Degenerate Principal Series and Local Theta Correspondence

In this paper we determine the structure of the natural Ũ(n, n) module Ωp,q(l) which is the Howe quotient corresponding to the determinant character det of U(p, q). We first give a description of the tempered distributions on Mp+q,n(C) which transform according to the character det−l under the linear action of U(p, q). We then show that after tensoring with a character, Ωp,q(l) can be embedded into one of the degenerate series representations of U(n, n). This allows us to determine the module structure of Ωp,q(l). Moreover we show that certain irreducible constituents in the degenerate series can be identified with some of these representations Ωp,q(l) or their irreducible quotients. We also compute the Gelfand-Kirillov dimensions of the irreducible constituents of the degenerate series.