## Resonances for the Laplacian on Riemannian symmetric spaces: The case of $\mathrm {SL}(3,\mathbb {R})/\mathrm {SO}(3)$

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- by J. Hilgert, A. Pasquale and T. Przebinda
- Represent. Theory
**21**(2017), 416-457 - DOI: https://doi.org/10.1090/ert/506
- Published electronically: October 11, 2017
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## Abstract:

We show that the resolvent of the Laplacian on $\mathrm {SL}(3,\mathbb {R})/\mathrm {SO}(3)$ can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb C$. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are infinite dimensional irreducible $\mathrm {SL}(3,\mathbb {R})$-representations. We determine their Langlands parameters and wave front sets. Also, we show that precisely one of these representations is unitarizable. Alternatively, they are given by the differential equations which determine the image of the Poisson transform associated with the resonance.## References

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## Bibliographic Information

**J. Hilgert**- Affiliation: Department of Mathematics, Paderborn University, Warburger Str. 100, D-33098 Paderborn, Germany
- Email: hilgert@math.uni-paderborn.de
**A. Pasquale**- Affiliation: Université de Lorraine, Institut Elie Cartan de Lorraine, UMR CNRS 7502, Metz, F-57045, France
- MR Author ID: 337961
- Email: angela.pasquale@univ-lorraine.fr
**T. Przebinda**- Affiliation: Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA
- MR Author ID: 257122
- Email: tprzebinda@ou.edu
- Received by editor(s): October 13, 2016
- Received by editor(s) in revised form: August 28, 2017
- Published electronically: October 11, 2017
- Additional Notes: The first and second author would like to thank the University of Oklahoma for their hospitality and financial support. The third author gratefully acknowledges hospitality and financial support from the Université de Lorraine and partial support from the NSA grant H98230-13-1-0205.
- © Copyright 2017 American Mathematical Society
- Journal: Represent. Theory
**21**(2017), 416-457 - MSC (2010): Primary 43A85; Secondary 58J50, 22E30
- DOI: https://doi.org/10.1090/ert/506
- MathSciNet review: 3710652