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!



Global symplectic (orthogonal) type of automorphic representation compels its type to all its local components?
Here is a proof of the claim using results from Arthur's monograph [The Endoscopic Classification of
Representations: Orthogonal and
Symplectic Groups](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.294.6966&rep=rep1&type=pdf).


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.


