Revista Matemática Complutense

, Volume 25, Issue 1, pp 139–155

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

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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. 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.

Keywords

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

Mathematics Subject Classification (2000)

14L24 14L30 20G20 

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References

  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) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bierstone, E., Milman, P.D.: Arc-analytic functions. Invent. Math. 101(2), 411–424 (1990) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Dadok, J., Kac, V.: Polar representations. J. Algebra 92(2), 504–524 (1985) CrossRefMATHMathSciNetGoogle 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) MATHMathSciNetGoogle 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) MATHMathSciNetGoogle 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) MATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHGoogle Scholar
  15. 15.
    Siciak, J.: A characterization of analytic functions of n real variables. Stud. Math. 35, 293–297 (1970) MATHMathSciNetGoogle Scholar
  16. 16.
    Thilliez, V.: On quasianalytic local rings. Expo. Math. 26(1), 1–23 (2008) MATHMathSciNetGoogle Scholar
  17. 17.
    Thilliez, V.: Smooth solutions of quasianalytic or ultraholomorphic equations. Monatshefte Math. 160(4), 443–453 (2010) CrossRefMATHMathSciNetGoogle 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

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© Revista Matemática Complutense 2011

Authors and Affiliations

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

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