The Fourier expansion of modular forms on quaternionic exceptional groups

Abstract

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type and with $G\left(\mathbf{R}\right)$ quaternionic (in the sense of Gross and Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G\left(\mathbf{R}\right)$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P$ of $G$.