Global symplectic (orthogonal) type of automorphic representation compels its type to all its local components? 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!
 A: 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 $\pi^\vee \cong \pi$.

*

*Since $\pi \cong \pi^\vee$, the Rankin-Selberg $L$-function $L(\pi \times \pi, s)$ has a simple pole at $s = 1$. We have the factorisation $L(\pi \times \pi, s) = L(\pi, S^2, s) L(\pi, \wedge^2, s)$ and both factors are non-vanishing at $s = 1$, so exactly one of them must have a pole; thus $\pi$ is orthogonal or symplectic, in the sense of the question, but never both.

*The discussion preceding Theorem 1.5.3 of Arthur shows that $\pi$ defines an element of his set $\tilde{\Phi}_{\mathrm{sim}}(N)$ of global parameters, and this lies in the subset $\tilde{\Phi}_{\mathrm{sim}}(G)$ for a uniquely determined quasi-split group $G$ whose Langlands dual $\widehat{G}$ is either $SO_{2n}$ or $Sp_{2n}$.

*Theorem 1.5.3 of Arthur shows that $\widehat{G}$ is symplectic if $\pi$ is of symplectic type, and $\widehat{G}$ is orthogonal if $\pi$ is of orthogonal type. (This is a very deep theorem, despite sounding like a tautology!)

*Theorem 1.4.2 of op.cit. now shows that there is a cuspidal automorphic representation $\sigma$ of $G$ such that, for every $v$, the Weil–Deligne representation associated to $\pi_v$ is the image in $GL_N$ of the ${}^L G$-valued parameter associated to $\sigma_v$. So, in particular, it is symplectic (resp. orthogonal) if $\pi$ is.

