I am reading a paper written by Reza Rezazadegan. On page 5 of the paper, he claimed something like: "the special orthogonal group $\operatorname{SO}(n)=\operatorname{SO}(n; \mathbb{R})$ acts on $\mathbb{C}^{n}$ with a moment map $\eta \colon \mathbb{C}^{n} → \operatorname{so}(n)^∗$ whose regular fiber is a sphere $S^{n-1}$." Here, the complex Euclidean space $\mathbb{C}^{n}$ is equipped with the standard symplectic structure $\omega_{0}={\sqrt{-1}}/{2}\sum_{j=1}^{n}dz_{j} \wedge d\overline{z}_{j}$. My questions are the following:
- What is this $\operatorname{SO}(n)$-action precisely?
- What is the moment map of this action?
So far I considered the $\operatorname{SO}(n)$-action on $\mathbb{C}^{n}$ as the restriction of the standard $\operatorname{SO}(n; \mathbb{C})$-action, but it turned out that this is not the case, that is, its moment map doesn't have the desired property.
It is definitely helpful if you answer the questions. I really appreciate your help.