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Type: Journal article
Title: A geometric realisation of tempered representations restricted to maximal compact subgroups
Author: Hochs, P.
Song, Y.
Yu, S.
Citation: Mathematische Annalen, 2020; 378(1-2):97-152
Publisher: Springer-Verlag
Issue Date: 2020
ISSN: 0025-5831
Statement of
Peter Hochs, Yanli Song and Shilin Yu
Abstract: Let G be a connected, linear, real reductive Lie group with compact centre. Let K<G be maximal compact. For a tempered representation π of G, we realise the restriction π|K as the K-equivariant index of a Dirac operator on a homogeneous space of the form G/H, for a Cartan subgroup H<G. (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of G, so that we obtain an explicit version of Kirillov’s orbit method for π|K. In a companion paper, we use this realisation of π|K to give a geometric expression for the multiplicities of the K-types of π, in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.
Description: Published: 16 May 2020
Rights: © Springer-Verlag GmbH Germany, part of Springer Nature 2020
DOI: 10.1007/s00208-020-02006-4
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