STABILITY OF AFFINE G-VARIETIES AND IRREDUCIBILITY IN REDUCTIVE GROUPS

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 ∈ 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 = GN. 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.