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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms
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by Patrick Delorme, Friedrich Knop, Bernhard Krötz and Henrik Schlichtkrull HTML | PDF
J. Amer. Math. Soc. 34 (2021), 815-908 Request permission

Abstract:

This paper lays the foundation for Plancherel theory on real spherical spaces $Z=G/H$, namely it provides the decomposition of $L^2(Z)$ into different series of representations via Bernstein morphisms. These series are parametrized by subsets of spherical roots which determine the fine geometry of $Z$ at infinity. In particular, we obtain a generalization of the Maass-Selberg relations. As a corollary we obtain a partial geometric characterization of the discrete spectrum: $L^2(Z)_{\mathrm {disc}}\neq \emptyset$ if $\mathfrak {h}^\perp$ contains elliptic elements in its interior.

In case $Z$ is a real reductive group or, more generally, a symmetric space our results retrieve the Plancherel formula of Harish-Chandra (for the group) as well as that of Delorme and van den Ban-Schlichtkrull (for symmetric spaces) up to the explicit determination of the discrete series for the inducing datum.

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Additional Information
  • Patrick Delorme
  • Affiliation: Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
  • MR Author ID: 198663
  • Email: patrick.delorme@univ-amu.fr
  • Friedrich Knop
  • Affiliation: Department Mathematik, Emmy-Noether-Zentrum, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
  • MR Author ID: 103390
  • ORCID: 0000-0002-4908-4060
  • Email: friedrich.knop@fau.de
  • Bernhard Krötz
  • Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Straße 100, 33098 Paderborn, Germany
  • Email: bkroetz@gmx.de
  • Henrik Schlichtkrull
  • Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
  • MR Author ID: 156155
  • ORCID: 0000-0002-4681-3563
  • Email: schlicht@math.ku.dk
  • Received by editor(s): October 29, 2020
  • Published electronically: March 25, 2021
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 34 (2021), 815-908
  • MSC (2020): Primary 20G20, 22E46, 22F30, 43A85, 53C35
  • DOI: https://doi.org/10.1090/jams/971
  • MathSciNet review: 4334192