Finite orbit modules for parabolic subgroups of exceptional groups

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and Pu its unipotent radical. We consider the adjoint action of P on the Lie algebra u of Pu. Each higher term u(l) of the descending central series of u is stable under this action. For classical G all instances when P acts on u(l) with a finite number of orbits were determined in [9], [10], [3] and [4]. In this note we extend these results to groups of type F4 and E6. Moreover, when P acts on u(l) with an infinite number of orbits, we determine whether P still acts with a dense orbit. For G of type E7 and E8 we investigate only the case of a Borel subgroup.We present a complete classification of all instances when u(l) is a prehomogeneous space for a Borel subgroup B of a reductive algebraic group for any l ? 0.