Actions of compact Lie groups on homogeneous spaces
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.
cohomology theory of transformation groups
scientific article; zbMATH DE number 3870244