My question is very simple and it is about the theta lift of unitary group in global situation.
Let $E/F$ be a quadratic number fields and $V,W$ be an $n$-dimensional and $1$-dimensional hermitian $E$-vector spaces respectively. Let $\pi$ be an automorphic cuspidal representation of $U(V)$.
Then there is some famous theorem (Rallis inner product theorem) which determines the nonvanishing of the $\Theta$-lift of $\pi$ such that if the critical value of some twisted $L$-function of $\pi$ is nonvanishing, there is some unique equivalent class of $1$-dimension hermitian space $W'$ such that the theta lift of $\pi$ to $U(W')$ is nonzero.
I think all $1$-dimension hermitian $E$-vector spaces are equivalent and so if the nonvanishing of the twisted $L$-function of $\pi$ is ensured, the the theta lift of $\pi$ to $U(W)$ for all 1-dimension hermitian spaces are nonzero.
My guess is right?
Any comment will be greatly helpful!