Let $F$ be a number field. For an irreducible cuspidal automorphic representation $\pi$ of $\operatorname{GL}_n(\mathbb{A}_F)$, we say that $\pi$ is symplectic (or orthogonal) if $L(s,\pi,\bigwedge^{2})$ (or $L(s,\pi,\operatorname{Sym}^2)$) has a pole at $s=1$.
I am wondering whether if $\pi=\bigotimes \pi_v$ is symplectic (or orthogonal), then $\pi_v$’s are also symplectic (or orthogonal) for all places $v$?
(Here, $\pi_v$ is symplectic (or orthogonal) means that its corresponding Weil–Deligne group representation by local Langlands correspondence is of such type.)
Any comments are welcome!
Best Answer
Here is a proof of the claim using results from Arthur's monograph The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups.
Let $N = 2n$ be an even integer, and $\pi$ a cuspidal automorphic representation satisfying $\pi^\vee \cong \pi$.