  A new approach to the quantization of constrained or otherwise reduced
classical mechanical systems is proposed. On the classical side, the
generalized symplectic reduction procedure of Mikami and Weinstein, as further
extended by Xu in connection with symplectic equivalence bimodules and Morita
equivalence of Poisson manifolds, is rewritten so as to avoid the use of
symplectic groupoids, whose quantum analogue is unknown. A theorem on
symplectic reduction in stages is given. This allows one to discern that the
`quantization' of the generalized moment map consists of an operator-valued
inner product on a (pre-) Hilbert space (that is, a structure similar to a
Hilbert $C^*$-module). Hence Rieffel's far-reaching operator-algebraic
generalization of the notion of an induced representation is seen to be the
exact quantum counterpart of the classical idea of symplectic reduction, with
imprimitivity bimodules and strong Morita equivalence of $C^*$-algebras falling
in the right place. Various examples involving groups as well as groupoids are
given, and known difficulties with both Dirac and BRST quantization are seen to
be absent in our approach.
