Cocharacter-closure and the rational Hilbert-Mumford Theorem

Abstract

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.

33 pages, v. 2 reference added, slight changes, v. 3 minor changes, v. 4 added new Thm 1.6 characterizing k-anisotropic reductive groups in terms of cocharacter-closure; v. 5 final version, further small changes; to appear in Math. Z