Conformally equivariant quantization: Existence and uniqueness

We prove the existence and the uniqueness of a conformally equivariant symb ol calculus and quantization on any conformally hat pseudo-riemannian manif old (I, g). In other words, we establish a canonical isomorphism between th e spaces of polynomials on T*M and of differential operators on tensor dens ities over M, both viewed as modules over the Lie algebra o(p + 1, q + 1) w here p + q = dim(M). This quantization exists for generic values of the wei ghts of the tensor densities and we compute the critical values of the weig hts yielding obstructions to the existence of such an isomorphism. In the p articular case of half-densities, we obtain a conformally invariant star-pr oduct.