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Fourier coefficients and small automorphic representations

Gourevitch, D., Gustafsson, H. P. A., Kleinschmidt, A., Persson, D., & Sahi, S. (in preparation). Fourier coefficients and small automorphic representations.

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Creators:
Gourevitch, Dmitry, Author
Gustafsson, Henrik P. A., Author
Kleinschmidt, Axel1, Author
Sahi, Siddhartha, Author
Affiliations:
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 reductive groups $G(\mathbb{A})$. Let $\pi$ be an automorphic representation of $G(\mathbb{A})$. It is well-known that Fourier coefficients of automorphic forms can be organized into nilpotent orbits $\mathcal{O}$ of $G$. We prove that any Fourier coefficient $\mathcal{F}_\mathcal{O}$ attached to $\pi$ is linearly determined by so-called 'Levi-distinguished' coefficients associated with orbits which are equal or larger than $\mathcal{O}$. When $G$ is split and simply-laced, and $\pi$ is a minimal or next-to-minimal automorphic representation of $G(\mathbb{A})$, we prove that any $\eta \in \pi$ is completely determined by its standard Whittaker coefficients with respect to the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $\mathrm{GL}_n$. In this setting we also derive explicit formulas expressing any maximal parabolic Fourier coefficient in terms of (possibly degenerate) standard Whittaker coefficients for all simply-laced groups. We provide detailed examples for when $G$ is of type $D_5$, $E_6$, $E_7$ or $E_8$ with potential applications to scattering amplitudes in string theory.

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Dates: 2018-11-14
Publication Status: Not specified
Pages: 52 pages
Publishing info: -
Rev. Method: -
Identifiers: arXiv: 1811.05966
URI: http://arxiv.org/abs/1811.05966
Degree: -

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