# Computing Green functions in small characteristic

## TL;DR

This paper introduces a computational method to determine Green functions for certain exceptional Lie type groups in small characteristics, solving previously open cases using reduction and computer algebra techniques.

## Contribution

It presents a new general approach that reduces complex cases to prime q and employs computer algebra, successfully resolving open problems in exceptional groups.

## Key findings

- All open cases in ${^2 ext{E}_6}$ and $E_7$ are solved.
- At least one open case in $E_8$ is resolved.
- The method generalizes to small characteristic cases for exceptional groups.

## Abstract

Let $G(q)$ be a finite group of Lie type over a field with $q$ elements, where $q$ is a prime power. The Green functions of $G(q)$, as defined by Deligne and Lusztig, are known in \textit{almost} all cases by work of Beynon--Spaltenstein, Lusztig und Shoji. Open cases exist for groups of exceptional type ${^2\!E}_6$, $E_7$, $E_8$ in small characteristics. We propose a general method for dealing with these cases, which procedes by a reduction to the case where $q$ is a prime and then uses computer algebra techniques. In this way, all open cases in type ${^2\!E}_6$, $E_7$ are solved, as well as at least one particular open case in type $E_8$.