On atypical representations of classical Lie superalgebras of defect one.

摘要

In this paper, we extend the reduction method proposed by V. Serganova in [S1] for sl(m/n) to all basic classical Lie superalgebras of defect one. For an atypical central character chi, this method suggests the construction of a special quotient Uchi of the enveloping algebra which helps to study representations admitting character chi. First, we show that the construction of Uchi can be carried out for all basic classical Lie superalgebras of defect one. Further, we obtain necessary conditions for elements of the center of Uchi that hold for all such Lie superalgebras. Finally, we describe the center Z(U chi) of Uchi explicitly for osp(2|2n), osp(3|2), and D(2,1; alpha). Our results support the conjecture made by V. Serganova in [S1] that Z(U chi) separates almost all highest weight irreducible representations which admit a generic atypical central character chi.