Unisingular representations in arithmetic and Lie theory

Abstract

Let G be a subgroup of GL(V ), where V is finite dimensional vector space over a finite field of characteristic p > 0. If det(g − 1) = 0 for all g ∈ G then we call G a fixed-point subgroup of GL(V ). Motivated in parallel by questions in arithmetic and linear group theory, we classify all irreducible fixed-point subgroups of Sp 8 (2) and give new infinite series of irreducible fixed-point subgroups of symplectic groups Sp m (2) for various m arising from certain representations of groups of Lie type.