Automorphic forms with degenerate Fourier coefficients

The main theme of this paper is that singular automorphic forms on classical groups are given by theta series liftings. We establish several inequalities relating the automorphic multiplicities of a given representation and that of its abstract theta lift. Our methods allow one to lift noncuspidal, square integrable automorphic forms when the dual pair involved is in the stable range. In this way we construct new families of singular automorphic forms, many of which are clearly unipotent. In fact, starting from one-dimensional representations and repeating the procedure (of lifting in the stable range), one may obtain all automorphic forms which are quadratic unipotent in the sense of Moeglin.