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Affine Deligne–Lusztig varieties at infinite level

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Abstract

We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for \({{\,\mathrm{GL}\,}}_n\) and its inner forms, Lusztig’s semi-infinite Deligne–Lusztig construction is isomorphic to an affine Deligne–Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet–Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig’s 1979 conjecture in this setting for minimal admissible characters.

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Notes

  1. \((b\sigma )^c \varpi ^d({\mathscr {L}}_0)\) coincides with the operator defined in [32, Equation (4.3)].

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Acknowledgements

The first author was partially supported by NSF Grants DMS-0943832 and DMS-1160720, the ERC starting grant 277889, the DFG via P. Scholze’s Leibniz Prize, and an NSF Postdoctoral Research Fellowship, Award No. 1802905. In addition, she would like to thank the Technische Universität München and Universität Bonn for their hospitality during her visits in 2016 and 2018. The second author was partially supported by European Research Council Starting Grant 277889 “Moduli spaces of local G-shtukas”, by a postdoctoral research grant of the DFG during his stay at University Paris 6 (Jussieu), and by the DFG via P. Scholze’s Leibniz Prize. The authors thank Eva Viehmann for very enlightening discussions on this article, especially for the explanations concerning connected components, and also thank Laurent Fargues for his observation concerning the scheme structure on semi-infinite Deligne–Lusztig sets. Finally, the authors thank the anonymous referee for numerous careful and insightful comments.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Charlotte Chan

  2. Mathematisches Institut Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Alexander Ivanov

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  1. Charlotte Chan
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  2. Alexander Ivanov
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Correspondence to Charlotte Chan.

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Communicated by Wei Zhang.

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Chan, C., Ivanov, A. Affine Deligne–Lusztig varieties at infinite level. Math. Ann. 380, 1801–1890 (2021). https://doi.org/10.1007/s00208-020-02092-4

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