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

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt

Dokumenter

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.
OriginalsprogEngelsk
TidsskriftJournal of the American Mathematical Society
Vol/bind34
Udgave nummer3
Sider (fra-til)815-908
ISSN0894-0347
DOI
StatusUdgivet - 2021

ID: 260665586