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Nonemptiness of single affine Deligne-Lusztig varieties

Dong Gyu Lim

Abstract

Affine Deligne-Lusztig varieties with various level structures show up in the study of Shimura varieties and moduli spaces of shtukas. Among is the Iwahori level structure which is the most refined one. We study the nonemptiness problem of single affine Deligne-Lusztig varieties at Iwahori level in the basic case. Under a genericity condition (the ``shrunken Weyl chambers'' condition), an explicit criterion is known. However, no explicit criterion has been available without the condition even conjecturally. We conjecture a new criterion in full generality, and prove it except for finitely many cases. As an application, the nonemptiness problem for special cases and a new conjectural dimension formula are discussed.

Nonemptiness of single affine Deligne-Lusztig varieties

Abstract

Affine Deligne-Lusztig varieties with various level structures show up in the study of Shimura varieties and moduli spaces of shtukas. Among is the Iwahori level structure which is the most refined one. We study the nonemptiness problem of single affine Deligne-Lusztig varieties at Iwahori level in the basic case. Under a genericity condition (the ``shrunken Weyl chambers'' condition), an explicit criterion is known. However, no explicit criterion has been available without the condition even conjecturally. We conjecture a new criterion in full generality, and prove it except for finitely many cases. As an application, the nonemptiness problem for special cases and a new conjectural dimension formula are discussed.
Paper Structure (31 sections, 34 theorems, 35 equations, 1 figure)

This paper contains 31 sections, 34 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Mazur's Inequality
  4. Single affine Deligne-Lusztig variety at Iwahori level
  5. A side remark on explicit criteria
  6. Main Conjecture
  7. New Ideas and Sketch of Proof
  8. Applications and future works
  9. The set $B(G)_x$ and cordial elements
  10. Future works on new dimension formulas
  11. Organization
  12. Basics on single affine Deligne-Lusztig varieties
  13. Iwahori-Weyl group and $B(G)$
  14. Iwahori-Weyl group $\widetilde{W}$
  15. $B(G)$ with Newton map and Kottwitz map
  16. ...and 16 more sections

Key Result

Lemma 1.1

Assume that $\kappa_G(x)=\kappa_G(b)$. If $\widetilde{\mathop{\mathrm{supp}}\nolimits_\sigma}(x)\neq\widetilde{\mathbb{S}}$ then $X_x(b)\neq\emptyset$.

Figures (1)

  • Figure 1: An apartment of the Bruhat--Tits building of $\mathrm{PGL}_3$.

Theorems & Definitions (80)

  • Lemma 1.1
  • proof
  • Conjecture 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Stronger \ref{['ghna']}
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • ...and 70 more