Positive definite spherical functions on Ol$'$shanskii domains

Let $G$ be a simply connected complex Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{h}$ a real form of $\mathfrak{g}$, and $H$ the analytic subgroup of $G$ corresponding to $\mathfrak{h}$. The symmetric space ${\mathcal{M}}=H\backslash G$ together with a $G$-invariant partial order $\le$ is referred to as an Ol$'$shanskii space. In a previous paper we constructed a family of integral spherical functions $\phi _{\mu }$ on the positive domain ${\mathcal{M}}^{+} := \{Hx\colon Hx\ge H\}$ of ${\mathcal{M}}$. In this paper we determine all of those spherical functions on ${\mathcal{M}}^{+}$ which are positive definite in a certain sense.