**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
,
then *P* can be viewed as the stabilizer
of a (partial) flag in *k*^{n}.
We consider the action of *P*on its unipotent radical *P*_{u} via conjugation and on
,
the Lie algebra of *P*_{u}, via the adjoint representation.

In 1974 R.W. Richardson proved that *P*admits an open dense orbit on *P*_{u}, 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 *P*_{u}.

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.