On the distinguished spectrum of Sp(2n) with respect to Sp(n) x Sp(n)

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) x Sp(n)). In the first case we give a complete description by using results of Jacquet and Rallis as well as Offen and Yamana. In the second case we give an upper bound, generalizing vanishing results of Ash, Ginzburg, and Rallis, and a lower bound, extending results of Ginzburg, Rallis, and Soudry.