## Abstract

We give a classification of the Harish-Chandra modules generated by the pullback to \(\mathrm{SL}_2(\mathbb {R})\) of harmonic Maass forms for congruence subgroups of \(\mathrm{SL}_2(\mathbb {Z})\) with exponential growth allowed at the cusps. We assume that the weight is integral but include vector-valued forms. Due to the weak growth condition, these modules do not need to be irreducible. Elementary Lie algebra considerations imply that there are nine possibilities, and we show, by giving explicit examples, that all of them arise from harmonic Maass forms. Finally, we briefly discuss the case of forms that are not harmonic but rather are annihilated by a power of the Laplacian, where much more complicated Harish-Chandra modules can arise. We hope that our classification will prove useful in understanding harmonic Maass forms from a representation theoretic perspective and that it will illustrate in the simplest case the phenomenon of extensions occurring in the space of automorphic forms.

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## Notes

In the case \(k=1\), the principal series,

*V*(1) in his notation, is a direct sum \({\mathrm{LDS}}^{+}(0)\oplus {\mathrm{LDS}}^{-}(0)\), in our notation. So his indecomposable in this case should be taken to mean the indecomposable II(b) in our Theorem 5.2 rather than*V*(1).We sometimes write \(\beta _k(x)\) in place of \(W_k(-x/2)\), as this notation occurs in many places in the literature.

A function in this space is a finite linear combination of functions satisfying (3.3) for various weights

*k*.Hence the superscript ‘sm’.

Here we write

*s*for \(\nu \) and take \(\varepsilon =0\).Note that we have changed the sign compared to [13].

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## Acknowledgements

This project was begun during a visit by the second author to the University of Cologne in of 2014 and completed during visits to Oberwolfach, TU Darmstadt, and ETH, Zürich, in the summer of 2016. He would like to thank these institutions for their support and stimulating working environments. The research of the first author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 335220—AQSER. The authors thank Dan Bump, Stephan Ehlen, Olav Richter, Rainer Schulze-Pillot, and Martin Westerholt-Raum for useful comments on an earlier version of this paper.

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Communicated by Toby Gee.

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Bringmann, K., Kudla, S. A classification of harmonic Maass forms.
*Math. Ann.* **370**, 1729–1758 (2018). https://doi.org/10.1007/s00208-017-1563-x

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DOI: https://doi.org/10.1007/s00208-017-1563-x