I am reading the paper "Ricci-flat metrics on the complexification of a compact rank one symmetric space" by Stenzel (here is the link:"http://www.math.osu.edu/~stenzel.3/research/publications/ricci-flat.pdf") and I have some questions.

On page 2 he makes the assumption that the linear isotropy group $Ad_{G}(K)$ acts transitively on the unit sphere in $\mathfrak{p}$. Why does he make this assumption? Why is this assumption good for the further proof?

On page 3 (on top) he says that "if $G/K$ is a compact, rank one, globally symmetric space, the orbits of $G$ in the Stein manifold $G_{\mathbb{C}} / K_{\mathbb{C}}$ are very easy to describe: they are hypersurfaces diffeomorphic to the sphere bundle in $T^{*}G/K$, and an exceptional orbit diffeomorpic to $G/K$.". Why is this so? Is there any reference?

On page 5 there is Theorem 3. The proof of Theorem 3 is in the preprint "An adapted complex structure on the cotangent bundle of a compact Riemannian homogeneous space". Where is this preprint? I cant find it. Can someone give a link or a similar reference where there is a proof?

Why do we need for this proof rank one ? What happens in higher rank ?

I hope that someone could answer some of these questions. Thanks. bernard

havenow solved the higher rank case). For Point 2, which is rather easy, check Helgason. About 3, ask the author. – Robert Bryant Apr 10 '13 at 17:25