MULTIPLICITIES FORMULA FOR GEOMETRIC QUANTIZATION, PART II

1. Introduction. Let P be a compact manifold. Let H be a compact Lie group acting on the right on P. We assume that the stabilizer of each element y e P is a finite subgroup of H. The space M P/H is an orbifold and every orbifold can be presented this way. If H acts freely, then M is a manifold. If aE-graded space of H-invariant solutions of the H-horizontal Dirac operator on P twisted by the line bundle Let #: M-o 9" be the moment map for the G-action. Assume 0 is a regular value of #. Let Mred be the reduced orbifold of M; that is, mred #-(O)/G. Consider the reduced orbifold line bundle rd lu-lto)/G on M,oa. In the ease where both G andH are torus, we prove here the formula

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