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 $B_I:L^2(Z_I{)}_{\mathrm{disc}}\to L^2(Z)$. We show that $B:={⨁}_I{B}_I$ 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 $L^2(Z{)}_{\mathrm{disc}}\ne \varnothing$ provided that ${\mathfrak{h}}^{\perp }$ contains elliptic elements in its interior.

101 pages. Final version. Accepted to J. Amer. Math. Soc