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

TL;DR

This paper investigates the automorphic spectrum of symplectic groups relative to certain subgroups, providing complete descriptions in some cases and bounds in others, advancing understanding of distinguished automorphic representations.

Contribution

It introduces the notion of the $H$-distinguished automorphic spectrum and analyzes it for specific pairs, extending previous results with new bounds and descriptions.

Findings

Complete description of the spectrum for $(GL_{2n},Sp_n)$

Upper bounds for the spectrum of $(Sp_{2n},Sp_n\times Sp_n)$

Lower bounds extending Ginzburg--Rallis--Soudry results

Abstract

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.