  The first obstacle in building a Geometric Quantization theory for nilpotent
orbits of a real semisimple Lie group has been the lack of an invariant
polarization. In order to generalize the Fock space construction of the quantum
mechanical oscillator, a polarization of the symplectic orbit invariant under
the maximal compact subgroup is required.
  In this paper, we explain how such a polarization on the orbit arises
naturally from the work of Kronheimer and Vergne. This occurs in the context of
hyperkaehler geometry. The polarization is complex and in fact makes the orbit
into a (positive) Kaehler manifold. We study the geometry of this Kaehler
structure, the Vergne diffeomorphism, and the Hamiltonian functions giving the
symmetry. We indicate how all this fits into a quantization program.
