Abstract
Motivated by Lusztig’s G-stable pieces, we consider the combinatorial pieces: the pairs (w, K) for elements w in the Weyl group and subsets K of simple reflections that are normalized by w. We generalize the notion of cyclic shift classes on the Weyl groups to the set of combinatorial pieces. We show that the partial cyclic shift classes of combinatorial pieces associated with minimal-length elements have nice representatives. As applications, we prove the left-right symmetry and the compatibility of the induction functors of the parabolic character sheaves.
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Acknowledgements
XH is partially supported by the New Cornerstone Science Foundation through the New Cornerstone Investigator Program and the Xplorer Prize, by Hong Kong RGC grant 14300221, and by funds connected with Choh-Ming Chair at CUHK. This paper is motivated by a question of Zhiwei Yun on the parabolic character sheaves, which is used in [11]. We thank him for explaining to me the question and related materials in [11]. We also thank George Lusztig for the helpful discussions. We thank the referee for the useful comments and suggestions.
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He, X. A generalization of cyclic shift classes. Sel. Math. New Ser. 29, 83 (2023). https://doi.org/10.1007/s00029-023-00885-4
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DOI: https://doi.org/10.1007/s00029-023-00885-4