We propose that geometric quantization of symplectic manifolds is the arrow
part of a functor, whose object part is deformation quantization of Poisson
manifolds. The `quantization commutes with reduction' conjecture of Guillemin
and Sternberg then becomes a special case of the functoriality of quantization.
In fact, our formulation yields almost unlimited generalizations of the
Guillemin--Sternberg conjecture, extending it, for example, to arbitrary Lie
groups or even Lie groupoids. Technically, this involves symplectic reduction
and Weinstein's dual pairs on the classical side, and Kasparov's bivariant
K-theory for C*-algebras (KK-theory) on the quantum side.Comment: 15 pages. Proc. Bialowieza 200