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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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  • 1. Ö. Akin and H. Leutwiler. On the invariance of the Weinstein equation under Möbiustransformations. In: Classical and modern potential theory and applications. Proceedings of the NATO advanced research workshop, Chateau de Bonas, France, July 25-31, 1993. NATO ASI Ser. C, Math. Phys. Sci. 430, Kluwer, Dordrecht, 1994, 19-29. MR 1321603 (96k:31008)
  • 2. L. V. Ahlfors. Möbius transformations in $ \mathbb{R}^{n}$ expressed through $ 2\times 2$ matrices of Clifford numbers, Complex Variables 5 (1986), 215-224. MR 846490 (88a:15052)
  • 3. Brackx, F., Delanghe R. and Sommen, F. Clifford Analysis. Pitman Res. Notes 76, Boston-London-Melbourne, 1982. MR 697564 (85j:30103)
  • 4. E. Bulla, D. Constales, R.S. Kraußhar and J. Ryan. Dirac type operators for arithmetic subgroups of generalized modular groups. J. Reine Angew. Math. 643 (2010), 1-19. MR 2658187
  • 5. K. Gürlebeck, U. Kähler, J. Ryan and W. Sprößig. Clifford analysis over unbounded domains. Adv. Appl. Math. 19 No. 2 (1997), 216-239. MR 1459499 (98h:30069)
  • 6. D. Constales and R.S. Kraußhar. Bergman spaces of higher dimensional hyperbolic polyhedron type domains I, Mathematical Methods in the Applied Sciences 29 No. 1 (2006), 85-98. MR 2185635 (2006g:30083)
  • 7. D. Constales, R.S. Kraußhar and J. Ryan. $ k$-hypermonogenic automorphic forms. J. Number Theory 126 (2007), 254-271. MR 2354932 (2009f:11043)
  • 8. R. Delanghe, F. Sommen and V. Souček. Clifford Algebra and Spinor Valued Functions. Kluwer, Dordrecht-Boston-London, 1992. MR 1169463 (94d:30084)
  • 9. J. Elstrodt, F. Grunewald and J. Mennicke. Eisenstein series on three-dimensional hyperbolic space and imaginary quadratic number fields. J. Reine Angew. Math. 360 (1985), 160-213. MR 799662 (87c:11052)
  • 10. J. Elstrodt, F. Grunewald and J. Mennicke. Vahlen's group of Clifford matrices and spin-groups. Math. Z. 196 (1987), 369-390. MR 913663 (89b:11031)
  • 11. J. Elstrodt, F. Grunewald and J. Mennicke. Arithmetic applications of the hyperbolic lattice point theorem. Proc. London Math. Soc. III 57 No.2 (1988), 239-288. MR 950591 (89g:11033)
  • 12. J. Elstrodt, F. Grunewald and J. Mennicke. Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces, Invent. Math. 101 No. 3 (1990), 641-668. MR 1062799 (91j:11038)
  • 13. S.-L. Eriksson-Bique. $ k$-hypermonogenic functions. In: Progress in analysis, Proceedings of the 3rd International ISAAC Congress I, edited by H. Begehr et al, World Sci. Publishing, River Edge, New Jersey, 2003, 337-348. MR 2032702 (2004m:30072)
  • 14. S.-L. Eriksson. Integral formulas for hypermonogenic functions. Bull. Belg. Math. Soc. 11 No. 5 (2004), 705-717. MR 2130634 (2006b:30084)
  • 15. S.-L. Eriksson-Bique, Möbius transformations in several function classes, Univ. Joensuu Dept. Math. Rep. Ser 7, (2004), 213-226. MR 2103713 (2005j:30064)
  • 16. E. Freitag. Hilbert Modular Forms, Springer, Berlin-Heidelberg-New York, 1990. MR 1050763 (91c:11025)
  • 17. V. Gritsenko. Arithmetic of quaternions and Eisenstein series. J. Sov. Math. 52 No.3 (1990); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 160 (1987), 82-90. MR 906846 (88m:11035)
  • 18. K. Gürlebeck and W. Sprössig. Quaternionic analysis and elliptic boundary value problems. Birkhäuser, Basel, 1990. MR 1096955 (91k:35002b)
  • 19. T. Hempfling. Beiträge zur modifizierten Clifford-Analysis. Ph.D. Thesis, Universität Erlangen-Nürnberg, Erlangen, 1997.
  • 20. U. Kähler. Elliptic boundary value problems in bounded and unbounded domains. In: Dirac operators in analysis (eds. J. Ryan et al.), Pitman Res. Notes Math. Ser 394 (1998), 122-140. MR 1845955 (2002c:35082)
  • 21. R. S. Kraußhar. Eisenstein Series in Clifford Analysis. Ph.D. Thesis RWTH Aachen, Aachener Beiträge zur Mathematik 28, Wissenschaftsverlag Mainz, Aachen, 2000.
  • 22. R. S. Kraußhar. Generalized analytic automorphic forms in hypercomplex spaces. Frontiers in Mathematics, Birkhäuser, Basel, 2004. MR 2037622 (2005b:32051)
  • 23. R. S. Kraußhar. Generalized analytic automorphic forms for some arithmetic congruence subgroups of the Vahlen group on the $ n$-dimensional hyperbolic space. Bull. Belg. Math. Soc. Simon Stevin 11 No. 5, (2004), 759-774. MR 2130637 (2005m:11072)
  • 24. A. Krieg. Modular Forms on Half-Spaces of Quaternions. Springer Verlag, Berlin-Heidelberg, 1985. MR 807947 (87f:11033)
  • 25. A. Krieg. Eisenstein series on real, complex and quaternionic half-spaces. Pac. J. Math. 133 No.2 (1988), 315-354. MR 941926 (89f:11074)
  • 26. A. Krieg. Eisenstein-series on the four-dimensional hyperbolic space. Journal of Number Theory 30 (1988), 177-197. MR 961915 (90a:11059)
  • 27. A. Krieg. Eisenstein Series on Kähler's Poincaré Group. In: Erich Kähler, Mathematische Werke, edited by R. Berndt and O. Riemenschneider, Walter de Gruyter, Berlin, 2003, 891-906.
  • 28. G. Laville and I. Ramadanoff. Holomorphic Cliffordian functions. Adv. Appl. Clifford Algebr. 8 No. 2 (1998), 323-340. MR 1697976 (2000e:30088)
  • 29. G. Laville and I. Ramadanoff. Elliptic Cliffordian functions. Complex Variables 45 No. 4 (2001), 297-318. MR 1866497 (2002j:30081)
  • 30. G. Laville and I. Ramadanoff. Jacobi elliptic Cliffordian functions. Complex Variables 47 No. 9 (2002), 787-802. MR 1925175 (2004b:30085)
  • 31. H. Leutwiler, Best constants in the Harnack inequality for the Weinstein equation. Aequationes Mathematicae 34 (1987), 304-305. MR 921108 (88m:35040)
  • 32. H. Leutwiler. Modified Clifford analysis. Complex Variables 17 (1991), 153-171. MR 1147046 (93g:30067)
  • 33. H. Maaß. Automorphe Funktionen von mehreren Veränderlichen und Dirichletsche Reihen. Abh. Math. Sem. Univ. Hamb. 16 (1949), 53-104. MR 0033317 (11:421c)
  • 34. C. Maclachlan and A. W. Reid. The Arithmetic of Hyperbolic $ 3$-Manifolds, Springer, New York, 2003. MR 1937957 (2004i:57021)
  • 35. Y. Qiao, S. Bernstein, S.-L. Eriksson and J. Ryan. Function theory for Laplace and Dirac Hodge operators in hyperbolic space. Journal d'Analyse Mathématique 98 (2006), 43-64. MR 2254479 (2007f:58026)
  • 36. I. Ramadanoff. Monogenic, hypermonogenic and holomorphic Cliffordian functions - a survey. Sekigawa, Kouei (ed.) et al., Trends in Differential Geometry, Complex Analysis and Mathematical Physics. Proceedings of 9th International Workshop on Complex Structures, Integrability and Vector Fields, Sofia, Bulgaria, August 25-29, 2008. Hackensack, NJ: World Scientific, 2009, 199-209.
  • 37. J. Ryan. Conformal Clifford manifolds arising in Clifford analysis, Proc. R. Ir. Acad., Sect. A 85 1985, 1-23. MR 821418 (87e:30058)
  • 38. B. Schoeneberg. Elliptic Modular Functions. Die Grundlagen der mathematischen Wissenschaften 201, Springer, Berlin-Heidelberg-New York, 1974. MR 0412107 (54:236)

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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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