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 1 K ( G ) of K-invariant integrable functions on G is commu-tative 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.