Stability of Affine G-varieties and Irreducibility in Reductive Groups

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 ∈ X in terms of intrinsically defined closed subgroups Hx of G, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-Röhrle 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.