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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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
Published electronically: October 11, 2017


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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Bibliographic Information
  • J. Hilgert
  • Affiliation: Department of Mathematics, Paderborn University, Warburger Str. 100, D-33098 Paderborn, Germany
  • Email:
  • A. Pasquale
  • Affiliation: Université de Lorraine, Institut Elie Cartan de Lorraine, UMR CNRS 7502, Metz, F-57045, France
  • MR Author ID: 337961
  • Email:
  • T. Przebinda
  • Affiliation: Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA
  • MR Author ID: 257122
  • Email:
  • 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:
  • MathSciNet review: 3710652