It's all in the title: Are Wolf spaces flag manifolds? 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 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?


2 Answers 2


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 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 $\mathrm{SO}(6)/\mathrm{SO}(2)\mathrm{SO}(4)$.


Let me just remark that there is a 1-1 correspondence between compact connected homogeneous complex contact manifolds and compact simply connected quaternionic symmetric spaces. It is actually described in the original paper by Wolf, theorem 6.1. The correspondence is given by bundles with fibers being 2-spheres:

$$ G/L\cdot S^1 \to G/L\cdot Sp(1) $$


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