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 σ 1, . . . , σ p . Schwarz (Topology 14:63–68, 1975) proved that each G-invariant C ∞-function f on V has the form f = F(σ 1, . . . , σ p ) for a C ∞-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(σ 1, . . . , σ p ), if N is strongly regular and satisfies$$\sup_{k \in \mathbb{N}_{ > 0}}\left(\frac{M_{km}}{N_k}\right)^{\frac{1}{k}} < \infty,$$where m is an (explicitly known) integer depending only on the representation. In particular, each G-invariant (1 + δ)-Gevrey function f (with δ > 0) has the form f = F(σ 1, . . . , σ p ) for a (1 + δm)-Gevrey function F. Applications to equivariant functions and basic differential forms are given.