Restriction of modular forms on $E_{7,3}$ to $Sp_6$

TL;DR

This paper investigates how certain modular forms on the exceptional group $E_{7,3}$ behave when restricted to the symplectic group $Sp_6$, providing explicit descriptions for small weights and exploring their classification.

Contribution

It provides explicit formulas for the restriction of modular forms from $E_{7,3}$ to $Sp_6$, including small weight cases and their relation to known lifts and CAP forms.

Findings

Explicit restriction formulas for small weight modular forms.

Identification of Miyawaki lifts and genuine forms within the restriction.

Compatibility of the forms with Arthur's classification.

Abstract

In this paper, we study the restriction of modular forms such as Ikeda type lifts and the Eisenstein series on the exceptional group of type $E_{7,3}$ to the symplectic group $Sp_6$ (rank 3). As an application, we explicitly write down the restriction when modular forms have small weight. The restriction may contain Miyawaki lifts of type I,II (CAP forms) and genuine forms whose description is compatible with Arthur's classification.