Affine Deligne-Lusztig varieties beyond the minute case

TL;DR

This paper extends the class of affine Deligne-Lusztig varieties beyond the well-understood minute case by introducing a notion of depth for Shimura data, enabling a broader and still manageable geometric description.

Contribution

It generalizes the minute condition for ADLVs by defining a depth parameter, expanding the class of varieties with computable geometric properties.

Findings

Broader class of ADLVs with explicit geometric descriptions

Introduction of depth as a key parameter in Shimura data

Extension beyond the depth ≤ 1 case to depth < 2

Abstract

Affine Deligne-Lusztig varieties in the fully Hodge-Newton decomposable (or minute) case are the only larger class of ADLVs which could be described completely in the past. Instances of them play important roles in arithmetic geometry, from Harris-Taylor's proof of the local Langlands correspondence to applications in the Kudla program. We study generalizations for many of the equivalent conditions characterizing them to obtain in this way a larger class of ADLVs that still have a similarly good and computable description of their geometry. To generalize the minute condition itself, we introduce the notion of depth for a Shimura datum - the minute cases being those of depth bounded by 1, the cases we study being the ones of depth less than 2.