Toda frames, harmonic maps and extended Dynkin diagrams

We consider a natural subclass of harmonic maps from a surface into G/T, namely cyclic primitive maps. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and both are chosen so that there is a Coxeter automorphism on GC/TC which restricts to give a k-symmetric space structure on G/T. When G is compact, any Coxeter automorphism restricts to the real form. It was shown in [3] that cyclic primitive immersions into compact G/T correspond to solutions of the affine Toda field equations and all those of a genus one surface can be constructed by integrating a pair of commuting vector fields on a finite dimensional vector subspace of a Lie algebra. We generalise these results, removing the assumption that G is compact. The first major obstacle is that a Coxeter automorphism may not restrict to a non-compact real form. We characterise, in terms of extended Dynkin diagrams, those simple real Lie groups G and Cartan subgroups T such that G/T has a k-symmetric space structure induced from a Coxeter automorphism. A Coxeter automorphism preserves the real Lie algebra g if and only if any corresponding Cartan involution defines a permutation of the extended Dynkin diagram for gC=g⊗C; we show that every involution of the extended Dynkin diagram for a simple complex Lie algebra gC is induced by a Cartan involution of a real form of gC.