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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
Abstract

We study affine Deligne–Lusztig varieties Xw(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 Xw(b) if and only if b satisfies a certain Hodge–Newton indecomposability condition. Our main result is that for such w and b, Xw(b) has a simple geometric structure: the σ-centralizer of b acts transitively on the set of irreducible components of Xw(b); and each irreducible component is an iterated fibration over a classical Deligne–Lusztig variety of Coxeter type, and the iterated fibers are either 𝔸1 or 𝔾m.

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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