It's all in the title: **Are [Wolf spaces](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space) [flag manifolds](https://en.wikipedia.org/wiki/Generalized_flag_variety)**? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general case. However, I recently read this [post](https://mathoverflow.net/questions/52396/are-there-topological-restrictions-to-the-existence-of-almost-quaternionic-struc) by Moroianu, where he says that 



> 
> the quaternionic projective spaces $ℍP\_n$ are quaternion-Kähler, but have no almost complex structure
> 
> 
> 


I now think my assumption is probably false. Can someone please help "de-confuse" me? 



Are Wolf spaces flag manifolds?
Flag manifolds have the form $G/C(S)$ where $C(S)$ is the centralizer (in $G$) of its center $S$ (a torus).


Most $G/H$ in the [list](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space) don't have this form, for $H$ has discrete center in each case except the complex Grassmannians $\mathrm{SU}(p+2)/\mathrm S(\mathrm{U}(p)\times\mathrm{U}(2))$ and $\mathrm{SO}(6)/\mathrm{SO}(2)\mathrm{SO}(4)$.


