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# Lifting Quasianalytic Mappings over Invariants

Published:2011-07-15
Printed: Apr 2012
• Armin Rainer,
Fakultät für Mathematik, Universität Wien, A-1090 Wien, Austria
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## Abstract

Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. 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 that 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 that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.
 Keywords: lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation
 MSC Classifications: 14L24 - Geometric invariant theory [See also 13A50] 14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 20G20 - Linear algebraic groups over the reals, the complexes, the quaternions 22E45 - Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}