Moduli of local systems on Deligne-Mumford stacks
Seminar Room 1, Newton Institute
Most aspects of the theory of moduli of local systems on smooth projective varieties extend to smooth (or even just normal) proper DM-stacks, via a little covering lemma. For other singularities or simplicial varieties we meet a phenomenon of weight filtration.
The DM-stack case also allows us to approach the question of open varieties while avoiding the more difficult technical aspects there, and it provides a convenient formalism for finite group actions. In the example of a root stack over the projective line, the moduli space can have components containing no representations into a compact group.