Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

TL;DR

This paper studies Fourier coefficients of minimal and next-to-minimal automorphic forms on simply-laced groups, showing they are determined by Whittaker coefficients and deriving explicit formulas, with implications for the non-existence of cusp forms.

Contribution

It establishes that automorphic forms in minimal and next-to-minimal representations are uniquely determined by their Whittaker coefficients and provides explicit formulas for their Fourier coefficients.

Findings

Automorphic forms in these representations are determined by Whittaker coefficients.

Explicit formulas relate Fourier coefficients to Whittaker coefficients.

Cusp forms do not exist in the minimal and next-to-minimal automorphic spectrum.

Abstract

In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simply-laced group. Let $\pi$ be a minimal or next-to-minimal automorphic representation of $G$. We prove that any $\eta\in \pi$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient in terms of these Whittaker coefficients. A consequence of our results is the non-existence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to scattering amplitudes in string theory.