Fourier coefficients and small automorphic representations

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