Conformal symbols and the action of contact vector fields over the superline

Let K be the Lie superalgebra of contact vector fields on the supersymmetric line. We compute the action of K on the modules of differential and pseudodifferential operators between spaces of tensor densities, in terms of their conformal symbols. As applications we deduce the geometric subsymbols, 1-cohomology, and various uniserial subquotients of these modules. We also outline the computation of the K-equivalences and symmetries of their subquotients.Comment: 48 page

Linear differential operators on contact manifolds

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus

Centers and characters of Jacobi group-invariant differential operator algebras

We study the algebras of differential operators invariant with respect to the scalar slash actions of real Jacobi groups of arbitrary rank. These algebras are non-commutative and are generated by their elements of orders 2 and 3. We prove that their centers are polynomial in one variable and are generated by the Casimir operator. For slash actions with invertible indices we also compute the characters of the IDO algebras: in rank exceeding 1 there are two, and in rank 1 there are in general five. In rank 1 we compute in addition all irreducible admissible representations of the IDO algebras.Comment: 16 page

Annihilators of tensor density modules

AbstractWe 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)

Relative extremal projectors

AbstractThis paper proves the existence of relative extremal projectors. An infinite factorization is given as well as a summation formula