Abstract
Let G be a connected semisimple algebraic group over an algebraically closed field of characteristic zero, and let θ be an automorphism of G. We give a characterization of spherical θ-twisted conjugacy classes in G by a formula for their dimensions in terms of certain elements in the Weyl group of G, generalizing a result of N. Cantarini, G. Carnovale, and M. Costantini when θ is the identity automorphism. For G simple and θ an outer automorphism of G, we also classify the Weyl group elements that appear in the dimension formula.
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Acknowledgments
The author would like to thank G. Carnovale, K. Y. Chan and G. Lusztig for discussions. The author is especially grateful to G. Carnovale and M. Costantini for pointing out some of the results in [6]. She would also like to thank the referee for very helpful comments. Research in this paper was partially supported by HKRGC grants 703405 and 703707 and by the Seed Fund for Basic Research from the HKU (Project code 10400942).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Lu, JH. On a dimension formula for spherical twisted conjugacy classes in semisimple algebraic groups. Math. Z. 269, 1181–1188 (2011). https://doi.org/10.1007/s00209-010-0776-4
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DOI: https://doi.org/10.1007/s00209-010-0776-4