Weil representations as globally irreducible representations

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 recently by N. ELKIES and T. SHIODA using Mordell-Weil lattices of elliptic curves. It has been observed by R. Gow and GROSS that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.