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Mackey analogy as deformation of \({\mathcal {D}}\)-modules

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Given a real reductive linear Lie group \(G_{\mathbb {R}}\), the Mackey analogy is a bijection between the set of irreducible tempered representations of \(G_{\mathbb {R}}\) and the set of irreducible unitary representations of its Cartan motion group, established by Higson and Afgoustidis. We show that this bijection arises naturally from families of twisted \({\mathcal {D}}\)-modules over the ag variety of \(G_{\mathbb {R}}\).

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Acknowledgements

The author would like to thank Jonathan Block, Justin Hilburn, Nigel Higson and Tony Pantev for numerous discussions. The author appreciates Alexandre Afgoustidis for sharing the draft of his paper at early stage and for the detailed explanations of his work. The author also wants to express gratitude to Dragan Miličić, Wilfried Schmid and David Vogan for their great patience with the author’s elementary questions about representation theory. The hospitality of Jeffrey Adams and David Vogan during the author’s visit to University of Maryland and Massachusetts Institute of Technology respectively is gratefully acknowledged.

Special thanks go to Junyan Cao, who offered the floor of his hotel room to the author during the ‘Algebraic Geometry 2015’ conference at University of Utah, where the main idea of this paper came up.

The author was partially supported by the Direct Grants and Research Fellowship Scheme from The Chinese University of Hong Kong, the US National Science Foundation (Award No. 1564398 and 1700021), the China NSFC grants (Project No. 12001453 and 1213000100) and Fundamental Research Funds for the Central Universities (Project No. 20720200067 and 20720200071).

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  1. Department of Mathematics, School of Mathematical Sciences, Xiamen University, Fujian, 36100, China

    Shilin Yu

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Dedicated to the memory of Krzysztof Wysocki.

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Yu, S. Mackey analogy as deformation of \({\mathcal {D}}\)-modules. Math. Ann. 385, 421–457 (2023). https://doi.org/10.1007/s00208-021-02332-1

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