Please look at this problem:
Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix.
$\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ identifies both spaces.
Let $T=\lbrace diag(\exp(i\tau_1),...,\exp(i\tau_n))\rbrace$ a maximal torus in $U(n)$, $\mathfrak{t}\simeq \mathbb{R}^n$ his Lie algebra.
$\textbf {1.}$ Let $A\in \mathcal{H}$, show that the moment map $\mu_T$ of the $T$-action on $\mathcal{O}_A:= Ad_{U(n)}A$ is $$\mu_T(B)=(B_{11},...,B_{nn}) \in \mathbb{R}^n\simeq \mathfrak{t}$$
So we have to show that $\mu$ is T-equivariant, what is clear since T acts trivially on his Lie algebra The second condition to be a moment map comes when computing $d<\mu\vert_B(\xi),\eta>$ and finding $\omega_B^{\mathfrak{u}_n}(\eta,\xi)=-Tr(B [\eta,\xi]).$ So i'm ok with this question now.
$\textbf{2.}$ Show that the fixed points of this action are $\lbrace diag(\lambda_{\sigma(1)},...,\lambda_{\sigma(n)})\rbrace_{\sigma \in \mathfrak{S}_n}$ where the $\lambda_i$'s are the eigenvalues of $A$.
Comes easily from a simple calculus, or with a bit of deeper knowledge on the adjoint action of maximum tori in compact groups.
$\textbf{3.}$ Conclude that $(A_{11},...,A_{nn})$ lies in the convex span of $(\lambda_{\sigma(1)},...,\lambda_{\sigma(n)})\rbrace_{\sigma \in \mathfrak{S}_n}$
I found in the Mc. Duff-Salamon's book (introduction on symplectic topology) something on it, but it's a strong theorem due to Atiyah-Guillemin-Sternberg, i think there is smth easier here... It's suppose to be an exercise!
Thank you for whatever can help me!