Comparison of some modules of the Lie algebra of vector fields

The Lie algebra of vector fields of a smooth manifold M acts by Lie derivatives on the space D-k(p) Of differential operators of order less than or equal to p on the fields of densities of degree k of M. If dim M greater than or equal to 2 and p greater than or equal to 3, the dimension of the space of linear equivariant maps from D-k(p) into D-l(p) is shown to be O, I or 2 according to whether (k, l) belongs to 0, 1 or 2 of the lines of R(2) of equations k = 0, k = -1, k = l and k + l + 1 = 0. This answers a question of C. Duval and V. Ovsienko who have determined these spaces for p less than or equal to 2 [2].