Abstract view
Lifting Quasianalytic Mappings over Invariants


Published:20110715
Printed: Apr 2012
Armin Rainer,
Fakultät für Mathematik, Universität Wien, A1090 Wien, Austria
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 blowups 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.
MSC Classifications: 
14L24, 14L30, 20G20, 22E45 show english descriptions
Geometric invariant theory [See also 13A50] Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] Linear algebraic groups over the reals, the complexes, the quaternions Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
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}
