Kähler potential for $SU(n)$ coadjoint orbit

Question

The coadjoint orbit of the Lie group $SU(n)$ is given by $\mathcal{O}_{SU(n)} = SU(n)/\mathbb{T}_{n-1} \cong \mathbb{C}\mathbb{P}^{n-1} \ltimes \mathcal{O}_{SU(n-1)}$.

It is well known that this orbit $\mathcal{O}$ is a homogeneous Kähler manifold, which implies the existence of a Kähler potential on it. In general, what is the Kähler potential for this orbit?

For instance, for $n=2$, where the coadjoint orbit is the two-sphere, the Kähler potential is $2\log(1+|z|^2)$. For $n=3$, it is provided in this link.

I am interested in finding the Kähler potential for general $n$. Any references or clues?

Answer 1

The paper "The Duistermaat-Heckman integration formula on flag manifolds" by Picken has a detailed discussion of how to calculate the Kähler potential for $SU(n)$ coadjoint orbits. As far as I can tell, it gives the general method, but not a general formula. But for example, equation (3.40) gives the potential for $SU(4)$. Also note that the you've just mentioned the maximal dimension coadjoint orbit, but there are also more degenerate orbits (which Picken also discusses).