I am going through Corlette's paper "Flat G-bundles with canonical metrics", and I am having some difficulty with the computations of Proposition 2.1.
Let $P$ be a principal $SL(n, \mathbb{C})$-bundle over a compact Riemannian manifold $M$, and $\mathcal{C}$ be the space of $L^2$ connections on $P$. Let the unitary gauge group $\mathcal{U}$ act on $\mathcal{C}$. In particular, the space of skew-adjoint $ad(P)$-valued 1-forms is the lie algebra $\mathfrak{u}$ of $\mathcal{U}$.
The claim is that for any $D \in \mathcal{C}$ and $\xi \in \mathfrak{u},$ the map $\Phi: \mathcal{C} \rightarrow \mathfrak{u}^*$ given by $$\Phi_D(\xi) = -i \int_M Tr((D^+)^* \theta)\xi dvol = -i \langle (D^+)^* \theta, \xi \rangle$$ is a moment map. Here, $D^+$ and $\theta$ are the skew and self-adjoint parts of $D$ respectively. My problem essentially boils down to the first few lines of the computation. For $\xi \in \mathfrak{u}$ and $D \in \mathcal{C}$, let $f_\xi(D) = \Phi_D(\xi)$. Then for any $\eta + i \eta \in T_D(\mathcal{C}) = \Gamma(ad(P) \wedge TM^*),$
\begin{align*}(df_{\xi})_D(\eta + i \eta) &= - i \int_M tr((D^+)^*\eta - \star [i\eta', \star \theta])\xi dvol \\ &= -i \langle (D^+)^* \eta - \star[i\eta', \star \theta], \xi\rangle \\ &= -i Im(\langle \eta, D^+ \xi\rangle + \langle i\eta', [\theta, \xi]\rangle) \end{align*}
I think Line 1 is okay. We differentiate $f_\xi$ using a curve $c(t)$ such that $c'(0) = \eta + i\eta$ and $c(0) = D$, and keep track of the skew and self-adjoint parts. Line 2 follows by definition. Why do we only get the imaginary part in line 3? Thanks in advance.