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


Authors: J. Hilgert, A. Pasquale and T. Przebinda
Journal: Represent. Theory 21 (2017), 416-457
MSC (2010): Primary 43A85; Secondary 58J50, 22E30
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.


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Additional 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
Email: angela.pasquale@univ-lorraine.fr

T. Przebinda
Affiliation: Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA
Email: tprzebinda@ou.edu

DOI: https://doi.org/10.1090/ert/506
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.
Article copyright: © Copyright 2017 American Mathematical Society

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