Cohomology automorphisms of some homogeneous spaces (Q1091644)

Let \(G\) be a compact connected simple Lie group and let \(U\) be a closed subgroup of maximal rank. The authors show, by studying the Lie algebra of the group \(\Aut H^*(G/U, \mathbb{C})\) of degree-preserving algebra automorphisms of the cohomology ring of \(G/U\) with complex coefficients, that the identity component of \(\Aut H^*(G/U,\mathbb{C})\) is equal to the subgroup of grading automorphisms, \(x\to c^{|x|/2}x\), \(c\in \mathbb{C}\), and thus is isomorphic to \(\mathbb{C}^*\). Combining this theorem with an analysis of invariant polynomials of Weyl groups, they prove that, if \(G\) is not of type \(D_n\) then \(\Aut H^*(G/U,k)\) is generated by \(N_{W(G)}(W(U))/W(U)\) and grading automorphisms and that, if \(G\) is of type \(A_n\) or \(E_6\) then \(\Aut H^*(G/U,k)\) is isomorphic to the direct product \(N_{W(G)}(W(U))/W(U)\times k^*\). This gives an affirmative answer to a conjecture made by \textit{H. H. Glover} and \textit{G. Mislin} [Enseign. Math., II. Sér. 27, 211--219 (1981; Zbl 0498.55004)].