Let $M$ be a quaternionic-Kähler manifold, with fundamental form $\omega$, and let $L$ be the Lefschetz operator of $\omega$. In the Kähler and, more generally, symplectic cases, there is a mysterious relation between the Lefschetz $\frak{sl}_2$-decomposition and the Hodge operator due to Weil (see Huybrechts 1.2.31)

Does there exist a quaternionic-Kähler analogue?



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