Symplectic Surgery and the Spinc-Dirac Operator

Abstract

LetGbe a compact connected Lie group, and (M, ω) a compact HamiltonianG-space, with moment mapJ:M→g'. Under the assumption that these data are pre-quantizable, one can construct an associated Spinc-Dirac operator[formula], whose equivariant index yields a virtual representation ofG. We prove a conjecture of Guillemin and Sternberg that if 0 is a regular value ofJ, the multiplicityN(0) of the trivial representation in the index space[formula], is equal to the index of the Spinc-Dirac operator for the symplectic quotientM0=J-1(0)/G. This generalizes previous results for the case thatG=Tis abelian, i.e., a torus. © 1998 Academic Press.