We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is "multiplicatively integrable", i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the Bohr-Sommerfeld conditions. We apply this procedure to the case of a family of Poisson structures on CP_n, seen as Poisson homogeneous spaces of the standard Poisson-Lie group SU(n+1). We show that a bihamiltoniam system on CP_n defines a multiplicative integrable model on the symplectic groupoid; we compute the Bohr-Sommerfeld groupoid and show that it satisfies the needed properties for applying Renault theory. We recover and extend Sheu's description of quantum homogeneous spaces as groupoid C*-algebras.
我们在黎曼流形上引入了一类新的Poisson结构。这类中的泊松结构称为Killing-Poisson结构。Killing-Poisson结构类包含辛结构类、在{ \it（Differential Geometry and its Applications，{ \bf Vol.20，Issue 3}（2004），279- 291）}中研究的Poisson结构类以及由黎曼流形上的某些无穷小李代数作用诱导的Poisson结构类。我们证明了辛流形上的一些经典结果（李代数体结构与辛结构的可积性，紧流形上辛结构的非精确性）对正则Killing-Poisson结构仍然有效.
