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

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