Olanzapine in non-refractory schizophrenia: Effects on neuropsychology and psychopathology

Abstract

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∗Rn, 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 of GR 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 ℷR 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 ℷR 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 GR. Jordan algebras play a key role in the geometry and the quantization.