Symplectic reduction and a weighted multiplicity formula for twisted Spin${}^c$-Dirac operators

We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin${}^c$-complex under consideration is allowed to be further twisted by certain natural exterior power bundles. The main result is a weighted quantization formula in the presence of commuting Hamiltonian actions. The corresponding Morse type inequalities in holomorphic situations are also established.