Stabilizers of irreducible components of affine Deligne--Lusztig varieties

TL;DR

This paper investigates the stabilizers of top-dimensional irreducible components of affine Deligne--Lusztig varieties, confirming a conjecture and linking these components to the structure of Shimura varieties.

Contribution

It proves that stabilizers are maximal-volume parahoric subgroups, confirming X. Zhu's conjecture and describing components in Shimura varieties.

Findings

Stabilizers are maximal-volume parahoric subgroups.

Confirmed a conjecture of X. Zhu.

Described irreducible components in Shimura varieties.

Abstract

We study the $J_b(F)$-action on the set of top-dimensional irreducible components of affine Deligne--Lusztig varieties in the affine Grassmannian. We show that the stabilizer of any such component is a parahoric subgroup of $J_b(F)$ of maximal volume, verifying a conjecture of X.~Zhu. As an application, we give a description of the set of top-dimensional irreducible components in the basic locus of Shimura varieties.