Title: Chevalley's theorem in class <i>C<sup>r</sup></i>
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 . Let r be a positive integer and [ r / h ] be the integer part of r / h . There exists a linear mapping $\mathcal{C}^r(\mathbb{R}^n)^W\ni f\mapsto F\in\mathcal{C}^{[r/h]}(\mathbb{R}^n)$ such that f = F ∘ P , which is continuous for the natural Fréchet topologies. A general counter-example shows that this result is the best possible. The proof uses techniques of division by linear forms and a study of compensation phenomena. An extension to P −1 (ℝ n ) of invariant formally holomorphic regular fields is needed.