On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the space $SU(m+1)/T^{m}$? Does there exist more than one almost complex structure, and if so, can we classify them?



Equivariant Almost Complex Structures on the Full Flag Manifolds
Actually, though this may seem pedantic, there are *two* almost-complex structures on $\mathbb{CP}^m$ that are invariant under $\mathrm{SU}(m{+}1)$, namely the 'standard' one and its conjugate. Of course, they are equivalent, but only by using an *outer* automorphism of $\mathrm{SU}(m{+}1)$.


The reason this is of more than pedantic interest is that, when you go to $\mathrm{SU}(m{+}1)/T^m$, what you find is that there are a very large number of $\mathrm{SU}(m{+}1)$-invariant almost-complex structures on this flag manifold. In fact, the number is $2^{m(m+1)/2}$, where I am counting conjugate pairs as distinct (otherwise, you should divide this number by $2$). 


The majority of these almost-complex structures are not integrable, though, and, for large $m$ there is a very large number of diffeomorphism-inequivalent ones. The number of *integrable* ones turns out to be the number of ways to split the roots (defined relative to the maximal torus $T^m$) into positive and negative subsets. The integrable ones corresponding to each such choice are distinct, but, up to an (possibly outer) automorphism of $\mathrm{SU}(m{+}1)$, they are all equivalent.


For more detail, I might suggest that you look at the final section of my paper *Lie groups and twistor spaces*, Duke Mathematical Journal **52** (1985), 223–261.


