Cyclic Higgs bundles and the Toledo invariant

Let G be a complex semisimple Lie group and g its Lie algebra. In this paper, we study a special class of cyclic Higgs bundles constructed from a Z-grading g = ₉=₁-₌^m-1 gⱼ by using the natural representation G₀ GL (g₁ g₁-₌), where G₀ G is the connected subgroup corresponding to g₀. The resulting Higgs pairs include G^ R-Higgs bundles for G^ R G a real form of Hermitian type (in the case m=2) and fixed points of the C^*-action on G-Higgs bundles (in the case where the Higgs field vanishes along g₁-₌). In both of these situations a topological invariant with interesting properties, known as the Toledo invariant, has been defined and studied in the literature. This paper generalises its definition and properties to the case of arbitrary (G₀, g₁ g₁-₌) -Higgs pairs, which give rise to families of cyclic Higgs bundles. The results are applied to the example with m=3 that arises from the theory of quaternion-K\"ahler symmetric spaces.