

Nonbasic cases controlled by straight elements and Levi subgroups.

Verification of Langlands-Rapoport conjecture in residually split cases.

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.