Abstract
Let G be a classical group in the following families SO_e(p,q), U(p,q), Sp(p,q), Sp(2n,R), O^*(2n), and O be a nilpotent K_C-orbit with g-height less than or equal to 2, we develop a technique to determine the global sections of certain line bundles over this orbit. Then we apply this method to the line bundle associated to the admissible data. In particular, this method also could be used to determine the ring of regular functions of this orbit.

Furthermore, if the boundary ∂O has complex codimension at least 2 in Ō, we use theta lifting to construct certain irreducible unitary representations and show these are the unitary representations "attached" to O. The existences of such representations are conjectured by Vogan.