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Affine Deligne-Lusztig varieties beyond the minute case

Felix Schremmer, Eva Viehmann

TL;DR

This work generalizes the minute (fully Hodge-Newton decomposable) setting for affine Deligne-Lusztig varieties by introducing a depth parameter for Shimura data and classifying depth$<2$ cases. It establishes that many non-minute ADLVs retain tractable geometry: explicit EO/KR stratifications, equidimensionality, and computable dimension formulas, especially in type $A$ (with two sporadic exceptions). Central technical tools include Deligne-Lusztig reduction, the L1BC/CL1BC properties, and the ING condition, which together yield universal structure theorems for single ADLVs and their EO-strata, and provide a pathway to dimension and component counts in a broad class of groups and parahoric levels. The results illuminate how Newton stratifications interact with EKOR/EO stratifications, enabling uniform dimension formulas and component counts that tie representation-theoretic data (Deligne-Lusztig varieties of Coxeter type) to the geometry of ADLVs beyond the minute case. Collectively, these findings deepen the connection between group-theoretic invariants (depth, Coxeter type) and the arithmetic geometry of Shimura varieties and local shtukas, with potential applications to local Langlands and related programs.

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.

Affine Deligne-Lusztig varieties beyond the minute case

TL;DR

This work generalizes the minute (fully Hodge-Newton decomposable) setting for affine Deligne-Lusztig varieties by introducing a depth parameter for Shimura data and classifying depth cases. It establishes that many non-minute ADLVs retain tractable geometry: explicit EO/KR stratifications, equidimensionality, and computable dimension formulas, especially in type (with two sporadic exceptions). Central technical tools include Deligne-Lusztig reduction, the L1BC/CL1BC properties, and the ING condition, which together yield universal structure theorems for single ADLVs and their EO-strata, and provide a pathway to dimension and component counts in a broad class of groups and parahoric levels. The results illuminate how Newton stratifications interact with EKOR/EO stratifications, enabling uniform dimension formulas and component counts that tie representation-theoretic data (Deligne-Lusztig varieties of Coxeter type) to the geometry of ADLVs beyond the minute case. Collectively, these findings deepen the connection between group-theoretic invariants (depth, Coxeter type) and the arithmetic geometry of Shimura varieties and local shtukas, with potential applications to local Langlands and related programs.

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.

Paper Structure

This paper contains 26 sections, 49 theorems, 109 equations.

Table of Contents

  1. Notation and preliminaries
  2. The notion of depth
  3. Definition and basic properties
  4. Compatibility with Weil restrictions of scalars
  5. The depth with respect to Levi subgroups
  6. Classification
  7. Non-quasi-split groups
  8. Ekedahl-Oort strata
  9. Geometric properties
  10. Proof of Theorem \ref{['thm:depth2PositiveCoxeterType']}
  11. Geometry of single ADLV in flag varieties
  12. The Deligne-Lusztig reduction method
  13. The Length 1 Bruhat Cover property
  14. Geometric consequences
  15. (CL1BC) and the depth is less than 2
  16. ...and 11 more sections

Key Result

Theorem A

Let $\mu\in X_\ast(T)$ be a dominant cocharacter such that $\mathop{\mathrm{depth}}\nolimits(G,\mu)<2$, and let $x\in\mathop{\mathrm{Adm}}\nolimits(\mu)^K$, where $K$ is hyperspecial and $\sigma$-stable.

Theorems & Definitions (112)

  • Definition
  • Theorem A: Cf. Theorem \ref{['thm:EOGeometry']}
  • Theorem B
  • Theorem C: Cf. Theorem \ref{['thm:depth2GLn']}
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 102 more