Green Functions in Small Characteristic

TL;DR

This paper completes the determination of Green functions for all groups of Lie type by solving the remaining cases for exceptional groups E8(q) over fields with characteristic 2, 3, or 5.

Contribution

It provides the missing Green function values for E8(q) groups in small characteristics, finishing a long-standing classification effort.

Findings

Green functions for E8(q) in small characteristics are now fully determined.

The results fill the last gaps in the classification of Green functions for Lie type groups.

This advances the understanding of representation theory for these exceptional groups.

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 $5$.