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

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. [ABSTRACT FROM AUTHOR]