Miguel González

Let G be a semisimple complex algebraic group. We study the ℂ*-action on the moduli space of strongly parabolic G-Higgs bundles over a punctured curve with weights in the interior of the standard Weyl alcove. We describe its fixed points and, for the components with generically regular Higgs field, we classify the fixed points that have closed Białynicki-Birula upward flows, called very stable. The classification is stated in terms of a naturally associated divisor valued in the extended affine Weyl group of G which encodes the information of the fixed point. Under this point of view, very stable fixed points naturally coincide with divisors where the coefficients are minimal under the Bruhat order. Furthermore, when the divisor is supported at the parabolic punctures, the corresponding very stable upward flow is a section of the Hitchin map. For these upward flows, we propose a conjectural mirror line bundle over the dual moduli space, and verify its validity for G=PGLn(ℂ).

We give a classification of very stable G-Higgs bundles in the generically regular Higgs field case for G an arbitrary connected semisimple complex group. This extends the classification for G=GL_n(C) and fixed point type (1,1,...,1) given by Hausel and Hitchin.

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 = \oplus_j=1-m^m-1 g_j by using the natural representation G_0 \to GL(g_1 ⊕g_1-m), where G_0 \le G is the connected subgroup corresponding to g_0. The resulting Higgs pairs include G^R-Higgs bundles for G^R \le 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 g_1-m). 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 (G_0,g_1 ⊕g_1-m)-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-Kähler symmetric spaces.

We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GLn case, we classify the type (1,…, 1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, 4n-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.