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
- J. Amer. Math. Soc. 34 (2021), 815-908
- DOI: https://doi.org/10.1090/jams/971
- Published electronically: March 25, 2021
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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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Bibliographic 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