Conjugating Representations in $PGL(k,\mathbb{C})$ into $PGL(k,\mathbb{R})$

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 Teichmüller theory. For a closed surface, classical Teichmüller space is a connected component of the moduli space of representations into $PSL(2,\mathbb{R})$ and [Fock:2006] showed that the space of positive representations into $PSL(k,\mathbb{R})$ coincides with the Hitchin component. In this paper we study representations of finitely generated groups into $PGL(k,\mathbb{C})$ and determine necessary and sufficient conditions for such a representation to be conjugate into $PGL(k,\mathbb{R})$. In this way, we identify representations in the larger representation variety which are conjugate in $PGL(k,\mathbb{C})$ to a representation in $\mathrm{hom}({\pi }_1(\mathrm{\Sigma }),PGL(k,\mathbb{R}))\mathrm{/}PGL(k,\mathbb{R})$.