Affine Deligne–Lusztig varieties via the double Bruhat graph, I : Semi-infinite orbits

Schremmer, Felix

doi:10.2140/ant.2025.19.1973

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Affine Deligne–Lusztig varieties via the double Bruhat graph, I: Semi-infinite orbits

Felix Schremmer

Vol. 19 (2025), No. 10, 1973–2014

DOI: 10.2140/ant.2025.19.1973

Abstract

We introduce a new language to describe the geometry of affine Deligne–Lusztig varieties in affine flag varieties. This first part of a two-paper series develops the definition and fundamental properties of the double Bruhat graph by studying semi-infinite orbits. This double Bruhat graph was originally introduced by Naito and Watanabe to study periodic $R$ -polynomials. We use it to describe the geometry of many affine Deligne–Lusztig varieties, overcoming a previously ubiquitous regularity condition.

Keywords

affine Weyl group, affine Deligne–Lusztig variety, double Bruhat graph, semi-infinite orbit, Shimura variety, Langlands program

Mathematical Subject Classification

Primary: 11G25, 20G25

Milestones

Received: 12 June 2023

Revised: 8 August 2024

Accepted: 18 October 2024

Published: 5 September 2025

Authors

Felix Schremmer
Department of Mathematics and New Cornerstone Laboratory The University of Hong Kong Hong Kong

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Vol. 19, No. 10, 2025