L. Hille et G. Rohrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, TRANSFORM G, 4(1), 1999, pp. 35-52
Let G be a classical algebraic group defined over an algebraically closed f
ield. We classify all instances when a parabolic subgroup P of G acts on it
s unipotent radical P-u, or on p(u), the Lie algebra of P-u, with only a fi
nite number of orbits.
The proof proceeds in two parts. First we obtain a reduction to the case of
general linear groups. In a second step, a solution for these is achieved
by studying the representation theory of a particular quiver with certain r
elations.
Furthermore, for general linear groups we obtain a combinatorial formula fo
r the number of orbits in the finite cases.