Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph

TL;DR

This paper advances the understanding of affine Deligne-Lusztig varieties by weakening key hypotheses, leading to broader applicability of dimension formulas, Newton point descriptions, and group relations within the affine flag variety.

Contribution

It significantly relaxes the superregularity assumption in the study of affine Deligne-Lusztig varieties, extending existing results and strengthening theoretical foundations.

Findings

Weakened superregularity hypothesis broadens applicability.

Extended dimension formulas for affine Deligne-Lusztig varieties.

Enhanced description of Newton points in Iwahori double cosets.

Abstract

In this paper, we consider affine Deligne-Lusztig varieties $X_w(b)$ and their certain union $X(\mu,b)$ inside the affine flag variety of a reductive group. Several important results in the study of affine Deligne-Lusztig varieties have been established under the so-called superregularity hypothesis. Such results include a description of generic Newton points in Iwahori double cosets of loop groups, covering relation in associated Iwahori-Weyl group and dimension formula for $X(\mu,b)$. We show that one can considerably weaken the superregularity hypothesis and sometimes completely eliminate it, thus strengthening these existing results.