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

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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.
Original languageEnglish
JournalJournal of the American Mathematical Society
Volume34
Issue number3
Pages (from-to)815-908
ISSN0894-0347
DOIs
Publication statusPublished - 2021

ID: 260665586