On lie algebras of K-invariant functions

Abstract

Let $G$ be a locally compact group and let $K$ be a compact subgroup of $Aut(G)$, the group of automorphisms of $G$. $(G,K)$ is a Gelfand pair if the algebra $L_{K}^{1}(G)$ of K-invariant integrable functions on $G$ is commutative under convolution. In this paper, we give some charactezations of this algebra in the nilpotent case, which generalize some results obtained by C. Benson, J. Jenkins, G. Ratcliff in [1] and obtain a new criterion for Gelfand pairs.