Affineness of Deligne-Lusztig varieties for minimal length elements

TL;DR

This paper proves that Deligne-Lusztig varieties linked to minimal length elements in their twisted class are affine, offering a new proof inspired by regular elements and Broué's conjectures.

Contribution

It introduces a novel proof demonstrating the affineness of these varieties, differing from previous methods by He and Orlik-Rapoport.

Findings

Deligne-Lusztig varieties for minimal length elements are affine

New proof technique inspired by regular elements and Broué's conjectures

Enhances understanding of the structure of these varieties

Abstract

We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Brou\'e's conjectures.