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 -