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 $(K\ltimes N,K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space $\bR^d$. The identification of the spherical transforms of $K$-invariant Schwartz functions on $N$ with the restrictions to the spectrum of Schwartz functions on $\bR^d$ 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 $K={\rm SO}_3$ \cite{ADR1, ADR2, FiRi}. 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 $\fn$ of $N$ for all Gelfand pairs $(K\ltimes N,K)$ in Vinberg's list \cite{V2, Y2}.