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

Gourevitch, D., Gustafsson, H. P. A., Kleinschmidt, A., Persson, D., & Sahi, S. (2020). Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups. Canadian Journal of Mathematics. doi:10.4153/S0008414X20000711.

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Gourevitch, Dmitry, Author
Gustafsson, Henrik P. A., Author
Kleinschmidt, Axel1, Author              
Persson, Daniel, Author
Sahi, Siddhartha, Author
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1Quantum Gravity and Unified Theories, AEI Golm, MPI for Gravitational Physics, Max Planck Society, ou_24014              

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Free keywords: Mathematics, Number Theory, math.NT,High Energy Physics - Theory, hep-th,Mathematics, Representation Theory, math.RT
 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.

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 Dates: 2019-08-212020-09-21
 Publication Status: Published online
 Pages: 44 pages, this paper builds upon and extends the results of the second half of arXiv:1811.05966v1, which was split into two parts. The first part (with new title) is arXiv:1811.05966v2 and the present paper is an extension of the second part
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Title: Canadian Journal of Mathematics
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