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

TL;DR

This paper derives explicit formulas for bounded spherical functions on free two-step nilpotent Lie groups, analyzes the sub-Laplacian, and computes the radial Plancherel measure, advancing harmonic analysis on these groups.

Contribution

It provides explicit expressions for spherical functions and computes the Plancherel measure for free two-step nilpotent Lie groups, which was previously not fully characterized.

Findings

Explicit formulas for bounded spherical functions.

Results on the (Kohn) sub-Laplacian.

Calculation of the radial Plancherel measure.

Abstract

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