Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond

Abstract

We introduce and study invariant differential operators acting on the space $\mathcal{H}(\Omega )$ 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 $\Omega$, the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$ in terms of such invariant differential operators. These formulas are given in form of factorial series which depend holomorphically on a complex deformation parameter $\hslash$ and lead to asymptotic expansions of the star products in powers of $\hslash$.

Dedicated to David Minda; final version to appear in Journal d'Analyse Mathématique