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

Authors :
Delorme, Patrick
Knop, Friedrich
Krötz, Bernhard
Schlichtkrull, Henrik
Source :
Journal of the American Mathematical Society; Jul2021, Vol. 34 Issue 3, p815-908, 94p
Publication Year :
2021

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. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
08940347
Volume :
34
Issue :
3
Database :
Complementary Index
Journal :
Journal of the American Mathematical Society
Publication Type :
Academic Journal
Accession number :
153372850
Full Text :
https://doi.org/10.1090/jams/971