Cartan--Helgason theorem for quaternionic symmetric and twistor spaces

Abstract

Let $(\mathfrak{g},\mathfrak{k})$ be a complex quaternionic symmetric pair with $\mathfrak{k}$ having an ideal $\mathfrak{sl}(2,\mathbb{C})$, $\mathfrak{k}=\mathfrak{sl}(2,\mathbb{C})+{\mathfrak{m}}_c$. Consider the representation $S^m({\mathbb{C}}^2)={\mathbb{C}}^{m+1}$ of $\mathfrak{k}$ via the projection onto the ideal $\mathfrak{k}\to \mathfrak{sl}(2,\mathbb{C})$. We study 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 $m(\lambda ,m)=\dim \mathrm{Hom}(V(\lambda ){|}_{\mathfrak{k}},S^m({\mathbb{C}}^2)).$ We consider also 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 compact symmetric space of the compact real form $G_c$ of $G_{\mathbb{C}}$, $\text{Lie}(G_{\mathbb{C}})=\mathfrak{g}$, 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 $(\mathfrak{g},\mathfrak{k})$ and Schlichtkrull's theorem for Hermitian symmetric spaces where one-dimensional representations of $\mathfrak{k}$ are considered.

updated version, correcting some inaccurate arguments in the proof of Proposition A1 in the earlier version