Prehomogeneous spaces for parabolic group actions in classical 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 of Pu. Richardsonʹs dense orbit theorem says that there is a dense P-orbit in . We consider some instances when P acts with a dense orbit on terms of the descending central series of . In particular, we show (in good characteristic) that a Borel subgroup B of a classical group acts on with a dense orbit for all l. Further we give some families of parabolic subgroups P such that contains a dense P-orbit for all l.