## Kyoto Journal of Mathematics

- Kyoto J. Math.
- Volume 58, Number 1 (2018), 101-171.

### On the distinguished spectrum of ${Sp}_{2n}$ with respect to ${Sp}_{n}\times {Sp}_{n}$

Erez Moshe Lapid and Omer Offen

#### 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 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.

#### Article information

**Source**

Kyoto J. Math., Volume 58, Number 1 (2018), 101-171.

**Dates**

Received: 5 June 2016

Accepted: 19 July 2016

First available in Project Euclid: 10 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1507600817

**Digital Object Identifier**

doi:10.1215/21562261-2017-0019

**Mathematical Reviews number (MathSciNet)**

MR3776281

**Zentralblatt MATH identifier**

06873130

**Subjects**

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

**Keywords**

automorphic forms distinguished representations period integrals symplectic group

#### Citation

Lapid, Erez Moshe; Offen, Omer. On the distinguished spectrum of $\operatorname{Sp}_{2n}$ with respect to $\operatorname{Sp}_{n}\times\operatorname{Sp}_{n}$. Kyoto J. Math. 58 (2018), no. 1, 101--171. doi:10.1215/21562261-2017-0019. https://projecteuclid.org/euclid.kjm/1507600817