Globally irreducible representations of SL₃(q) and SU₃(q)

The notion of globally irreducible representations of finite groups was introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered by N. Elkies and T. Shioda using Mordell-Weil lattices of elliptic curves. In this paper we classify all globally irreducible representations coming from protective complex representations of the finite simple groups PSL₃(q) and PSU₃(q). The main result is that these representations are essentially those discovered by Gross.