Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds

A very natural Dirac operator for Wolf spaces is discussed in Köhler, K, Weingart, G., Quaternionic analytic torsion, Adv. Math. 178 (2003), 375–395. It acts on subcomplexes of the de Rham complex, in analogy with the Kähler situation.

EDIT: There is a hierarchy of groups $$Spin(n)\hookrightarrow Spin^c(n)=Spin(n)\cdot U(1)\hookrightarrow Spin^h(n)=Spin(n)\cdot Sp(1)\twoheadrightarrow SO(n)\;.$$ Here "$\cdot$" means "product divided by the diagonal $\mathbb Z/2$ action". If there is a principal bundle $P$ with structure group $Spin^x$ together with an equivariant map to the $SO(n)$ frame bundle of an oriented Riemannian manifold, then $P$ is called a spin$^x$ structure ($x$ being either void, or $c$, or $h$). Associated to $P$ is a real, complex, or quaternionic spinor bundle, respectively.

Every QK manifold comes with a natural spin$^h$ structure. For the Wolf spaces, these are automatically equivariant, see this answer. One of the Dirac operators in [op. cit.] acts on the corresponding quaternionic spinor bundles. Some Wolf spaces are spin, e.g. $\mathbb HP^k$, $G_2(\mathbb C^n)$ or $G_4(\mathbb R^n)$, the last two for $n$ even. Some are spin$^c$ but not spin, e.g. $G_2(\mathbb C^n)$ , and some are not even spin$^c$, e.g. $G_4(\mathbb R^n)$ for odd $n$, if I am not mistaken.

It is a result of Salamon that $8n$-dimensional quaternion-Kähler manifolds are spin.