Quantising proper actions on $\mathrm{Spin}^c$-manifolds

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to $\mathrm{Spin}^c$-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 $\mathrm{Spin}^c$-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.

Keywords

$\mathrm{Spin}^c$-manifolds, geometric quantisation, quantisation commutes with reduction, proper Lie group actions, noncompact index theorem

2010 Mathematics Subject Classification

Primary 53C27. Secondary 53D20, 53D50, 58J20, 81S10.

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Received 2 July 2015

Published 25 August 2017