Spectral synthesis of the invariant Laplacian and complexified spherical harmonics

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