A Plancherel formula for $L^2(G/H )$ for almost symmetric subgroups

Bent Ørsted and Birgit Speh

Vol. 283 (2016), No. 1, 157–170

DOI: 10.2140/pjm.2016.283.157

Abstract

We study the Plancherel formula for a new class of homogeneous spaces for real reductive Lie groups; these spaces are fibered over non-Riemannian symmetric spaces, and they exhibit a phenomenon of uniform infinite multiplicities. The proof for this is new but rather elementary, and we give all details. As an application we use several results from the recent literature studying possible nontemperedness of homogeneous spaces; thus we provide examples of nontempered representations of the group appearing in the Plancherel formula for our homogeneous spaces. Several classes of examples are given, each building on different techniques and new results from the theory of symmetric spaces.

Keywords

reductive Lie group, tempered representations, Plancherel formula

Mathematical Subject Classification 2010

Primary: 22E46, 43A85

Secondary: 22E30

Milestones

Received: 10 December 2014

Revised: 14 July 2015

Accepted: 11 October 2015

Published: 14 June 2016