Modality for Parabolic Group Actions

Gerhard Röhrle

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany.

roehrle@mathematik.uni-bielefeld.de

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 $G = {\rm GL}_n(k)$, then P can be viewed as the stabilizer of a (partial) flag in kⁿ. We consider the action of Pon its unipotent radical P_u via conjugation and on ${\rm Lie} P_u = {\mathfrak p}_u$, the Lie algebra of P_u, via the adjoint representation.

In 1974 R.W. Richardson proved that Padmits an open dense orbit on P_u, similarly for the adjoint action of P on ${\mathfrak p}_u$. Thus there is a natural dichotomy: the instances when P acts on ${\mathfrak p}_u$with a finite number of orbits versus the cases when ${\rm mod} P$ is positive. Likewise for the action on P_u.

I shall give a complete description of all instances in classical goups when the number of P-orbits on ${\mathfrak p}_u$is finite. Moreover, I shall present a complete combinatorial description of the relation of the P-orbit closures in ${\mathfrak p}_u$ in these finite instances for $G = {\rm GL}_n(k)$.

Time permitting we shall discuss the more general concept of the modality of the action of P on ${\mathfrak p}_u$, denoted by ${\rm mod}(P : {\mathfrak p}_u)$. This is the maximal number of parameters upon which a family of P-orbits on ${\mathfrak p}_u$ depends. Note that ${\rm mod} (P : {\mathfrak p}_u) = 0$ precisely when P acts on ${\mathfrak p}_u$ with a finite number of orbits.