A limit formula for even nilpotent orbits

Let G(R) be a real form of a complex, semisimple Lie group G. Assume O is an even nilpotent coadjoint G(R)-orbit. We prove a limit formula, expressing the canonical measure on O as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with O.