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Zero-dimensional affine Deligne--Lusztig varieties

Xuhua He, Sian Nie, Qingchao Yu

Abstract

In this paper, we study the affine Deligne--Lusztig variety $X(μ,b)_K$ and classify all quadruples $(\mathbf{G}, μ, b, K)$ with $\dim X(μ, b)_K=0$. This question was first asked by Rapoport in 2005, who also made an explicit conjecture in the hyperspecial level. We prove that $\dim X(μ,b)_K=0$ if and only if, up to certain Hodge-Newton decomposition condition, the pair $(\mathbf{G}, \{μ\})$ is of extended Lubin-Tate type. We also give a combinatorial description of this condition by the essential gap function on $B(\mathbf{G})$ and the $μ$-ordinary condition for the generic Newton stratum.

Zero-dimensional affine Deligne--Lusztig varieties

Abstract

In this paper, we study the affine Deligne--Lusztig variety and classify all quadruples with . This question was first asked by Rapoport in 2005, who also made an explicit conjecture in the hyperspecial level. We prove that if and only if, up to certain Hodge-Newton decomposition condition, the pair is of extended Lubin-Tate type. We also give a combinatorial description of this condition by the essential gap function on and the -ordinary condition for the generic Newton stratum.
Paper Structure (24 sections, 15 theorems, 57 equations, 1 figure)

This paper contains 24 sections, 15 theorems, 57 equations, 1 figure.

Table of Contents

  1. Motivation
  2. Main result
  3. Preliminary
  4. The reductive group.
  5. The $\sigma$-conjugacy classes
  6. The best integral approximation
  7. The acceptable elements
  8. The essential gap function
  9. Definition
  10. The $\mathrm{GL}_n$ case.
  11. The general case
  12. Affine Deligne--Lusztig varieties
  13. (Single) affine Deligne--Lusztig variety
  14. Certain union of affine Deligne--Lusztig varieties
  15. Hodge-Newton decomposition
  16. ...and 9 more sections

Key Result

Theorem 1

The following conditions are equivalent (1) $\dim X(\mu, b)_K = 0$; (2) The datum $(\mathbf G, \{\mu\})$ is of extended Lubin-Tate type after applying the Hodge-Newton decomposition; (3) The maximum element $[b_{\mu,\max}]$ of $B(\mathbf G,\{\mu\})$ is $\mu$-ordinary and the essential gap between $[

Figures (1)

  • Figure 1:

Theorems & Definitions (35)

  • Theorem 1
  • Lemma 2.1
  • proof
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 25 more