Stratifications of affine Deligne-Lusztig varieties

TL;DR

This paper compares different stratifications of affine Deligne-Lusztig varieties, showing their equivalence in Coxeter type cases, which enhances understanding of their geometric and arithmetic structure.

Contribution

It proves the equivalence of Chen-Viehmann and Bruhat-Tits stratifications for affine Deligne-Lusztig varieties of Coxeter type, clarifying their relationship.

Findings

Stratifications coincide in all Coxeter type cases

Enhances understanding of affine Deligne-Lusztig varieties

Links geometric stratifications to arithmetic properties

Abstract

Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in the context of affine flag varieties and affine Grassmannians. They are closely related to moduli spaces of $p$-divisible groups in positive characteristic, and thus to arithmetic properties of Shimura varieties. We compare stratifications of affine Deligne-Lusztig varieties attached to a basic element $b$. In particular, we show that the stratification defined by Chen and Viehmann using the relative position to elements of the group $J_b$, the $\sigma$-centralizer of $b$, coincides with the Bruhat-Tits stratification in all cases of Coxeter type, as defined by X. He and the author.