The most continuous part of the Plancherel decomposition for a real spherical space

TL;DR

This paper provides a detailed description of the most continuous part of the Plancherel decomposition for a real spherical space, explicitly constructing the relevant functionals and inner products.

Contribution

It offers an explicit construction of H-invariant functionals on principal series and refines the understanding of the inner products in the decomposition.

Findings

Explicit construction of H-invariant functionals

Multiplicity space equals the full space of H-invariant functionals for generic data

Refined inner products via Maass-Selberg relations

Abstract

In this article we give a precise description of the Plancherel decomposition of the most continuous part of $L^{2}(Z)$ for a real spherical homogeneous space $Z$. Our starting point is the recent construction of Bernstein morphisms by Delorme, Knop, Kr\"otz and Schlichtkrull. The most continuous part decomposes into a direct integral of unitary principal series representations. We give an explicit construction of the $H$-invariant functionals on these principal series. We show that for generic induction data the multiplicity space equals the full space of $H$-invariant functionals. Finally, we determine the inner products on the multiplicity spaces by refining the Maass-Selberg relations.