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Affine Deligne–Lusztig varieties in affine flag varieties

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

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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

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