A geometric realisation of tempered representations restricted to maximal compact subgroups
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Mathematische Annalen
A geometric realisation of tempered representations restricted to maximal compact subgroups Peter Hochs1
· Yanli Song2 · Shilin Yu3
Received: 21 August 2018 / Revised: 20 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background and motivation . . . . . . . . . . . . . . . 1.2 The main result . . . . . . . . . . . . . . . . . . . . . 1.3 Relation with geometric quantisation of coadjoint orbits 1.4 Ingredients of the proof . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Communicated by Thomas Schick.
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Peter Hochs [email protected] Yanli Song [email protected] Shilin Yu [email protected]
1
University of Adelaide, Adelaide, SA, Australia
2
Washington University in St Louis, St Louis, MO, USA
3
School of Mathematical Sciences, Xiamen University, Fujian 361005, China
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P. Hochs et al. 2 Tempered representations . . . . . . . . . . . . . . . . . . . . 2.1 Limits of discrete series . . . . . . . . . . . . . . . . . . . 2.2 The Knapp–Zuckerman classification . . . . . . . . . . . . 3 Indices of deformed Dirac operators . . . . . . . . . . . . . . . 3.1 Deformed Dirac operators . . . . . . . . . . . . . . . . . . 3.2 Properties of the index . . . . . . . . . . . . . . . . . . . 3.3 The discrete series case . . . . . . . . . . . . . . . . . . . 3.4 An almost complex structure . . . . . . . . . . . . . . . . 3.5 Tempered representations; the main result . . . . . . . . . 3.6 Relation with the orbit method and geometric quantisation 3.7 Example: G = SL(2, R) . . . . . . . . . . . . . . . . . . 4 Linearising the index . . . . . . . . . . . . . . . . . . . . . . . 4.1 The linearised index . . . . . . . . . . . . . . . . . . . . . 4.2 Linearising G/H . . . . . . . . . . . . . . . . . . . . . . 4.3 Linearising almost complex structures . . . . . . . . . . . 4.4 Linearising
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