Distinguished unipotent elements and multiplicity-free subgroups of simple algebraic groups

For G a simple algebraic group over an algebraically closed field of characteristic 0, we determine the irreducible representations ρ:G→I(V), where I(V) denotes one of the classical groups SL(V), Sp(V), SO(V), such that ρ sends some distinguished unipotent element of G to a distinguished element of I(V). We also settle a base case of the general problem of determining when the restriction of ρ to a simple subgroup of G is multiplicity-free.