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} $ ?

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