On The Projective Classification of the Modules of Differential Operators on ℝ^m

Abstract

The Lie algebra sl _m+1 of infinitesimal projective linear transformations acts via Lie derivatives on the space D _λ,μ of differential operators between densities on IR^m of weights λ and μ. In most of the cases this module is isomorphic to the graded module associated to the filtration by the order of differentiation. This is, in particular, the case when λ equals μ and this leads to a sl _m+1-equivariant quantization. The modules D _λ,μ are more generally classified by two sets of integers. For a given difference δ = μ — λ, there are finitely many isomorphisms classes.