Prehomogeneous spaces for the coadjoint action of a parabolic group

Let k be an algebraically closed field and let G be a reductive linear algebraic group over k. Let P be a parabolic subgroup of G, Pu its unipotent radical and the Lie algebra of Pu. A fundamental result of R. Richardson says that P acts on with a dense orbit (see [R.W. Richardson, Bull. London Math. Soc. 6 (1974) 21–24]). The analogous result for the coadjoint action of P on is already known for chark=0 (see [A. Joseph, J. Algebra 48 (1977) 241–289]). In this note we prove this result for arbitrary characteristic. Our principal result is that is a prehomogeneous space for a Borel subgroup B of G. From this we deduce that a parabolic subgroup P of G acts on with a dense orbit for any P-submodule of P. Further, we determine when the orbit map for such an orbit is separable.