Annihilators of tensor density modules

We describe the two-sided ideals in the universal enveloping algebras of the Lie algebras of vector fields on the line and the circle which annihilate the tensor density modules. Both of these Lie algebras contain the projective subalgebra, a copy of sl2. The restrictions of the tensor density modules to this subalgebra are duals of Verma modules (of sl2) for Vec(R) and principal series modules (of sl2) for Vec(S1). Thus our results are related to the well-known theorem of Duflo describing the annihilating ideals of Verma modules of reductive Lie algebras. We find that, in general, the annihilator of a tensor density module of Vec(R) or Vec(S1) is generated by the Duflo generator of its annihilator over sl2 (the Casimir operator minus a scalar) together with one other generator, a cubic element of U(Vec(R)) not contained in U(sl2).