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Let $F(n)$ be a connected and simply connected free 2-step nilpotent liegroup and $K$ be a compact subgroup of Aut($F(n)$). We say that $(K,F(n))$ is aGelfand pair when the set of integrable $K$-invariant functions on $F(n)$ formsan abelian algebra under convolution. In this paper, we consider the case when$K=O(n)$. In this case, the Gelfand space $(O(n),F(n)$ is equipped with theGodement-Plancherel measure, and the spherical transform$\land:L_{O(n)}^{2}(F(n))\rightarrow L^{2}(\Delta (O(n),F(n)))$ is an isometry.I will prove the Gelfand space $\Delta (O(n),F(n))$ is equipped with theGodement-Plancherel measure and the inversion formula. Both of which havesomething related to its correspond Heisenberg group. The main result in thispaper provides a complete characterization of the set $\varphi_{O(n)}(F(n))^{\wedge}$=$\{\widehat{f}\mid f\in \varphi_{O(n)} (F(n))\}$ of sphericaltransforms of $O(n)$-invariant Schwartz functions on $F(n)$. I show that afunction $F$ on $\Delta (O(n),F(n))$ belongs to $\varphi_{O(n)}(F(n))^{\wedge}$ if and only if the functions obtained from $F$ via applicationof certain derivatives and difference operators satisfy decay conditions.

The spherical transform of a Schwartz function on the free two step nilpotent lie group