Hauschild, Volker Actions of compact Lie groups on homogeneous spaces. (English) Zbl 0546.57017 Math. Z. 189, 475-486 (1985). 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. Cited in 1 ReviewCited in 1 Document 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 Keywords:reflection groups; flag manifolds; endomorphisms of cohomology rings; actions on compact homogeneous spaces; compact simple Lie group; cohomology theory of transformation groups PDFBibTeX XMLCite \textit{V. Hauschild}, Math. Z. 189, 475--486 (1985; Zbl 0546.57017) Full Text: DOI EuDML References: [1] Borel, A.: Topics in the homology theory of fiber bundles, LNM No 36, Berlin-Heidelberg-New York: Springer 1967 · Zbl 0158.20503 [2] Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math.77, 778-782 (1955) · Zbl 0065.26103 · doi:10.2307/2372597 [3] Hauschild, V.: Über den Symmetriegrad der rational azyklischen kompakten homogenen Räume. Manuscr. Math.32, 365-379 (1980) · Zbl 0459.57021 · doi:10.1007/BF01299610 [4] Hsiang, W.Y.: Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 85. Berlin-Heidelberg-New York: Springer 1975 · Zbl 0429.57011 [5] Hsiang, W.Y., Tomter, P.: Transformation groups on complex Stiefel manifolds. Acta math.152: 1-2, 107-126 (1984) · Zbl 0542.57030 · doi:10.1007/BF02392193 [6] Slodowy, P.: Simply singularities and simple algebraic groups. Lecture Notes in Math. 815. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0441.14002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.