Invariant Functions in Denjoy–carleman Classes

Let V be a real finite dimensional representation of a compact Lie group G. It is well-known that the algebra R[V ] of G-invariant polynomials on V is finitely generated, say by σ1, . . . , σp. Schwarz [38] proved that each G-invariant C-function f on V has the form f = F (σ1, . . . , σp) for a Cfunction F on R. 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 (with M = (Mk)). For finite groups G and (more generally) for polar representations V we show that for each Ginvariant f of class C there is an F of class C such that f = F (σ1, . . . , σp), if N is strongly regular and satisfies sup k∈N>0 “Mkm Nk ” 1 k < ∞, 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 (σ1, . . . , σp) for a (1 + δm)-Gevrey function F . Applications to equivariant functions and basic differential forms are given.