A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical

Let G be a classical algebraic group defined over an algebraically closed field. We classify all instances when a parabolic subgroup P of G acts on its unipotent radical P-u, or on p(u), the Lie algebra of P-u, with only a finite 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 relations. Furthermore, for general linear groups we obtain a combinatorial formula for the number of orbits in the finite cases.