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

Abstract

Given a unimodular real spherical space $Z=G/H$ we construct for each boundary degeneration $Z_I=G/H_I$ of $Z$ a Bernstein morphism $B_I: L^2(Z_I)_{\rm disc }\to L^2(Z)$. We show that $B:=\bigoplus_I B_I$ provides an isospectral $G$-equivariant morphism onto $L^2(Z)$. Further, the maps $B_I$ are finite linear combinations of orthogonal projections which translates in the known cases where $Z$ is a group or a symmetric space into the familiar Maass-Selberg relations. As a corollary we obtain that $L^2(Z)_{\rm disc }\neq \emptyset$ provided that ${\mathfrak h}^\perp$ contains elliptic elements in its interior.