Higher level cusp forms on the exceptional group of type $E_{7}$

TL;DR

This paper constructs higher level cusp forms on the exceptional group $E_{7,3}$ using new techniques, generalizing previous level one results and revealing new phenomena such as handling non-trivial central characters and dependence only on $SL_2$ restrictions.

Contribution

It introduces a novel Ikeda type lift for $E_{7,3}$ that works at higher levels and with non-trivial central characters, expanding the scope of automorphic form constructions.

Findings

Constructed higher level cusp forms on $E_{7,3}$ using degenerate Whittaker functions.

The lift depends only on the restriction to $SL_2$, making it invariant under twists.

The method generalizes previous level one results to higher levels with new phenomena.

Abstract

By using new techniques with the degenerate Whittaker functions found by Ikeda-Yamana, we construct higher level cusp form on $E_{7,3}$, called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercuspidal local components. This generalizes the previous results on level one forms. But there are new phenomena in higher levels; first, we can handle non-trivial central characters. Second, the lift depends only on the restriction of the Hecke cusp form to $SL_2$. Hence any twist of the cusp form gives rise to the same lift. However for square free levels with the trivial central character, there is no such ambiguity.