A generalization of the Cartan-Helgason theorem for Riemannian symmetric spaces of rank one

Let $U/K$ be a compact Riemannian symmetric space with $U$ simply connected and $K$ connected. Let $G/K$ be the noncompact dual space, with $G$ and $U$ analytic subgroups of the simply connected complexification $G^{\mathbb{C}}$. Let $G=KAN$ be an Iwasawa decomposition of $G$, and let $M$ be the centralizer of $A$ in $K$. For $\d\in\widehat{U}$, let $\m$ be the highest restricted weight of $\d$, and let $\s$ be the $M$-type acting in the highest restricted weight subspace of $H_{\d}$. Fix a $K$-type $\tau$. In \cite{CAMPO} the following result was proved. Assume $U/K$ has rank one. Then $\d|_K$ contains $\tau$ if and only if 1) $\tau|_M$ contains $\s$ and 2) $\m\in\m_{\s,\tau}+\L_{sph}$, where $\L_{sph}$ is the set of highest restricted spherical weights, and $\m_{\s,\tau}$ is a suitable element of $\mathfrak{a}^{\ast}$ uniquely determined by $\s$ and $\tau$. In this paper we obtain an explicit formula for this element in the case of $U/K=S^n,\,P^n(\mathbb{C}),\,P^n(\mathbb{H})$. This gives a generalization of the Cartan-Helgason theorem to arbitrary $K$-types on these rank one symmetric spaces.