  Paradan and Vergne generalised the quantisation commutes with reduction
principle of Guillemin and Sternberg from symplectic to 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
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.
