We introduce and study invariant differential operators acting on the space \(\cal{H}(\Omega)\) of holomorphic functions on the complement \(\Omega=\{(z,w)\in \hat{\mathbb{C}}^{2}:z\cdot w \ne 1 \}\) of the “complexified unit circle” \(\{(z,w)\in \hat{\mathbb{C}}^{2}:z\cdot w = 1 \}\) . We obtain recursion identities, describe the behaviour under change of coordinates and find the generators of the corresponding operator algebra. We illustrate how this provides a unified framework for investigating conformally invariant differential operators on the unit disk \(\mathbb{D}\) and the Riemann sphere \(\hat{\mathbb{C}}\) , which have been studied by Peschl, Aharonov, Minda and many others, within their conjecturally natural habitat. We apply the machinery to a problem in deformation quantization by deriving explicit formulas for the canonical Wick-type star products on Ω, the unit disk \(\mathbb{D}\) and the Riemann sphere \(\hat{\mathbb{C}}\) in terms of such invariant differential operators. These formulas are given in the form of factorial series which depend holomorphically on a complex deformation parameter \(\bar{h}\) and lead to asymptotic expansions of the star products in powers of \(\bar{h}\) .
