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.

在本文中，我们分析了 adelic 分裂简单带状群的有限覆盖G上的自守形式的傅立叶系数。令 $\pi $ 是G的最小或次最小自守表示。我们证明任何 $\eta \in \pi $ 都完全由其关于固定 Borel 子群的单能部首的（可能退化的）特征的 Whittaker 系数决定，类似于尖点形式的 Piatetski-Shapiro-Shalika 公式 $\运营商名称 {GL}_n$ . 我们还根据这些 Whittaker 系数推导出表达形式的显式公式，以及它的所有最大抛物线傅立叶系数。我们的结果的一个结果是在最小和次小自守谱中不存在尖点形式。我们为 $D_5$ 和 $E_8$ 类型的G提供了详细的示例，以期在弦论中应用散射幅度。