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Paraphrasing from Cortes' notes:

The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a parallel subbundle $Q$ which is locally spanned by $3$ anticommuting skew-symmetric almost complex structures.

Now the Grassmannians $\operatorname{Gr}[p+2,2]$ are examples of quaternionic Kähler manifolds, and of homogeneous spaces. Is $Q(\operatorname{Gr}[p+2,2])$ an $U(p+2)$-equivariant vector bundle?

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  • $\begingroup$ These spaces are Wolf spaces [en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space]. The subbundle $Q$ is associated to the $SU(2)$ factor of the isotropy group, so yes, it is equivariant. $\endgroup$
    – Sebastian Goette
    Commented Oct 6, 2015 at 15:02
  • $\begingroup$ Thanks a lot for your answer. I now realize I was a little vague with my question, by equivariant I meant w.r.t. SU(p+2) (or U(p+2)). If I correctly understand your comment "The subbundle Q is associated to the SU(2) factor of the isotropy group" as "Q is the bundle corresponding to a rep of the isotropy subgroup of the form triviall rep on the SU(p+2) factor and some non-trivial rep on the SU(2) factor " then I guess this is the case. $\endgroup$
    – Pavel Katzo
    Commented Oct 6, 2015 at 16:12
  • $\begingroup$ Yes, that is what I meant. The isotropy representation on U(p)xU(2) is given by tensoring the standard representations. Because the actions commute, you can view the action of U(p) as a quaternionic action on the product. The bundle Q is associated to the tracefree part of the endomorphisms of C^2, and this is precisely the Lie algebra su(2). So, Q=U(p+2)x_{U(p)U(2)}(C\otimes su(2)). $\endgroup$
    – Sebastian Goette
    Commented Oct 6, 2015 at 18:10
  • $\begingroup$ Thanks a lot, please put you answer so I can vote it up. $\endgroup$
    – Pavel Katzo
    Commented Oct 6, 2015 at 19:10

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These spaces are Wolf spaces of the form $G/K$, where $K$ or a double cover equals $H\times SU(2)$. Here, $G=SU(p+2)$ and $H=U(p)$. The isotropy representation here is the exterior tensor product of some $H$-representation with the standard representation of $SU(2)$ (in this special case, $U(p)$ acts by its standard representation as well). The action of $H$ on this tensor product can thus be regarded as quaternionic. The bundle $Q$ is associated to the tracefree antiselfadjoint endomorphisms of the standard representation, hence to the Lie algebra $su(2)$. It can be written as a real $SU(p+2)$-equivariant vector bundle $$Q=G\times_{U(p)\cdot U(2)}(\mathbb R\otimes_{\mathbb R} su(2))$$

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