Localization of the Riemann-Roch character

Abstract

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. © 2001 Elsevier Science.