Rigidity properties of compact Lie groups modulo maximal tori (Q1070595)

Let \(M=G/T\) be the flag manifold \((G=compact\) connected Lie group). Main results: the group of rational (resp. integral) cohomology automorphisms of M is isomorphic to the normalizer of the Weyl group in the general linear group of the natural rational structure of L(T), resp. to the automorphism group of the root system of G [to be compared with the results for classifying spaces due to \textit{J. F. Adams} and \textit{Z. Mahmud}, Invent. Math. 35, 1-41 (1976; Zbl 0306.55019)]. The explicitness of the result has various consequences. First application: given any metric on M, and any isometry f, there exists an f-invariant geodesic; moreover (excepting a few cases which may be explicitly listed) there are infinitely many such geodesics, which are geometrically distinct (similar results were obtained by \textit{K. Grove} and \textit{S. Halperin} [Publ. Math., Inst. Hautes Étud. Sci. 56, 171-178 (1982; Zbl 0508.55013)] for the case of odd-dimensional, resp. rationally hyperbolic Riemannian manifolds). Further applications are addressed to the problem of the abundance of the fixed point-free self-maps of M and to the problem of the determination of the homotopy type of M by its localizations. Similar results, holding for general flag manifolds \(M=G/K\) (rank K\(=rank G)\), will appear elsewhere.