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.