Journal of Lie Theory

We consider a family of Gelfand pairs $(K \ltimes N, N)$ (in short $(K,N)$) where $N$ is a two step nilpotent Lie group, and $K$ is the group of orthogonal automorphisms of $N$. This family has a nice analytic property: almost all these 2-step nilpotent Lie group have square integrable representations. In these cases, following Moore-Wolf's theory, we find an explicit expression for the inversion formula of $N$, and as a consequence, we decompose the regular action of $K \ltimes N$ on $L^{2}(N)$. This explicit expression for the Fourier inversion formula of $N$, specialized to a class of commutative nilmanifolds described by J.\,Lauret, sharpens the analysis of J.\,A.\,Wolf in Section 14.5 in {\it Harmonic Analysis on Commutative Spaces} [Mathematical Surveys and Monographs 142, American Mathematical Society, Providence (2007)], and in {\it On the analytic structure of commutative nilmanifolds} [J. Geometric Analysis 26 (2016) 1011--1022], concerning the regular action of $K \ltimes N$ on $L^2(N)$. When $N$ is the Heisenberg group, we obtain the decomposition of $L^{2}(N)$ under the action of $K \ltimes N$ for all $K$ such that $(K,N)$ is a Gelfand pair. Finally, we also give a parametrization for the generic spherical functions associated to the pair $(K,N)$, and we give an explicit expression for these functions in some cases.