Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hi!

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!

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.