Observable actions of algebraic groups

Abstract

Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X\to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X{]}^G$ such that f(Y) =0. We characterize this condition geometrically as follows. The action $G\times X\to X$ is observable if and only if (1) there is a nonempty open subset $U\subseteq X$ consisting of closed orbits, and (2) the field $K(X{)}^G$ of G-invariant rational functions on X is equal to the quotient field of $K[X{]}^G$. In case G is reductive, we conclude that there exists a unique, maximal, G-stable, closed subset $X_{\soc}$ of $X$ such that $G\times X_{\soc} \to X_{\soc}$ is observable. Furthermore, the canonical map $X_{\soc}// G \to X//G$ is finite and bijective.

16 pages; v2. some proofs improved, change order of results in sect. 3, citations improved

References in corpus (2)

• Local structure of algebraic monoids
• Anti-affine algebraic groups

Cited by in corpus (1)

• The ring of regular functions of an algebraic monoid