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.
Bibliographical note33 pages, v. 2 reference added, slight changes
The authors acknowledge the financial support of EPSRC Grant EP/L005328/1 and of Marsden Grants UOC1009 and UOA1021. Part of the research for this paper was carried out while the authors were staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. Also, part of this paper was written during mutual visits to Auckland, Bochum and York. We are grateful to the referees for their careful reading of the paper and for helpful suggestions.
- Affine G-variety
- Cocharacter-closed orbit