Asian Journal of Mathematics

Volume 21 (2017)

Number 4

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

Pages: 631 – 686



Peter Hochs (School of Mathematical Sciences, University of Adelaide, SA, Australia)

Varghese Mathai (School of Mathematical Sciences, University of Adelaide, SA, Australia)


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.


$\mathrm{Spin}^c$-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.

Received 2 July 2015

Published 25 August 2017