Affine Deligne-Lusztig varieties and the action of 𝐽

Chen, Miaofen; Viehmann, Eva

doi:10.1090/jag/693

Affine Deligne-Lusztig varieties and the action of $J$

Authors: Miaofen Chen and Eva Viehmann
Journal: J. Algebraic Geom. 27 (2018), 273-304
DOI: https://doi.org/10.1090/jag/693
Published electronically: September 29, 2017
MathSciNet review: 3764277
Full-text PDF

Abstract | References | Additional Information

Abstract: We propose a new stratification of the reduced subschemes of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties that highlights the relation between the geometry of these spaces and the action of the associated automorphism group. We show that this provides a joint group-theoretic interpretation of well-known stratifications which only exist for special cases such as the Bruhat-Tits stratification of Vollaard and Wedhorn, the semi-module stratification of de Jong and Oort, and the locus where the $a$-invariant is equal to $1$.

References

Bhargav Bhatt and Peter Scholze, Projectivity of the Witt vector affine Grassmannian, Invent. Math. 209 (2017), no. 2, 329–423. MR 3674218, DOI https://doi.org/10.1007/s00222-016-0710-4
Miaofen Chen, Mark Kisin, and Eva Viehmann, Connected components of affine Deligne-Lusztig varieties in mixed characteristic, Compos. Math. 151 (2015), no. 9, 1697–1762. MR 3406443, DOI https://doi.org/10.1112/S0010437X15007253
A. J. de Jong and F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), no. 1, 209–241. MR 1703336, DOI https://doi.org/10.1090/S0894-0347-99-00322-7
I. M. Gel′fand, R. M. Goresky, R. D. MacPherson, and V. V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. in Math. 63 (1987), no. 3, 301–316. MR 877789, DOI https://doi.org/10.1016/0001-8708%2887%2990059-4
I. M. Gel′fand and V. V. Serganova, Combinatorial geometries and the strata of a torus on homogeneous compact manifolds, Uspekhi Mat. Nauk 42 (1987), no. 2(254), 107–134, 287 (Russian). MR 898623
Ulrich Görtz, Thomas J. Haines, Robert E. Kottwitz, and Daniel C. Reuman, Dimensions of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 3, 467–511 (English, with English and French summaries). MR 2265676, DOI https://doi.org/10.1016/j.ansens.2005.12.004
Ulrich Görtz and Xuhua He, Basic loci of Coxeter type in Shimura varieties, Camb. J. Math. 3 (2015), no. 3, 323–353. MR 3393024, DOI https://doi.org/10.4310/CJM.2015.v3.n3.a2
Paul Hamacher, The dimension of affine Deligne-Lusztig varieties in the affine Grassmannian, Int. Math. Res. Not. IMRN 23 (2015), 12804–12839. MR 3431637, DOI https://doi.org/10.1093/imrn/rnv081
Urs Hartl and Eva Viehmann, The Newton stratification on deformations of local $G$-shtukas, J. Reine Angew. Math. 656 (2011), 87–129. MR 2818857, DOI https://doi.org/10.1515/CRELLE.2011.044
Xuhua He and Sian Nie, Minimal length elements of extended affine Weyl groups, Compos. Math. 150 (2014), no. 11, 1903–1927. MR 3279261, DOI https://doi.org/10.1112/S0010437X14007349
Christian Kaiser, Ein getwistetes fundamentales Lemma für die ${\rm GSp}_4$, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 303, Universität Bonn, Mathematisches Institut, Bonn, 1997 (German). Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1997. MR 1612837
Robert E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201–220. MR 809866
Stephen S. Kudla and Michael Rapoport, Cycles on Siegel threefolds and derivatives of Eisenstein series, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 5, 695–756 (English, with English and French summaries). MR 1834500, DOI https://doi.org/10.1016/S0012-9593%2800%2901051-X

L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur $\mathbb P^1$, II. Exemple, L. Szpiro (ed.), Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque 86 (1981), 125–140.

Frans Oort, Which abelian surfaces are products of elliptic curves?, Math. Ann. 214 (1975), 35–47. MR 364264, DOI https://doi.org/10.1007/BF01428253

M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439

M. Rapoport and Th. Zink, A finiteness theorem in the Bruhat-Tits building: an application of Landvogt’s embedding theorem, Indag. Math. (N.S.) 10 (1999), no. 3, 449–458. MR 1819901, DOI https://doi.org/10.1016/S0019-3577%2899%2980035-4

Eva Viehmann, Moduli spaces of $p$-divisible groups, J. Algebraic Geom. 17 (2008), no. 2, 341–374. MR 2369090, DOI https://doi.org/10.1090/S1056-3911-07-00480-8

Eva Viehmann, The global structure of moduli spaces of polarized $p$-divisible groups, Doc. Math. 13 (2008), 825–852. MR 2466182

Eva Viehmann, The dimension of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 3, 513–526 (English, with English and French summaries). MR 2265677, DOI https://doi.org/10.1016/j.ansens.2006.04.001

Eva Viehmann, Truncations of level 1 of elements in the loop group of a reductive group, Ann. of Math. (2) 179 (2014), no. 3, 1009–1040. MR 3171757, DOI https://doi.org/10.4007/annals.2014.179.3.3

Inken Vollaard, The supersingular locus of the Shimura variety for ${\rm GU}(1,s)$, Canad. J. Math. 62 (2010), no. 3, 668–720. MR 2666394, DOI https://doi.org/10.4153/CJM-2010-031-2

Inken Vollaard and Torsten Wedhorn, The supersingular locus of the Shimura variety of ${\rm GU}(1,n-1)$ II, Invent. Math. 184 (2011), no. 3, 591–627. MR 2800696, DOI https://doi.org/10.1007/s00222-010-0299-y

Torsten Wedhorn, Bruhat strata for Shimura varieties of PEL type, Math. Z. 277 (2014), no. 3-4, 725–738. MR 3229962, DOI https://doi.org/10.1007/s00209-013-1274-2

Xinwen Zhu, Affine Grassmannians and the geometric Satake in mixed characteristic, Ann. of Math. (2) 185 (2017), no. 2, 403–492. MR 3612002, DOI https://doi.org/10.4007/annals.2017.185.2.2

Additional Information

Miaofen Chen
Affiliation: Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, No. 500, Dong Chuan Road, Shanghai, 200241, People’s Republic of China
MR Author ID: 1020475
Email: mfchen@math.ecnu.edu.cn

Eva Viehmann
Affiliation: Technische Universität München, Zentrum Mathematik – M11, Boltzmannstr. 3, 85748 Garching, Germany
Email: viehmann@ma.tum.de

Received by editor(s): November 25, 2015
Received by editor(s) in revised form: May 2, 2016
Published electronically: September 29, 2017
Additional Notes: The first author was partially supported by NSFC grant No. 11301185, SRFDP grant No. 20130076120002, and STCSM grant No. 13dz2260400. The second author was partially supported by ERC starting grant 277889 “Moduli spaces of local $G$-shtukas”.
Article copyright: © Copyright 2017 University Press, Inc.