实极小幂零轨道的几何量子化
Geometric Quantization of Real Minimal Nilpotent Orbits
摘要：
在本文中，我们开始了实半单李群的幂零轨道的量子化程序。这些轨道及其覆盖推广了辛向量空间。Kronheimer-Vergne Kaehler结构给出了极大紧子群下极化轨道和不变量的复结构.我们概述了一个几何方案，量化的轨道相对于此极化。我们详细地计算了非厄米特情形下的极小幂零轨道。量子化的希尔伯特空间由轨道上的全纯半形式组成。我们构造再生核。李代数通过显式的伪微分算子作用于半形式，其中量子化哈密顿量的能量算子被反转。该李代数表示指数化以给出最小酉梯形表示。若当代数在几何学和量子化中起着关键的作用。
英文摘要：
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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