Latent Quaternionic Geometry

Abstract

We discuss the interaction between the geometry of a quaternion-K\"{a}hler manifold $M$ and that of the Grassmannian $\mathbb{G}_3(\mathfrak{g})$ of oriented $3$-dimensional subspaces of a compact Lie algebra $\mathfrak{g}$. This interplay is described mainly through the moment mapping induced by the action of a group $G$ of quaternionic isometries on $M$. We give an alternative expression for the imaginary quaternionic endomorphisms $I,J,K$ in terms of the structure of the Grassmannian's tangent space. This relies on a correspondence between the solutions of respective twistor-type equations on $M$ and $\mathbb{G}_3(\mathfrak{g})$.