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Green Functions in Small Characteristic

Frank Lübeck

TL;DR

This work resolves the last open cases for ordinary Green functions in finite groups of Lie type by completing the computation for the exceptional group $ ext{E}_8(q)$ in bad characteristics, namely when $q$ is a power of $2$, $3$, or $5$. The authors develop and execute a computational framework combining the Lusztig–Shoji algorithm with explicit parabolic-permutation character analysis, leveraging the Steinberg presentation and Chevalley basis technology to handle unipotent elements. A key outcome is that, aside from a single exceptional class in the case $q ot o 0 mod 3$, all relevant scalars in the Green-function expression equal $1$, allowing complete determination of Green functions for $ ext{E}_8(q)$ in bad characteristic. The results, together with previous work, establish Green functions for all groups of Lie type and provide explicit data for further use in character theory and related computational tools.

Abstract

The values of the ordinary Green functions are known for almost all groups of Lie type, a long term achievement by various authors. In this note we solve the last open cases, which are for exceptional groups $E_8(q)$ where $q$ is a power of $2$, $3$ or $5$.

Green Functions in Small Characteristic

TL;DR

This work resolves the last open cases for ordinary Green functions in finite groups of Lie type by completing the computation for the exceptional group in bad characteristics, namely when is a power of , , or . The authors develop and execute a computational framework combining the Lusztig–Shoji algorithm with explicit parabolic-permutation character analysis, leveraging the Steinberg presentation and Chevalley basis technology to handle unipotent elements. A key outcome is that, aside from a single exceptional class in the case , all relevant scalars in the Green-function expression equal , allowing complete determination of Green functions for in bad characteristic. The results, together with previous work, establish Green functions for all groups of Lie type and provide explicit data for further use in character theory and related computational tools.

Abstract

The values of the ordinary Green functions are known for almost all groups of Lie type, a long term achievement by various authors. In this note we solve the last open cases, which are for exceptional groups where is a power of , or .
Paper Structure (13 sections, 4 theorems, 21 equations)

This paper contains 13 sections, 4 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Computing in the unipotent subgroup
  4. Different Chevalley bases
  5. Commutator formula
  6. Green functions by the Lusztig-Shoji algorithm
  7. Permutation characters of parabolic subgroups
  8. Permutation characters by Deligne-Lusztig characters
  9. Permutation characters by counting fixed points
  10. Computing a character value
  11. Counting solutions
  12. Detecting cases with no solutions
  13. Application to $\mathbf{E_8(q)}$

Key Result

Proposition 3.1

Let $r_1, r_2, \ldots, r_N$ be the positive roots of $\mathbf{G}$ in any fixed order, we write $x_i(a) := x_{r_i}(a)$ for the corresponding root elements.

Theorems & Definitions (6)

  • Proposition 3.1
  • Proposition 4.1
  • Remark 4.2
  • Remark 5.1
  • Proposition 5.2
  • Theorem 6.1