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The space of representations of a surface group into a given simple Lie groupis a very active area of research and is particularly relevant to higherTeichm\"uller theory. For a closed surface, classical Teichm\"uller space is aconnected component of the moduli space of representations into $PSL(2,\mathbb{R})$ and [Fock:2006] showed that the space of positive representationsinto $PSL(k, \mathbb{R})$ coincides with the Hitchin component. In this paperwe study representations of finitely generated groups into $PGL(k, \mathbb{C})$and determine necessary and sufficient conditions for such a representation tobe conjugate into $PGL(k, \mathbb{R})$. In this way, we identifyrepresentations in the larger representation variety which are conjugate in$PGL(k, \mathbb{C})$ to a representation in $\hom ( \pi_1 (\Sigma), PGL(k,\mathbb{R}) ) / PGL(k, \mathbb{R})$.

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