Symmetric pairs, Gelfand pairs and pairs such that generating functions of zonal spherical functions define unitary operators—the case of the compact type

A new notion which is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair can be defined by generating functions. In the case of the compact type, this statement is shown. This notion also concerns an existence of a “good” parametrization for the spherical representations. These mean that generating functions can be dealt with systematically in the compact case.