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Let $(\mathfrak{g}, \mathfrak{k})$ be a complex quaternionic symmetric pairwith $\mathfrak{k}$ having an ideal $\mathfrak{sl}(2, \mathbb{C})$,$\mathfrak{k}=\mathfrak{sl}(2, \mathbb{C})+\mathfrak{m}_c$. Consider therepresentation $S^m(\mathbb{C}^2)=\mathbb{C}^{m+1}$ of $\mathfrak{k}$ via theprojection onto the ideal $\mathfrak{k}\to \mathfrak{sl}(2, \mathbb{C})$. Westudy the finite dimensional irreducible representations $V(\lambda)$ of$\mathfrak{g}$ which contain $S^m(\mathbb{C}^2)$ under $\mathfrak{k}\subseteq\mathfrak{g}$. We give a characterization of all such representations$V(\lambda)$ and find the corresponding multiplicity $m(\lambda,m)=\dim\operatorname{Hom} (V(\lambda)|_\mathfrak{k},S^m(\mathbb{C}^2)).$ We consideralso the branching problem of $V(\lambda)$ under$\mathfrak{l}=\mathfrak{u}(1)_{\mathbb{C}} + \mathfrak{m}_c\subseteq\mathfrak{k}$ and find the multiplicities. Geometrically the Lie subalgebra$\mathfrak{l}\subseteq \mathfrak{k}$ defines a twistor space over the compactsymmetric space of the compact real form $G_c$ of $G_{\mathbb{C}}$,$\text{Lie}(G_{\mathbb{C}})=\mathfrak{g}$, and our results give thedecomposition for the $L^2$-spaces of sections of certain vector bundles overthe symmetric space and line bundles over the twistor space. This generalizesCartan--Helgason's theorem for symmetric spaces $(\mathfrak{g}, \mathfrak{k})$and Schlichtkrull's theorem for Hermitian symmetric spaces whereone-dimensional representations of $\mathfrak{k}$ are considered.

Cartan--Helgason theorem for quaternionic symmetric and twistor spaces