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Connected components of affine Deligne–Lusztig varieties for unramified groups

Part of: Linear algebraic groups and related topics Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  17 August 2023

Sian Nie
Sian Nie*
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China [email protected] School of Mathematical Sciences, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, PR China
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Abstract

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

Keywords

affine Deligne–Lusztig varietiesconnected componentsreduction of Shimura varietiesLanglands–Rapoport conjecture

MSC classification

Primary: 20G25: Linear algebraic groups over local fields and their integers
Secondary: 14G35: Modular and Shimura varieties 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 10 , October 2023 , pp. 2051 - 2088
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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References

Bhatt, B. and Scholze, P., Projectivity of the Witt vector Grassmannian, Invent. Math. 209 (2017), 329–423.CrossRefGoogle Scholar
Caraiani, A. and Scholze, P., On the generic part of the cohomology of compact unitary Shimura varieties, Ann. of Math. (2) 186 (2017), 649–766.CrossRefGoogle Scholar
Chen, M., Kisin, M. and Viehmann, E., Connected components of affine Deligne–Lusztig varieties in mixed characteristic, Compos. Math. 151 (2015), 1697–1762.CrossRefGoogle Scholar
Chen, L. and Nie, S., Connected components of closed affine Deligne–Lusztig varieties, Math. Ann. 375 (2019), 1355–1392.CrossRefGoogle Scholar
Chen, L. and Nie, S., Connected components of closed affine Deligne–Lusztig varieties for ${\rm Res}_{E/F} {\rm GL}_n$, J. Algebra 546 (2020), 1–26.10.1016/j.jalgebra.2019.10.031CrossRefGoogle Scholar
Gashi, Q., On a conjecture of Kottwitz and Rapoport, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 1017–1038.CrossRefGoogle Scholar
Görtz, U., He, X. and Nie, S., $P$-alcoves and nonemptiness of affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), 647–665.CrossRefGoogle Scholar
Görtz, U., He, X. and Nie, S., Fully Hodge–Newton decomposable Shimura varieties, Peking Math. J. 2 (2019), 99–154.10.1007/s42543-019-00013-2CrossRefGoogle Scholar
Haines, T., The combinatorics of Bernstein functions, Trans. Amer. Math. Soc. 353 (2001), 1251–1278.CrossRefGoogle Scholar
Haines, T. and He, X., Vertexwise criteria for admissibility of alcoves, Amer. J. Math. 139 (2017), 769–784.CrossRefGoogle Scholar
Hamacher, P., The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type, J. Algebraic Geom. 28 (2019), 721–749.CrossRefGoogle Scholar
Hamacher, P., The almost product structure of Newton strata in the deformation space of a Barsotti–Tate group with crystalline Tate tensors, Math. Z. 287 (2017), 1255–1277.CrossRefGoogle Scholar
Hamacher, P. and Kim, W., $\ell$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients, Math. Ann. 375 (2019), 973–1044.CrossRefGoogle Scholar
He, X., Minimal length elements in some double cosets of Coxeter groups, Adv. Math. 215 (2007), 469–503.CrossRefGoogle Scholar
He, X., Minimal length elements in conjugacy classes of extended affine Weyl group, Preprint (2010), arXiv:1004.4040.Google Scholar
He, X., Geometric and homological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404.CrossRefGoogle Scholar
He, X., Kottwitz–Rapoport conjecture on unions of affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), 1125–1141.CrossRefGoogle Scholar
He, X. and Nie, S., Minimal length elements of extended affine Weyl groups, Compos. Math. 150 (2014), 1903–1927.10.1112/S0010437X14007349CrossRefGoogle Scholar
He, X. and Nie, S., On the $\mu$-ordinary locus of a Shimura variety, Adv. Math. 321 (2017), 513–528.CrossRefGoogle Scholar
He, X. and Rapoport, M., Stratifications in the reduction of Shimura varieties, Manuscripta Math. 152 (2017), 317–343.10.1007/s00229-016-0863-xCrossRefGoogle Scholar
He, X. and Zhou, R., On the connected component of affine Deligne–Lusztig varieties, Duke Math. J. 169 (2020), 2697–2765.CrossRefGoogle Scholar
Humphreys, J., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9 (Springer, New York, Berlin, 1972).CrossRefGoogle Scholar
Kim, W., On central leaves of Hodge-type Shimura varieties with parahoric level structure, Math. Z. 291 (2019), 329–363.CrossRefGoogle Scholar
Kisin, M., Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), 967–1012.CrossRefGoogle Scholar
Kisin, M., Mod $p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30 (2017), 819–914.CrossRefGoogle Scholar
Kisin, M., Madapusi Pera, K. and Shin, S., Honda–Tate theory for Shimura varieties, Duke Math. J. 171 (2022), 1559–1614.CrossRefGoogle Scholar
Kisin, M. and Pappas, G., Integral models of Shimura varieties with parahoric level structure, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 121–218.CrossRefGoogle Scholar
Kottwitz, R., Isocrystals with additional structure, Compos. Math. 56 (1985), 201–220.Google Scholar
Kottwitz, R., Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.CrossRefGoogle Scholar
Langlands, R., Some contemporary problems with origins in the Jugendtraum, in Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics, vol. XXVIII, Part1 (American Mathematical Society, Providence, RI, 1976), 401–418.CrossRefGoogle Scholar
Langlands, R. and Rapoport, M., Shimuravarietäten und Gerben, J. Reine Angew. Math. 378 (1987), 113–220.Google Scholar
Mantovan, E., On the cohomology of certain PEL-type Shimura varieties, Duke Math. J. 129 (2005), 573–610.CrossRefGoogle Scholar
Nie, S., Fundamental elements of an affine Weyl group, Math. Ann. 362 (2015), 485–499.CrossRefGoogle Scholar
Nie, S., Connected components of closed affine Deligne–Lusztig varieties in affine Grassmannians, Amer. J. Math. 140 (2018), 1357–1397.CrossRefGoogle Scholar
Pappas, G. and Rapoport, M., $p$-adic shtukas and the theory of global and local Shimura varieties, Preprint (2021), arXiv:2106.08270.Google Scholar
Rapoport, M., A guide to the reduction modulo $p$ of Shimura varieties, Astérisque 298 (2005), 271–318.Google Scholar
Rapoport, M. and Zink, T., Period spaces for p-divisible groups, Ann. Math. Studies, vol. 141 (Princeton University Press, 1996).CrossRefGoogle Scholar
Shen, X., Yu, C.-F. and Zhang, C., EKOR strata for Shimura varieties with parahoric level structure, Duke Math. J. 170 (2021), 3111–3236.CrossRefGoogle Scholar
van Hoften, P., Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou), Preprint (2020), arXiv:2010.10496v2.Google Scholar
Viehmann, E., Connected components of affine Deligne–Lusztig varieties, Math. Ann. 340 (2008), 315–333.CrossRefGoogle Scholar
Zhou, R., Mod $p$ isogeny classes on Shimura varieties with parahoric level structure, Duke Math. J. 169 (2020), 2937–3031.CrossRefGoogle Scholar
Zhu, X., Affine Grassmannians and the geometric Satake in mixed characteristic, Ann. of Math. (2) 185 (2017), 403–492.CrossRefGoogle Scholar