Revista Matemática Complutense

, Volume 25, Issue 1, pp 139–155 | Cite as

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

  • Mark Losik
  • Peter W. MichorEmail author
  • Armin Rainer


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 ℂ n ) and for any t 0∈ℝ, there exists a positive integer N such that tc(t 0±(tt 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.


Puiseux’s theorem Reductive group representations Invariants Regular lifting Ultradifferentiable Denjoy–Carleman 

Mathematics Subject Classification (2000)

14L24 14L30 20G20 


  1. 1.
    Alekseevsky, D., Kriegl, A., Losik, M., Michor, P.W.: Lifting smooth curves over invariants for representations of compact Lie groups. Transform. Groups 5(2), 103–110 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bierstone, E., Milman, P.D.: Arc-analytic functions. Invent. Math. 101(2), 411–424 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dadok, J., Kac, V.: Polar representations. J. Algebra 92(2), 504–524 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kriegl, A., Losik, M., Michor, P.W., Rainer, A.: Lifting smooth curves over invariants for representations of compact Lie groups. II. J. Lie Theory 15(1), 227–234 (2005) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Kriegl, A., Losik, M., Michor, P.W., Rainer, A.: Lifting smooth curves over invariants for representations of compact Lie groups. III. J. Lie Theory 16(3), 579–600 (2006) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Kriegl, A., Losik, M., Michor, P.W., Rainer, A.: Lifting mappings over invariants of finite groups. Acta Math. Univ. Comen. (New Ser.) 77(1), 93–122 (2008) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Kriegl, A., Michor, P.W., Rainer, A.: The convenient setting for non-quasianalytic Denjoy–Carleman differentiable mappings. J. Funct. Anal. 256, 3510–3544 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Luna, D.: Slices étales. Sur les groupes algébriques, pp. 81–105. Soc. Math. France, Paris (1973). Bull. Soc. Math. France, Paris, Mémoire 33 Google Scholar
  9. 9.
    Pereira, M. Pe: Nash problem for quotient surface singularities (2010). arXiv:1011.3792
  10. 10.
    Puiseux, V.: Recherches sur les fonctions algébriques. J. Math. Pures Appl. 15(207), 365–480 (1850) Google Scholar
  11. 11.
    Rainer, A.: Perturbation of complex polynomials and normal operators. Math. Nachr. 282(12), 1623–1636 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Rainer, A.: Quasianalytic multiparameter perturbation of polynomials and normal matrices. Trans. Am. Math. Soc. (2009, to appear). arXiv:0905.0837
  13. 13.
    Rainer, A.: Lifting quasianalytic mappings over invariants. Can. J. Math. (2010, to appear). arXiv:1007.0836
  14. 14.
    Schwarz, G.W.: Lifting smooth homotopies of orbit spaces. Inst. Hautes Études Sci. Publ. Math. 51, 37–135 (1980) CrossRefzbMATHGoogle Scholar
  15. 15.
    Siciak, J.: A characterization of analytic functions of n real variables. Stud. Math. 35, 293–297 (1970) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Thilliez, V.: On quasianalytic local rings. Expo. Math. 26(1), 1–23 (2008) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Thilliez, V.: Smooth solutions of quasianalytic or ultraholomorphic equations. Monatshefte Math. 160(4), 443–453 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Vinberg, È.B., Popov, V.L.: Invariant Theory. Algebraic Geometry, vol. 4 (Russian), pp. 137–314, 315. Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1989) Google Scholar

Copyright information

© Revista Matemática Complutense 2011

Authors and Affiliations

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

Personalised recommendations