Conjugating Representations in PGL(k, C) into PGL(k, R)

The space of representations of a surface group into a given simple Lie group is a very active area of research and is particularly relevant to higher Teichmuller theory. For a closed surface, classical Teichmuller space is a connected component of the moduli space of representations into PSL(2, R) and [Fock-Goncharov:2006] showed that the space of positive representations into PSL(k, R) coincides with the Hitchin component. In this paper we study representations of finitely generated groups into PGL(k, C) and determine necessary and sufficient conditions for such a representation to be conjugate into PGL(k, R). In this way, we identify representations in the larger representation variety which are conjugate in PGL(k, C) to a representation in hom(pi_1 (Sigma), PGL(k, R)) / PGL(k, R).