Rational homotopy equivalences of Lie type

Abstract

Consider K⊂G, a proper pair of equal rank compact connected Lie groups. It is known (see [6], proof of theorem 1·1) that the group of self-homotopy equivalences of the rationalization of G/K is (anti)isomorphic, in a natural way, to the group of graded algebra automorphisms of H*(G/K; ℚ). On one side of the matter there is the fact that AutH*(G/K; ℚ) almost gives the integral picture of the group of homotopy classes of the self-homotopy equivalences of G/K, that is, up to a finite ambiguity and up to grading automorphisms of H*(G/K; ℚ) (i.e. those acting on H2i as λi.id, for some non-zero λ∈ℚ): see also [6]. On the other side there is the classical description by Borel[2] of H*(G/K; ℚ) in terms of invariants of Weyl groups, which gives hope for a satisfactory understanding of AutH*(G/K; ℚ).