Large dimensional classical groups and linear spaces

Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least 25, then $G$ acts transitively on the set of flags of $\mathcal{S}$ and hence the action is known. For particular families of classical groups our results hold for dimension smaller than 25. The group theoretic methods used to prove the result (described in Section 3) are robust and general and are likely to have wider application in the study of almost simple groups acting on finite linear spaces.Comment: 32 pages. Version 2 has a new format that includes less repetition. It also proves a slightly stronger result; with the addition of our "Concluding Remarks" section the result holds for dimension at least 2

On characters of Chevalley groups vanishing at the non-semisimple elements

Let G be a finite simple group of Lie type. In this paper we study characters of G that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of G. Such characters can be viewed as a natural generalization of the Steinberg character. For groups G of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.Comment: Dedicated to Lino Di Martino on the occasion of his 65th birthda

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

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.Comment: To appear on Journal of Group Theor