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This paper is the third of the series concerning the localization of theindex of Dirac-type operators. In our previous papers we gave a formulation ofindex 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 theends of the open manifolds. We introduce two equivariant versions of thelocalization. As an application we give a proof of Guillemin-Sternberg'squantization conjecture in the case of torus action.

Torus fibrations and localization of index III