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Convex elements and Steinberg's cross-sections

Sian Nie, Panjun Tan, Qingchao Yu

Abstract

In this paper, we study convex elements in a (twisted) Weyl group introduced by Ivanov and the first named author. We show that each conjugacy class of the twisted Weyl group contains a convex element, and moreover, the Steinberg cross-sections exist for all convex elements. This result strictly enlarges the cases of Steinberg cross-sections from a new perspective, and will play an essential role in the study of higher Deligne-Lusztig representations.

Convex elements and Steinberg's cross-sections

Abstract

In this paper, we study convex elements in a (twisted) Weyl group introduced by Ivanov and the first named author. We show that each conjugacy class of the twisted Weyl group contains a convex element, and moreover, the Steinberg cross-sections exist for all convex elements. This result strictly enlarges the cases of Steinberg cross-sections from a new perspective, and will play an essential role in the study of higher Deligne-Lusztig representations.

Paper Structure

This paper contains 13 sections, 13 theorems, 25 equations.

Table of Contents

  1. Background
  2. Main result
  3. Structure of the paper
  4. Convex Elements
  5. Convex elements
  6. The isomorphism $\Xi_{{\dot x}}$
  7. Transversality
  8. Inductive Construction of Convex Elements
  9. Good position elements
  10. Good position elements
  11. A result by He-Lusztig
  12. A counterexample
  13. Coxeter elements

Key Result

Theorem 1

Theorems & Definitions (35)

  • Theorem 1
  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • proof
  • Proposition 1.6
  • proof
  • ...and 25 more