Variation on a theme of Richardson Original Research Article

We consider the structure of parabolic subgroups P in general linear groups. The group P acts on its unipotent radical Pu and on all members of the descending central series Pu(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 Pu(l) for lgreater-or-equal, slanted1 using only most elementary methods. Despite the fact that for special P it is the case that P operates on Pu(l) with such a dense orbit for all lgreater-or-equal, slanted0, in general, however, this fails; we present a counterexample in GL15(k). Besides the general linear groups, we also study this question for other reductive algebraic groups.