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

Abstract

This paper lays the foundation for Plancherel theory on real spherical spaces Z = G / H Z=G/H , namely it provides the decomposition of L 2 ( Z ) 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 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 ) d i s c ≠ ∅ L^2(Z)_{\mathrm {disc}}\neq \emptyset if h ⊥ \mathfrak {h}^\perp contains elliptic elements in its interior.

In case Z 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.