Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I: rank-one actions on the centre

The spectrum of a Gelfand pair of the form ${(Kltimes N,K)}$ , where N is a nilpotent group, can be embedded in a Euclidean space ${{mathbb R}^d}$ . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on ${{mathbb R}^d}$ has been proved already when N is a Heisenberg group and in the case where N?=?N _3,2 is the free two-step nilpotent Lie group with three generators, with K?=?SO₃ (Astengo et?al. in J Funct Anal 251:772–791, 2007; Astengo et?al. in J Funct Anal 256:1565–1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143–2168, 2009). We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra ${{mathfrak n}}$ of N for all Gelfand pairs ${(Kltimes N,K)}$ in Vinberg’s list (Vinberg in Trans Moscow Math Soc 64:47–80, 2003; Yakimova in Transform Groups 11:305–335, 2006).