There is a realisation of the canonical fibrations of flag manifolds that serves to introduce a twistor space. For this, assume that G is of adjoint type (i.e. has trivial centre) and let ΩG denote the infinite-dimensional manifold of based loops in G: the loop group. In fact ΩG is a * Kähler manifold* and may be viewed as a flag manifold G

^{C}/P where G

^{C}is the manifold of loops in G

^{C}and P is the subgroup of those that extend holomorphically to the disc. We have various fibrations ρ

_{λ}: ΩG → G given by evaluation at λ ∈ S

^{1}and in some ways ρ

_{−1}behaves like a canonical fibration making ΩG into a universal twistor space for G. It is a theorem of

*that any harmonic map of S*

**Uhlenbeck**^{2}into G is of the form ρ

_{−1}◦ Φ for some “super-horzontal” holomorphic map Φ : S

^{2}→ ΩG.

The flag manifolds of G embed in ΩG as conjugacy classes of geodesics and we find a particular embedding of this kind using the canonical element. Indeed, our assumption that G be centre-free means that exp 2πξ = e for any canonical element ξ. Thus if F = G/H = G^{C}/P is a flag manifold with ξ the canonical element of p, we may define a map Γ: F → ΩG by setting

Γ(eH) = (e^{√−1t} → exp tξ)

and extending by equivariance. Moreover, if N is the inner symmetric space associated to F, we have a totally geodesic immersion γ : N → G defined by setting γ(x) equal to the element of G that generates the involution at x. We now have:

Γ: F → ΩG is a totally geodesic, holomorphic, isometric immersion and the following diagram commutes

where π_{1} is a canonical fibration. Thus we have a realisation of the canonical fibrations as the trace of ρ_{−1} on certain conjugacy classes of geodesics.