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

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 $\chi=1$.