Variation on a theme of Richardson

We consider the structure of parabolic subgroups P in general linear groups. The group P acts on its unipotent radical P-u and on all members of the descending central series P-u((l)) via conjugation. By a fundamental theorem due to Richardson P acts on Pu with an open dense orbit. In fact, this density theorem holds for any reductive algebraic group. In this note we investigate the question of the existence of a dense P-orbit on P-u((l)) for 1 greater than or equal to 1 using only most elementary methods. Despite the fact that for special P it is the case that P operates u with such a dense orbit for all 1 greater than or equal to 0, in general, however, this fails; we present a on P-u((l)) counterexample, in GL(15)(k). Besides the general linear groups, we also study this question for other reductive algebraic groups. (C) 2002 Elsevier Science Inc. All rights reserved.