On a certain metaplectic Eisenstein series and the twisted symmetric square L-function

TL;DR

This paper refines the understanding of the analytic properties of the twisted symmetric square L-function for GL(r), showing it is entire except for potential simple poles at s=0 and s=1, based on Eisenstein series analysis.

Contribution

It precisely determines the pole structure of the twisted symmetric square L-function using Eisenstein series on the double cover of GL(r).

Findings

The L-function is entire except possibly at s=0 and s=1.

It extends previous results by identifying potential poles.

The paper improves the analytic understanding of twisted symmetric square L-functions.

Abstract

In our earlier paper, based on a paper by Bump and Ginzburg, we used an Eisenstein series on the double cover of GL(r) to obtain an integral representation of the twisted symmetric square L-function of GL(r). Using that, we showed that the (incomplete) twisted symmetric square L-function of GL(r) is holomorphic for Re(s) > 1. In this paper, we will determine the possible poles of this Eisenstein series more precisely and show that the (incomplete) twisted symmetric square L-function is entire except possible simple poles at s = 0 and s = 1.