Counting Irreducible Components of the Semistable Global Nilpotent Cone

Source abstract

We discuss various methods for parametrizing and counting irreducible components of the global nilpotent cone of semistable $G$-Higgs bundles on a smooth complex curve. By restricting our study to the case of generically regular components following previous work of Faergeman and Raskin, we obtain explicit counts of semistable generically regular irreducible components for $G = \mathrm{SL}_n$, and give the first examples of such counts outside type A. These numbers are obtained by counting points in some natural polytopes associated to $G$, but our method requires a finer analysis in cases where the center of the group $G$ is disconnected.