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Gelfand pairs admit an Iwasawa decomposition

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Abstract

Every Gelfand pair (G, K) admits a decomposition \(G=KP\), where \(P<G\) is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous. Applications include the complete classification of non-positively curved Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand–Godement for general Gelfand pairs.

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Acknowledgements

I am grateful to Andy Zucker for sending me his preprint and to Pierre-Emmanuel Caprace for several insightful comments on a preliminary version.

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  1. EPFL, Lausanne, Switzerland

    Nicolas Monod

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  1. Nicolas Monod
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Correspondence to Nicolas Monod.

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Communicated by Andreas Thom.

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Monod, N. Gelfand pairs admit an Iwasawa decomposition. Math. Ann. 378, 605–611 (2020). https://doi.org/10.1007/s00208-020-02034-0

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  • DOI: https://doi.org/10.1007/s00208-020-02034-0

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