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Actions of compact Lie groups on homogeneous spaces. (English) Zbl 0546.57017

Let G be a compact simple Lie group acting on the homogeneous space \(X=G/T\) (T a maximal torus) with finitely many orbit types. Then it is shown that G acts always in the natural way up to conjugation. This follows in particular from the fact that a graded homomorphism \(H^*(G/T;{\mathbb{R}})\to H^*(G/T;{\mathbb{R}})\) is either trivial or an isomorphism in connection with classical results in the cohomology theory of transformation groups.

MSC:

57S15 Compact Lie groups of differentiable transformations
57T15 Homology and cohomology of homogeneous spaces of Lie groups
55N91 Equivariant homology and cohomology in algebraic topology
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References:

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