Homogeneous representations of compact manifolds

Question

There is a classification of effective transitive groups actions on the sphere by compact connected Lie groups, compare Besse "Einstein manifolds" 7.13 Examples.

Are there similar results for $\mathbb{C}P^n$ and $\mathbb{H}P^n$?

Answer 1

It seems that Table 3, p. 185 of

Gorbatsevich, V. V.; Onishchik, A. L. Lie transformation groups. Lie groups and Lie algebras, I, 95–235, Encyclopaedia Math. Sci., 20, Springer, Berlin, 1993.

contains the answer to your question. It contains Onishchik's classification of transitive actions of compact Lie groups on manifolds of rank 1. According to this table, only $SU(n+1)$ and, for $n$ odd, $Sp(\frac12(n+1))$ act transitively on $\mathbb CP^n$. On $\mathbb HP^n$ only $Sp(n+1)$ is acting transitively.