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Steinberg’s cross-section of Newton strata

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Abstract

In this note, we introduce a natural analogue of Steinberg’s cross-section in the loop group of a reductive group \(\textbf{G}\). We show this loop Steinberg’s cross-section provides a simple geometric model for the poset \(B(\textbf{G})\) of Frobenius-twisted conjugacy classes (referred to as Newton strata) of the loop group. As an application, we confirms a conjecture by Ivanov on decomposing loop Delgine–Lusztig varieties of Coxeter type. This geometric model also leads to new and direct proofs of several classical results, including the converse to Mazur’s inequality, Chai’s length formula on \(B(\textbf{G})\), and a key combinatorial identity in the study affine Deligne–Lusztig varieties with finite Coxeter parts.

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Acknowledgements

We would like to thank Xuhua He, Alexander Ivanov and Qingchao Yu for helpful comments and discussions. We thank the referee for careful reading of the manuscript and many helpful suggestions. The author is partially supported by National Key R &D Program of China, No. 2020YFA0712600, CAS Project for Young Scientists in Basic Research, Grant No. YSBR-003, and National Natural Science Foundation of China, No. 11922119, No. 12288201 and No. 12231001.

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Sian Nie

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing, 100049, China

    Sian Nie

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Nie, S. Steinberg’s cross-section of Newton strata. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02872-2

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