Connected components of affine Deligne-Lusztig varieties in mixed characteristic

TL;DR

This paper characterizes the connected components of affine Deligne-Lusztig varieties in mixed characteristic, extending known results to non-minuscule cases and unramified Rapoport-Zink spaces, with implications for arithmetic geometry.

Contribution

It provides a complete description of connected components for minuscule cases and partial results for non-minuscule cases in mixed characteristic settings.

Findings

Connected components of minuscule affine Deligne-Lusztig varieties are determined.

Partial results for non-minuscule affine Deligne-Lusztig varieties are obtained.

Connected components of unramified Rapoport-Zink spaces are characterized.

Abstract

We determine the set of connected components of minuscule affine Deligne-Lusztig varieties for special maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne-Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport-Zink spaces.