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Parabolic conjugacy class in finite reductive groups and its additive analogue

GyeongHyeon Nam

Abstract

In this paper, we answer the question posed by Goodwin and Röhrle for reductive groups and their parabolic subgroups. In addition, we consider an additive analogue of this problem. By studying this additive analogue, we identify similar properties between the Deligne-Lusztig character of a finite reductive group and the Harish-Chandra induction over the corresponding finite Lie algebra.

Parabolic conjugacy class in finite reductive groups and its additive analogue

Abstract

In this paper, we answer the question posed by Goodwin and Röhrle for reductive groups and their parabolic subgroups. In addition, we consider an additive analogue of this problem. By studying this additive analogue, we identify similar properties between the Deligne-Lusztig character of a finite reductive group and the Harish-Chandra induction over the corresponding finite Lie algebra.

Paper Structure

This paper contains 16 sections, 17 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Counting formula for reductive group
  4. A formula of $k(P^F,G^F)$
  5. Counting formula for Lie algebra
  6. Parabolic subalgebra
  7. Nilpotent subalgebra
  8. Types of elements
  9. Group case
  10. Additive case
  11. Group case result
  12. Types in $\mathcal{R}(P^F,G^F)$
  13. Conclusion
  14. Additive case result
  15. Types in $\mathcal{R}(\mathfrak{p}^F,\mathfrak{g}^F)$
  16. ...and 1 more sections

Key Result

Theorem 1

(Theorem thm:result-group) We have where $\xi$ is a pseudo-Levi subgroup of $G$, $W_\xi(T)$ the Weyl group of $\xi$ over a maximal torus $T$, $T_w$ a $w$-twisted torus over a split maximal torus $T$, and Furthermore, this satisfies polynomial count on residue class (PORC) property.

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 3
  • Theorem 4: Theorem \ref{['thm:GR-additive-analogue']}
  • Remark 5
  • Lemma 6
  • Proposition 7
  • Corollary 8
  • Lemma 9
  • proof
  • Proposition 10
  • ...and 20 more