Characterization of the Gelfand pair: Quasi Gelfand pair

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.