Higgs bundles and abelianization

Speaker: Ana Peon Nieto, Ruprecht-Karls-Universität Heidelberg

Given a reductive Lie group G, we consider   M(G), the moduli space of G-Higgs bundles over a Riemann surface X. Topologically, this corresponds to the moduli space of representations of the fundamental group of X into G.  This space admits a morphism onto an affine space. This so called Hitchin map was described for complex groups by Hitchin. He described the fibers in terms of Prym varieties over the spectral curve for groups of  type A, B, C, D. In particular, they are abelian varieties. The general case follows from the work of Donagi--Gaitsgory, which allows to interpret the fibers in terms of torsors of tori over the cameral cover.

We will explain how the results of DG generalise to the case of real forms of Lie groups. This allows to identify the class of real forms for which abelianization holds.  We will illustrate this by studying the cameral data for the group SU(p+1,p), which allows, in particular, to count the connected components of the corresponding moduli space.