Let $G_\mathbb{C}$ be a semisimple Lie group and $G$ an compact real form of it. For example, take $G_\mathbb{C} = SL(n,\mathbb{C})$ and $G = SU(n)$. Pick a Borel subgroup of $G_\mathbb{C}$. The intersection $T = B \cap G$ will be a maximal torus in $G$. In the above case, one can pick $B = \{ \text{upper triangular matrices in} \ SL(n,\mathbb{C}) \} $ and then $T$ will be $\{ \text{diagonal matrices in} \ SL(2,\mathbb{C}) \}$.
By Iwasawa decomposition, the inclusion $G \hookrightarrow G_\mathbb{C}$ induces diffeomorphism $G/T \cong G_\mathbb{C}/ B \cong \mathcal{B}$ and hence $T^\ast (G/T) \cong T^\ast (G_\mathbb{C}/B)$. Denote $\mathfrak{g}_\mathbb{C}$, $\mathfrak{g}$ the Lie algebra of $G_\mathbb{C}$, $G$ and similar notations hold for other Lie groups and Lie algebras.
By general symplectic geometry argument, there're moment maps $$\mu_\mathbb{C} : T^\ast (G_\mathbb{C}/T) \longrightarrow \mathfrak{g}_\mathbb{C}^\ast, \ \mu : T^\ast (G/B) \longrightarrow \mathfrak{g}^\ast.$$
Standard results shows that $$T^\ast(G_\mathbb{C}/B) \cong G \times_B \mathfrak{b}^\perp \cong G \times_B \mathfrak{n},$$ where $\mathfrak{n} = [\mathfrak{b},\mathfrak{b}]$ and under this identification $\mu_\mathbb{C}(x,n) = Ad_x n$. (c.f. Representation Theory and Complex Geometry by Christ and Ginzburg.) Here we identify $\mathfrak{g}_\mathbb{C}$ and $\mathfrak{g}_\mathbb{C}^\ast$ by the Killing form of $\mathfrak{g}_\mathbb{C}$.
I have two questions, an article I'm reading has a similar identification $T^\ast(G/T) \cong G \times_T \mathfrak{t}^\perp \cong G \times_T i \mathfrak{t}$. Which identification could be this so that $\mathfrak{t}^\perp \cong i \mathfrak{t}$? Also, how can I see that $\mu$ is the same as $\mu_\mathbb{C}$ composite with $n \mapsto n + n^\ast$ in $SL(n,\mathbb{C})$ case and what will be the relation between $\mu$ and $\mu_\mathbb{C}$ in general?