Are the quaternionic Grassmannians quaternionic Kaehler manifolds? The complex Grassmannians $\mathrm{Gr}(n,r)$, of $r$-planes in $\mathbb{C}^n$ are Kaehler manifolds. What about the quaternionic Grassmannians of $r$-planes in $\mathbb{H}^n$ are they quaternionic Kaehler manifolds?
 A: Perhaps the OP really wants to know why quaternionic Grassmannians other than the quaternionic projective spaces are not considered to be 'quaternion-Kähler'.
The reason goes back to Berger's classification of the possible holonomy groups of Riemannian manifolds:
In the 1920s, Élie Cartan classified the irreducible Riemannian symmetric spaces, of which the quaternionic Grassmannians are examples, and computed the holonomy of these spaces.
In 1929, Schouten and his student van Dantzig pubished a paper on the metrics in $2n$-dimensions admitting a parallel complex structure (which need not be locally symmetric spaces), whose holonomy was therefore contained in $\mathrm{U}(n)\subset\mathrm{O}(2n)$, but it was Erich Kähler whose 1933 paper on these metrics popularized them, and hence these metrics  are now known as Kähler metrics.
In 1954, Marcel Berger set himself the task of determining the possible holonomies of irreducible (i.e., non-product) Riemannian metrics.  He showed that, other than the cases of irreducible Riemannian locally symmetric metrics (which were all known), there were only a few possibilities for the identity component $H\subset\mathrm{SO}(n)$ of the holonomy.  Up to conjugacy, $H$ had to be one of $\mathrm{SO}(n)$, $\mathrm{U}(n/2)$, $\mathrm{SU}(n/2)$, $\mathrm{Sp}(n/4)$, $\mathrm{Sp}(n/4){\cdot}\mathrm{Sp}(1)$, $\mathrm{G}_2$ (when $n=7$), $\mathrm{Spin}(7)$ (when $n=8$), or $\mathrm{Spin}(9)$ (when $n=16$).
The cases $H\simeq\mathrm{U}(n/2)\subset\mathrm{SO}(n)$ ($n$ has to be even) were already known as Kähler metrics, and, eventually, analogous names were proposed for several of the other cases (by Eugenio Calabi, I think).  In particular, quaternion-Kähler was proposed for metrics in $4n$-dimensions with holonomy contained in $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$ (but not contained in $\mathrm{Sp}(n)$), and this nomenclature was widely adopted.
The quaternionic projective spaces $\mathbb{HP}^n$ do have holonomy satisfying this condition, but the other quaternionic Grassmanians do not. (For example, the quaternionic Grassmannian of quaternion $2$-planes in $\mathbb{H}^4$ is $\mathrm{Sp(4)}/\bigl(\mathrm{Sp}(2){\times}\mathrm{Sp}(2)\bigr)$ and the $\mathrm{Sp(4)}$-invariant metric on it has holonomy $\mathrm{Sp}(2){\cdot}\mathrm{Sp}(2)\subset\mathrm{SO}(16)$, which is not conjugate to a subgroup of $\mathrm{Sp}(4){\cdot}\mathrm{Sp}(1)$.)  Hence, they are not considered to be quaternion-Kähler.
