Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, ${\cal S}$ the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on ${\cal S}$. This cohomology space is closely related to the Vect(M)-modules, ${\cal D}$λ(M), of linear differential operators on the space of tensor densities on M of degree λ.