Invariant functions in Denjoy-Carleman classes

TL;DR

This paper studies how invariant functions in Denjoy-Carleman classes can be represented through generators of invariant polynomials, extending Schwarz's classical result to broader function classes with explicit regularity bounds.

Contribution

It extends Schwarz's theorem to Denjoy-Carleman classes, providing explicit conditions under which invariant functions can be expressed via polynomial invariants with controlled regularity.

Findings

Invariant functions in Denjoy-Carleman classes can be represented through polynomial invariants with increased regularity.

Explicit bounds relate the regularity of the original function to that of the representing function.

Applications include equivariant functions and differential forms with controlled regularity.

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.