Symplectic fibrations and Riemann-Roch numbers of reduced spaces

In this article we give formulas for the Riemann-Roch number of a symplectic quotient arising as the reduced space corresponding to a coadjoint orbit (for an orbit close to 0) as an evaluation of cohomology classes over the reduced space at 0. This formula exhibits the dependence of the Riemann-Roch number on the Lie algebra variable which specifies the orbit. We also express the formula as a sum over the components of the fixed point set of the maximal torus. Our proof applies to Hamiltonian G-manifolds even if they do not have a compatible Kahler structure, using the definition of quantisation in terms of the Spin-C Dirac operator.