Cartan--Helgason theorem for quaternionic symmetric and twistor spaces

TL;DR

This paper extends the Cartan--Helgason theorem to quaternionic symmetric and twistor spaces, characterizing certain irreducible representations and their multiplicities, with geometric applications to vector bundles and line bundles over symmetric and twistor spaces.

Contribution

It provides a new characterization of irreducible representations containing specific $ ext{sl}(2, ext{C})$-modules and analyzes their branching rules, generalizing classical theorems to quaternionic settings.

Findings

Characterization of irreducible representations containing $S^m( ext{C}^2)$

Explicit multiplicity formulas for these representations

Decomposition results for $L^2$-spaces over 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 $m(\lambda,m)=\dim \operatorname{Hom} (V(\lambda)|_\mathfrak{k},S^m(\mathbb{C}^2)).$ 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.