We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly noncompact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to noncompact Hamiltonian torus manifolds to define geometric quantization from the viewpoint of index theory. We give two applications. The first one is a proof of a [Q,R]=0 type theorem, which can be regarded as a proof of the Vergne conjecture for abelian case. The other is a Danilov-type formula for toric case in the noncompact setting, which is a localization phenomenon of geometric quantization in the noncompact setting. The proofs are based on the localization of index to lattice points.

我们给出了狄拉克算子沿可能非紧流形上的群作用轨道的变形的公式，以获得等变指数和表示该指数的K-同调循环。我们将此框架应用于非紧哈密顿环面流形，以从指数理论的角度定义几何量化。我们给出两个应用程序。第一个是[Q,R]=0类型定理的证明，可以看作是对阿贝尔情况的Vergne猜想的证明。另一种是非紧集下复曲面的Danilov型公式，是非紧集下几何量化的局部化现象。证明是基于索引到格点的本地化。