Let $M:=U(n)/T^n$ be a complete flag variety, where $U(n)$ is an unitary group and $T^n \simeq (S^1)^n$ consists of its diagonal matrices. I have heard the following construction of a symplectic moment map $\mu: M \to t^\*$. Let $B=\operatorname(b\_1, \ldots, b\_n)$ be a diagonal matrix with different $b\_i \in \mathbb R$. Then $M$ may be identified with a $U(n)$-orbit $M\_B$ of $B$ in hermitian matrices $Herm\_n$ under conjugation, namely
$$i: M \to M\_B \subset Herm\_n, \quad X \mapsto XBX^{-1},$$
and one can see that the stabilizer of $B$ is exactly $T^n$. 
It is said that there is a symplectic form on $M\_B$ such that a natural action of $T^n$ by conjugation is Hamiltonian with a moment map
$$\mu: M\_B \to t^\*, \quad Y \mapsto (y\_{11}, \ldots, y\_{nn}),$$
that is it maps matrix to its diagonal. Could you help me to see this symplectic form?



Moment map for complete flags variety
The form on the orbit of coadjoint action is called Kirillov-Kostant-Souriau form (thank to Friedrich Knop for reference!), it descends from a form $\omega\_\alpha: T^\*\_\alpha \mathfrak g^\* \times T^\*\_\alpha \mathfrak g^\*=\mathfrak g \times \mathfrak g \to \mathbb C$ that in a point $\alpha \in \mathfrak g^\*$ is given by $\omega\_\alpha(x, y)=\alpha([x, y])$.


The details are, for example, in [Chriss, Ginzburg *Representation Theory and Complex Geometry*](https://books.google.ru/books?id=OZlCAAAAQBAJ&pg=PA23), proposition 1.1.5 and claim 1.1.6.


