Abstract
We use the representation theory of \(SL(2, \mathbb{R})\) to construct examples of functions with transformation properties associated to classical modular forms and Maaß wave forms. We show that for special eigenvalues of the Laplacian, a Maaß wave form may be associated naturally with both a weak harmonic Maaß form and a classical modular form, leading to examples of weak harmonic Maaß forms for all even negative integer weights.
This paper is dedicated to Joseph Wolf
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Acknowledgements
The authors wish to thank Alan Huckleberry for his efforts in organizing this volume, the referee of this paper for helpful comments, and especially Joseph Wolf for his constant encouragement for younger colleagues, a life full of excellent mathematics, and more good work yet to come.
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Kostelec, P., Treneer, S., Wallace, D. (2013). Weak Harmonic Maaß Forms and the Principal Series for \(SL(2, \mathbb{R})\) . In: Huckleberry, A., Penkov, I., Zuckerman, G. (eds) Lie Groups: Structure, Actions, and Representations. Progress in Mathematics, vol 306. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7193-6_9
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DOI: https://doi.org/10.1007/978-1-4614-7193-6_9
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