Affine Deligne-Lusztig varieties and folded galleries governed by chimneys

TL;DR

This paper characterizes the nonemptiness and dimension of affine Deligne-Lusztig varieties using positively folded galleries related to chimneys, and proves nonemptiness for specific cases when the associated parabolic subgroup has rank 1.

Contribution

It introduces a gallery-based approach to analyze affine Deligne-Lusztig varieties and explicitly constructs galleries to prove nonemptiness in rank 1 cases.

Findings

Characterization of nonemptiness and dimension via galleries

Explicit construction of galleries for rank 1 parabolic subgroups

Proof of nonemptiness for certain elements in affine flag varieties

Abstract

We characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety $X_x(b)$ in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of b has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements x by explicitly constructing such galleries.