A new class of hypercomplex analytic cusp forms

TL;DR

This paper introduces a new class of Clifford algebra valued automorphic forms called $k$-holomorphic Cliffordian automorphic forms, explores their properties, and constructs non-trivial cusp forms using Poincaré series, expanding the understanding of automorphic forms in hypercomplex analysis.

Contribution

It defines the class of $k$-holomorphic Cliffordian automorphic forms, relates them to hyperbolic harmonic forms, and constructs explicit cusp forms via Poincaré series, providing a new framework in hypercomplex automorphic form theory.

Findings

Established a decomposition theorem for the space of $k$-holomorphic Cliffordian automorphic forms.

Constructed non-trivial cusp forms using Poincaré series within this new class.

Connected these forms to solutions of the Weinstein equation and hyperbolic harmonic automorphic forms.

Abstract

In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator $D \Delta^{k/2}$ for some even $k \in {\mathbb{Z}}$. They will be called $k$-holomorphic Cliffordian automorphic forms. $k$-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions have also the property that they are solutions to the homogeneous and inhomogeneous Weinstein equation. This function class includes the set of $k$-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing $k$-hypermonogenic cusp forms for $k \neq 0$, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms we turn to the construction of Poincar\'e series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of $k$-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of $k$-hypermonogenic Eisenstein series and of $k$-holomorphic Cliffordian cusp forms.