A Chevalley's theorem in class C^r

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.