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, P,, its unipotent radical and p(u) the Lie algebra of P-u. A fundamental result of R. Richardson says that P acts on p(u) 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 p(u)* is already known for char k = 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 b(u)* is a prehomogeneous space for a Borel subgroup B of G. From this we deduce that a parabolic subgroup P of G acts on n* with a dense orbit for any P-sumnodule n of P. Further, we determine when the orbit map for such an orbit is separable. (C) 2004 Elsevier Inc. All rights reserved.