Cocharacter-closure and spherical buildings

Let k be a field, let G be a reductive k-group and V an affine k-variety on which G acts. In this note we continue our study of the notion of cocharacterclosed G.k/-orbits in V . In earlier work we used a rationality condition on the point stabilizer of a G-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding G.k/-orbit in V . In the present paper we employ building-theoretic techniques to derive analogous results.