Quantum Co-Adjoint Orbits of the Group of Affine Transformations of the Complex Straight Line

We construct start-products on the co-adjoint orbit of Lie group $\Aff({\bf C})$ of affine transformations of the complex straight line and apply them to obtain the irreducible unitary representations of this group. These results show effectiveness of the Fedosov quantization even for groups which are neither nilpotent nor exponential. Together with the result for the group $\Aff({\bf R})$ in math.QA/9905002, we have thus a description of quantum $\bar{MD}$ co-adjoint orbits.