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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
\((b\sigma )^c \varpi ^d({\mathscr {L}}_0)\) coincides with the operator defined in [32, Equation (4.3)].
References
Beazley, E.T.: Codimensions of Newton strata for SL\(_3\)(F) in the Iwahori case. Math. Z. 263, 499–540 (2009)
Bhatt, B., Scholze, P.: Projectivity of the Witt vector affine Grassmannian. Invent. Math. 209(2), 329–423 (2017)
Borel, A.: Linear algebraic groups, Graduate Texts in Mathematics, vol. 126. Springer, New York (1991)
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21. Springer, Berlin (1990)
Boyarchenko, M.: Deligne–Lusztig constructions for unipotent and \(p\)-adic groups (preprint) (2012). arXiv:1207.5876
Boyarchenko, M., Weinstein, J.: Geometric realization of special cases of local Langlands and Jacquet–Langlands correspondences (preprint) (2013). arXiv:1303.5795
Boyarchenko, M., Weinstein, J.: Maximal varieties and the local Langlands correspondence for GL\((n)\). J. Am. Math. Soc. 29, 177–236 (2016)
Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)
Bushnell, C.J., Henniart, G.: The essentially tame local Langlands correspondence, II: totally ramified representations. Compos. Math. 141(4), 979–1011 (2005)
Chan, C.: Deligne–Lusztig constructions for division algebras and the local Langlands correspondence. Adv. Math. 294, 332–383 (2016)
Chan, C.: Deligne–Lusztig constructions for division algebras and the local Langlands correspondence. II. Selecta Math. 24(4), 3175–3216 (2018)
Chan, C.: The cohomology of semi-infinite Deligne–Lusztig varieties. J. Reine Angew. Math. t(to appear) (2019). arXiv:1606.01795, v2 or later
Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. 103(1), 103–161 (1976)
Digne, F., Michel, J.: Representations of Finite Groups of Lie Type. Cambridge University Press, Cambridge (1991)
Feigin, B., Frenkel, E.: Affine Kac–Moody algebras and semi-infinite flag manifolds. Commun. Math. Phys. 128(1), 161–189 (1990)
Görtz, U., He, X.: Basic loci in Shimura varieties of Coxeter type. Camb. J. Math. 3, 323–353 (2015)
Görtz, U., Haines, T., Kottwitz, R.E., Reuman, D.: Affine Deligne–Lusztig varieties in affine flag varieties. Compos. Math. 146(5), 1339–1382 (2010)
Hazewinkel, M.: Formal groups and applications, Pure and Applied Mathematics, vol. 78. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers] (1978)
Henniart, G.: Correspondance de Langlands–Kazhdan explicite dans le cas non ramifié. Math. Nachr. 158(7–26) (1992)
Henniart, G.: Correspondance de Jacquet–Langlands explicite. I. Le cas modéré de degré premier. In: S’eminaire de Théorie des Nombres, Paris, 1990-91, Progr. Math., vol. 108, pp. 85–114. Birkhäuser Boston, Boston (1993)
Ivanov, A.: Affine Deligne–Lusztig varieties of higher level and the local Langlands correspondence for GL\({}_2\). Adv. Math. 299, 640–686 (2016)
Ivanov, A.: Ramified automorphic induction and zero-dimensional affine Deligne–Lusztig varieties. Math. Z. 288, 439–490 (2018)
Ivanov, A.: Ordinary GL\({}_2\)(F)-representations in characteristic two via affine Deligne–Lusztig constructions. Math. Res. Lett. (to appear) (2019). arXiv:1802.02456
Langlands, R.P., Shelstad, D.: On the definition of transfer factors. Math. Ann. 278(1–4), 219–271 (1987)
Lusztig, G.: Some remarks on the supercuspidal representations of \(p\)-adic semisimple groups. In Automorphic forms, representations and L-functions, Proc. Symp. Pure Math., vol. 33. Part 1 (Corvallis, Ore., 1977), pp. 171–175 (1979)
Lusztig, G.: Representations of reductive groups over finite rings. Represent. Theory 8, 1–14 (2004)
Moy, A., Prasad, G.: Jacquet functors and unrefined minimal k-types. Comment. Math. Helv. 71(1), 98–121 (1996)
Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. 219, 118–198 (2008)
Rapoport, M.: A guide to the reduction modulo \(p\) of Shimura varieties. Astérisque 298, 271–318 (2005)
Rapoport, M., Zink, T.: Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies. Princeton University Press (1996)
Stasinski, A.: Unramified representations of reductive groups over finite rings. Represent. Theory 13, 636–656 (2009)
Viehmann, E.: Moduli spaces of \(p\)-divisible groups. J. Algebr. Geom. 17, 341–374 (2008)
Yu, J.-K.: Smooth models associated to concave functions in Bruhat–Tits theory. In: Autour des schémas en groupes. Vol. III, vol. 47. Panor. Syntheses (2015)
Zhu, X.: Affine Grassmannians and the geometric Satake in mixed characteristic. Ann. Math. 185(2), 403–492 (2017)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Wei Zhang.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-020-02092-4