 10135 Gerard P. Barbanson
 Reflection Groups and Composite Mappings in class C^r
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Aug 31, 10

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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 rregular 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 1regularity 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.
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