On Inductive Limit of Commutative Triples

Abstract

In this paper, we extend Olshanski's work on Gelfand pairs  to commutative triples. We introduce the notion of spherical triples as generalization of commutative triples. We prove that inductive limit of  increasing sequence of commutative triples is a spherical triple which shows that the former is also a generalization of spherical pairs. Furthermore we define spherical functions associated with this spherical triples. At end, we characterize this spherical functions by a functionnal equation and we give some of its properties.