Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany.
In this talk I shall discuss several aspects of the operation of parabolic subgroups of reductive groups.
Let G be a linear reductive algebraic group defined over an algebraically closed field k and let P be a parabolic subgroup of G. For instance if , then P can be viewed as the stabilizer of a (partial) flag in kn. We consider the action of Pon its unipotent radical Pu via conjugation and on , the Lie algebra of Pu, via the adjoint representation.
In 1974 R.W. Richardson proved that Padmits an open dense orbit on Pu, similarly for the adjoint action of P on . Thus there is a natural dichotomy: the instances when P acts on with a finite number of orbits versus the cases when is positive. Likewise for the action on Pu.
I shall give a complete description of all instances in classical goups when the number of P-orbits on is finite. Moreover, I shall present a complete combinatorial description of the relation of the P-orbit closures in in these finite instances for .
Time permitting we shall discuss the more general concept of the modality of the action of P on , denoted by . This is the maximal number of parameters upon which a family of P-orbits on depends. Note that precisely when P acts on with a finite number of orbits.