Quantising proper actions on SpinC-manifolds

Abstract

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to SpinC-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The result for cocompact actions is stated in terms of K-theory of group C*-algebras, and the result for non-cocompact actions is an equality of numerical indices. In the non-cocompact case, the result generalises to SpinC-Dirac operators twisted by vector bundles. This yields an index formula for Braverman's analytic index of such operators, in terms of characteristic classes on reduced spaces.