Differential Representations of the Lie Superalgebra C(n+1)

In this paper, we realize Verma modules and the vector-coherent-state (VCS) representations of the orthosymplectic Lie superalgebra osp(2 vertical bar 2n) as differential operators with vector coefficients. We also characterize simple sub-representations of VCS representations as kernels of some finite number of differential operators. The singular vectors of the atypical representation of osp(2 vertical bar 2n) are explicitly given.