Cocharacter-closure and the rational Hilbert-Mumford Theorem

TL;DR

This paper introduces a rational version of the Hilbert-Mumford Theorem using cocharacter-closure of orbits, providing new tools for geometric invariant theory over arbitrary fields and exploring applications and examples.

Contribution

It develops a rational analogue of the Hilbert-Mumford Theorem based on cocharacter-closure, extending geometric invariant theory to non-algebraically closed fields.

Findings

Established a criterion for G to be k-anisotropic over perfect fields.

Illustrated differences between cocharacter-closure and Zariski-closure with examples.

Initiated applications of the rational Hilbert-Mumford Theorem in invariant theory.

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.