Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T23:01:26.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Connected components of affine Deligne–Lusztig varieties in mixed characteristic

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

Published online by Cambridge University Press:  05 May 2015

Miaofen Chen ,
Mark Kisin  and
Eva Viehmann
Miaofen Chen
Affiliation:
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, No. 500, Dong Chuan Road, Shanghai, 200241, PR China email mfchen@math.ecnu.edu.cn
Mark Kisin
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, MA 02138, USA email kisin@math.harvard.edu
Eva Viehmann
Affiliation:
Fakultät für Mathematik der Technischen Universität München – M11, Boltzmannstr. 3, 85748 Garching, Germany email viehmann@ma.tum.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.

Keywords

affine Deligne–Lusztig varietyaffine GrassmannianRapoport–Zink space

MSC classification

Primary: 14G35: Modular and Shimura varieties 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 9 , September 2015 , pp. 1697 - 1762
Copyright
© The Authors 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Borel, A. and Tits, J., Groupes réductifs, Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–150.CrossRefGoogle Scholar
Borovoi, M., Abelian Galois cohomology of reductive groups, Memoirs of the American Mathematical Society 626 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Bourbaki, N., Lie groups and Lie algebras. Chapters 4–6 (Springer, Berlin, 2002), Translated from the 1968 French original by Andrew Pressley.CrossRefGoogle Scholar
Broshi, M., Moduli of finite flat group schemes with $G$-structure, Thesis, University of Chicago (2008).Google Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local I, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251.CrossRefGoogle Scholar
Chen, M., Composantes connexes géométriques d’espaces de modules de groupes p-divisibles, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 723–764.CrossRefGoogle Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., Fibrés quadratiques et composantes connexes réelles, Math. Ann. 244 (1979), 105–134.CrossRefGoogle Scholar
de Jong, A. J., Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. Inst. Hautes Études Sci. 82 (1995), 5–96.Google Scholar
Grothendieck, A. and Dieudonné, J., Elèments de géometrie algèbrique I, II, III, IV, Publ. Math. Inst. Hautes Études Sci. 4, 8, 11, 17, 20, 24, 28, 32 (1961–67).Google Scholar
Gashi, Q., On a conjecture of Kottwitz and Rapoport, Ann. Sci. Éc. Norm. Supér. 43 (2010), 1017–1038.CrossRefGoogle Scholar
Görtz, U., Haines, T. J., Kottwitz, R. E. and Reuman, D. C., Dimensions of some affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. 39 (2006), 467–511.CrossRefGoogle Scholar
Haboush, W., Infinite dimensional algebraic geometry: algebraic structures on p-adic groups and their homogeneous spaces, Tohoku Math. J. (2) 57 (2005), 65–117.CrossRefGoogle Scholar
Haines, T. and Rapoport, M., On parahoric subgroups, Adv. Math. 219 (2008), 188–198.CrossRefGoogle Scholar
Humphreys, J. E., Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43 (American Mathematical Society, Providence, RI, 1995).Google Scholar
Katz, N., Slope filtrations for F-crystals, Astérisque 63 (1979), 113–163.Google Scholar
Kisin, M., Mod $p$points on Shimura varieties of abelian type, Preprint (2013).Google Scholar
Kottwitz, R. E., Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), 287–300.CrossRefGoogle Scholar
Kottwitz, R. E., Isocrystals with additional structure, Compositio Math. 56 (1985), 201–220.Google Scholar
Kottwitz, R. E., Isocrystals with additional structure. II, Compositio Math. 109 (1997), 255–339.CrossRefGoogle Scholar
Kottwitz, R. E., On the Hodge–Newton decomposition for split groups, Int. Math. Res. Not. IMRN 26 (2003), 1433–1447.CrossRefGoogle Scholar
Kottwitz, R. E. and Rapoport, M., On the existence of F-crystals, Comment. Math. Helv. 78 (2003), 153–184.CrossRefGoogle Scholar
Kreidl, M., On p-adic lattices and Grassmannians, Math. Z. 276 (2014), 859–888.CrossRefGoogle Scholar
Lusztig, G., Unipotent almost characters of simple p-adic groups, Astérisque, to appear. Preprint (2012).Google Scholar
Mantovan, E., On non-basic Rapoport–Zink spaces, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 671–716.CrossRefGoogle Scholar
Mantovan, E. and Viehmann, E., On the Hodge–Newton filtration for p-divisible 𝓞-modules, Math. Z. 266 (2010), 193–205.CrossRefGoogle Scholar
Nori, M., On the representations of the fundamental group, Compositio Math. 33 (1976), 29–41.Google Scholar
Rapoport, M., A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2000), 153–166.CrossRefGoogle Scholar
Rapoport, M. and Richartz, M., On the classification and specialization of F-isocrystals with additional structure, Compositio Math. 103 (1996), 153–181.Google Scholar
Raynaud, M. and Gruson, L., Critères de platitude et de projectivité, Invent. Math. 13 (1971), 1–89.CrossRefGoogle Scholar
Rapoport, M. and Zink, T., Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141 (Princeton University Press, Princeton, NJ, 1996).CrossRefGoogle Scholar
Demazure, M. and Grothendieck, A., Séminaire de Géométrie Algébrique du Bois Marie - 1962–64 - Schémas en groupes (SGA3), vol. 3, Lecture Notes in Mathematics, vol. 153 (Springer, Berlin, 1970).Google Scholar
Scholze, P. and Weinstein, J., Moduli of p-divisible groups, Cambridge J. Math. 1 (2013), 145–237.CrossRefGoogle Scholar
Shen, X., On the Hodge–Newton filtration for p-divisible groups with additional structures, Int. Math. Res. Not. 2014(13) (2014), 3582–3631.CrossRefGoogle Scholar
Springer, T. A., Twisted conjugacy in simply connected groups, Trans. Groups 11 (2006), 539–545.CrossRefGoogle Scholar
Tits, J., Reductive groups over local fields, in Automorphic forms, representations and L-functions (Corvallis 1977), Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 29–69.CrossRefGoogle Scholar
Viehmann, E., Connected components of affine Deligne–Lusztig varieties, Math. Ann. 340 (2008), 315–333.CrossRefGoogle Scholar
Viehmann, E. and Wedhorn, T., Ekedahl-Oort and Newton strata for Shmura varieties of PEL type, Math. Ann. 356 (2013), 1493–1550.CrossRefGoogle Scholar
Wintenberger, J.-P., Existence de F-cristaux avec structures supplémentaires, Adv. Math. 190 (2005), 196–224.CrossRefGoogle Scholar
Zink, T., Windows for displays of p-divisible groups, in Moduli of abelian varieties (Text 1999), Progress in Mathematics, vol. 195 (Birkhäuser, Basel, 2001), 491–518.CrossRefGoogle Scholar