Positive definite spherical functions on Ol'shanskii domains

Let G be a simply connected complex Lie group with Lie algebra g, h a real form of g, and H the analytic subgroup of G corresponding to h. The symmetr ic space M = H\G together with a G-invariant partial order less than or equ al to is referred to as an Ol'shanskii space. In a previous paper we constr ucted a family of integral spherical functions phi(mu) on the positive doma in M+ := {Hx: Hx greater than or equal to H} of M. In this paper we determi ne all of those spherical functions on M+ which are positive definite in a certain sense.