On sl(2,R)-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). 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^2(Vect(R),sl(2);D_{\lambda,\mu}). 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).