Bad irreducible subgroups and singular locus for character varieties in PSL(p,C)

We give the centralizers of irreducible representations from a finitely generated group $\Gamma$ to $PSL(p,\mathbb{C})$ where p is a prime number. This leads to a description of the singular locus (te (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $\chi^i(\Gamma,PSL(p,\mathbb{C}))$. When $\Gamma$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of algebraic singularities of the irreducible part of the character variety.