Projectively equivariant symbol calculus

The spaces of linear differential operators cal D-lambda(R-n) acting on lambda-densities on R-n and the space Pol(T* R-n) of functions on T* R-n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R-n) of vector fields of R-n. However, these modules are isomorphic as sl(n + 1, R)-modules where sl(n + 1, R) subset of Vect(R-n) is the Lie algebra of infinitesimal projective transformations. In addition, such an sl(n+1)-equivariant bijection is unique (up to normalization). This leads to a notion of projective equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the sl(n+1)-equivariant symbol map to study the Vect(M)-modules D-lambda(k)(M) of kth-order linear differential operators acting on lambda-densities, for an arbitrary manifold M and classify the quotient-modules D-lambda(k)(M)/D-lambda(l)(M).