Gelfand transforms of SO(3)-invariant Schwartz functions on the free nilpotent group N_{3,2}

The spectrum of a Gelfand pair (K\ltimes N,K), where N is a nilpotent group, can be embedded in a Euclidean space. We prove that, in general, the Schwartz functions on the spectrum are Gelfand transforms of Schwartz K-invariant functions on N. We also show the converse in the case of the Gelfand pair (SO_3\ltimes N_{3,2},SO_3), where N_{3,2} is the free 2-step nilpotent Lie group with 3 generators. This extends recent results for the Heisenberg group.