Revista Matemática Complutense

, Volume 25, Issue 1, pp 139–155

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


DOI: 10.1007/s13163-011-0062-y

Cite this article as:
Losik, M., Michor, P.W. & Rainer, A. Rev Mat Complut (2012) 25: 139. doi:10.1007/s13163-011-0062-y


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. CM) curve c:ℝ→V//G in the categorical quotient V//G (viewed as affine variety in some ℂn) and for any t0∈ℝ, there exists a positive integer N such that tc(t0±(tt0)N) allows a smooth (resp. CM) lift to the representation space near t0. (CM denotes the Denjoy–Carleman class associated with M=(Mk), 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 CM curve in V//G is actually CM. There are applications to polar representations.


Puiseux’s theoremReductive group representationsInvariantsRegular liftingUltradifferentiableDenjoy–Carleman

Mathematics Subject Classification (2000)


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© Revista Matemática Complutense 2011

Authors and Affiliations

  1. 1.Saratov State UniversitySaratovRussia
  2. 2.Fakultät für MathematikUniversität WienViennaAustria