Affine Deligne–Lusztig varieties with finite Coxeter parts

He, Xuhua; Nie, Sian; Yu, Qingchao

doi:10.2140/ant.2024.18.1681

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Affine Deligne–Lusztig varieties with finite Coxeter parts

Xuhua He, Sian Nie and Qingchao Yu

Vol. 18 (2024), No. 9, 1681–1714

DOI: 10.2140/ant.2024.18.1681

Abstract

We study affine Deligne–Lusztig varieties $X_{w (b)}$ when the finite part of the element $w$ in the Iwahori–Weyl group is a partial $σ$ -Coxeter element. We show that such $w$ is a cordial element and $X_{w (b)} \neq \emptyset$ if and only if $b$ satisfies a certain Hodge–Newton indecomposability condition. Our main result is that for such $w$ and $b$ , $X_{w (b)}$ has a simple geometric structure: the $σ$ -centralizer of $b$ acts transitively on the set of irreducible components of $X_{w (b)}$ ; and each irreducible component is an iterated fibration over a classical Deligne–Lusztig variety of Coxeter type, and the iterated fibers are either $𝔸^{1}$ or $𝔾_{m}$ .

Keywords

affine Deligne–Lusztig varieties, Coxeter elements

Mathematical Subject Classification

Primary: 11G25, 20G25

Milestones

Received: 23 October 2022

Revised: 28 June 2023

Accepted: 12 October 2023

Published: 19 September 2024

Authors

Xuhua He
Department of Mathematics University of Hong Kong Pokfulam Hong Kong

Sian Nie
Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China	School of Mathematical Sciences University of Chinese Academy of Sciences Chinese Academy of Sciences Beijing China

Qingchao Yu
Beijing International Center for Mathematical Research Beijing University Beijing China

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Vol. 18, No. 9, 2024