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

Abstract

Let $\rho :G\to \mathrm{GL}(V)$ be a rational representation of a reductive linear algebraic group $G$ defined over $\mathbb{C}$ on a finite dimensional complex vector space $V$. We show that, for any generic smooth (resp. $C^M$) curve $c:\mathbb{R}\to V//G$ in the categorical quotient $V//G$ (viewed as affine variety in some ${\mathbb{C}}^n$) and for any $t_0\in \mathbb{R}$, there exists a positive integer $N$ such that $t\mapsto c(t_0\pm (t-t_0{)}^N)$ allows a smooth (resp. $mathbb{C}^M$) 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 $C^{\infty }$ 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.

14 pages, LaTeX, small changes to make it consistent with the published version

References in corpus (7)

• On quasianalytic local rings
• Smooth solutions of quasianalytic or ultraholomorphic equations
• Lifting smooth curves over invariants for representations of compact Lie groups. II
• Lifting smooth curves over invariants for representations of compact Lie groups, III
• Lifting mappings over invariants of finite groups
• Lifting quasianalytic mappings over invariants
• Addendum to: "Lifting smooth curves over invariants for representations of compact Lie groups, III" [J. Lie Theory 16 (2006), No. 3, 579-600.]

Cited by in corpus (5)

• Perturbation theory for normal operators
• Lifting quasianalytic mappings over invariants
• Addendum to: "Lifting smooth curves over invariants for representations of compact Lie groups, III" [J. Lie Theory 16 (2006), No. 3, 579-600.]
• Optimal Sobolev regularity of roots of polynomials
• Sobolev Lifting over Invariants