Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

Let be a complex quaternionic symmetric pair with having an ideal , . Consider the representation of via the projection onto the ideal . We study the finite dimensional irreducible representations of which contain under . We give a characterization of all such representations and find the corresponding multiplicity, the dimension of We consider also the branching problem of under and find the multiplicities. Geometrically the Lie subalgebra defines a twistor space over the compact symmetric space of the compact real form of , , and our results give the decomposition for the -spaces of sections of certain vector bundles over the symmetric space and line bundles over the twistor space. This generalizes Cartan–Helgason’s theorem for symmetric spaces and Schlichtkrull’s theorem for Hermitian symmetric spaces where one-dimensional representations of are considered.