On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators

Let Vect(R) be the Lie algebra of smooth vector fields on R. the space of symbols Pol(T*R) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(R)-module that becomes trivial once the action is restricted to sl(2) subset of Vect(R). The deformations of Pol(T*R), which become trivial once the action is restricted to sl(2) and such that the Vect(R)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of H(diff)(2)(Vect(R), sl(2);D(lambda,mu)), where H(diff)(i) denotes the differential cohomoology (i.e, we consider only cochains that are given by differential operators) and where D(lambda,mu) = Hom(diff)(F(lambda),F(mu)) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning H(2)(g;D(lambda,mu)) for g = Vect(R) and sl(2).