Integrability of quaternion-K\"ahler symmetric spaces

TL;DR

This paper establishes a necessary condition for Lie group actions by quaternionic automorphisms on integrable quaternionic manifolds and classifies symmetric spaces with invariant integrable quaternionic structures.

Contribution

It introduces a representation-based necessary condition for such actions and characterizes all Riemannian symmetric spaces of dimension 4n with invariant integrable quaternionic structures.

Findings

Invariant integrable quaternionic structures exist only on specific symmetric spaces.

Classified symmetric spaces with such structures as quaternionic vector space, hyperbolic space, or projective space.

Provided a representation-theoretic criterion for quaternionic automorphism actions.

Abstract

We find a necessary condition for the existence of an action of a Lie group $G$ by quaternionic automorphisms on an integrable quaternionic manifold in terms of representations of $\mathfrak{g}$. We check this condition and prove that a Riemannian symmetric space of dimension $4n$ for $n\geq 2$ has an invariant integrable almost quaternionic structure if and only if it is quaternionic vector space, quaternionic hyperbolic space or quaternionic projective space.