On tori periods of Weil representations of unitary groups

We determine the restriction of Weil representations of unitary groups to maximal tori. In the local case, we show that the Weil representation contains a pair of compatible characters if and only if a root number condition holds. In the global case, we show that a torus period corresponding to a maximal anisotropic torus of the global theta lift of a character does not vanish if and only if the local condition is satisfied everywhere and a central value of an $L$-function does not vanish. Our proof makes use of the seesaw argument and of the well-known theta lifting results from $\mathrm{U}\left(1\right)$ to $\mathrm{U}\left(1\right)$. Our results are used in other papers to construct Arthur packets for $G_2$.