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The dimension of some affine Deligne-Lusztig varieties. (English) Zbl 1108.14036
M. Rapoport [Astérisque 298, 271–318 (2005; Zbl 1084.11029)] conjectured a formula for the dimensions of affine Deligne-Lusztig varieties, $$X_{\mu}(b)$$, in the affine Grassmanian case. U. Görtz et al. [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 3, 467–511 (2006; Zbl 1108.14035)] proved that it is enough to prove the formula for superbasic $$b \in \text{GL}_n$$. The aim of this paper is to prove the conjectured dimension formula in this case, thus completing the proof of the conjecture. We refer the reader to the review of the above mentioned paper of Görtz et al. for details regarding notations and the statement of the conjecture.
The proof involves decomposition of $$X_{\mu}(b)$$ into finitely many locally closed subschemes whose dimensions can be written down combinatorially depending on certain discrete invariants, called extended semi-modules. The dimension formula is then proved by estimating these dimensions.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M15 Grassmannians, Schubert varieties, flag manifolds 20G25 Linear algebraic groups over local fields and their integers 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties
##### Keywords:
Deligne-Lusztig varieties; reductive groups
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##### References:
 [1] Görtz U. , Haines Th.J. , Kottwitz R.E. , Reuman D.C. , Dimensions of some affine Deligne-Lusztig varieties , Ann. Scient. Éc. Norm. Sup. 39 ( 3 ) ( 2006 ), math.AG/0504443 . Numdam | MR 2265676 | Zbl 1108.14035 · Zbl 1108.14035 [2] de Jong A.J. , Oort F. , Purity of the stratification by Newton polygons , J. Amer. Math. Soc. 13 ( 2000 ) 209 - 241 . MR 1703336 | Zbl 0954.14007 · Zbl 0954.14007 [3] Kottwitz R.E. , Isocrystals with additional structure , Comp. Math. 56 ( 1985 ) 201 - 220 . Numdam | MR 809866 | Zbl 0597.20038 · Zbl 0597.20038 [4] Kottwitz R.E. , Dimensions of Newton strata in the adjoint quotient of reductive groups , math.AG/0601196 . arXiv · Zbl 1109.11033 [5] Kottwitz R.E. , Rapoport M. , On the existence of F -crystals , Comment. Math. Helv. 78 ( 2003 ) 153 - 184 . MR 1966756 | Zbl 01990090 · Zbl 1126.14023 [6] Rapoport M. , A positivity property of the Satake isomorphism , Manuscripta Math. 101 ( 2 ) ( 2000 ) 153 - 166 . MR 1742251 | Zbl 0941.22006 · Zbl 0941.22006 [7] Rapoport M. , A guide to the reduction modulo p of Shimura varieties , Astérisque 298 ( 2005 ) 271 - 318 . MR 2141705 | Zbl 1084.11029 · Zbl 1084.11029 [8] Reuman D.C. , Determining whether certain affine Deligne-Lusztig sets are empty , PhD thesis , Chicago 2002, math.RT/0211434 . arXiv [9] Reuman D.C. , Formulas for the dimensions of some affine Deligne-Lusztig varieties , Michigan Math. J. 52 ( 2004 ) 435 - 451 . Article | MR 2069809 | Zbl 1053.22010 · Zbl 1053.22010 [10] Viehmann E. , Moduli spaces of p -divisible groups , math.AG/0502320 . arXiv · Zbl 1144.14040
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