Abstract: \font\msbm=msbm10 \def\bbR{\hbox{\msbm R}} \def\bbC{\hbox{\msbm C}} We construct star-products on the co-adjoint orbit of the Lie group $ Aff(\bbC)$ of affine transformations of the complex line and apply them to obtain the irreducible unitary representations of this group. These results show the effectiveness of the Fedosov quantization even for groups which are neither nilpotent nor exponential. Together with the result for the group $ Aff(\bbR)$ (see [DH]), we thus have a description of quantum $\overline MD$ co-adjoint orbits. \item{[DH]} Do Ngoc Diep; Nguyen Viet Hai: Quantum half-plane via Deformation Quantization. Math. QA/9905002, 2 May 1999.
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