Luna Slice Theorem

Author: Eiko

Tags: algebraic geometry, luna, slice theorem, reductive group, group action

Time: 2025-01-06 07:12:05 - 2025-01-06 07:12:05 (UTC)

Idea

Let $G$ be a reductive complex algebraic group, this includes all finite groups, tori, semisimple groups, classical groups. Consider a finite $C$ -space $V$ where $G$ acts linearly.

We know that when we have a reductive group action on affine spaces, there will be a lot of orbits, some of them are closed. If $G v$ is a closed orbit, we know that $G_{v}$ is also reductive by Matsushima’s theorem.

This means, $V$ can be decomposed into direct sum of $G_{v}$ -modules, since $T_{v} G v \subset V$ we have $V = T_{v} G v \oplus N_{v}$ , where the normal space $N_{v}$ is the slice representation at $v$ .

A basic idea of slice theorem is that, the $G$ action near the closed orbit $G v$ should be determined by the $G_{v}$ action on the slice representation $N_{v}$ .