Paraphrasing from Cortes' [notes](https://www2.math.hu-berlin.de/gradkoll/Cortes_vorlesung1_handout.pdf):



> 
> 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?



Homogeneous Quaternionic-Kähler Structure of the Grassmannians?
These spaces are [Wolf spaces](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space) 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 $$Q=G\times\_{U(p)\cdot U(2)}(\mathbb R\otimes\_{\mathbb R} su(2))$$


