Lifting quasianalytic mappings over invariants

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.Comment: 17 pages, 1 table, minor corrections, to appear in Canad. J. Mat

Regular infinite dimensional Lie groups

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply connected and the image group is regular.Comment: AmSTeX, using diag.tex with fonts lams?.ps, 38 page

Differentiable perturbation of unbounded operators

If $A(t)$ is a C^{1,\al}-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized $C^1$ in $t$. If $A$ is $C^\infty$ then the eigenvalues can be parameterized twice differentiable.Comment: amstex 9 pages. Some misprints correcte

The Convenient Setting for Quasianalytic Denjoy--Carleman Differentiable Mappings

For quasianalytic Denjoy--Carleman differentiable function classes $C^Q$ where the weight sequence $Q=(Q_k)$ is log-convex, stable under derivations, of moderate growth and also an $\mathcal L$-intersection (see 1.6), we prove the following: The category of $C^Q$-mappings is cartesian closed in the sense that $C^Q(E,C^Q(F,G))\cong C^Q(E\times F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^Q$-diffeomorphisms is a regular $C^Q$-Lie group but not better.Comment: 29 pages. Some typos corrected; J. Functional Analysis (2011