  We give a formulation of a deformation of Dirac operator along orbits of a
group action on a possibly non-compact manifold to get an equivariant index and
a K-homology cycle representing the index. We apply this framework to
non-compact Hamiltonian torus manifolds to define geometric quantization from
the view point 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 non-compact setting, which shows that this geometric quantization is
independent of the choice of polarization. The proofs are based on the
localization of index to lattice points.
