The Minimal Primitive Spectrum of the Enveloping Algebra of the Lie Superalgebra osp(1, 2l)

Abstract

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 substitute of the centre is the centralizer of the even part of the enveloping algebra. This algebra shares some nice properties with the centre. As a consequence of the annihilation theorem we obtain the description of the minimal primitive spectrum of the enveloping algebra of the Lie superalgebra osp(1, 2l). We also deduce a criterium for a osp(1, 2l)-Verma module to be a direct sum of sp(2l)-Verma modules.