Lifting quasianalytic mappings over invariants

TL;DR

This paper proves that quasianalytic mappings over invariants of reductive groups can be lifted to the original space after desingularization, with the lift belonging to the same class or to SBV_loc, especially for compact Lie groups.

Contribution

It establishes conditions under which quasianalytic mappings over invariants admit lifts of the same class after resolution of singularities, extending previous results to broader classes of functions.

Findings

Lifts exist in quasianalytic classes after desingularization.

Mappings can be lifted to SBV_loc functions.

Stronger results hold for real representations of compact Lie groups.

Abstract

Let $\rho : G \to \operatorname{GL}(V)$ 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 $SBV_{\operatorname{loc}}$ (i.e.\ special functions of bounded variation). If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.