  Given a reductive group $G$ and a reductive subgroup $H$, both defined over a
number field $F$, we introduce the notion of the $H$-distinguished automorphic
spectrum 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 description
using results of Jacquet--Rallis, Offen and Yamana. In the second case we give
an upper bound, generalizing vanishing results of Ash--Ginzburg--Rallis and a
lower bound, extending results of Ginzburg--Rallis--Soudry.
