Observable actions of algebraic groups

TL;DR

This paper characterizes observable actions of affine algebraic groups on varieties, linking geometric conditions with invariant theory, and explores properties for reductive groups including a maximal observable subset.

Contribution

It provides a geometric characterization of observable actions and describes the structure of maximal observable subsets for reductive groups.

Findings

Observable actions are characterized by open subsets of closed orbits and invariant rational functions.

For reductive groups, a unique maximal observable subset exists with finite, bijective quotient map.

The paper establishes a connection between geometric properties and invariant theory in algebraic group actions.

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.