Oberseminar Funktionentheorie, The dark side of the unit disk III: Invariant Differential Operators

We generalize the concept of conformally invariant derivatives introduced by Peschl to the category of smooth functions of several variables on complex manifolds. We analyze the structure of the algebra of these Peschl-Minda-type differential operators, prove a global linearization theorem, and explicitly determine their kernels. As an application we discuss an explicit formula for the star product on Ω, on the unit disk and on the Riemann sphere. We also introduce the concept of dual Peschl-Minda operators and explain their role in analyzing the Möbius-invariant eigenspaces of the Laplace operators on Ω, the unit disk and the Riemann sphere.