Fourier coefficients and small automorphic representations

Dmitry Gourevitch, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson, Siddhartha Sahi

Published 2018-11-14Version 1

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.