Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

Abstract

Let $(g,k)$ be a complex quaternionic symmetric pair with $k$ having an ideal $\mathrm{sl}(2,ℂ)$, $k=\mathrm{sl}(2,ℂ)+m_c$. Consider the representation $S^m\left({ℂ}^2\right)={ℂ}^{m+1}$ of $k$ via the projection $k\to \mathrm{sl}(2,ℂ)$ onto the ideal $\mathrm{sl}(2,ℂ)$. We study the finite dimensional irreducible representations $V\left(\lambda \right)$ of $g$ which contain $S^m\left({ℂ}^2\right)$ under $k\subseteq g$. We give a characterization of all such representations $V\left(\lambda \right)$ and find the corresponding multiplicity, the dimension of $Hom(V\left(\lambda \right){|}_k,S^m\left({ℂ}^2\right)).$ We consider also the branching problem of $V\left(\lambda \right)$ under $l=u{\left(1\right)}_{ℂ}+m_c\subset k$ and find the multiplicities. Geometrically the Lie subalgebra $l\subset k$ defines a twistor space over the compact symmetric space of the compact real form $G_u$ of $G_{ℂ}$, $\text{Lie}\left(G_{ℂ}\right)=g$, and our results give the decomposition for the $L^2$-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.