On normal abelian subgroups in parabolic groups

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical P-u, and A a closed connected subgroup of P-u which is normalized by P. We show that P acts on A with finitely many orbits provided A is abelian. This generalizes a well-known finiteness result, namely the case when A is central in P-u. We also obtain an analogous result for the adjoint action of P on invariant linear subspaces of the Lie algebra of P-u which are abelian Lie algebras. Finally, we discuss a connection to some work of Mal'cev on maximal abelian subalgebras of the Lie algebra of G.