A generalization of Puiseux’s theorem and lifting curves over invariants

Abstract

Let ρ:G→GL (V) be a rational representation of a reductive linear algebraic group G defined over ℂ on a finite dimensional complex vector space V. We show that, for any generic smooth (resp. C ^M) curve c:ℝ→V//G in the categorical quotient V//G (viewed as affine variety in some ℂⁿ) and for any t ₀∈ℝ, there exists a positive integer N such that t↦c(t ₀±(t−t ₀)^N) allows a smooth (resp. C ^M) lift to the representation space near t ₀. (C ^M denotes the Denjoy–Carleman class associated with M=(M _k ), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V//G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C ^∞ curve which represents a lift of a germ of a quasianalytic C ^M curve in V//G is actually C ^M. There are applications to polar representations.