On irreducible subgroups of simple algebraic groups

Let G be a simple algebraic group over an algebraically closed field K of characteristic p ⩾ 0 , let H be a proper closed subgroup of G and let V be a nontrivial irreducible KG -module, which is p -restricted, tensor indecomposable and rational. Assume that the restriction of V to H is irreducible. In this paper, we study the triples ( G , H , V ) of this form when G is a classical group and H is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples ( G , H , V ) to the case where G is an orthogonal group, V is a spin module and H normalizes an orthogonal decomposition of the natural KG -module.