Real Reductive Groups and Relative Roots

In this talk, we discuss the classification of reductive groups over R. When working over C, the classification of reductive groups reduces to checking root data, a combinatorial problem. Over R, the classification is more subtle. Two non-isomorphic real reductive groups may share root data--e.g. SL_n(R) and SU(n). The solution to this phenomenon is to consider so-called relative root data, essentially complex root data together with a Galois action. We discuss the interaction of conjugation with root systems and how it manifests in the classification, structure theory, and geometry of real reductive groups.