Spectral synthesis of the invariant Laplacian and complexified spherical harmonics

TL;DR

This paper constructs an orthogonal basis of eigenfunctions for the Laplacian on a complex domain, linking holomorphic functions to classical spherical harmonics and revealing new invariance properties.

Contribution

It introduces a Schauder basis of eigenfunctions for the Laplacian on a complex domain and connects these to classical spherical harmonics via biholomorphic mapping.

Findings

Identification of the classical spherical harmonics as restrictions of holomorphic functions in the complex domain.

Establishment of invariance of zonal harmonics under automorphisms induced by M"obius transformations.

Proof of a combinatorial identity using hypergeometric functions.

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 the 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 M\"obius 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.