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I have the following problem : Let $(G,G')$ and $(H,H')$ be a pair of dual reductive pairs in a symplectic space $Sp(W)$ (in a global setting) forming a seesaw pair (with $H \subset G$ and $G' \subset H'$).

I consider an automorphic form $f$ of $G'$ and lift it to $G$, using some function $\phi$ in the Bruhat-Schwartz space. I denote by $\theta_\phi^f$ this lift ; one thus has : $$ \theta_\phi^f(g) = \int_{G'(\mathbb{k}) \backslash \tilde{G}'(A)} \Theta(\omega(gg') \phi) f(g') dg', $$ where $\omega$ is the global Weil representation and $\Theta$ is the usual distribution on the Bruhat-Schwartz space.

The Rallis inner product formula is useful to determine whether this global theta lift vanishes or not. My question is: how can one study the restriction of $\theta_\phi^f$ to $H$ ?

In particular, is it possible to see this restriction itself as a global theta lift from an automorphic form in $H'$ ?

Thank you.

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