Realizing $\mathbb{CP}^n$ as a coadjoint orbit of $SU(n+1)$

Question

The natural action of $SU(n+1)$ is transitive on $\mathbb{CP}^n$. This suggests that the latter can be realized as a coadjoint orbit of the former. I am trying make this explicit.

I have been able to show that $\mathbb{CP}^n\cong SU(n+1)/S(U(1)\times U(n))$.

On the other hand, for the coadjoint orbit $\theta_f$ of, say, $f\in\mathfrak{g}^*$, we know that $\theta_f=G/H$, where $H$ is the stabilizer/isotropy group of $f$ (where the action is the coadjoint action).

So, in order to establish $\mathbb{CP}^n$ as a coadjoint orbit of $SU(n+1)$, what we need to do is find an element $f\in\mathfrak{g}^*$ whose isotropy group is $H=S(U(1)\times U(n))$.

By definition, that means $f\in\mathfrak{g}^*$ such that, for $g\in H, Y\in\mathfrak{g},[Ad^*_g(f)](Y)=f(Y)=f(Ad_g(Y))=f(gYg^{-1})$.

So our task amounts to finding some kind of class function on $H$? I'm kind of stuck here. Are there any other viable approaches?

P.S.: It may be of relevance that I showed $\mathbb{CP}^n\cong SU(n+1)/S(U(1)\times U(n))$ through projection maps, i.e., $P_A$ which projects onto the subspace $A\in\mathbb{CP}^n$. It is easy to see that for each $B, A\in\mathbb{CP}^n$, there is a $g\in SU(n+1)$ such that $P_B=gP_Ag^{-1}$ (something that looks a lot like conjugation, but I'm not quite able to put the pieces together).

Answer 1

The isotropy group $H=S(U(1)\times U(n))$ can also be just identified with $H=U(n)$, and the inclusion $H\to G$ is $$ U(n)\to SU(n+1);\quad h\mapsto \text{diag}(h, \det(h)^{-1}). $$

If I identify ${\mathfrak g}^*$ with ${\mathfrak g}$ by the Killing form, then the coadjoint action is just the adjoint action. A generic element $x\in {\mathfrak g}$ would have its isotropy group just as the maximal torus of $U(1)^n\to SU(n+1)$. To have a bigger isotropy group such as $H$, we need a more special element.

Just make it as much as a scalar matrix as possible for $x\in {\mathfrak g}={\mathfrak s}{\mathfrak u}(n+1)$. For example, take $$ x=\text{diag}(i, \dots, i, -ni). $$ Then the isotropy group is $U(n)\to SU(n+1)$, and the adjoint orbit is ${\mathbb C}P^n$.