Acta Mathematica

Volume 218 (2017)

Number 2

The tempered spectrum of a real spherical space

Pages: 319 – 383



Friedrich Knop (Department Mathematik, Emmy-Noether Zentrum, FAU Erlangen-Nürnberg, Erlangen, Germany)

Bernhard Krötz (Institut für Mathematik, Universität Paderborn, Germany)

Henrik Schlichtkrull (Department of Mathematics, University of Copenhagen, Denmark)


Let $G/H$ be a unimodular real spherical space which is either absolutely spherical, i.e. the real form of a complex spherical space, or of wave-front type. It is shown that every tempered representation for $G/H$ embeds into a twisted discrete series for a boundary degeneration of $G/H$. If $G/H$ is of wave-front type it follows that the tempered representation is parabolically induced by a twisted discrete series representation for a real spherical space formed by a Levi subgroup.

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The second author was supported by ERC Advanced Investigators Grant HARG 268105.

Received 30 September 2015

Received revised 12 August 2016

Published 27 November 2017