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 p(u) of P-u. Each higher term p(u)((1)) descending central series of p(u) is stable under this action. For classical G all instances when P acts on p(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 F-4 and E-6. Moreover, when P acts on p(u)((l)) with an infinite number of orbits, we determine whether P still acts with a dense orbit. For G of type E-7 and E-8 we investigate only the case of a Borel subgroup. We present a complete classification of all instances when b(u)((l)) is a prehomogeneous space for a Borel subgroup B of a reductive algebraic group for any l greater than or equal to 0.