Nilpotent Gelfand Pairs and Spherical Transforms of Schwartz Functions I. Rank-One Actions on the Centre

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The spectrum of a Gelfand pair of the form (K lx N, K), where N is a nilpotent group, can be embedded in a Euclidean space Rd. The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on Rd has been proved already when N is a Heisenberg group and in the case where N = N3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11]. 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 n of N for all Gelfand pairs (K lx N, K) in Vinberg's list [27, 30]. (The references numbers refers to the bibliography at the end of the article)