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

In this paper, we analyze Fourier coefficients of automorphic forms on a finite coverGof an adelic split simply-laced group. Let p be a minimal or next-to-minimal automorphic representation of G. We prove that any eta epsilon 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. Aconsequenceofour results is thenonexistence of cusp forms intheminimal 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.