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

Abstract

Let M be a smooth manifold endowed with a flat conformal structure and F λ(M) the space of densities of degree λ on M. We study the space D λ,μ 3(M) of third-order differential operators from F λ(M) to F μ(M) as a module over the conformal Lie algebra o( p+1, q+1). We prove that D λ,μ 3(M) is isomorphic to the corresponding module of third-order polynomials on T ∗(M) for almost all values of δ= μ− λ, 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.