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Transactions of the American Mathematical Society

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A new class of hypercomplex analytic cusp forms

Authors: D. Constales, D. Grob and R. S. Kraußhar
Journal: Trans. Amer. Math. Soc. 365 (2013), 811-835
MSC (2010): Primary 11F03, 11F30, 11F55, 30G35, 35J05
Published electronically: August 16, 2012
MathSciNet review: 2995374
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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 also have the property that they are solutions to the homogeneous and inhomogeneous Weinstein equations. 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 Poincaré 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.

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Additional Information

D. Constales
Affiliation: Department of Mathematical Analysis, Ghent University, Building S-22 – and – Laboratory for Chemical Technology, Ghent University, Building S-5; both at Krijgslaan 281, B-9000 Gent, Belgium

D. Grob
Affiliation: Lehrstuhl A für Mathematik, RWTH Aachen, D-52056 Aachen, Germany

R. S. Kraußhar
Affiliation: Arbeitsgruppe Algebra, Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstraße 7, D-64289 Darmstadt, Germany

Keywords: Hypercomplex cusp forms, Poincaré series, hyperbolic harmonic functions, Maaß wave forms, Dirac type operators, Clifford algebras.
Received by editor(s): March 25, 2011
Published electronically: August 16, 2012
Additional Notes: Financial support from BOF/GOA 01GA0405 of Ghent University and from the Long Term Structural Methusalem Funding by the Flemish Government gratefully acknowledged.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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