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 $\sigma$?


1 Answer 1


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.

  • 4
    $\begingroup$ "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. $\endgroup$ Aug 14, 2010 at 21:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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