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Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms

Authors: Patrick Delorme, Friedrich Knop, Bernhard Krötz and Henrik Schlichtkrull
Journal: J. Amer. Math. Soc. 34 (2021), 815-908
MSC (2020): Primary 20G20, 22E46, 22F30, 43A85, 53C35
Published electronically: March 25, 2021
MathSciNet review: 4334192
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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

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

Bernhard Krötz
Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Straße 100, 33098 Paderborn, Germany

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

Received by editor(s): October 29, 2020
Published electronically: March 25, 2021
Article copyright: © Copyright 2021 American Mathematical Society