Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T21:07:04.197Z Has data issue: false hasContentIssue false
  • English
  • Français

Affine Deligne–Lusztig varieties in affine flag varieties

Part of: Algebraic groups Linear algebraic groups and related topics Algebraic number theory: local and $p$-adic fields (Co)homology theory

Published online by Cambridge University Press:  07 July 2010

Ulrich Görtz ,
Thomas J. Haines ,
Robert E. Kottwitz  and
Daniel C. Reuman
Ulrich Görtz
Affiliation:
Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Ellernstrasse 29, 45326 Essen, Germany (email: ulrich.goertz@uni-due.de)
Thomas J. Haines
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015, USA (email: tjh@math.umd.edu)
Robert E. Kottwitz
Affiliation:
Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637, USA (email: kottwitz@math.uchicago.edu)
Daniel C. Reuman
Affiliation:
Imperial College London, Silwood Park Campus, Buckhurst Road, Ascot, Berkshire SL5 7PY, UK (email: d.reuman@imperial.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies affine Deligne–Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, and extends previous conjectures concerning their dimensions. We generalize the superset method, an algorithmic approach to the questions of non-emptiness and dimension. Our non-emptiness results apply equally well to the p-adic context and therefore relate to moduli of p-divisible groups and Shimura varieties with Iwahori level structure.

Keywords

affine Deligne–Lusztig varietiesσ-conjugacy classesMoy–Prasad filtrationsBruhat–Tits building

MSC classification

Secondary: 14L05: Formal groups, $p$-divisible groups 11S25: Galois cohomology 20G25: Linear algebraic groups over local fields and their integers 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 5 , September 2010 , pp. 1339 - 1382
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Beazley, E. T., Codimensions of Newton strata for SL 3(F) in the Iwahori case, Math. Z. 263 (2009), 499–540.CrossRefGoogle Scholar
[2]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (Springer, New York, 1991).CrossRefGoogle Scholar
[3]Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. I, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251.CrossRefGoogle Scholar
[4]Chai, C.-L., Newton polygons as lattice points, Amer. J. Math. 122 (2000), 967–990.CrossRefGoogle Scholar
[5]Gashi, Q., On a conjecture of Kottwitz and Rapoport, Preprint (2008), arXiv:0805.4575v2, Ann. Sci. École Norm. Sup. (4), to appear.Google Scholar
[6]Geck, M. and Pfeiffer, G., Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Mathematical Society Monographs, vol. 21 (Oxford University Press, Oxford, 2000).CrossRefGoogle Scholar
[7]Görtz, U., Haines, T., Kottwitz, R. and Reuman, D., Dimensions of some affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), 467–511.CrossRefGoogle Scholar
[8]Haines, T., Introduction to Shimura varieties with bad reduction of parahoric type, Clay Math. Proc. 4 (2005), 583–642.Google Scholar
[9]Haines, T. and Ngô, B. C., Alcoves associated to special fibers of local models, Amer. J. Math. 124 (2002), 1125–1152.CrossRefGoogle Scholar
[10]Hartl, U. and Viehmann, E., The Newton stratification on deformations of local G-shtuka, Preprint (2008), arXiv:0810.0821v2, J. Reine Angew. Math., to appear.Google Scholar
[11]Kottwitz, R., Isocrystals with additional structure, Compositio Math. 56 (1985), 201–220.Google Scholar
[12]Kottwitz, R., Isocrystals with additional structure. II, Compositio Math. 109 (1997), 255–339.CrossRefGoogle Scholar
[13]Kottwitz, R., On the Hodge–Newton decomposition for split groups, Int. Math. Res. Not. 26 (2003), 1433–1447.CrossRefGoogle Scholar
[14]Kottwitz, R., Dimensions of Newton strata in the adjoint quotient of reductive groups, Pure Appl. Math. Q. 2 (2006), 817–836.CrossRefGoogle Scholar
[15]Kottwitz, R. and Rapoport, M., On the existence of F-crystals, Comment. Math. Helv. 78 (2003), 153–184.CrossRefGoogle Scholar
[16]Labesse, J.-P., Fonctions élémentaires et lemme fondamental pour la changement de base stable, Duke Math J. 61 (1990), 519–530.CrossRefGoogle Scholar
[17]Lucarelli, C., A converse to Mazur’s inequality for split classical groups, J. Inst. Math. Jussieu 3 (2004), 165–183.CrossRefGoogle Scholar
[18]Mantovan, E. and Viehmann, E., On the Hodge–Newton filtration for p-divisible O-modules, Math. Z., to appear, arXiv:0701.4194v2.Google Scholar
[19]Moy, A. and Prasad, G., Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), 393–408.CrossRefGoogle Scholar
[20]Pappas, G. and Rapoport, M., Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), 118–198.CrossRefGoogle Scholar
[21]Rapoport, M., A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271–318.Google Scholar
[22]Reuman, D., Determining whether certain affine Deligne–Lusztig sets are non-empty, PhD thesis, University of Chicago (2002), arXiv:math/0211434v1.Google Scholar
[23]Reuman, D., Formulas for the dimensions of some affine Deligne–Lusztig varieties, Michigan Math. J. 52 (2004), 435–451.CrossRefGoogle Scholar
[24]Rousseau, G., Euclidean buildings, in Géométries à courbure négative ou nulle, groupes discrets et rigidités, Séminaires et Congrès, vol. 18 (Société Mathématique de France, Paris, 2009), 77–116 first presented at the Summer School on ‘Non-positively curved geometries, discrete groups and rigidity’, Grenoble, 14 June to 2 July 2004.Google Scholar
[25]Viehmann, E., The dimension of affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), 513–526.CrossRefGoogle Scholar
[26]Viehmann, E., Connected components of closed affine Deligne–Lusztig varieties, Math. Ann. 340 (2008), 315–333.CrossRefGoogle Scholar
[27]Viehmann, E., Truncations of level 1 of elements in the loop group of a reductive group, Preprint (2009), arXiv:0907.2331v1.Google Scholar
[28]Wintenberger, J.-P., Existence de F-cristaux avec structures supplémentaires, Adv. Math. 190 (2005), 196–224.CrossRefGoogle Scholar