Hypergroups, Symmetric Spaces, and Wrapping Maps

This paper attempts an exposition of the connection between symmetric spaces and hypergroup theory. This connection is not new to those involved in hypergroups, as the classical symmetric spaces have always provided a very rich source of examples for the theory. However for those working in symmetric spaces, the usefulness of the hypergroup point of view may not be well known. It provides a new way of looking at some of the fundamental aspects of harmonic analysis on these spaces. In addition the language of hypergroups forces us to come to grips with concrete formulae for convolutions (often involving special functions) which the general theory often allows us to finesse. On the other hand, the rich and explicit structures underlying Lie theory allow us to write down explicit convolution structures for certain hypergroups that appear which suggest new directions for hypergroup theory as well.