Let $G$ be a compact connected Lie group. We denote by $\mathfrak{g}$ the Lie algebra of $G$ and by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $\mathcal{O}_r: = G\cdot r$ be a generic coadjoint orbit of $G$.
The coadjoint orbit $\mathcal{O}_r$ endowed with the Kirillov–Kostant–Souriau $\omega$ is a symplectic manifold. I've read that it is also a Kähler manifold; meaning that there exists a unique almost complex structure $J$ on $\mathcal{O}_r$ which is compatible with $\omega$ and such that the form $g(\cdot,\cdot):= \omega(\cdot,J\cdot )$ is a Riemannian metric on $\mathcal{O}_r$.
Given an element $\beta \in \mathcal{O}_r $, then the tangent space of $\mathcal{O}_r$ at $\beta$ is $T_\beta \mathcal{O}_r = \lbrace \xi_{\mathcal{O}_r}(\beta), \xi \in \mathfrak{g}\rbrace$ , where $\xi_{\mathcal{O}_r}(\beta) = \frac{d}{dt}\rvert_ {t=0} e^{-t \xi}\cdot\beta$.
What is $J(\xi_{\mathcal{O}_r}(\beta) )$, $\xi \in \mathfrak{g} $ ?
Best Answer
Put $T = G_r$. We may, and do, assume that $\beta = r$, and simply describe a $T$-invariant complex structure on $\operatorname T_{\mathcal O_r}(r)$.
Instead of having one 1-dimensional subspace of $\mathfrak g$ for every root $\alpha$, we get a $2$-dimensional subspace $\{X_\alpha + \overline{X_\alpha} \mathrel: X_\alpha \in (\mathfrak g_{\mathbb C})_\alpha\}$ for every pair of roots $\{\alpha, \overline\alpha = -\alpha\}$.
Our Kähler structure treats this $2$-dimensional space, which I will provocatively call $\mathfrak g_{\pm\alpha}$ since its complexification is $(\mathfrak g_{\mathbb C})_\alpha \oplus (\mathfrak g_{\mathbb C})_{-\alpha}$, as a $\mathbb C$-vector space via the (isomorphic) projection to $(\mathfrak g_{\mathbb C})_\alpha$, and then rotates by $i$—but we must choose $i$ appropriately to get a negative definite metric. After our discussion in the comments, I think I have finally cleaned up the relevant signs.
Fix a root $\alpha$ of $T$ in $\mathfrak g_{\mathbb C}$. Put $i_\alpha = -\lambda\lvert\lambda\rvert^{-1}$, where $\lambda = r(\mathrm d\alpha^\vee(1))$ ($H_\alpha \mathrel{:=} \mathrm d\alpha^\vee(1)$ is sometimes called the coroot, but I prefer to reserve that terminology for $\alpha^\vee$ itself), so that $i_\alpha$ is a square root of $-1$. Then $J$ carries $\xi_{\mathcal O_r}(r)$, where $\xi = X_\alpha + \overline{X_\alpha}$, to $\xi'_{\mathcal O_r}(r)$, where $\xi' = i_\alpha(X_\alpha - \overline{X_\alpha})$, for every $X_\alpha \in (\mathfrak g_{\mathbb C})_\alpha$.