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

Abstract

This paper lays the foundation for Plancherel theory on real spherical spacesZ=G/HZ=G/H, namely it provides the decomposition ofL2(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 ofZZat 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:L2(Z)disc≠∅L^2(Z)_{\mathrm {disc}}\neq \emptysetifh⊥\mathfrak {h}^\perpcontains elliptic elements in its interior.In caseZZis 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.