The characterization of theta-distinguished representations of $GL_n$

TL;DR

This paper characterizes irreducible generic quotients of tensor products of exceptional metaplectic representations of $GL_n$, linking their properties to poles of symmetric square L-functions and introducing a new metaplectic Shalika model.

Contribution

It provides a complete characterization of these quotients, introduces a globalization technique, and establishes a new metaplectic Shalika model for $ heta$.

Findings

Irreducible generic quotients correspond to poles of symmetric square L-functions.

In the square-integrable case, quotients are characterized by distinguished data.

$ heta$ admits a new metaplectic Shalika model.

Abstract

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".