# On the values of unipotent characters of finite Chevalley groups of type $E_6$ in characteristic 3

## TL;DR

This paper computes the values of unipotent characters of finite Chevalley groups of type E6 over fields of characteristic 3, resolving unknown scalars in Lusztig's character sheaf framework using Hecke algebra representations.

## Contribution

It determines the scalars needed to compute unipotent character values for E6 groups in characteristic 3, advancing Lusztig's theory application in this specific case.

## Key findings

- Calculated scalars for E6 in characteristic 3
- Enhanced understanding of character sheaves in bad characteristic
- Provided explicit character value computations

## Abstract

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power of the bad prime $p=3$, by exploiting known facts about the representation theory of the Hecke algebra associated with $G$.