On the Schur indices of cuspidal unipotent characters

TL;DR

This paper determines the Schur indices of all cuspidal unipotent characters for certain exceptional Lie type groups, resolving most of the previously open cases under specific conditions.

Contribution

It proves that Schur indices are 1 for cuspidal unipotent characters in F4 and E8 groups over good characteristic fields, and bounds them for E7.

Findings

Schur indices for F4 and E8 are 1 under good characteristic.

Schur indices for E7 are at most 2.

Four of six previously open cases are settled.

Abstract

In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type $F_4$, $E_7$ and $E_8$. In this paper, we show that the Schur indices of all cuspidal unipotent characters for type $F_4$ and $E_8$ are~1, assuming that the group is defined over a field of ``good'' characteristic. This settles four out of the six open cases. For type $E_7$, we show that the Schur indices are at most~2.