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Cusp forms on the exceptional group of type $E_{7}$

Part of: Lie groups Discontinuous groups and automorphic forms Linear algebraic groups and related topics

Published online by Cambridge University Press:  07 September 2015

Henry H. Kim  and
Takuya Yamauchi
Henry H. Kim
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada Korea Institute for Advanced Study, Seoul, Korea email henrykim@math.toronto.edu
Takuya Yamauchi
Affiliation:
Department of Mathematics, Faculty of Education, Kagoshima University, Korimoto 1-20-6, Kagoshima 890-0065, Japan Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada email yamauchi@edu.kagoshima-u.ac.jp, tyama@math.toronto.edu
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Abstract

Let $\mathbf{G}$ 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 ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

Keywords

exceptional group of type E7Ikeda liftEisenstein seriesLanglands functoriality

MSC classification

Primary: 11F55: Other groups and their modular and automorphic forms (several variables)
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings 20G41: Exceptional groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 2 , February 2016 , pp. 223 - 254
Copyright
© The Authors 2015 

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