Spectral synthesis of the invariant Laplacian and complexified spherical harmonics
Abstract
We show that the space $\mathcal {H}(\Omega )$ of holomorphic functions $F:\Omega \to \mathbb C$, where $\Omega =\{(z,w)\in \widehat {\mathbb C}^2: z\cdot w\neq 1\}$, possesses an orthogonal Schauder basis consisting of distinguished eigenfunctions of the canonical Laplacian on $\Omega $. Mapping $\Omega $ 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 elements in $\mathcal {H}(\Omega )$ to the real two-sphere analogue in $\Omega $. In particular, we show that the zonal harmonics correspond to those functions in $\mathcal {H}(\Omega )$ that are invariant under automorphisms of $\Omega $ induced by Möbius transformations. The proof of the Schauder basis result is based on an involved combinatorial identity for binomial coefficients which we prove with the aid of generalized hypergeometric functions, and which could be of independent interest.
