On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups

Let W be a vector space over an algebraically closed field k. Let H be a quasisimple group of Lie type of characteristic p≠char(k) acting irreducibly on W. Suppose also that G is a classical group with natural module W, chosen minimally with respect to containing the image of H under the associated representation. We consider the question of when H can act irreducibly on a G-constituent of W and study its relationship to the maximal subgroup problem for finite classical groups.