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?


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.


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