Differentiability of quantum moment maps

We study quantum moment maps of $G$-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a $G$-invariant star product is differentiable. This property gives us a new method for the classification of $G$-invariant star products on regular coadjoint orbits of compact semisimple Lie groups.