In this paper we study invariant differential operators on manifolds with a given parabolic structure. The model for the parabolic geometry is the quotient of the orthogonal group by a maximal parabolic subgroup corresponding to crossing of the k-th simple root of the Dynkin diagram. In particular, invariant differential operators discussed in the paper correspond (in a flat model) to the Dirac...

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1, 2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not gene...

In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov–Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U (

A well known theorem of Duflo, the “annihilation theorem”, claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is centrally generated. For the Lie superalgebra osp(1, 2l), this result does not hold. In this article, we introduce a “correct” analogue of the centre for which the annihilation theorem does hold in the case osp(1, 2l). This sub...

Given a weight of sl(n,C), we derive a system of variable-coefficient secondorder linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group Sn on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exac...

Let g be a complex semisimple Lie algebra, with a Bore1 subalgebra b c g and Cartan subalgebra h c b. In classifying the finite dimensional representations of g, Cartan showed that any simple finite dimensional g-module has a generating element u, annihilated by n = [b, b], on which h acts by a linear form I E h*. Such an element is called a primitive vector (for the module). Harish-Chandra [9]...

The problem of describing the singular vectors of W 3 and W (2) 3 Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of A (1) 2. Singular vectors of an A (1) 2 Verma module are mapped into W algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of th...

For any additive subgroup G of an arbitrary field F of characteristic zero, there corresponds a generalized Heisenberg-Virasoro algebra L[G]. Given a total order of G compatible with its group structure, and any h, hI , c, cI , cLI ∈ F, a Verma module M̃(h, hI , c, cI , cLI) over L[G] is defined. In the this note, the irreducibility of Verma modules M̃(h, hI , c, cI , cLI) is completely determined.

On the space of homomorphisms from a Verma module to an indecomposable tilting module of the BGG-category O we define a natural filtration following Andersen [And97] and establish a formula expressing the dimensions of the filtration steps in terms of coefficients of Kazhdan-Lusztig polynomials.