ADHM construction and Momentum Map while I was reading about ADHM construction I had some troubles with precise geometrical identification of the various quantities. My doubts is well manifest in these two Wikipedia pages
1) ADHM Construction of Instantons:
2) Moment(um) map.
In fact, it's well known that moment(um) map $\mu$ on a symplectic manifold $(M,\omega)$ on which acts a group $G$ is a map between
$$
\mu: M\rightarrow \mathfrak{g}^*;
$$
so it's a object that, when evaluated in a point "eats" an element of Lie algebra $\mathfrak{g}$ and gives some number.
But in the ADHM Wikipedia page, from definition it seems that
$$
\mu\in\mathrm{End}(V),
$$
where $V$ is a vector space of dimension $k$. I cannot how the definition of moment map and the use made in Wikipedia pages (as well as in many other references) can match
 A: The space of ADHM quadruples $(B_1,B_2,I,J) \in \mathbb{C}^{k\times k}\times\mathbb{C}^{k\times k}\times\mathbb{C}^{k\times N}\times\mathbb{C}^{N\times k}$ carries a group action of $\mathrm{GL}(k,\mathbb{C})$ as
$$ (B_1,B_2,I,J)\mapsto (UB_1 U^{-1},UB_2U^{-1},IU^{-1},UJ)$$
for $U\in\mathrm{GL}(k,\mathbb{C})$. This descends to a $\mathrm{U}(k)$ action on the quadruples $M_\text{ADHM}$ and gives that 
$$ \mu : M_\text{ADHM}\to \mathfrak{u}(k), (B_1,B_2,I,J)\mapsto [B_1,B_1^\dagger]+[B_2,B_2^\dagger] + II^\dagger - J^\dagger J$$
is a map from this manifold into the Lie algebra of $\mathrm{U}(k)$ (by inspection, the image of $\mu$ is always Hermitian). If it bothers you that this doesn't land in the "dual", just consider the Lie algebra acting on itself through the Killing form.
$M_\text{ADHM}$ is a symplectic manifold because the space of complex $k\times k$ matrices is Kähler and the space $\mathbb{C}^{n\times k}\times\mathbb{C}^{k\times n}$ is also Kähler, and you may check that the $\mu$ is indeed the moment map for the action of the $\mathrm{U}(k)$.
