On the $\mathrm{Sp}_{2n}$-distinguished automorphic spectrum of $\mathrm{U}_{2n}$

TL;DR

This paper introduces and analyzes the $H$-distinguished automorphic spectrum for the pair $(\mathrm{U}_{2n}, \mathrm{Sp}_{2n})$, deriving a formula for period integrals and providing bounds that advance understanding of automorphic representations.

Contribution

It defines the $H$-distinguished automorphic spectrum for $(\mathrm{U}_{2n}, \mathrm{Sp}_{2n})$ and derives a formula for period integrals, extending prior work on related pairs.

Findings

Derived a formula for period integrals of pseudo-Eisenstein series.

Provided an upper bound for the distinguished spectrum.

Expected a non-trivial lower bound based on the formula.

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 pair $(\mathrm{U}_{2n},\mathrm{Sp}_{2n})$. We derived a formula for period integrals of pseudo-Eisenstein series of $\mathrm{U}_{2n}$ in analogy with the main result of Lapid and Offen in their work analyzing the pair $(\mathrm{Sp}_{4n},\mathrm{Sp}_{2n} \times \mathrm{Sp}_{2n})$. We give an upper bound of the distinguished spectrum with the formula. A non-trivial lower bound of the discrete distinguished spectrum is expected from the formula, given the previous work.