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In this paper we show how to predict relative trace identities from the computation of Jacquet modules of the Weil representations. Many previously considered special cases of relative trace identities fit the principle we develop here, including those with important applications on L-functions. We also show how to prove these identities using the Weil representation. We give a proof of the relative trace identities between the distributions on SO(n + 1, n) and $\widetilde{\textit{Sp}}(m)$$(n\geq m)$. The proof should serve as a model to the other cases conjectured in the paper.

Let $G$ be either a split ${\rm SO}(2n+2)$, or a split adjoint group of type $E_n, (n=6,7,8)$, over a $p$-adic field. In this article we study correspondences arising by restricting the minimal representation of $G$ to various dual pairs in $G$.

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Jacquet modules of the Weil representations and families of relative trace identities

Exceptional $\Theta$-Correspondences I

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Jacquet modules of the Weil representations and families of relative trace identities

Exceptional $\Theta$-Correspondences I