Dimensions of affine Deligne-Lusztig varieties in affine flag varieties

TL;DR

This paper investigates the non-emptiness and dimension of affine Deligne-Lusztig varieties in affine flag varieties, confirming conjectures for split classical groups and basic classes, advancing understanding of their geometric structure.

Contribution

It proves a conjecture on non-emptiness of affine Deligne-Lusztig varieties for split groups and basic classes, and verifies the dimension formula for classical groups in most cases.

Findings

Confirmed non-emptiness conjecture for affine Deligne-Lusztig varieties.

Validated the dimension formula for classical groups with trivial basic class.

Extended understanding of the geometric properties of affine Deligne-Lusztig varieties.

Abstract

Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic $\sigma$-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic $\sigma$-conjugacy class is the class of $b=1$, we also prove the dimension formula predicted in op. cit. in almost all cases.