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A geometric realisation of tempered representations restricted to maximal compact subgroups

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Abstract

Let G be a connected, linear, real reductive Lie group with compact centre. Let \(K<G\) be maximal compact. For a tempered representation \(\pi \) of G, we realise the restriction \(\pi |_K\) 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 \(\pi |_K\). In a companion paper, we use this realisation of \(\pi |_K\) to give a geometric expression for the multiplicities of the K-types of \(\pi \), in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.

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Notes

  1. See http://www.liegroups.org/software/.

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Acknowledgements

The authors are grateful to Maxim Braverman, Paul-Émile Paradan and David Vogan for their hospitality and inspiring discussions at various stages. Peter Hochs was partially supported by the European Union, through Marie Curie fellowship PIOF-GA-2011-299300. He thanks Dartmouth College for funding a visit there. Yanli Song is supported by NSF grant 1800667. Shilin Yu was supported by the Direct Grants and Research Fellowship Scheme from the Chinese University of Hong Kong.

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Authors and Affiliations

  1. University of Adelaide, Adelaide, SA, Australia

    Peter Hochs

  2. Washington University in St Louis, St Louis, MO, USA

    Yanli Song

  3. School of Mathematical Sciences, Xiamen University, Fujian, 361005, China

    Shilin Yu

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  1. Peter Hochs
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  2. Yanli Song
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  3. Shilin Yu
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Correspondence to Peter Hochs.

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Hochs, P., Song, Y. & Yu, S. A geometric realisation of tempered representations restricted to maximal compact subgroups. Math. Ann. 378, 97–152 (2020). https://doi.org/10.1007/s00208-020-02006-4

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