  Let $\theta$ and $\theta'$ be a pair of exceptional representations in the
sense of Kazhdan and Patterson [KP], of a metaplectic double cover of $GL_n$.
The tensor $\theta\otimes\theta'$ is a (very large) representation of $GL_n$.
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".
