Affine Deligne-Lusztig varieties associated to additive affine Weyl group elements

TL;DR

This paper establishes a method to prove non-emptiness of affine Deligne-Lusztig varieties for certain affine Weyl group elements, advancing understanding of their geometric properties and confirming conjectures in specific cases.

Contribution

It introduces a new approach using cuspidal conjugacy classes and combinatorics of fully commutative elements to prove non-emptiness under length additivity conditions.

Findings

Non-emptiness holds for affine Deligne-Lusztig varieties in specific cases.

Provides a partial converse to existing emptiness results.

Connects combinatorics of Weyl groups with geometric properties.

Abstract

Affine Deligne-Lusztig varieties can be thought of as affine analogs of classical Deligne-Lusztig varieties, or Frobenius-twisted analogs of Schubert varieties. We provide a method for proving a non-emptiness statement for affine Deligne-Lusztig varieties inside the affine flag variety associated to affine Weyl group elements satisfying a certain length additivity hypothesis. In particular, we prove that non-emptiness holds whenever it is conjectured to do so for alcoves in the shrunken dominant Weyl chamber, providing a partial converse to the emptiness results of Goertz, Haines, Kottwitz, and Reuman. Our technique involves the work of Geck and Pfeiffer on cuspidal conjugacy classes, in addition to an analysis of the combinatorics of certain fully commutative elements in the finite Weyl group.