  Let G be a locally compact group and let K be a compact subgroup of Aut(G),
the group of automorphisms of G. The pair $(G, K )$ is a Gelfand pair if the
algebra $L^{1}_{K}(G)$ of K-invariant integrable functions on G is commutative
under convolution. In \cite{toure2008lie}, the charactezations of this algebra
in the nilpotent case were studied, which generalize some results obtained by
C. Benson, J. Jenkins, G. Ratcliff in \cite{benson1990gel} and obtained a new
criterion for Gelfand pairs. In this paper we describe the spherical function
associated with this type of pair.
