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Given a reductive group $G$ and a reductive subgroup $H$, both defined over anumber field $F$, we introduce the notion of the $H$-distinguished automorphicspectrum of $G$ and analyze it for the pairs $(GL_{2n},Sp_n)$ and$(Sp_{2n},Sp_n\times Sp_n)$. In the first case we give a complete descriptionusing results of Jacquet--Rallis, Offen and Yamana. In the second case we givean upper bound, generalizing vanishing results of Ash--Ginzburg--Rallis and alower bound, extending results of Ginzburg--Rallis--Soudry.

On the distinguished spectrum of $Sp_{2n}$ with respect to $Sp_n\times Sp_n$