Abstract
We develop a unified geometric framework for formal deformation quantization of coadjoint orbits of a Lie group $ G $ with Lie algebra $ \mathfrak{g} $, in which the \emph{internal} deformation parameters themselves range over the unitary dual $ \widehat{G} $. For each coadjoint orbit $ \mathcal{O}_{\lambda_0} \subset \mathfrak{g}^* $, the dual vector space of the Lie algebra $ \mathfrak{g} $, one constructs a Fedosov-type flat connection on the analogue of the single-parameter Weyl algebra bundle
\[D = \nabla - \delta + \sum_{I=1}^N \frac{1}{\lambda_I} \operatorname{ad}(r_I)\]where $ N = \dim G - \dim \mathcal{O}_{\lambda_0}$
and the formal variables $ \lambda = (\lambda_1, \dots, \lambda_N) \in \widehat{G} $ label transverse directions to the orbit. We call the underlying bundle a $ \mathfrak{g}[[\lambda]] $-bundle. Imposing holonomy quantization on the compactified parameter torus forces $ \lambda $ to lie in the discrete set of actual unitary parameters, so that the resulting star product algebra $ C^\infty(\mathcal{O}_{\lambda_0})[[\lambda ]], *\lambda $ specializes—without additional projection—to the matrix algebra of the corresponding genuine irreducible representation of $ G $.
At the same time, the Fedosov twist \(\mathcal{F} = \exp\left( -\sum_{I=1}^N \frac{1}{\lambda_I} \operatorname{ad}(r_I) \right)\) deforms the coproduct of $ U(\mathfrak{g}) $ into a multiparameter quantum group $ U_\lambda(\mathfrak{g}) $, and one checks that $ C^\infty(\mathcal{O}_{\lambda_0})[ [ \lambda ] ]$
becomes a natural $ U_\lambda(\mathfrak{g}) $-module algebra. Our construction thus marries three pillars—symplectic geometry, Hopf-algebra deformation, and exact matching to the group’s unirreps—into a single, coherent deformation-quantization picture.