Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

Abstract

Let (g,k) be a complex quaternionic symmetric pair with k having an ideal sl(2,ℂ), k=sl(2,ℂ)+mc. Consider the representation Sm(ℂ2)=ℂm+1 of k via the projection k→sl(2,ℂ) onto the ideal sl(2,ℂ). We study the finite dimensional irreducible representations V(λ) of g which contain Sm(ℂ2) under k⊆g. We give a characterization of all such representations V(λ) and find the corresponding multiplicity, the dimension of Hom(V(λ)|k,Sm(ℂ2)). We consider also the branching problem of V(λ) under l=u(1)ℂ+mc⊂k and find the multiplicities. Geometrically the Lie subalgebra l⊂k defines a twistor space over the compact symmetric space of the compact real form Gu of Gℂ, Lie(Gℂ)=g, and our results give the decomposition for the L2-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 (g,k) and Schlichtkrull’s theorem for Hermitian symmetric spaces where one-dimensional representations of k are considered.