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

Fischer, Veronique; Ricci, Fulvio

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

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Description

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 the 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 two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.
28 pages