A generalization of Puiseux’s theorem and lifting curves over invariants
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 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.
KeywordsPuiseux’s theorem Reductive group representations Invariants Regular lifting Ultradifferentiable Denjoy–Carleman
Mathematics Subject Classification (2000)14L24 14L30 20G20
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- 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.Pereira, M. Pe: Nash problem for quotient surface singularities (2010). arXiv:1011.3792
- 10.Puiseux, V.: Recherches sur les fonctions algébriques. J. Math. Pures Appl. 15(207), 365–480 (1850) Google Scholar
- 12.Rainer, A.: Quasianalytic multiparameter perturbation of polynomials and normal matrices. Trans. Am. Math. Soc. (2009, to appear). arXiv:0905.0837
- 13.Rainer, A.: Lifting quasianalytic mappings over invariants. Can. J. Math. (2010, to appear). arXiv:1007.0836
- 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