Fourier-Jacobi expansion of automorphic forms generating quaternionic discrete series

TL;DR

This paper develops a Fourier-Jacobi expansion theory for automorphic forms on certain groups, demonstrating growth conditions and spectral properties for forms generating quaternionic discrete series, with implications for automorphic representation theory.

Contribution

It introduces a new Fourier-Jacobi expansion framework respecting Heisenberg parabolics and proves growth and spectral properties for quaternionic discrete series automorphic forms.

Findings

Automorphic forms satisfy moderate growth conditions except for specific groups.

Fourier expansion terms with non-trivial central characters do not contribute to the discrete spectrum.

The theory extends the K"ocher principle to quaternionic discrete series automorphic forms.

Abstract

We provide a theory of the Fourier-Jacobi expansion for automorphic forms on simple adjoint groups of some general class. This theory respects the Heisenberg parabolic subgroups, whose unipotent radicals are the Heisenberg groups uniformly explained in terms of the notion of cubic norm structures. Based on this theory of the Fourier expansion, we prove that automorphic forms generating quaternionic discrete series representations automatically satisfy the moderate growth condition except for the cases of the group of $G_2$-type and special orthogonal groups of signature $(4,N)$. This should be called ``K\"ocher principle'' verified already for the case of the quaternion unitary group $Sp(1,q)$ for $q>1$ by the author. We also prove that every term of the Fourier expansion with a non-trivial central character for cusp forms generating quaternionic discrete series has no contribution by the discrete spectrum of the Jacobi group, which is a non-reductive subgroup of the Heisenberg parabolic subgroup. This is obtained by showing that generalized Whittaker functions of moderate growth for the Schr\"odinger representations are zero under some assumption of the separation of variables, which suffices for our purpose to establish such consequence.