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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 $\sigma$?

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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.

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"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

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