A Godement-Jacquet type integral and the metaplectic Shalika model

TL;DR

This paper introduces a new integral representation for certain automorphic L-functions using the metaplectic Shalika model, revealing new insights into periods of residual representations and establishing key uniqueness and formula results.

Contribution

It develops a novel integral for automorphic L-functions involving the metaplectic Shalika model, including uniqueness proofs and a Casselman-Shalika type formula.

Findings

New integral representation for automorphic L-functions.

Proved uniqueness of the metaplectic Shalika model over various fields.

Established a Casselman-Shalika type formula for the model.

Abstract

We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.