Almost cyclic elements in cross-characteristic representations of finite groups of Lie type

TL;DR

This paper classifies certain projective irreducible representations of finite simple groups of Lie type, focusing on those with elements represented by almost cyclic matrices, completing the understanding beyond Weil representations.

Contribution

It provides a comprehensive classification of cross-characteristic projective irreducible representations of finite quasi-simple groups of Lie type with almost cyclic element images.

Findings

Complete classification for all such representations of finite groups of Lie type.

Extension of previous work on Weil representations to broader classes.

Clarification of the structure of representations with almost cyclic matrices.

Abstract

This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix $M$ of size $n$ over a field $F$ with the property that there exists $\alpha\in F$ such that $M$ is similar to $diag(\alpha \cdot Id_k, M_1)$, where $M_1$ is cyclic and $0\leq k\leq n$). While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general picture, the present paper provides a conclusive answer for all cross-characteristic projective irreducible representations of the finite quasi-simple groups of Lie type and their automorphism groups.