Stability of Affine G-varieties and Irreducibility in Reductive Groups

TL;DR

This paper develops a stability criterion for affine G-varieties using intrinsic subgroup structures, connecting it with classical numerical criteria and generalizing results related to representation varieties and non-abelian Hodge theory.

Contribution

It introduces a new stability criterion based on intrinsic subgroup data, extending classical stability concepts to affine G-varieties and linking them with existing criteria.

Findings

Established a stability criterion using intrinsic subgroups.

Connected the criterion with Mumford's numerical stability and other classical criteria.

Generalized results on complex representation varieties and non-abelian Hodge theory.

Abstract

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-R\"ohrle criteria, in the case $X=G^{N}$. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions.