Abstract
Let G be a compact connected simple Lie group acting non-transitively, non-trivially on itself. Hsiang (Cohomology theory of topological transformation groups, (1975) (New York: Springer)) conjectured that the principal isotropy subgroup type must be the maximal torus and the action must be cohomologically similar to the adjoint action and the orbit space must be a simplex. But Bredon (Bull AMS 83(4) (1977) 711–718) found a simple counterexample, where the principal isotropy subgroup is not a maximal torus and which has no fixed point. In this work, we prove that if \(SO(n)\), (\(n\ge 34\)) or \(SU(3)\) acts smoothly (and nontrivially) on itself with non-empty fixed point set, then the principal isotropy subgroups are maximal tori.
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Acknowledgements
This study was financially supported by Tübitak-Bideb 2211 and Çukurova University (Project No. FDK-2015-4603). The authors would like to thank the anonymous referee for pointing out some errors and gaps in their arguments and also for suggesting improvements, especially in the proof of non-existence of orbits of type \(\mathbb {C}\mathrm {P}^2\) in the last theorem.
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Communicating Editor: Parameswaran Sankaran
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Çobankaya, A., Dönmez, D. Actions of some simple compact Lie groups on themselves. Proc Math Sci 129, 66 (2019). https://doi.org/10.1007/s12044-019-0502-z
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DOI: https://doi.org/10.1007/s12044-019-0502-z