On an order on the set of partial flag varieties

In 1974 R.W. Richardson have constructed a map from the set of parabolic
subgroups of a given reductive group G to the set of nilpotent co-adjoint
orbits in g*. Natural inclusion-in-closure order on the set of nilpotent
co-adjoint orbits induces, via the Richardson map, a partial order on the
set of parabolic subgroups of a given reductive group G.

Classes of conjugacy of parabolic subgroups in a simple classical Lie
group can be identified with (isotropic) flag varieties. Thus Richardson
map induces a partial order on the set of (isotropic) partial flag
varieties.

In my talk I will describe this order on the set of flag varieties in
terms of Young diagrams and apply this description to a classification of
spherical actions on flag varieties (this is some class of G-actions with
finitely many G-orbits). This classification was obtained in a joint work
with R.~Avdeev and heavily depends on a recent result of I.~Losev.