The twisted symmetric square $L$-function of $GL(r)$

TL;DR

This paper studies the analytic properties of the twisted symmetric square $L$-function for automorphic representations of $GL(r)$, establishing holomorphy except at specific points and identifying conditions for potential poles.

Contribution

It extends previous work by Bump and Ginzburg to include twisted $L$-functions, showing their holomorphy and pole conditions for general Hecke characters.

Findings

$L^S(s,,Sym^2\u2286 ext)$ is holomorphic except at $s=0$ and $s=1$.

Poles occur only when $\u03c9^r\u03c9^2=1$, where  is the representation and  its central character.

The proof adapts methods from Bump and Ginzburg to the twisted case.

Abstract

In this paper, we consider the (partial) symmetric square $L$-function $L^S(s,\pi,Sym^2\otimes\chi)$ of an irreducible cuspidal automorphic representation $\pi$ of $\GL_r(\A)$ twisted by a Hecke character $\chi$. In particular, we will show that the $L$-function $L^S(s,\pi,Sym^2\otimes\chi)$ is holomorphic except at $s=0$ and $s=1$, and moreover the possible poles could occur only when $\chi^r\omega^2=1$, where $\omega$ is the central character of $\pi$. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case $\chi=1$.