Symplectic surgery and the Spinc-Dirac operator

Let G be a compact connected Lie group, and (M, omega) a compact Hamiltonian G-space, with moment map J: M --> g*. Under the assumption that these data are pre-quantizable, one can construct an associated Spin(c)-Dirac operator partial derivative(C), whose equivariant index yields a virtual representation of G. We prove a conjecture of Guillemin and Sternberg that if 0 is a regular value of J, the multiplicity N(0) of the trivial representation in the index space ind(partial derivative(C)), is equal to the index of the Spin-Dirac operator for the symplectic quotient M-0 = J(-1)(0)/G. This generalizes previous results for the case that G = T is abelian, i.e., a torus. (C) 1998 Academic Press.