On Riemann-Roch Formulas for Multiplicities

A theorem of Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kähler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectic quotient. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. The result extends to non-Kählerian settings, if one defines the representation by the equivariant index of the $\text{Spin}^c$-Dirac operator associated to the quantizing line bundle.