The integral representations of harmonic polynomials in the case of \({\mathfrak{sp}}(p,1)\) (Q1970880)

Let \({\mathfrak g}\) be a complex reductive Lie algebra and \({\mathfrak g}_\mathbb{R}\) a noncompact real form of \({\mathfrak g}\). Let \({\mathfrak g}_\mathbb{R}={\mathfrak k}_\mathbb{R}+{\mathfrak p}_\mathbb{R}\) be a Cartan decomposition of \({\mathfrak g}_\mathbb{R}\) and \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) the direct sum obtained by complexifying \({\mathfrak k}_\mathbb{R}\) and \({\mathfrak p}_\mathbb{R}\). After recalling some results on the space of harmonic polynomials of degree \(n\) on \(\mathbb{C}^q\), the author obtains explicit integral representation formulas of harmonic polynomials and reproducing kernels of these formulas in the classical real rank one case \({\mathfrak g}_\mathbb{R}={\mathfrak sp} (p,1)\), i.e. harmonic polynomials on \({\mathfrak p}\) for \({\mathfrak g}_\mathbb{R}={\mathfrak sp} (p,1)\) are represented by an integral on some \(K_\mathbb{R}\)-orbits, where \(K_\mathbb{R}= \exp\text{ad} {\mathfrak k}_\mathbb{R}\) acts on the space \({\mathfrak p}\).