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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