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 cocharacter-closed $G(k)$-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.
For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V, we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We
initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure. When k is perfect, we give a criterion in terms of closed orbits for G to be k-anisotropic, answering a question of Borel.
Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G^n, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of
the closed H-orbits in G^n, generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serres notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.
