Connectedness of affine Deligne-Lusztig varieties for unramified groups

TL;DR

This paper characterizes the connected components of affine Deligne-Lusztig varieties for unramified groups, enabling verification of key conjectures and structures related to Shimura varieties and their mod p reductions.

Contribution

It determines the connected components of affine Deligne-Lusztig varieties for unramified groups, confirming several conjectures and structural properties in this setting.

Findings

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

Verification of He-Rapoport axioms in the unramified case.

Confirmation of the Newton strata's almost product structure.

Abstract

For unramified reductive groups, we determine the connected components of affine Deligne-Lusztig varieties in the partial affine flag varieties. Based on the work of Hamacher-Kim and Zhou, this result allows us to verify, in the unramified group case, the He-Rapoport axioms, the ``almost product structure" of Newton strata, and the precise description of mod $p$ isogeny classes predicted by the Langlands-Rapoport conjecture, for the Kisin-Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.