Period of the Ikeda type lift for $E_{7,3}$

TL;DR

This paper establishes an explicit formula for the Petersson norm of an Ikeda type lift on the exceptional group $E_{7,3}$, linking it to special values of symmetric square L-functions, advancing understanding of automorphic forms on large groups.

Contribution

It provides a new explicit formula for the period of the Ikeda type lift on $E_{7,3}$, overcoming technical challenges due to the group's complexity.

Findings

Explicit formula for the Petersson norm in terms of L-values

Overcoming technical difficulties from the group's size

Extension of previous work on automorphic lifts

Abstract

In our previous work, he second and the third named authors constructed the Ikeda type lift for the exceptional group $E_{7,3}$ from an elliptic modular cusp form. In this paper, we prove an explicit formula for the period or the Petersson norm of the Ikeda type lift in terms of the product of the special values of the symmetric square $L$-function of the elliptic modular form. There are similar works done by the first author with his collaborator, but new technical inputs are required and developed to overcome some difficulties coming from the hugeness of $E_{7,3}$.