Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups
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 
$\operatorname {GL}_n$
 . 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 nonexistence 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 toward applications to scattering amplitudes in string theory.