Geometric Quantization of Real Minimal Nilpotent Orbits

In this paper, we begin a quantization program for nilpotent orbits OR of a real semisimple Lie group GR. These orbits arise naturally as the coadjoint orbits of GR which are stable under scaling, and thus they have a canonical symplectic structure ω where the GR-action is Hamiltonian. These orbits and their covers generalize the oscillator phase space T R, which occurs here when GR = Sp(2n,R) and OR is minimal. A complex structure J polarizing OR and invariant under a maximal compact subgroup KR ofGR is provided by the Kronheimer-Vergne Kaehler structure (J, ω). We argue that the Kaehler potential serves as the Hamiltonian. Using this setup, we realize the Lie algebra gR of GR as a Lie algebra of rational functions on the holomorphic cotangent bundle T Y where Y = (OR,J). Thus we transform the quantization problem on OR into a quantization problem on T Y . We explain this in detail and solve the new quantization problem on T Y in a uniform manner for minimal nilpotent orbits in the non-Hermitian case. The Hilbert space of quantization consists of holomorphic half-forms on Y . We construct the reproducing kernel. The Lie algebra gR acts by explicit pseudodifferential operators on half-forms where the energy operator quantizing the Hamiltonian is inverted. The Lie algebra representation exponentiates to give a minimal unitary ladder representation of a cover of GR. Jordan algebras play a key role in the geometry and the quantization. §