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 $F\in \mathfrak{U}_{\delta}^{1}(N)$.