Degenerate principal series and local theta correspondence

In this paper we determine the structure of the natural $\widetilde{U}(n,n)$ module ${\Omega^{p,q}(l)}$ which is the Howe quotient corresponding to the determinant character $\det^l$ of $U(p,q)$. We first give a description of the tempered distributions on $M_{p+q,n}(\mathbb C)$ 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, ${\Omega^{p,q}(l)}$ can be embedded into one of the degenerate series representations of $U(n,n)$. This allows us to determine the module structure of ${\Omega^{p,q}(l)}$. Moreover we show that certain irreducible constituents in the degenerate series can be identified with some of these representations ${\Omega^{p,q}(l)}$ or their irreducible quotients. We also compute the Gelfand-Kirillov dimensions of the irreducible constituents of the degenerate series.