It has been known for some time  that an irreducible representation of a
Fuchsian group (e.g. a surface group) into complex simple Lie groups is a
non-singular point of the representation variety. However, on the
character variety, this might lead to orbifold singularities. In order to
study those particular singularities, one needs to understand centralizers
of irreducible representations, we will explain what happens when the Lie
group is PSL(n,C).