Invariant functions in Denjoy-Carleman classes

الأصل · EN

Let V be a real finite dimensional representation of a compact Lie group G. It is well-known that the algebra R[V]ᵍ of G-invariant polynomials on V is finitely generated, say by σ₁,...,σₚ. Schwarz proved that each G-invariant C∞-function f on V has the form f=F(σ₁,...,σₚ) for a C∞-function F on Rᵖ. 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 Cₘ (with M=(Mₖ)). For finite groups G and (more generally) for polar representations V we show that for each G-invariant f of class Cₘ there is an F of class Cₙ such that f=F(σ₁,...,σₚ), if N is strongly regular and satisfies Nₖ ≥ Mkm ᵏ⁺¹, for all k, with m an (explicitly known) integer depending only on the representation and ε>0 independent of k. In particular, each G-invariant (1+δ)-Gevrey function f has the form f=F(σ₁,...,σₚ) for a (1+δm)-Gevrey function F. Applications to equivariant functions and basic differential forms are given.