Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$ .

假设$G$是$\mathbf{Q}$上的简单约简群，具有特殊的 Dynkin 类型和$G(\mathbf{R})$四元数（在 Gross–Wallach 的意义上）。在之前的一篇论文中，我们给出了$G$上沿海森堡抛物线的单能根的模形式的傅里叶展开的显式形式。在本文中，我们给出了四元数$E_{8}$上最小模形式$\unicode[STIX]{x1D703}_{Gan}$的傅里叶展开和一些应用。$Sym^{8} ( V_{2})$值自守函数$\unicode[STIX]{x1D703}_{Gan}$是$E_{8}$上的权重为 4 的一级模形式 , 甘曾研究过。我们给出的应用是在四元数$E_{7},E_{6}$和$G_{2}$上构造特殊的模形式。我们还讨论了群$G$上的简并 Heisenberg Eisenstein 级数族，它可以被认为类似于群$\operatorname{GSp}_{2n}$上的全纯 Siegel Eisenstein 级数的四元数异常群.