Geometric Quantization of Real Minimal Nilpotent Orbits

Details Geometric Quantization of Real Minimal Nilpotent Orbits Geometric Quantization of Real Minimal Nilpotent Orbits

1998-11-06

arxiv.org/abs/math/9811033v1

DGA, vol. 9 (1998), 5-58

Mathematics

Symplectic Geometry

Scientific paper

In this paper, we begin a quantization program for nilpotent orbits of a real semisimple Lie group. These orbits and their covers generalize the symplectic vector space. A complex structure polarizing the orbit and invariant under a maximal compact subgroup is provided by the Kronheimer-Vergne Kaehler structure. We outline a geometric program for quantizing the orbit with respect to this polarization. We work out this program in detail for minimal nilpotent orbits in the non-Hermitian case. The Hilbert space of quantization consists of holomorphic half-forms on the orbit. We construct the reproducing kernel. The Lie algebra 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. Jordan algebras play a key role in the geometry and the quantization.

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