A Chevalley's theorem in class C^r

TL;DR

This paper extends Chevalley's theorem to class C^r functions under finite reflection groups, establishing a continuous linear mapping from invariant functions to functions composed with Chevalley's polynomial map, with optimal regularity results.

Contribution

It introduces a new continuous linear mapping for C^r invariant functions, generalizing Chevalley's theorem to non-analytic smooth functions with optimal regularity.

Findings

Existence of a continuous linear map from (C^r(R^n))^W to C^[r/h](R^n)

Optimality of the regularity result demonstrated by a counterexample

Use of division by linear forms and compensation phenomena in the proof

Abstract

Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in P.There exists a linear mapping from (C^r(R^n))^W to C^[r/h](R^n), f\mapsto F such that f=F \circ P, continuous for the natural Fr\'echet topologies. A general counterexample shows that this result is the best possible. The proof uses techniques of division by linear forms and a study of compensation phenomenons. An extension to P^{-1}(R^n) of invariant formally holomorphic regular fields is needed.