Geometric quantization of real minimal nilpotent orbits

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In this paper, we begin a quantization program for nilpotent orbits O R of a real semisimple Lie group G R . These orbits arise naturally as the coadjoint orbits of G R which are stable under scaling, and thus they have a canonical symplectic structure ω where the G R -action is Hamiltonian. These orbits and their covers generalize the oscillator phase space T ∗R n , which occurs here when G R = Sp(2 n, R) and O R is minimal. A complex structure J polarizing O R and invariant under a maximal compact subgroup K R of G R 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 ℷ R of G R as a Lie algebra of rational functions on the holomorphic cotangent bundle T ∗Y where Y = ( O R , J). Thus we transform the quantization problem on O R 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 ℷ R acts by explicit pseudo-differential 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 G R . Jordan algebras play a key role in the geometry and the quantization.