Vector Fields and Luna Strata

TL;DR

This paper investigates whether the Luna stratification of the categorical quotient of a G-module is preserved under automorphisms, demonstrating that it is indeed intrinsic for most cases using vector fields.

Contribution

The authors prove that the Luna stratification is intrinsic for almost all G-modules, extending previous results and employing vector fields to analyze automorphisms.

Findings

Luna stratification is intrinsic for most G-modules.

Automorphisms preserve the stratification.

Vector fields are key to proving stratification invariance.

Abstract

Let V be a G-module where G is a complex reductive group. Let Z:=V//G denote the categorical quotient. One can ask if the Luna stratification of Z is intrinsic. That is, if phi : Z\to Z is any automorphism, does phi send strata to strata? In a paper of Kuttler and Reichstein the answer was shown to be yes for V a direct sum of sufficiently many copies of a G-module W. We show that the answer is yes for almost all V. The key is to consider the vector fields on Z. Our methods also show that complex analytic automorphisms preserve the stratification.