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Dimension formula for the affine Deligne–Lusztig variety \(X(\mu , b)\)

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Abstract

The study of certain union \(X(\mu , b)\) of affine Deligne–Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit dimension formula for \(X(\mu , b)\) associated to sufficiently large dominant coweight \(\mu \).

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Notes

  1. The meaning of depth is different from the one used in [1].

  2. There was a typo for type \(E_6\) in [13], which we corrected here.

References

  1. Bagno, E., Biagioli, R., Novick, M., Woo, A.: Depth in classical Coxeter groups. J. Algebr. Comb. 44(3), 645–676 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bourbaki, N.: Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics (Berlin). Springer, Berlin (2002)

    MATH  Google Scholar 

  3. Carter, R.W.: Conjugacy classes in the weyl group. Compos. Math. 25, 1–59 (1972)

    MathSciNet  MATH  Google Scholar 

  4. Fomin, S., Gelfand, S., Postnikov, A.: Quantum Schubert polynomials. J. Am. Math. Soc. 10(3), 565–596 (1997)

    Article  MathSciNet  Google Scholar 

  5. Görtz, U., Haines, T., Kottwitz, R., Reuman, D.: Dimensions of some affine Deligne-Lusztig varieties. Ann. Sci. École Norm. Sup. 39, 467–511 (2006)

    Article  MathSciNet  Google Scholar 

  6. Görtz, U., Haines, T., Kottwitz, R., Reuman, D.: Affine Deligne-Lusztig varieties in affine flag varieties. Compos. Math. 146, 1339–1382 (2010)

    Article  MathSciNet  Google Scholar 

  7. Görtz, U., He, X., Nie, S.: Fully Hodge-Newton decomoposable Shimura varieties. Peking Math. J. 2(2), 99–154 (2019)

    Article  MathSciNet  Google Scholar 

  8. Görtz, U., He, X. and Rapoport, M.: Extremal cases of Rapoport-Zink spaces. arXiv:1909.00423

  9. Görtz, U., Yu, C.-F.: Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties. J. Inst. Math. Jussieu 9, 357–390 (2010)

    Article  MathSciNet  Google Scholar 

  10. Haines, T., He, X.: Vertexwise criteria for admissibility of alcoves. Am. J. Math. 139, 769–784 (2017)

    Article  MathSciNet  Google Scholar 

  11. Haines, T., Ngô, B.C.: Alcoves associated to special fibers of local models. Am. J. Math. 124, 1125–1152 (2002)

    Article  MathSciNet  Google Scholar 

  12. He, X.: Minimal length elements in some double cosets of Coxeter groups. Adv. Math. 215, 469–503 (2007)

    Article  MathSciNet  Google Scholar 

  13. He, X.: A subalgebra of \(0\)-Hecke algebra. J. Algebra 322, 4030–4039 (2009)

    Article  MathSciNet  Google Scholar 

  14. He, X.: Geometric and homological properties of affine Deligne-Lusztig varieties. Ann. Math. 179, 367–404 (2014)

    Article  MathSciNet  Google Scholar 

  15. He, X.: Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties. Ann. Sci. Ecole Norm. Sup. 49, 1125–1141 (2016)

    Article  MathSciNet  Google Scholar 

  16. He, X.: Hecke Algebras and \(p\)-Adic Groups, Current Developments in Mathematics, pp. 73–135. International Press, Somerville (2016)

    MATH  Google Scholar 

  17. He, X.: Cordial elements and dimensions of affine Deligne-Lusztig varieties. arXiv:2001.03325

  18. He, X., Rapoport, M.: Stratifications in the reduction of Shimura varieties. Manuscripta Math. 152, 317–343 (2017)

    Article  MathSciNet  Google Scholar 

  19. Kottwitz, R.: Isocrystals with additional structure. Compos. Math. 56, 201–220 (1985)

    MathSciNet  MATH  Google Scholar 

  20. Kottwitz, R.: Isocrystals with additional structure. II. Compos. Math. 109, 255–339 (1997)

    Article  MathSciNet  Google Scholar 

  21. Kottwitz, R., Rapoport, M.: On the existence of \(F\)-crystals. Comment. Math. Helv. 78, 153–184 (2003)

    Article  MathSciNet  Google Scholar 

  22. Lam, T., Shimozono, M.: Quantum cohomology of \(G/P\) and homology of affine Grassmannian. Acta Math. 204(1), 49–90 (2010)

    Article  MathSciNet  Google Scholar 

  23. Lusztig, G.: From conjugacy classes in the Weyl group to unipotent classes. Represent. Theory 15, 494–530 (2011)

    Article  MathSciNet  Google Scholar 

  24. Milićević, E.: Maximal Newton points and the quantum Bruhat graph. arXiv:1606.07478

  25. Milićević, E. and Viehmann, E.: Generic Newton points and the Newton poset in Iwahori double cosets. arXiv:1902.02415

  26. Postnikov, A.: Quantum Bruhat graph and Schubert polynomials. Proc. Am. Math. Soc. 133(3), 699–709 (2005)

    Article  MathSciNet  Google Scholar 

  27. Rapoport, M.: A guide to the reduction modulo \(p\) of Shimura varieties. Astérisque 298, 271–318 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Rapoport, M., Richartz, M.: On the classification and specialization of \(F\)-isocrystals with additional structure. Compos. Math. 103(2), 153–181 (1996)

    MathSciNet  MATH  Google Scholar 

  29. Viehmann, E.: The dimension of affine Deligne-Lusztig varieties. Ann. Sci. École Norm. Sup. 4(39), 513–526 (2006)

    Article  MathSciNet  Google Scholar 

  30. Wintenberger, J.-P.: Existence de \(F\)-cristaux avec structures supplémentaires. Adv. Math. 190, 196–224 (2005)

    Article  MathSciNet  Google Scholar 

  31. Zhu, X.: Affine Grassmannians and the geometric Satake in mixed characteristic. Ann. Math 185(2), 403–192 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank Jeff Adams and David Vogan for their help on the computation for exceptional groups. We thank George Lusztig for pointing out an intriguing connection between the reflection length (and thus the dimension formula in Theorem 1.1) with the dimension of certain Springer fiber. We thank Ulrich Görtz and Elizabeth Milićević for many useful comments on a preliminary version of the paper. We thank the referee for his/her valuable suggestions. XH is partially supported by a start-up grant and by funds connected with Choh-Ming Chair at CUHK, and by Hong Kong RGC grant 14300220. QY is supported by the National Natural Science Foundation of China (Grant No. 11621061).

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Authors and Affiliations

  1. The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, China

    Xuhua He

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Haidian, Beijing, China

    Qingchao Yu

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  1. Xuhua He
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  2. Qingchao Yu
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Correspondence to Xuhua He.

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Communicated by Wei Zhang.

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He, X., Yu, Q. Dimension formula for the affine Deligne–Lusztig variety \(X(\mu , b)\). Math. Ann. 379, 1747–1765 (2021). https://doi.org/10.1007/s00208-020-02102-5

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