A geometric realisation of tempered representations restricted to maximal compact subgroups

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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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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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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