Strict quantization of coadjoint orbits

We obtain a family of strict $\hat G$-invariant products on the space of holomorphic functions on a semisimple coadjoint orbit of a complex connected semisimple Lie group $\hat G$. By restriction, we also obtain strict $G$-invariant products $*_\hbar$ on a space $A(O)$ of certain analytic functions on a semisimple coadjoint orbit $O$ of a real connected semisimple Lie group $G$. The space $A(O)$ endowed with one of the products $*_\hbar$ is a Fr\'echet algebra, and the formal expansion of the products around $\hbar = 0$ determines a formal deformation quantization of $O$, which is of Wick type if $G$ is compact. We study a generalization of a Wick rotation, which provides isomorphisms between the quantizations obtained for different real orbits with the same complexification. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules, and complex analytic results on the extension of holomorphic functions.