Osp(1, 2l) -verma Modules Annihilators

In [Zh], R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l + 1) and the Lie superalgebra osp(1, 2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebrasà la Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie superalgebra osp(1, 2l) is generated by its intersection with the centralizer of the even part of the enveloping algbra. A well known theorem of Duflo claims that the annihilator of a Verma module over a complex semi-simple Lie algebra is generated by its intersection with the centre of the enveloping algbra. In [GL2] we show that in order for this theorem to hold in the case of the Lie superalgebra osp(1, 2l) one has to replace the centre by the centralizer of the even part of the enveloping algbra. The purpose of this article is to show how quantum groups can illucidate this phenomemon. Let k, g be respectively the complex simple Lie algebra o(2l + 1) and the complex superalgebra osp(1, 2l). From many point of views, the algebras g and k are very similar objects. For instance, identifying properly Cartan subalgebras of k and g, the root systems ∆ k , ∆ g are contained one into the other, and the set of irreductible roots of ∆ g is ∆ k. Moreover, given a simple finite dimensional g-module, the corresponding simple k-module of the same highest weight is also finite dimensional and has the same formal character (and even the same crystal graph). Nevertheless, there is no obvious direct way to link the algebras g, k. To bridge the gap, one has to go through the quantum level: in his article published in 1992, R. B. Zhang (see [Zh], 3) found a recipe to pass from a certain formal deformation of U(k) to a formal deformation of U(g). In this article we present a reformulation of the Zhang transformation in the more algebraic context of the quantizationsà la Drinfeld-Jimbo. The idea is to start with the Drinfeld-Jimbo quantum enveloping algebra U := U √ q (o(2l + 1)) and to extend the torus by the finite group Γ := ∆ g /2∆ g. In other words, we introduce the semi-direct product by elements of Γ, we build a subalgebra …