The characterization of theta-distinguished representations of GL(n)

Let θ and θ’ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP84], of a metaplectic double cover of GL_n. The tensor θ ⊗ θ’ 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 contragredients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of θ. As a corollary, θ is shown to admit a new “metaplectic Shalika model”.