Instantons and Kaehler Geometry of Nilpotent Orbits

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.