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In this paper, we give the expressions for the bounded spherical functions,or equivalently the spherical functions of positive type, for the free two-stepnilpotent Lie groups endowed with the actions of orthogonal groups or theirspecial subgroups. Next we deduce some results about the (Kohn) sub-Laplacian,and we compute the radial Plancherel measure.

The bounded spherical functions for the free two step nilpotent Lie group