Almost cyclic elements in Weil representations of finite classical groups

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-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 α∈F such that M is similar to diag(α⋅Idk,M1), where M1 is cyclic and 0≤k≤n). The paper focuses on the Weil representations of finite classical groups, as there is strong evidence that these representations play a key role in the general picture.