Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms
Research output: Contribution to journal › Journal article › Research › peer-review
Standard
Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms. / Delorme, Patrick; Knop, Friedrich; Krötz, Bernhard; Schlichtkrull, Henrik.
In: Journal of the American Mathematical Society, Vol. 34, No. 3, 2021, p. 815-908.Research output: Contribution to journal › Journal article › Research › peer-review
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms
AU - Delorme, Patrick
AU - Knop, Friedrich
AU - Krötz, Bernhard
AU - Schlichtkrull, Henrik
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
U2 - 10.1090/jams/971
DO - 10.1090/jams/971
M3 - Journal article
VL - 34
SP - 815
EP - 908
JO - Journal of the American Mathematical Society
JF - Journal of the American Mathematical Society
SN - 0894-0347
IS - 3
ER -
ID: 260665586