Representations of Algebraic Groups Containing Matrices with Large Jordan Blocks

For a classical algebraic group G of rank n, irreducible rational representations ϕ the images of which contain matrices with at least one Jordan block of size at least dim ϕ/n are determined provided that the characteristic of a ground field is not equal to 2 if G is not of type An. For arbitrary simple algebraic groups the question is reduced to infinitesimally irreducible representations.