α-Primitive Elements in the Generalized Verma Modules

We present explicit formulas for -primitive elements in the generalized Verma module, which generalize the known formulas for primitive elements in the Verma modules.     

Quantum Deformations of Imaginary Verma Modules

We construct quantum deformations of imaginary Verma modulesover and show that, for generic q, imaginary Verma modules over can be deformed to those over the quantum group in such a way that the dimensions of the weightspaces are invariant under the deformation. We also prove thePBW theorem for with respect to the triangular decomposition induced from the root partitioncorresponding to the imaginary Verma modules. 1991 MathematicsSubject Classification: 17B67, 17B65, 17B10.     

On Projective Modules in Category \mathcal{O}_{int} of Quantum \mathfrak{sl}_{2}

The restriction of a Verma module of ${\bf U}(\mathfrak{sl}_3)$ to ${\bf U}(\mathfrak{sl}_2)$ is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of ${\bf U}(\mathfrak{sl}_2)$ . The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category $\mathcal O_{\rm{int}}$ of quantum $\mathfrak{sl}_2$ is given.     

Differential-operator representations of Weyl group and singular vectors in Verma modules

Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.     

Verma modules over generalized Virasoro algebras Vir[G]

Let F be a field of characteristic 0, not necessarily algebraically closed, and G be an additive subgroup of F. For any total order on G which is compatible with the group addition, and for any , a Verma module over the generalized Virasoro algebra Vir[G] is defined. In the present paper, the irreducibility of Verma modules is completely determined.     

Combinatorial proof of the inversion formula on the Kazhdan–Lusztig R-polynomials

In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\) -polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equidistribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma’s equidistribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element.     

Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs

Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of \(SL(2,\mathbb {R})\). We address a general problem to find explicit formulæ  for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose, we use a new method (F-method) developed in Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9) and based on the algebraic Fourier transform for generalized Verma modules.The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulæ  of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin–Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.     

Determinant formula and a realization for the Lie algebra <Emphasis Type="Italic">W</Emphasis> (2, 2)

In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of certain vaccum module for the algebra W(2, 2) via theWeyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).     

Twists and Singular Vectors in \widehat{sl}\left( {2\left| 1 \right.} \right) Representations

We propose new formulas for singular vectors in Verma modules over the affine Lie superalgebra . We analyze the coexistence of singular vectors of different types and identify the twisted modules arising as submodules and quotient modules of Verma modules. We show that with the twists (spectral flow transformations) properly taken into account, a resolution of irreducible representations can be constructed consisting of only the modules.     

Weight sl(3)-modules generated by semiprimitive elements

Generalized Verma modules over the Lie algebra sl(3, ) that contain no highest vector are studied. Such modules are generated by semiprimitive elements. The composition structure of these modules is studied, an irreducibility criterion is given. Explicit formulas for semiprimitive elements are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 281–285, February, 1991.     

Andersen Filtration and Hard Lefschetz

On the space of homomorphisms from a Verma module to an indecomposable tilting module of the BGG-category we define a natural filtration following Andersen [A] and establish a formula expressing the dimensions of the filtration steps in terms of coefficients of Kazhdan–Lusztig polynomials. Received: May 2006, Revision: July 2007, Accepted: July 2007     

Verma type modules for toroidal lie algebras

We study the structure of imaginary Verma modules induced from the"natural"Borel subalgebra of a toroidal Lie algebra. In particular, we establish a criterion of irreducibility for imaginary Verma modules and describe their submodules and irreducible quotients. We also describe the structure of Verma type modules in the case of sl(2)-toroidal Lie algebra over two variables.     

Closed product formulas for extensions of generalized Verma modules

We give explicit combinatorial product formulas for the polynomials encoding the dimensions of the spaces of extensions of -generalized Verma modules, in the cases when corresponds to an indecomposable classic Hermitian symmetric pair. The formulas imply that these dimensions are combinatorial invariants. We also discuss how these polynomials, defined by Shelton, are related to the parabolic -polynomials introduced by Deodhar.

     

Fock Space Realizations of Imaginary Verma Modules

This work expands to the setting of the results of H. Jakobsen and V. Kac and independently D. Bernard and G. Felder on the realization of , in terms of infinite sums of partial differential operators. We note in the paper that, in the generic case, these geometric constructions are just realizations of the imaginary Verma modules studied by V. Futorny. Presented by A. VerschorenMathematics Subject Classifications (2000) Primary: 17B67, 81R10.     

Application of Vertex Algebras to the Structure Theory of Certain Representations Over the Virasoro Algebra

In this paper, we discuss the structure of the tensor product \(V_{\alpha,\beta }^{\prime}\otimes L(c,h)\) of an irreducible module from an intermediate series and irreducible highest-weight module over the Virasoro algebra. We generalize Zhang’s irreducibility criterion from Zhang (J Algebra 190:1–10, 1997), and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operators for modules over vertex operator algebra yields reducibility of \(V_{\alpha ,\beta}^{\prime}\otimes L(c,h)\) , which is a completely new point of view to this problem. As an example, the complete structure of the tensor product with minimal models c?=???22/5 and c?=?1/2 is presented.     

On the Reducibility of Scalar Generalized Verma Modules of Abelian Type

A parabolic subalgebra \(\mathfrak {p}\) of a complex semisimple Lie algebra \(\mathfrak {g}\) is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the parameters for scalar generalized Verma modules attached to parabolic subalgebras of abelian type such that the modules are reducible. The proofs use Jantzen’s simplicity criterion, as well as the Enright-Howe-Wallach classification of unitary highest weight modules.     

On Ext-transfer for Reductive Lie Algebras

Let G be a connected reductive algebraic group over an algebraically closed field of prime characteristic p and ?? be the Lie algebra of G. In this paper, we study the representations of ?? when p-character has standard Levi form. An Ext-transfer from the Ext-groups of induced ??-modules to its Levi subalgebras is obtained. Furthermore, we reduce the computation of the multiplicities of simple factors in baby Verma modules over ?? to its Levi subalgebras.     

Graded Limits of Minimal Affinizations over the Quantum Loop Algebra of Type <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

The aim of this paper is to study the graded limits of minimal affinizations over the quantum loop algebra of type G ₂. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and obtain defining relations of them. As an application, we obtain a polyhedral multiplicity formula for the decomposition of minimal affinizations of type G ₂ as a \(U_{q}(\mathfrak {g})\)-module, by showing the corresponding formula for the graded limits. As another application, we prove a character formula of the least affinizations of generic parabolic Verma modules of type G ₂ conjectured by Mukhin and Young.