Singular Conformal Oscillator Representations of Orthosymplectic Lie Superalgebras

In our earlier paper, we generalize the one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebras arising from conformal transformations to those of orthosymplectic Lie superalgebras, and determine the irreducible condition. This paper deals with the cases when the irreducible condition fails. We prove that if n - m - 1 > 0 and c is an integer satisfying 1 <= c <= n - m - 1, the representation of osp(2n + 2|2m) has a composition series of length 2, and when n - m - 1 >= 0 and c is an element of -N, the representation of osp(2n + 2|2m) has a composition series of length 3, where N is the set of nonnegative integers. Moreover, we show that if c is an element of (max{n - m, 0} - 1/2 - N) ? (-N), the representation of osp(2n + 3|2m) has a composition series of length 2. In particular, we obtain an explicit presentation of the irreducible module with highest weight l lambda(2) - lambda(1), where l is any positive integer and it is not a generalized Verma module.