Equivariant Operators between some Modules of the Lie Algebra of Vector Fields

Abstract

The space D(k,p) of differential operators of order at most k, from the differential forms of degree p of a smooth manifold M into the functions of M, is a module over the Lie algebra of vector fields of M, when it's equipped with the natural Lie derivative. In this paper, we compute all equivariant i.e. intertwining operators from D(k,p) into D(k',p') and conclude that the preceding modules of differential operators are never isomorphic. We also answer a question of P. Lecomte, who observed that the restriction to D(k,p) of some homotopy operator, introduced in one of his works, is equivariant for small values of k and p.