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Let $F$ be a number field and $\mathbb{A}$ its adele ring.

For a dual reductive pair $G$ and $H$, let $\pi$ be a cuspidal irreducible representation of $G(\mathbb{A})$. Let $\Theta(\pi)$ be the global theta lift of $\pi$ to $H(\mathbb{A})$. Then if $\Theta(\pi)$ is non-zero irreducible cuspidal, it is known that $\Theta(\pi)=\otimes_v \theta(\pi_v)$, where $\theta(\pi_v)$ is the local theta lift of $\pi_v$.

I am wondering whether this also holds when $F$ is a global function field $F_q(t)$, where $q$ is $p$-power. I searched some references but in the function field case, it seems that even the global theta functions are not defined yet.

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  • $\begingroup$ Maybe arxiv.org/pdf/math/0701170.pdf is a reference for the symplectic-orthogonal theta correspondence in the function field case. $\endgroup$
    – Erica
    Feb 15, 2023 at 12:28
  • $\begingroup$ @Erica, thank you very much for nice reference! I will look at it! $\endgroup$
    – Monty
    Feb 16, 2023 at 13:42

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