The tempered spectrum of a real spherical space

TL;DR

This paper investigates the structure of tempered representations of unimodular real spherical spaces, showing their embeddings into boundary degenerations and their induction from lower-dimensional spaces, advancing understanding of harmonic analysis on these spaces.

Contribution

It establishes that all tempered representations of certain real spherical spaces embed into boundary degenerations and are induced from lower-dimensional discrete series, providing new structural insights.

Findings

Tempered representations embed into boundary degenerations.

Tempered representations are induced from lower-dimensional spaces.

Results apply to both absolutely spherical and wave-front types.

Abstract

Let G/H be a unimodular real spherical space which is either absolutely spherical or wave-front. It is shown that every tempered representation of G/H embeds into a relative discrete series of a boundary degeneration of G/H. If in addition G/H is of wave-front type it follows that the tempered representation is parabolically induced from a discrete series representation of a lower dimensional real spherical space. Final version. To appear in Acta Math.