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.
