MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over $E$ of dimension $2,$ and let $V$ be a skew-hermitian space of dimension $3$ over $E.$ Consider the associated unitary groups $H:=U(W)$ and $G:=U(V).$ Let $\sigma$ be an irreducible, cuspidal, automorphic representation of $H(\mathbb{A}_F).$ Let $\pi=\Theta(\sigma,\psi,\gamma)$ be a theta lift of $\sigma$ to $G(\mathbb{A}_F)$. ($\psi:\mathbb{A}_F/F\to \mathbb{C}^\times$ and $\gamma:\mathbb{A}_E^\times/E^\times\to\mathbb{C}^\times$ are the splitting data necessary to define the theta-lift for unitary groups.)

My question is, how do automorphic $L$-functions (standard, adjoint, etc.) for $\pi$ relate to those for $\sigma$?

share|cite|improve this question
up vote 2 down vote accepted

This question is answered in a paper of Gan, Gross, and D. Prasad. Here's a link:

The relation between L-parameters of representations and their theta-lifts (at least locally) is discussed in section 7 of the paper.

share|cite|improve this answer
"At least locally" deserves to be emphasized: the case of the Shimura correspondence teaches us that there are non-trivial global obstructions to the theta-lift being non-zero. – Victor Protsak Aug 14 '10 at 21:27

Your Answer


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

Not the answer you're looking for? Browse other questions tagged or ask your own question.