Topology of the moduli space via arithmetic geometry and representation theory

Let G be a complex reductive group and X a Riemann surface. It is known that the following moduli spaces are isomorphic: 

1. Solutions to the Hitchin (aka 2d Yang-Mills) equations 
2. Flat G-bundles  
3. Higgs bundle 
4. Representations of the fundamental group (aka character variety)

Thus, there has been a huge interest in understanding the topology of this space by people from diverse parts of mathematics. Despite some recent breakthroughs (e.g. Schiffmann’s formula for the Betti numbers of moduli space for G=GL_n in the non-singular case), much remains to be done. For instance, we know very little about the singular case or when G=Sp_4.

Inspired by the pioneering work of Hausel and Rodriguez-Villegas  we count points over finite fields and use Weil’s conjecture to determine the Euler characteristic of the moduli space for G=GL_n in the singular case. The point of departure for us is that we interpret the word “space” to mean “stack” as oppose to “variety”. The approach will work for general group G, provided one can extract the relevant information from representations theory of the finite group G(F_q). 

Based on a joint project with David Baraglia.