User:TakuyaMurata/Isotropy representation

Let X be a complex manifold and G a complex reductive Lie group acting on it as diffeomorphisms. Differentiating the orbit map $$\sigma_x: G \to X, x \mapsto g x$$ at the identity gives a map $$\mathfrak{g} \to T_x X$$ still denoted by $$\sigma_x$$. Now what?