The characterization of theta-distinguished representations of $G{L}_n$

Abstract

Let $\theta$ and ${\theta }^{\prime }$ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP], of a metaplectic double cover of $G{L}_n$. The tensor $\theta \otimes {\theta }^{\prime }$ is a (very large) representation of $G{L}_n$. We characterize its irreducible generic quotients. In the square-integrable case, these are precisely the representations whose symmetric square L-function has a pole at s=0. Our proof of this case involves a new globalization result. In the general case these are the representations induced from distinguished data or pairs of representations and their contragradients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of $\theta$. As a corollary, $\theta$ is shown to admit a new "metaplectic Shalika model".