Fourier coefficients of minimal and next-to-minimal automorphic representations 
of simply-laced groups

Gourevitch, Dmitry; Gustafsson, Henrik P. A.; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha

doi:10.4153/S0008414X20000711

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Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

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Kleinschmidt, Axel
Quantum Gravity and Unified Theories, AEI Golm, MPI for Gravitational Physics, Max Planck Society;

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Citation

Gourevitch, D., Gustafsson, H. P. A., Kleinschmidt, A., Persson, D., & Sahi, S. (2022). Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups. Canadian Journal of Mathematics, 74(1), 122-169. doi:10.4153/S0008414X20000711.

Cite as: https://hdl.handle.net/21.11116/0000-0004-8C29-1

Abstract

In this paper we analyze Fourier coefficients of automorphic forms on adelic
split simply-laced reductive groups $G(\mathbb{A})$. Let $\pi$ be a minimal or
next-to-minimal automorphic representation of $G(\mathbb{A})$. We prove that
any $\eta\in \pi$ is completely determined by its Whittaker coefficients with
respect to (possibly degenerate) characters of the unipotent radical of a fixed
Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp
forms on $GL_n$. We also derive explicit formulas expressing the form, as well
as all its maximal parabolic Fourier coefficient in terms of these Whittaker
coefficients. A consequence of our results is the non-existence of cusp forms
in the minimal and next-to-minimal automorphic spectrum. We provide detailed
examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to
scattering amplitudes in string theory.