Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms
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- PLANCHEREL THEORY FOR REAL SPHERICAL SPACES
Accepted author manuscript, 1.07 MB, PDF document
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.
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.
Original language | English |
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Journal | Journal of the American Mathematical Society |
Volume | 34 |
Issue number | 3 |
Pages (from-to) | 815-908 |
ISSN | 0894-0347 |
DOIs | |
Publication status | Published - 2021 |
ID: 260665586