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

Abstract

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 Teichm\"uller theory. For a closed surface, classical Teichm\"uller 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 $\hom ( \pi_1 (\Sigma), PGL(k, \mathbb{R}) ) / PGL(k, \mathbb{R})$.