The integral representations of harmonic polynomials in the case of \({\mathfrak su}(p,1)\) (Q1267145)

If \(\mathfrak g\) is a complex reductive Lie algebra and \(\mathfrak g=\mathfrak k+\mathfrak p\) is the complexification of a Cartan decomposition of \(\mathfrak g_{\mathbf R}\), where \(\mathfrak g_{\mathbf R}\) is a noncompact real form of \(\mathfrak g\), \textit{B. Kostant} and \textit{S. Rallis} [Am. J. Math. 93, 753-809 (1971; Zbl 0224.22013)] showed that polynomials on \(\mathfrak p\) are the tensor product of harmonic polynomials and K-invariant polynomials, where \(K=\exp( ad\mathfrak k)\). Classical harmonic functions on \(\mathbf C^p\) correspond to the harmonic functions on \(\mathfrak p\) for the case \(\mathfrak g_{\mathbf R}=\mathfrak s\mathfrak o(p,1)\). Analogous results to the classical ones (stated in an appendix) are obtained in which harmonic polynomials for \(\mathfrak su(p,1)\) are expressed as double integrals on some \(K_{\mathbf R}\)-orbits.