Gerard P. BARBANSON
A CHEVALLEY'S THEOREM IN CLASS ${\cal C}^r$.
(253K, pdf)

ABSTRACT.  Let $W$ be a finite reflection group acting orthogonally on 
 ${\bf R}^n$, $P$ 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: 
 ${\cal C}^r(R^n)^W \ni f \to F\in {\cal C}^{[r/h]}(R^n)$ 
 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 by induction on $h$ uses techniques 
 of division by linear forms and a study of 
 compensation phenomenons. An extension to $P^{-1}({\bf R}^n)$ 
 of invariant formally holomorphic regular fields is needed. 
 This is a revised version of # 06-20 posted earlier on this site with the same name.