Symplectic reduction of Sasakian manifolds

When a complex semisimple group $G$ acts holomorphically on a K\"ahler manifold $(X,\omega)$ such that a maximal compact subgroup $K\subset G$ preserves the symplectic form $\omega$, a basic result of symplectic geometry says that the corresponding categorical quotient $X/G$ can be identified with quotient of the zero-set of the moment map by the action of $K$. We extend this to the context of a semisimple group acting on a Sasakian manifold.