Rational visibility of a Lie group in the monoid of self-homotopy equivalences of a homogeneous space

Let $G$ be a connected Lie group and $M$ a homogeneous space admitting a left translation by $G$. Let aut$_1(M)$ denote the identity component of the monoid of self-homotopy equivalences of $M$. Then the action of $G$ on $M$ gives rise to a map $\lambda : G \to$   aut$_1(M)$. The purpose of this article is to investigate the injectivity of the homomorphism which $\lambda$ induces on the rational homotopy. In particular, the visible degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik. The main tool for the study is an elaborate rational model for an evaluation map, which is constructed in [B-M] and [Ku] with the function space model due to Brown and Szczarba [B-S].

[B-S] Brown Jr, E. H. and Szczarba, R. H.: Rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349(1997), 4931-4951.

[B-M]U. Buijs and A. Murillo, Basic constructions in rational homotopy theory of function spaces, Ann. Inst. Fourier (Grenoble) 56(2006), 815-838.

[Ku] K. Kuribayashi, A rational model for the evaluation map, Georgian Mathematical Journal 13(2006), 127-141.