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Basic Loci of Positive Coxeter Type for $GL_n$

Ryosuke Shimada

Abstract

Motivated by the problem of giving an explicit description of the basic locus in the reduction of Shimura varieties, Görtz, He and Nie studied the cases where the basic affine Deligne-Lusztig variety, which serves as its group-theoretic model, is a union of classical Deligne-Lusztig varieties associated to Coxeter elements. In this paper, we study a natural generalization of this stratification in the case of $GL_n$.

Basic Loci of Positive Coxeter Type for $GL_n$

Abstract

Motivated by the problem of giving an explicit description of the basic locus in the reduction of Shimura varieties, Görtz, He and Nie studied the cases where the basic affine Deligne-Lusztig variety, which serves as its group-theoretic model, is a union of classical Deligne-Lusztig varieties associated to Coxeter elements. In this paper, we study a natural generalization of this stratification in the case of .
Paper Structure (25 sections, 28 theorems, 85 equations)

This paper contains 25 sections, 28 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Affine Deligne-Lusztig Varieties
  5. The $\mathbb J$-stratification
  6. Deligne-Lusztig Reduction Method
  7. Length Positive Elements
  8. Classification
  9. The Ekedahl-Oort Stratification for $\omega_2+\omega_{n-2}$
  10. Classification
  11. The cases of $\omega_2$ and $2\omega_1$
  12. The case of $\omega_2$ when $n$ is odd
  13. The case of $\omega_2$ when $n$ is even
  14. The case of $2\omega_1$ when $n$ is odd
  15. The case of $2\omega_1$ when $n$ is even
  16. ...and 10 more sections

Key Result

Theorem A

Let $G=\mathop{\mathrm{GL}}\nolimits_n$ and let $\mu\in X_*(T)_+$. Then the following assertions are equivalent. Here $\omega_k$ denotes the cocharacter of the form $(1,\ldots,1,0,\ldots,0)$ in which $1$ is repeated $k$ times. If $\mu$ satisfies the equivalent conditions above, then $X_{\preceq\mu}(\tau_\mu)$ is universally homeomorphic to a union of the product of a Deligne-Lusztig variety of Co

Theorems & Definitions (47)

  • Theorem A: See Theorem \ref{['classification theorem']} and Theorem \ref{['geometric structure']}
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • ...and 37 more