I am having trouble in understanding the section of this paper http://www-math.mit.edu/~etingof/zlecnew.pdf where the author introduces the Calogero-Moser system as the reduction of a manifold $M$ on which is acting a group $G$ along the coadjoint orbit $\mathcal{O}$ of $G$:$\mathcal{C}_n \doteq R(M,G,\mathcal{O})$.

We have the manifold $M = T^\ast\text{Mat}_n(\mathbb{C})$ which, I suppose, is the cotangent fiber of the set of all possible $\mathbb{C}$-valued matrices. Then we define a symplectic structure on it which is "the usual trace form": $$\omega =\text{Tr}\ ( dX \wedge dY).$$ Now, as I see it, this must is some sense be similar to what we do when we define the usual cotangent coordinates $q_i$ and $p_i$ on a cotangent fiber and get our canonical form $\sum_i dq_i\wedge dp_i$. However I do not quite understand what is the exterior derivative of a matrix $X$ or $Y$ and how such an $\omega$ actually defines a symplectic form.

Given all this, we can identify $$ \mathfrak{g}^\ast \simeq \mathfrak{g}$$ $$ M \simeq \text{Mat}_n(\mathbb{C}) \oplus \text{Mat}_n(\mathbb{C}).$$ How...?

Finally we have some functions $$ H_i = \text{Tr}(Y^i), \qquad i=1,\ldots , n$$ which are claimed to be in involution with each other $\{H_i, H_j\}=0$; but to verify that one would have to compute the corresponding hamiltonian vectors $X_i$ and $X_j$ and check that $\omega(X_i, X_j)=0.$ How do we do this?

I don't ask for the whole solution: just for a few good hints or, in alternative, for a good reference where I should look this stuff up.

(My background: basics on symplectic geometry such as Moment Maps, Coadjoint Representation, Reduction and so on)