Invariant functions in Denjoy-Carleman classes

Abstract

Let $V$ be a real finite dimensional representation of a compact Lie group $G$. It is well-known that the algebra $\mathbb R[V]^G$ of $G$-invariant polynomials on $V$ is finitely generated, say by $\sigma_1,...,\sigma_p$. Schwarz proved that each $G$-invariant $C^\infty$-function $f$ on $V$ has the form $f=F(\sigma_1,...,\sigma_p)$ for a $C^\infty$-function $F$ on $\mathbb R^p$. We investigate this representation within the framework of Denjoy-Carleman classes. One can in general not expect that $f$ and $F$ lie in the same Denjoy-Carleman class $C_M$ (with $M=(M_k)$). For finite groups $G$ and (more generally) for polar representations $V$ we show that for each $G$-invariant $f$ of class $C_M$ there is an $F$ of class $C_N$ such that $f=F(\sigma_1,...,\sigma_p)$, if $N$ is strongly regular and satisfies $N_k \ge M_{km} \ep^{k+1}$, for all $k$, with $m$ an (explicitly known) integer depending only on the representation and $\epsilon>0$ independent of $k$. In particular, each $G$-invariant $(1+\delta)$-Gevrey function $f$ has the form $f=F(\sigma_1,...,\sigma_p)$ for a $(1+\delta m)$-Gevrey function $F$. Applications to equivariant functions and basic differential forms are given.