Nonemptiness of single affine Deligne-Lusztig varieties

TL;DR

This paper investigates the nonemptiness of single affine Deligne-Lusztig varieties at Iwahori level, proposing a new conjectural criterion that extends previous results and discussing implications for related problems.

Contribution

It introduces a new conjectural criterion for nonemptiness of affine Deligne-Lusztig varieties at Iwahori level without genericity assumptions, and proves it in most cases.

Findings

Proposed a new conjectural criterion for nonemptiness.

Proved the criterion in all but finitely many cases.

Discussed applications to dimension formulas and special cases.

Abstract

Affine Deligne-Lusztig varieties with various level structures show up in the study of Shimura varieties and moduli spaces of shtukas. Among is the Iwahori level structure which is the most refined one. We study the nonemptiness problem of single affine Deligne-Lusztig varieties at Iwahori level in the basic case. Under a genericity condition (the ``shrunken Weyl chambers'' condition), an explicit criterion is known. However, no explicit criterion has been available without the condition even conjecturally. We conjecture a new criterion in full generality, and prove it except for finitely many cases. As an application, the nonemptiness problem for special cases and a new conjectural dimension formula are discussed.