Reductive group actions over nonclosed fields of characteristic zero

We present a structure theory for actions of reductive groups which, to some extent, is parallel to the classical theory of reductive groups by Satake and Borel–Tits. In particular, properties of reductive groups actions on varieties with enough rational points are controlled by a restricted root system and an anisotropic kernel. The theory is most effective for varieties on which a minimal parabolic has an open orbit (called k-spherical). Here we construct wonderful completions. For local fields they turn out to be genuine compactification of the rational points. This is joint work with Bernhard Krötz.