  We establish, via geometric quantization of the supercotangent bundle sM of
(M,g), a correspondence between its conformal geometry and those of the spinor
bundle. In particular, the Kosmann Lie derivative of spinors is obtained by
quantization of the comoment map, associated to the new Hamiltonian action of
conf(M,g) on sM. We study then the conf(M,g)-module structures induced on the
space of differential operators acting on spinor densities and on its spaces of
symbols (functions on sM). In the conformally flat case, we classify their
conformal invariants, including the conformally odd powers of the Dirac
operator.
