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#### A reduction principle for Fourier coefficients of automorphic forms

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1811.05966.pdf

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1811.05966v4.pdf

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Gourevitch2021_Article_AReductionPrincipleForFourierC.pdf

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##### Citation

Gourevitch, D., Gustafsson, H. P. A., Kleinschmidt, A., Persson, D., & Sahi, S. (2021).
A reduction principle for Fourier coefficients of automorphic forms.* Mathematische Zeitschrift*.
doi:10.1007/s00209-021-02784-w.

Cite as: https://hdl.handle.net/21.11116/0000-0002-82B6-D

##### 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.

Extended results and paper split into two parts with second part appearing soon. New title to reflect new focus of this part. v3: Minor corrections and updated reference to the second part that has appeared as arXiv:1908.08296 . v4: Minor corrections and reformulations

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.

Extended results and paper split into two parts with second part appearing soon. New title to reflect new focus of this part. v3: Minor corrections and updated reference to the second part that has appeared as arXiv:1908.08296 . v4: Minor corrections and reformulations