Localization of the Riemann–Roch Character

We present a K-theoretic approach to the Guillemin–Sternberg conjecture (V. Guillemin and S. Sternberg, Invent. Math.67 (1982), 515–538), about the commutativity of geometric quantization and symplectic reduction, which was proved by E. Meinrenken (J. Amer. Math. Soc.9 (1996), 373–389; Adv. Math.134, (1998), 240–277) and Tian–Zhang (Y. Tian and W. Zhang, Invent. Math.132 (1998), 229–259). Besides providing a new proof of this conjecture for the full non-Abelian group action case, our methods lead to a generalization for compact Lie group actions on manifolds that are not symplectic; these manifolds carry an invariant almost complex structure and an abstract moment map.