Degenerate principal series and local theta correspondence

Abstract:

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.