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Fundamental elements of an affine Weyl group

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Abstract

Fundamental elements are certain special elements of affine Weyl groups introduced by Görtz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne–Lusztig varieties. In this paper, we obtain characterizations of the fundamental elements and their natural generalizations. We also derive an inverse to a version of “Newton-Hodge decomposition” in affine flag varieties. As an application, we obtain a group-theoretic generalization of Oort’s results on minimal \(p\)-divisible groups, and we show that, in certain good reduction of PEL Shimura datum, each Newton stratum contains a minimal Ekedahl–Oort stratum. This generalizes a result of Viehmann and Wedhorn.

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Notes

  1. We can also consider the reductive group defined over the ring of Witt vectors over \(\mathbb F\). Since we use group-theoretic methods, all the results of the paper remain true.

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Goertz, U., He, X., Nie, S.: \(P\)-alcoves and non-emptiness of affine Deligne–lusztig varieties, arXiv:1211.3784

  3. He, X.: Minimal length elements in some double cosets of Coxeter groups. Adv. Math. 215(2), 469–503 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. He, X., Nie, S.: Minimal length elements of extended affine Weyl groups, II, arXiv:1112.0824, to appear in Compositio Math

  6. He, X., Nie, S.: \(P\)-alcoves, parabolic subalgebras and cocenter of affine Hecke algebras, arXiv:1310.3940

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

    MATH  MathSciNet  Google Scholar 

  8. Moonen, B., Wedhorn, T.: Discrete invariants of varieties in positive characteristic. Int. Math. Res. Not. No. 72, 3855–3903 (2004)

    Article  MathSciNet  Google Scholar 

  9. Oort, F.: Minimal \(p\)-divisible groups. Ann. Math. 161, 1021–1036 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Viehmann, E.: Truncations of level 1 of elements in the loop group of a reductive group. Ann. Math. 179, 1009–1040 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Viehmann, E., Wedhorn, T.: Ekedahl-Oort and Newton strata for Shimura varieties of PEL type. Math. Ann. 356, 1493–1550 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We would like to thank Xuhua He and Chao Zhang for helpful advice and discussions. We also thank the anonymous referee for the careful reading of the manuscript and many helpful suggestions.

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  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), Beijing, 100190, China

    Sian Nie

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  1. Sian Nie
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Correspondence to Sian Nie.

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Nie, S. Fundamental elements of an affine Weyl group. Math. Ann. 362, 485–499 (2015). https://doi.org/10.1007/s00208-014-1122-7

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  • DOI: https://doi.org/10.1007/s00208-014-1122-7

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