A PARAMETRIZATION OF δ-SPHERICAL FUNCTIONS ON COMMUTATIVE TRIPLES ASSOCIATED WITH NILPOTENT LIE GROUPS

Abstract

Let $N$ be a connected and simply connected nilpotent Lie group, $K$ be a compact subgroup of $Aut(N)$, the group of automorphisms of $N$ and $\\delta$ be a class of unitary irreducible representations of $K$. The triple $(N,K,\\delta)$ is a commutative triple if the convolution algebra $\\mathfrak{U}_{\\delta}^{1}(N)$ of $\\delta$-radial integrable functions is commutative. In this paper, we obtain first a parametrization of $\\delta$ spherical functions by means of the unitary dual $\\widehat{N}$ and then an inversion formula for the spherical transform of $F\\in \\mathfrak{U}_{\\delta}^{1}(N)$.