Lifting Quasianalytic Mappings over Invariants

Abstract

Abstract Let be a rational finite dimensional complex representation of a reductive linear algebraic group G, and let be a system of generators of the algebra of invariant poly- nomials ℂ[V]G. We study the problem of lifting mappings over the mapping of invariants . Note that ¾(V) can be identified with the cate- gorical quotientV//G and its points correspond bijectively to the closed orbits inV. We prove that if f belongs to a quasianalytic subclass satisfying some mild closedness properties that guarantee resolution of singularities in 𝒞 e.g., the real analytic class, then f admits a lift of the same class 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 SBVloc, i.e., special functions of bounded variation. If ρ is a real representation of a compact Lie group, we obtain stronger versions.