Spectral synthesis of the invariant Laplacian and complexified spherical harmonics

Abstract

We show that the space $\mathcal{H}(Ω)$ of holomorphic functions $F:Ω\to\mathbb{C}$, where ${Ω=\{(z,w)\in\widehat{\mathbb{C}}^2\,:\, z\cdot w\neq 1\}}$, possesses an orthogonal Schauder basis consisting of distinguished eigenfunctions of the canonical Laplacian on $Ω$. Mapping $Ω$ biholomorphically onto the complex two-sphere, we use the Schauder basis result in order to identify the classical three-dimensional spherical harmonics as restrictions of the elements in $\mathcal{H}(Ω)$ to the real two-sphere analogue in $Ω$. In particular, we show that the zonal harmonics correspond to those functions in $\mathcal{H}(Ω)$ that are invariant under automorphisms of $Ω$ induced by Möbius transformations. The proof of the Schauder basis result is based on a curious combinatorial identity which we prove with the help of generalized hypergeometric functions.