Ana Canas da Silva, Yael Karshon, and Susan Tolman
We prove several versions of ``quantization commutes with reduction'' for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin$^c$ structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, symplectic) manifold.