A fixed-point characterization of linearly reductive groups

Abstract

This chapter discusses a fixed-point characterization of unipotent groups, viz., if G is a connected linear algebraic group over the field k, then G is unipotent if and only if, for all proper geometrically connected G-schemes X over k, the fixed-point scheme XG is connected. To give an analogous characterization of linearly reductive groups, the chapter presents the proof for the following theorem: Let G be a linear algebraic group over the field k. Then G is linearly reductive if and only if for all smooth algebraic G-schemes X over k, the fixed-point scheme XG is smooth. A linear algebraic group over k, means a reduced affine group scheme of finite type over k. If G is a linear algebraic group over k, by a G-scheme X over k it is meant that a scheme X over k, plus a map of schemes α: G x X → X satisfies the usual axioms for an action of G on X. If X is a G-scheme, the fixed-point scheme XG is the largest closed subscheme of X on which G acts trivially.