Gerard P. Barbanson
Reflection Groups and Composite Mappings in class C^r
(53K, AMS-LaTeX)
ABSTRACT. Let W be a finite reflection group acting orthogonally on R^n, P=(p_1,p_2,...,p_n) be a set of basic polynomial invariants and h be the highest degree of the p is which is the largest Coxeter number of the irreducible components of W. We first briefly study the subspace of functions of the invariants in P*^(-1)(C^(hr)(R^n)^W), which is a subspace of multipliers in \mathcal{E}^r(P(R^n)). Then we show that the algebra of composite mappings P*(\mathcal{E}^r(P(R^n))) is a Frechet subspace of the space of r-regular jets of order hr. The algebra homomorphism P* identifies this Frechet space with the space of functions of class C^r of the polynomial invariants. This study needs the Whitney 1-regularity property of P(R^n). By lack of reference, we give in appendix a proof of this regularity by completing a result of Givental' with a lemma not proven for all Coxeter groups.