  This paper is the third of the series concerning the localization of the
index of Dirac-type operators. In our previous papers we gave a formulation of
index of Dirac-type operators on open manifolds under some geometric setting,
whose typical example was given by the structure of a torus fiber bundle on the
ends of the open manifolds. We introduce two equivariant versions of the
localization. As an application we give a proof of Guillemin-Sternberg's
quantization conjecture in the case of torus action.
