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 of differential operators of order ≤ p on the fields of densities of degree k of M. If dim M ≥ 2 and p ≥ 3, the dimension of the space of linear equivariant maps from into is shown to be 0, 1 or 2 according to whether (k, l) belongs to 0, 1 or 2 of the lines of 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 ≤ 22].