Lifting quasianalytic mappings over invariants

Abstract

Let $\rho :G\to \mathrm{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let ${\sigma }_1,{\sigma }_n$ be a system of generators of the algebra of invariant polynomials $\mathbb{C}[V{]}^G$. We study the problem of lifting mappings $f:{\mathbb{R}}^q\supseteq U\to \sigma (V)\subseteq {\mathbb{C}}^n$ over the mapping of invariants $\sigma =({\sigma }_1,{\sigma }_n):V\to \sigma (V)$. Note that $\sigma (V)$ can be identified with the categorical quotient $V//G$ and its points correspond bijectively to the closed orbits in $V$. We prove that, if $f$ belongs to a quasianalytic subclass $\mathcal{C}\subseteq C^{\infty }$ satisfying some mild closedness properties which guarantee resolution of singularities in $\mathcal{C}$ (e.g.\ the real analytic class), then $f$ admits a lift of the same class $\mathcal{C}$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift which belongs to $SB{V}_{\mathrm{loc}}$ (i.e.\ special functions of bounded variation). If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.

17 pages, 1 table, minor corrections, to appear in Canad. J. Math

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Cited by in corpus (2)

• A generalization of Puiseux's theorem and lifting curves over invariants
• Optimal Sobolev regularity of roots of polynomials