Cusp forms for exceptional group of type $E_{7}$

Abstract

Let $\bf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let $Γ=\bf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\ge 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to $Γ$ from a Hecke cusp form in $S_{2k-8}(SL_2(\mathbb{Z}))$. This lift is an analogue of Ikeda's construction.