Finite orbit modules for parabolic subgroups of exceptional groups

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and P_u its unipotent radical. We consider the adjoint action of P on the Lie algebra _u of P_u. 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.