Orbits of Real Semisimple Lie Groups on Real Loci of Complex Symmetric Spaces
Let G be a complex semisimple algebraic group and X be a complex symmetric homogeneous G-variety. Assume that both G, X as well as the G-action on X are defined over real numbers. Then G(ℝ) acts on X(ℝ) with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of Borel and Ji.
KeywordsSemisimple group symmetric space real point orbit Galois cohomology
MR(2010) Subject Classification14M27 14G05 11E72 20G10
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This work was initiated during Dmitry Timashev’s visit to the Ruhr University of Bochum in July 2016. The second author thanks the Transformation Groups Research team of Bochum for its hospitality and providing excellent working conditions. Both authors are grateful to Peter Heinzner for his support. Many thanks are due to the referee for helpful comments and suggestions aimed at improving the presentation.
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