Lifting Deligne-Lusztig Reduction and Geometric Coxeter Type Elements

TL;DR

This paper develops a framework to analyze affine Deligne-Lusztig varieties, introducing geometric Coxeter type elements, and provides criteria to identify when these varieties decompose into products of classical Deligne-Lusztig varieties and affine spaces.

Contribution

It introduces the concept of geometric Coxeter type elements and establishes a criterion for affine Deligne-Lusztig varieties to decompose into simpler components.

Findings

A uniform framework for studying affine Deligne-Lusztig varieties.

Introduction of geometric Coxeter type elements.

A criterion for product decompositions of affine Deligne-Lusztig varieties.

Abstract

Cases of Shimura varieties where the special fibre of a Rapoport-Zink space is simply the union of classical Deligne-Lusztig varieties are known as fully Hodge-Newton decomposable ones, and have been studied with great interest in the past. In recent times, the focus has shifted to identify tractable cases beyond the fully Hodge-Newton decomposable ones, and several instances have been identified where only products of classical Deligne-Lusztig varities with simpler spaces occur. In our paper, we provide a uniform framework to capture these phenomena. By studying liftings from the affine flag variety to the loop group and combining them with the Deligne-Lusztig reduction method, our main result is a powerful criterion to show that an affine Deligne-Lusztig variety is the product of a classical Deligne-Lusztig variety with affine spaces and pointed affine spaces. We introduce the class of elements that we call having geometric Coxeter type, strictly including previously studied notions such as positive Coxeter type or finite Coxeter type. These elements of geometric Coxeter type satisfy the conditions for our main result and also a condition on the Newton stratification introduced by Mili\'cevi\'c-Viehmann.