Comparison of some modules of the Lie algebra of vector fields (Q675786)

The authors study the action of the Lie algebra of vector fields of a smooth manifold \(M\) on the space \({\mathcal D}^p_k\) of differential operators of order \(\leq p\) on the fields of densities of degree \(k\) of \(M\). When \(\dim M\geq 2\) and \(p\geq 3\), the dimension of the space of linear equivariant maps from \({\mathcal D}^p_k\) to \({\mathcal D}^p_\ell\) is shown to be 0, 1, or 2, according to the number of the lines \(\{k=0\); \(k=-1\); \(k=\ell\); \(k+\ell +1=0\}\) that contain the point \((k,\ell)\).