The Fourier expansion of modular forms on quaternionic exceptional groups

TL;DR

This paper derives an explicit Fourier expansion for modular forms on quaternionic exceptional groups over $ extbf{Q}$, based on their quaternionic discrete series representations, advancing understanding of automorphic forms on these complex groups.

Contribution

It provides the first explicit Fourier expansion formula for modular forms on quaternionic exceptional groups, connecting representation theory and automorphic forms.

Findings

Explicit Fourier expansion formula derived

Connects modular forms with quaternionic discrete series

Enhances understanding of automorphic forms on exceptional groups

Abstract

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. 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 $P = MN$ of $G$.