Prehomogeneous spaces for parabolic group actions in classical 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. Richardson's dense orbit theorem says that there is a dense P-orbit in p(u). We consider some instances when P acts with a dense orbit on terms p(u)((l)) of the descending central series of p(u). In particular, we show (in good characteristic) that a Borel subgroup B of a classical group acts on b(u)((l)) with a dense orbit for all l. Further we give some families of parabolic subgroups P such that p(u)((l)) contains a dense P-orbit for all l. (C) 2004 Elsevier Inc. All rights reserved.