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
DOI:
https://doi.org/10.1090/jams/971
Published electronically:
March 25, 2021
MathSciNet review:
4334192
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Abstract | References | Similar Articles | Additional Information
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
Article copyright:
© Copyright 2021
American Mathematical Society