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

TL;DR

This paper develops a Plancherel theory for real spherical spaces by constructing Bernstein morphisms from boundary degenerations, revealing spectral properties and generalizing known cases like groups and symmetric spaces.

Contribution

It introduces a new construction of Bernstein morphisms for boundary degenerations of real spherical spaces, extending spectral analysis and unifying known special cases.

Findings

Constructed Bernstein morphisms for boundary degenerations.

Proved the isospectrality and G-equivariance of the combined morphism.

Established conditions for non-empty discrete spectrum in L^2 spaces.

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.