Modules of third-order differential operators on a conformally flat manifold

Let M be a smooth manifold endowed with a flat conformal structure and F-lambda(M) the space of densities of degree lambda on M. We study the space D-lambda,mu(3) (M) of third-order differential operators from F-lambda(M) to F-mu(M) as a module over the conformal Lie algebra o(p + 1, q + 1). We prove that D-lambda,mu(3) (M) is isomorphic to the corresponding module of third-order polynomials on T*(M) for almost all values of delta = mu - lambda, except for eight resonant values. The isomorphism is unique and will be given explicitly, yielding a conformally equivariant quantization. We also study the modules in the case of resonance. (C) 2001 Elsevier Science B.V. All rights reserved.