Paraphrasing from Cortes' [notes][1]:

> The quaternionic K\"ahler condition for a manifold $M$, means that End$(T(M))$ admits a
parallel subbundle $Q$ which is locally spanned by $3$
anticommuting skew-symm. almost complex  structures.

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


  [1]: https://www2.math.hu-berlin.de/gradkoll/Cortes_vorlesung1_handout.pdf