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Finite Coxeter groups, r-regular jets of order m, algebra of composite mappings, Whitney regularity property.
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\begin {document}
\title[]{Reflection groups and \\ Composite mappings in class $\mathcal{C}^r$}%
\author{G\'erard P. BARBAN\c CON}%
\address{University of Texas at Austin}%
\email{gbarbanson@yahoo.com }%
\thanks{}%
\subjclass{}%
\keywords{Finite Coxeter groups, $r$-regular jets of order $m$, algebra of composite mappings, Whitney regularity property.}%
%\date{October 2009}%
% ----------------------------------------------------------------
\begin{abstract}
Let $W$ be a finite reflection group acting orthogonally on
$\mathbb{R}^n$, $P=(p_1, \ldots ,p_n)$ be a set of basic polynomial
invariants and $h$ be the highest degree of the $p_i$s which is the
largest Coxeter number of the irreducible components of $W$. We
first study the subspace of functions of the invariants that
are invariant functions of class $C^{hr}$. This subspace, say
$(P^*)^{-1}(\mathcal{C}^{hr}(\mathbb{R}^n)^W)$ is a space
of ``compositors'' in $\mathcal{E}^r(P(\mathbb{R}^n))$. Then we show
that the algebra of composite mappings
$P^*(\mathcal{E}^r(P(\mathbb{R}^n)))$ is a Fr\'echet subspace of the
space of $r$-regular jets of order $hr$. The algebra homomorphism
$P^*$ identifies this Fr\'echet space with the space of functions of
class $\mathcal{C}^r$ of the polynomial invariants. This study needs
the Whitney 1-regularity property of $P(\mathbb{R}^n)$. By lack of a
reference, we give in appendix a proof of this regularity property
by completing a result given in ~\cite{7} with a lemma that was not
proved for all Coxeter groups.
\end{abstract}
\maketitle
% ----------------------------------------------------------------
\section{Introduction}
Let $W$ be a finite reflection group acting orthogonally on $\mathbb{R}^n$.
The algebra of $W$-invariant polynomials is generated by $n$
algebraically independent $W$-invariant homogeneous polynomials and
the degrees of these basic invariants are uniquely determined ~\cite {13}, ~\cite{5}.
Let $p_1, \ldots ,p_n$ be a set of basic invariants, we define the `Chevalley'
mapping
\[P:{\mathbb R}^n\To \mathbb{ R}^n,\quad P(x)=\big(p_1(x), \ldots , p_n(x)\big).\]
Glaeser's theorem ~\cite{9} shows that $W$-invariant functions of
class ${\mathcal C}^{\infty}$, may be expressed as functions of
class ${\mathcal C}^{\infty}$ of the basic invariants, and actually
that the subalgebra $P^*(\mathcal{C}^\infty(\mathbb{R}^n))$ of
composite mappings of the form $F\circ P$ with $F$ of class
$\mathcal{C}^{\infty}$, is closed in
$\mathcal{C}^{\infty}(\mathbb{R}^n)$. In finite class of
differentiability, the situation is not this simple. Let $h$ be the
highest degree of the coordinate polynomials in $P$, equal to the
greatest Coxeter number of the irreducible components of $W$, in
~\cite{4} it is shown that if a function $f\in
\mathcal{C}^{hr}(\mathbb{R}^n)$ is invariant by $W$, there exists an
$F$ of class $\mathcal{C}^r$, such that $f=F\circ P$. A general
counterexample shows that this result is the best possible. However
if $F$ is of class $\mathcal{C}^r$, in general $f=F\circ P$ is of
class $\mathcal{C}^r$ and not of class $\mathcal{C}^{hr}$. So the
following questions naturally arise: what is the space of functions
of the invariants that are invariant functions of class
$\mathcal{C}^{hr}$ on $\mathbb{R}^n$, and what is the space of
$W$-invariant functions that are functions of class $\mathcal{C}^r$
of the polynomial invariants?
We first complement the result given in ~\cite{4} by showing that
the functions $F$ of the invariants such that $f=F\circ P$ is
of class $\mathcal{C}^{hr}$, belong to a subspace
of rough compositors in $\mathcal{E}^r(P(\mathbb{R}^n))$, isomorphic
to $\mathcal{C}^{hr}(\mathbb{R}^n)^W$.
Then, we study the algebra of composite mappings
$P^*(\mathcal{C}^r(\mathbb{R}^n))$. The functions in this algebra
induce jets in a Fr\'echet space isomorphic to the space of
functions of class $\mathcal{C}^r$ of the invariants.
This study uses the Whitney 1-regularity of the image set
$P(\mathbb{R}^n)$.
\begin{defn} ~\cite{16}~\cite{14} A compact set $K\subset \mathbb{R}^n$,
connected by rectifiable arcs, is \emph{Whitney 1-regular} if the
geodesic distance in $K$ is equivalent to the Euclidean distance:
there exists a constant $k_K>0$ such that for all $(x,x')\in K^2$,
there is a rectifiable arc from $x$ to $x'$ in $K$ with length
$\ell(x,x')\le k_K \abs {x-x'}$.\end{defn} A closed set is Whitney
1-regular if it is a union of Whitney 1-regular compact sets.
The 1-regularity of $P(\mathbb{R}^n)$ was conjectured in ~\cite{4}
and already in ~\cite{7}. By lack of a better reference we state and
prove this property in the appendix, using the results of ~\cite{7}
and some results given in ~\cite{11} for the symmetric group that
apply for any Coxeter group, \emph{mutatis mutandis}.
\section{ The Chevalley mapping}
A detailed study may be found in \cite{5} or \cite{8}.
When $W$ is reducible, it is a direct product of its irreducible
components, say $W= W^0\times W^1 \times \ldots \times W^s $ and we may
write $\mathbb{R}^n$ as an orthogonal direct sum $\mathbb{R}^{n_0}\oplus
\mathbb{R}^{n_1}\oplus \ldots \oplus \mathbb{R}^{n_s}$. The first component $W^0$ is
the identity on the $W$-invariant subspace $\mathbb{R}^{n_0}$,
and for $i=1,\ldots, s$, $W^i$ is an irreducible finite Coxeter group
acting on $\mathbb {R}^{n_i}$.
We will choose coordinates that fit with this orthogonal direct sum.
If $w=w_0 w_1\ldots w_s\in W$ with $w_i\in W^i, \; 0\le i\le s$
we have $w(x)= w(x^0, x^1,\ldots , x^s)= (x^0, w_1(x^1),\ldots,
w_s(x^s))$ for all $x\in \mathbb{R}^{n}$. The
direct product of the identity $P^0$ on $\mathbb {R}^{n_0}$ and Chevalley
mappings $P^i$ associated with $W^i$ acting on $\mathbb{R}^{n_i},\; 1
\le i\le s$, is a Chevalley map associated with the action of $W$
on $\mathbb{R}^{n}$.
For an irreducible $W$ (or for an irreducible component)
we will assume as we may that the degrees of the coordinate
polynomials $p_1,\ldots ,p_n$ are in increasing order:
$2=k_1\le \ldots \le k_n=h$, where $h$ is the Coxeter number of $W$.
In the reducible case, if there is an invariant subspace $\mathbb{R}^{n_0}$, the corresponding $p_i$s say
$P^0_j, 1\le j\le n_0$ are of degree $k_0=1$. Then for each $i=1\ldots s$
the $P^i_j$ are of degree $k^i_j$, for $j=1$ to $n_i$. The $k^i_1$ are
equal to 2, and $k^i_{n_i}$ is the Coxeter number of $W^i$. We denote
with $h$ is the maximal Coxeter number (or highest degree of the coordinate
polynomials) of the irreducible components.
Let $\mathcal{R}$ be the set of reflections different from identity
in $W$. The number of these reflections is
$\mathcal{R}^{\#}=d=\sum_{i=1}^n(k_i-1)$. For each $\tau \in
\mathcal{R}$, let $ \lambda_{\tau}$ be a linear form the kernel of
which is the hyperplane $ H_{\tau}=\{ x\in {\bf R}^n |\tau (x) = x
\} $.
The jacobian matrix $J_P$ is block diagonal. Let $I_{n_0}, J_{P^1},
\ldots, J_{P^{n_s}}$ be the diagonal blocks. The $n_0\times n_0$
minor in the upper left corner is $1$ but the determinants
$\abs{J_{P^1}}, \ldots, \abs{J_{P^s}}$, all vanish on $\mathbb
{R}^{n_0}$. The jacobian determinant of $P$ is $\abs{J_P}=c
\prod_{\tau \in \mathcal{R}} \lambda_{\tau}$ for some constant
$c\neq 0$. The critical set is the union of the $ H_{\tau}$ when
$\tau$ runs through $\mathcal{R}$. The invariant
subspace is the intersection $\bigcap_{\tau\in \mathcal R}H_\tau$ of
the reflecting hyperplanes.
A Weyl Chamber $C$ is a connected component of the regular
set. The other connected components are obtained by the
action of $W$ and the regular set is $\bigcup_{w\in W} w(C)$.
There is a stratification of ${\mathbf R}^n$ by the regular set, the
reflecting hyperplanes $ H_{\tau}$ and their intersections.
The mapping $P$ is neither injective
nor surjective but it induces an analytic diffeomorphism of $C$ onto
the interior of $P({\mathbf R}^n)$ and an homeomorphism
that carries the stratification from the
fundamental domain $\overline{C}$ onto $P(\mathbf{R}^n)$.
Finally, let us recall that the Chevalley mapping is proper and separates the orbits.
\section{ Whitney Functions and $r$-regular jets of order $m$.}
For Whitney functions of class $\mathcal{C}^r$ one can see
~\cite{14}, the notations of which will be used freely. \vskip 5pt
A jet of order $m\in {\mathbf N}$, on a locally closed set $E\subset
{\mathbf R}^n$ is a collection $A=(a_k)_{k\in {\bf N^n}\atop \abs{k}\le m}$
of real valued functions $a_k$ continuous on $E$. At
each point $x\in E$ the jet $A$ determines a polynomial $A_x(X)$,
and we sometimes speak of continuous polynomial fields instead of
jets. As a function, $A_x$ acts upon vectors $x'-x$ tangent
to ${\bf R}^n$ at $x$ and we write somewhat inconsistently:
\[ A_x: x' \mapsto A_x(x')=\sum_k {1\over k!} a_k(x)\; (x'-x)^k.\]
The space $J^m(E)$ of jets of order $m$ on E is naturally provided
with the Fr\'echet topology induced by the family of semi-norms:
$\abs {A}_m^{K_s}= \sup_{x\in K_s \atop \abs {k}\leq m}
\abs{a_k(x)}$, where $K_s$ runs through a countable exhaustive
collection of compact sets of $E$.
\noindent By formal derivation of $A$ of order $q\in \mathbb
{N}^n,\; \abs {q} \le m$, we get jets of the form $(a_{q+k})_{\abs{
k} \le m-\abs{ q}}$ inducing polynomials
\[(D^qA)_x(x')=a_q(x)+\sum_{k>q\atop \abs{k}\le m}{1\over (k-q)!}
a_k(x)\; (x'-x)^{k-q}.\]
\noindent For $0\le \abs{ q} \leq r\le m$, we put:
\[(R_xA)^q(x')=(D^q A)_{x'}(x')-(D^q A)_x(x').\]
\begin{defn} Let $A$ be a jet of order $m$ on $E$. For $r\le m ,\; A$ is
$r$-regular on $E$, if and only if for all compact set $K$ in $E$,
for $(x, x')\in K^2$, and for all $q\in {\bf N}^n$ with $\abs{ q}
\leq r$, it satisfies the Whitney conditions.
\[(\mathcal {W}_q^r) \quad (R_xA)^q(x') = o (\abs{
x'-x}^{r-\abs{ q}}),\; {\rm when} \abs{ x-x'} \to 0.\]
\end{defn}
\begin{rem} \label{D^q} If $m> r$ there is no need to
consider the truncated field $A^r$ instead of $A$ in the conditions
$({\mathcal W}_q^r)$. Actually $(R_xA^r)^q(x')$ and $(R_xA)^q(x')$
differ by a sum of terms $[a_k(x)/(k-q)!] \, (x'-x)^{k-q}$, with
$a_k$ uniformly continuous on $K$ and $ \abs{ k} -\abs{ q}
> r-\abs{ q} $.\end{rem}
The space of $r$-regular jets of order $m$ on $E$, is naturally
provided with the Fr\'echet topology defined by the family of
semi-norms: \[\norm{ A}^{K_s}_{r,m}=\abs{A}^{K_s}_m
+\sup_{(x,x')\in K_s^2 \atop x\neq x',\; \abs{k}\leq r} \Big(
{\abs{ (R_xA)^k(x')} \over \abs{ x-x'}^{r-\abs{k}}}\Big).\] when
$K_s, s\in \mathbb{N}$ runs through an exhaustive family of compact
sets in $E$. This Fr\'echet space is denoted by $\mathcal
{E}^{r,m}(E)$.
If $r=m,\; \mathcal{E}^r(E)$ is the space of Whitney fields of order
$r$ or Whitney functions of class ${\mathcal C}^r$ on $E$. When $E$
is locally closed, we have:
\begin{thm} \label{Wextsion} {\rm Whitney extension theorem} ~\cite{15}.
The restriction mapping of the space ${\mathcal E}^r({\mathbf R}^n)$
of functions of class ${\mathcal C}^r$ on ${\mathbf R}^n $ to the
space ${\mathcal E}^r(E)$ of Whitney fields of order $r$ on $E$, is
surjective. There is a linear section, continuous when the spaces
are provided with their natural Fr\'echet topologies. \end{thm}
In general the semi-norms $\norm{.}^{K_s}_r$ and $\abs{.}^{K_s}_r$
are not equivalent on $\mathcal{E}^{r}(E)$. They are when $E$ is
open. In this case $\mathcal{E}^r(E)$ is the space of fields of
Taylor polynomials of functions in $\mathcal{C}^r(E)$ and the two
spaces $\mathcal{E}^r(E)$ and $\mathcal{C}^r(E)$ may be identified.
When $A\in \mathcal{E}^r(E)$ and $\abs {q}\le r$, the formal derivation $D^q A$
and the regular derivative $\displaystyle{{\partial^{\abs {q}} A / \partial x^q}}$
define the same polynomial
\[(D^qA)_x(x')= \left({\partial^{\abs {q}} A\over \partial x^q}\right)_x (x')\]
and they can be identified.
If $E$ is open and if for all $q$ with $\abs{q}\le m$,
$a_q$ is a continuous partial derivative, then $A$ is $m$-regular.
So, when $E$ is open, if $A\in \mathcal{E}^{r,m}(E)$ but $A\notin \mathcal{E}^{m}(E)$,
for $\abs{q}>r$ the $a_q$ may or may not be interpreted as partial derivatives.
The identification of $D^qA$ and $\displaystyle{{\partial^{\abs {q}} A / \partial x^q}}$
may not be possible and for $\abs{\ell}r$ are not necessarily partial derivatives
of order $\abs{q}$, or are not necessarily continuous (\emph{e.g.}
they are continuous on the interior of $E$ but not on its border.)
Conversely, assuming that the compact $K$ is connected by
rectifiable arcs (or is a finite union of sets connected by
rectifiable arcs), Glaeser has proved :
\begin{prop} {\rm ~\cite{8}, ~\cite{14}} \label{prop2} If the norms
$\norm{.}_1^K$ and $\abs{.}_1^K$
are equivalent on $\mathcal{E}^1(K)$, then $K$ is 1-regular.\end{prop}
In appendix A we will show that:
\begin{thm} The image of the Chevalley mapping $P(\mathbb{R}^n)$ is Whitney 1-regular.
\end{thm}
As a consequence we have:
\begin{cor} On $\mathcal{E}^r(P(\mathbb{R}^n))$ the families of semi-norms $\norm{.}^{K_s}_r$ and
$\abs{.}^{K_s}_r$ are equivalent. \end{cor} Besides a geometric
property of $P(\mathbb{R}^n)$ which is of interest by itself,
Whitney 1-regularity plays an important part when studying the
algebras of composite mappings invariant by reflection groups in
finite class of differentiability.
\section{Algebra of $W$-invariant functions of class
$\mathcal{C}^{hr}$}\label{W-invariant}
There exists a linear and continuous mapping (see ~\cite {4}
\footnote{Even though the proof would be basically the same, the
1-regularity makes it simpler since the $r$-continuity on
$P(\mathbb{R}^n)$ of the field $F$ brings its $r$-regularity and
there is no need for an extension to $P^{-1}(\mathbb{R}^n)$.}):
\[L: \mathcal{C}^{hr}(\mathbb{R}^n)^W \To,
\quad L(f)=F,\quad f=F\circ P.\]
Clearly, $L$ is injective and surjective on its image.
On the interior of $P(\mathbb{R}^n)$, $F$ is of class
$\mathcal{C}^{hr}$. Let $S$ be a stratum of $\overline{C}$,
intersection of $p$ reflecting hyperplanes. On $P(S)$ the loss of
differentiability is governed by $W_S$, the isotropy subgroup of
$S$, and $F$ is of class $[hr/h_S]$, where $h_S$ is the highest
degree of the $W_S$-invariants. Moreover we have:
\begin {prop} $L(\mathcal{C}^{hr}(\mathbb{R}^n)^W$ is a space of rough ``compositors''.
More specifically if $F\circ P=f$, the partial derivatives of $F$ of
order $\beta =(\beta_1,\ldots,\beta_n)$ with $\sum_{i=1}^n
\beta_ik_i \le hr$ are continuous on $P(\mathbb{R}^n)$.
When $\sum_{i=1}^n \beta_ik_i > hr$, the $\partial^{|\beta|} F/\partial
P^{\beta}$ do not exist on some strata of $P(\mathbb{R}^n)$ but
there are continuous weights $\varepsilon_\beta\ge 0$, such that
$\varepsilon_\beta (x)(\partial^{\abs{\beta}} F/\partial P^{\beta})
\circ P(x)$ tends to zero when $x\in C$ tends to a stratum $S$ of $\overline{C}$ where
$(\partial^{|\beta|} F/\partial P^{\beta}) \circ P$ does not exist.
If we put $\varepsilon_\beta (x)(\partial^{\abs{\beta}} F/\partial P^{\beta}) \circ P(x) =0$
at these points, the $\varepsilon_\beta (\partial^{\abs{\beta}} F/\partial P^{\beta}) \circ P$
are continuous on $\overline{C}$.
\end{prop}
The proof of the first part of the statement was almost complete in
~\cite{4}. Let us briefly recall that it is sufficient to study the
problem for each $W^i$ acting on $\mathbb{R}^{n_i}$ so that we may
assume $W$ to be irreducible. We are to solve the system:
\begin{equation}
\left( {\partial f\over \partial x}\right) = \left(\left(
{\partial p_i\over \partial x_j}\right)_{1\le i\le n\atop 1\le j\le n}\right)
\left({\partial F\over \partial p} \circ P\right). \label{sys1}
\end{equation}
For the derivation with respect to $p_j$ we get:
\begin{equation}
\left \{ c \;\displaystyle{(\prod_{\tau\in \mathcal{R}}
\lambda_{\tau})\; \;{\partial F\over
\partial p_j}\circ P =\sum_{i=1}^n
(-1)^{i+j}M_{i,j}{\partial f \over \partial
x_i}},\; j=1\ldots ,n \right. \label{soleq2}
\end{equation}
There is a loss of differentiability of $1+d-s_j = k_j$ units, where $s_j$ is the degree of $M_{i,j}$.
This loss of differentiability is clear at the
origin. In the neighborhood of a stratum $S$ in $\overline{C}$,
we write $P=Q\circ V$ with $Q$ invertible and $V$ a Chevalley
mapping for the isotropy subgroup $W_{S}$ of $S$. We have
$M_{i,j}=\sum_{k=1}^n V_{k,j}Q_{i,k}$. The $V_{k,j}$ and accordingly
the $M_{i,j}$ are $(s_p^j-1)$-flat on $S $, where $s_p^j$ is the degree of $V_{k,j}$.
Since $p$ is the number of reflections in $W_{S},\; 1-s_p^j+p$ is the degree of
the $j^{\rm th}$ polynomial of $V$ which is less than or equal to $k_j$.
The remaining part of the proof
is exactly as in ~\cite{4} where the details are given for the case
$j=n$. The continuity on $P(\mathbb{R}^n)$ of the partial derivatives of $F$ of order $\beta
=(\beta_1,\ldots,\beta_n)$ with $\sum_{i=1}^n \beta_ik_i \le hr$ follows at once.
The study of the adherence of the ideal generated by $\abs{J_P}= c\prod_{\tau\in \mathcal{R}} \lambda_{\tau}$
in $\mathcal{C}^{k}(\mathbb{R}^n)$ and in spaces of multipliers shows that even after compensation by the $M_{i,j}$,
the functions $(\partial F/\partial p_j) \circ P$ and more generally
$(\partial^{\abs{\beta}} F/\partial P^{\beta}) \circ P$ are multipliers (see ~\cite{10} and ~\cite{3}) and this brings the second part of the statement.
When $\sum_{i=1}^n \beta_ik_i \le hr$,
we take $\varepsilon_\beta =1$. When $\sum_{i=1}^n \beta_ik_i > hr$, the
$\varepsilon_\beta$ are determined by products of $\sum_{i=1}^n \beta_ik_i- hr$
factors $\abs {\lambda_\tau}$. Using the 1-regular separation of the linear strata of $\overline{C}$,
it would be possible to express these $\varepsilon_\beta $ as powers of distances to the strata.
The complete determination of the weights $\varepsilon_\beta$ is tedious.
We will not do it here.
When computing $\partial^{\abs{\ell}}(F\circ P)/\partial x^\ell$, $\abs{\ell}\le hr$,
by the Faa di Bruno formula, the derivative $(\partial^{\abs{\beta}} F/\partial P^{\beta}) \circ P$
is multiplied by the product $\prod_{j=1}^n(D^1p_j)^{\mu_1^j}\ldots(D^qp_j)^{\mu_q^j}$,
with $\mu_1^j+\ldots+\mu_q^j=\beta_j$, and $\sum_j(1\mu_1^j+\ldots+q\mu_q^j)=\abs{\ell}\le hr$.
Since the $p_j$ are homogeneous of degree $k_j$, up to a multiplicative numerical constant
$(\partial^{\abs{\beta}} F/\partial P^{\beta}) \circ P$ is multiplied by
$\abs{x}^{\sum_j\beta_jk_j-\abs{\ell}}\le \abs{x}^{\sum_j\beta_jk_j-hr}$
and the product tends to 0 with $x$.
When $x$ tends to a stratum $S$, we write as above $P=Q\circ V$.
Each $D^s(Q\circ V)$ in the Faa di Bruno formula, may in turn be expanded by the same formula,
and the $D^tV$ thus obtained provide the power of the distance $d(x, S)$ we need.
It is possible to determine the $\varepsilon_\beta$ from these considerations. Actually
a good applicant that satisfies all the requirements is: $\varepsilon_\beta=\prod_{j=1}^n(D^1p_j)^{\mu_1^j}\ldots(D^qp_j)^{\mu_q^j}$,
with $\mu_1^j+\ldots+\mu_q^j=\beta_j$ for $j=1,\ldots,n$ and $\sum_j(1\mu_1^j+\ldots+q\mu_q^j)= hr$.
If we define the semi-norms:
\[\norm{F}^{hr}_{P(K_s)}=\max_{\abs{\beta}\le hr\atop x\in K_s}\abs{{\partial^{\abs{\beta}} F\over \partial P^\beta}\circ P(x) \varepsilon_\beta(x)},\]
the Faa di Bruno formula shows directly that $\norm{f}^{hr}_{K_s}\le C_{K_s}\norm{F}^{hr}_{P(K_s)}$,
and through an induction it also shows that actually the norms are equivalent.
$L$ is a Fr\'echet isomorphism of $\mathcal{C}^{hr}(\mathbb{R}^n)^W$ onto the subspace of compositors in
$\mathcal{E}^r(P(\mathbb{R}^n))$ on which $P^*$ is a section for $L$.
\section{Algebra of composite mappings
$P^*(\mathcal{E}^r(P(\mathbb{R}^n)))$} \label{closed-alg}
We consider the homomorphism induced by $P$:
\[P^*:\mathcal{E}^r(P(\mathbb{R}^n))\To \mathcal{E}^r(\mathbb{R}^n),\quad P^*(F)= F\circ P.\]
For any $(a,x)\in \mathbb {R}^n\times\mathbb{R}^n$, by the Taylor formula for $F$ between $P(a)$ and $P(x)$, we have:
\begin{equation} F[P(x)]= F[P(a)]+\sum_{1\le \abs{ \beta}\leq r} {1\over \beta!} D^{\beta} F[P(a)]\; (P(x)-P(a))^{\beta} +o(\abs{P(x)-P(a)}^r),
\label{eqn2}\end{equation} either by using an extension of $F$ to
$\mathcal{E}^r(\mathbb{R}^n)$ given by theorem ~\ref{Wextsion}
\footnote{We might also consider $\mathcal{C}^r(\mathbb{R}^n)$ as
the source of $P^*$, instead of $\mathcal{E}^r(P(\mathbb{R}^n))$.}
or by using a Taylor integral remainder along an integrable path
satisfying the inequality given by the 1-regularity property
~\cite{16}.
Expanding $P(x)-P(a)$ by the polynomial Taylor formula we get a polynomial in $x-a$ of degree $h$. Hence for $f=F\circ P$:
\begin{equation}f(x)= f(a)+\sum_{1\le \abs{\alpha}\leq hr} {1\over \alpha!}f_{\alpha}(a)\; (x-a)^{\alpha} +o(\vert P(x)-P(a)\vert^r).
\label{eqn3}\end{equation} On a compact $K\subset \mathbb{R}^n$
containing $[a,x]$, there exists a constant $C_K$ such that $\abs{
P(x)-P(a)}^r\le C_K \abs {x-a}^r$, and $f\in
\mathcal{E}^{r,hr}(\mathbb{R}^n)$.
When $\abs{\alpha} \leq r$ the $f_\alpha$ are partial derivatives, but when $ r< \abs{ \alpha} \leq hr$,
the $f_\alpha$ are obtained by the composition process with $\abs{\beta} \le r$ and are not partial derivatives in general.
Nevertheless, they allow the estimate of $f_{x}(x)-f_a(x)=o(\abs{P(x)-P(a)}^r)$ instead of $f_{x}(x)-f_a(x)=o(\abs{x - a}^r)$
as we would have with the truncated $f^r$.
The derivatives of $f$ of order $\abs{\alpha} \leq r$ are in
$\mathcal{E}^{r-\abs{\alpha},h(r-\abs{\alpha})}(\mathbb{R}^n)$ by
reference to the derivatives of $F$ of the same order. For an
example:
\[{\partial f\over\partial x_i}(x)=\sum_{j=1}^n {\partial p_j\over \partial x_i}(a) [{\partial F\over \partial p_j} \circ P(a)+
\sum_{1\le \abs{ \beta}\leq r-1} {1\over \beta!}{\partial^{\abs{\beta}+1}\over \partial P^\beta\partial p_j} F[P(a)]\;
(P(x)-P(a))^{\beta}] \]\[+o(\abs{P(x)-P(a)}^{r-1}).\]
\begin{defn} We say that $f\in J^{hr}(\mathbb{R}^n)$ pointwise belongs to $P^*(J^r(P(\mathbb{R}^n)))$, if for each $x$
there exists an $F\in J^r(P(\mathbb{R}^n))$ such that $f_x=(F\circ
P)_x$. \end{defn}
This means that for each $x\in \mathbb{R}^n$, the
polynomial $f_x(X)$ is $W$-invariant. Of course it is necessary that
$f$ be $W$-invariant but it is not sufficient. Let $g\in
J^{r}(\mathbb{R}^n)^W$, from ~\cite{4} we know that there is an
$F\in J^r(P(\mathbb{R}^n))$ such that $g=(F\circ P)^r$. Here we
consider $f=F\circ P$ before truncation and it is of degree $hr$.
\begin{lem} \label{lem4} If $f\in J^{hr}(\mathbb{R}^n)$ pointwise belongs to $P^*(J^r(P(\mathbb{R}^n)))$,
and we write $f=F\circ P$, then for $0\le \abs{\beta}\le r$, $F_{\beta}\circ P (x)$ is a linear combination
of some $f_{\alpha}(x)$, $0\le \abs{\alpha}\le hr$. In particular the $F_\beta$ are continuous,
$F\in J^r(P(\mathbb{R}^n)))$ and $f\in P^*(J^r(P(\mathbb{R}^n)))$.
\end{lem}
\noindent Clearly $f_0(x)=F_0(P(x))$.
\noindent By induction, let us first assume $r=1$. We identify
\[f_x(x')=f_0(x)+\sum_{1\leq \abs{ \alpha} \leq h}
{1\over \alpha !} f_{\alpha}(x) (x'_1-x_1)^{\alpha_1} ... (x'_{\ell}-x_{\ell})^{\alpha_l},\;{\rm with}\]
\[f_x(x')=F_0(P(x))+\sum_1^l F_i\circ P(x) (\sum_{\abs{ \alpha}=1}^{k_i} {1\over \alpha !}
{\partial^{\abs{ \alpha}}p_i\over \partial x^{\alpha}}(x)(x'-x)^{\alpha})\]
where $F_i$ stands for $F_\beta,\; \beta_i=1, \beta_j=0 \;{\rm if}\; i\neq j$.
\noindent For $\abs{\alpha} = k_i$, $\alpha_1\geq \alpha_2 \geq ...
\geq \alpha_n$, we get:
\begin{equation} f_{\alpha}=F_i\circ P {\partial^{k_i}p_i\over
\partial x^{\alpha}}+ \sum_{s>i} F_s\circ P
{\partial^{k_i}p_s\over \partial x^{\alpha}}.\label{eqn4}\end{equation}
In particular for $\alpha$ with $\abs{\alpha} =h , \;
\alpha_1\geq \alpha_2 \geq ... \geq \alpha_n$, we have:
\[f_{\alpha}=F_n\circ P {\partial^{h}p_n\over \partial x^{\alpha}},\]
and since $p_n$ is of degree $h$, there is a $\partial^{h}p_n/ \partial x^{\alpha}$ which is not 0.
Hence the result for $F_n$.
\noindent Solving the equations \eqref{eqn4} in succession gives the
result for the $F_i\circ P,\;i=1, ..., n$.\medskip
For more explicit computations, observe that if $W$ is reducible, it would be sufficient to study each
irreducible component in each subspace $\mathbb{R}^{n_i}$ and gather the results at the end.
For an irreducible component, we may use the polynomial invariants given in
~\cite{12}. Disregarding $D_n$ for a while, for all the other groups the $k_i$
are distinct and there is an invariant set of real linear forms
$\{L_1,\ldots ,L_v\}$ such that their symmetric functions $\sum_{i=1}^v L_i^{k}$
are $W$-invariant, and we may take $p_i(X)=\sum_{j=1}^v [L_j(X)]^{k_i}$
with $k_i$ as determined in ~\cite{6}.
At least one of the $L_j(X)$ contains a monomial in $X_1$,
bringing in $p_i(X)$ a monomial in $X_1^{k_i}$ that cannot be
canceled since the $k_i$ are all even, with 2 exceptions: $A_{n}$
and $I_2(p)$. For $I_2(p)$ we may choose $p_1(X)=X_1^2+X_2^2$ and
$p_2(X)=\sum_{j=1}^p(X_1\cos 2j\theta+X_2\sin 2j\theta)^p$ in
which the coefficient of $X_1^p$ is $\sum_{j=1}^p (\cos 2j\theta)^p \neq 0$.
For $A_n$ we may take $L_i(X)=X_i, i=1,\ldots,n+1$ and there is no possible
cancelation either.
Finally, for $D_n$ if we choose as basic invariant
polynomials $p_j(x)=\sum_{i=1}^n x_i^{2j}$, $ j=1,\ldots, n-1$ and
$q(x)=x_1x_2\ldots x_n$\footnote{With this choice, the degree of $p_n$ is $n$.
The highest degree $h=2n-2$ is the degree of $p_{n-1}$.},
we may use the above method when $1\le j\le n-1$, and consider
$ \partial^n q/ \partial x_1 \ldots \partial x_n = 1$
to get the continuity of $(\partial F/\partial q)\circ P$.\vskip 5pt
Anyway $\partial^{k_n}p_n/ \partial x_1^{k_n} =k_n ! c_n$ for some coefficient
$c_n\neq 0$, while for $j< n$, $\partial^{k_n}p_j/ \partial x_1^{k_n}=0$,
since the greatest exponent of $x_1$ in $p_j(x)$ is $k_ji$, the $ F_s\circ P$ are linear combinations
of $f_{\alpha},\; \abs{\alpha}\le h$, since $p_i(x)$ contains a monomial in
$x_1^{k_i}$, we have $\partial^{k_i}p_i/ \partial x_1^{k_i}
= k_i ! c_i$ for some coefficient $c_i\neq 0$,
while as above for $j*i} F_s\circ P \; (1/ k_i !)
\; \partial^{k_i}p_s/ \partial x_1^{k_i}$.
\noindent By using the induction assumption we get the result for $F_i\circ P$, and
by decreasing induction for all the $F_j\circ P,\; j=1,\ldots, n$.
Let us assume that the lemma is true when $r\le k$. When $\abs{\beta}=k$, $F_\beta \circ P$ is a
linear combination of some $f_{\alpha}$s with $\abs{\alpha} \le hk$. By using the basis step of
the induction for the function $g(x)= F_\beta \circ P (x)=G\circ P(x)$, we get that the $ G_i\circ P$
are linear combinations of $g_{\alpha}, \abs{\alpha}\le h$ and using the induction assumption for $g$
we get that for $\abs{\gamma}=k+1$, $F_{\gamma}\circ P$ is a linear combination of $f_{\alpha}$s with
$\abs{\alpha} \le h(k+1)$. This achieves the induction and the proof of the first part of the lemma.
Since $f\in J^{hr}(\mathbb{R}^n)$, the $f_\alpha, \abs{\alpha}\le hr$ are continuous.
Hence the $F_\beta\circ P$ are also continuous for $\abs{\beta}\le r$ and since $P$ is proper,
the $F_\beta$ themselves are continuous, so that $F\in J^r(P(\mathbb{R}^n))$. $\square$
From the lemma, we get that there exists a constant $C_K$ depending
only on $K$ and $W$, such that $\abs{F}_r^{P(K)}\le C_K\abs{f}_{hr}^K$.
Then, by using the 1-regularity there is a constant
$C^1_K$ such that $\norm{F}_r^{P(K)}\le C^1_K\abs{f}_{hr}^K$.
\begin{thm} \label{thm5} $P^*(\mathcal{E}^r(P(\mathbb{R}^n))$ is closed in $\mathcal{E}^{r,rh}(\mathbb{R}^n)$.
$P^*$ is an isomorphism of Fr\'echet spaces from $\mathcal{E}^r(P(\mathbb{R}^n)$ onto its image. \end{thm}
The algebra homomorphism:
$P^*:\mathcal{E}^r(P(\mathbb{R}^n)\To \mathcal{E}^{r,rh}(\mathbb{R}^n)$, is injective and surjective onto its
image. The inverse $(P^*)^{-1}$ exists and is continuous by the above inequality.
Let $(f_k)_{k\in N}$ be a Cauchy sequence in $P^*(\mathcal{E}^r(P(\mathbb{R}^n))$. This sequence converges in
$\mathcal{E}^{r,rh}(\mathbb{R}^n)$. For each $k$ there exists an $F_k\in \mathcal{E}^r(P(\mathbb{R}^n))$ such that
$f_k=F_k\circ P$, and since $\norm{F}^r_{P(K)}\le C^1_K\abs{f}^{hr}_K$, the sequence $(F_k)_{k\in N}$ is a
Cauchy sequence for the topology defined by the semi-norms $\norm{F}^r_{P(K)}$. Hence $(F_k)_{k\in N}$ converges
to an $F\in \mathcal{E}^r(P(\mathbb{R}^n)$. If we take the limit in $f_k=F_k\circ P$ we see that $f=F\circ P$ is also in
$P^*(\mathcal{E}^r(P(\mathbb{R}^n))$.
So $P^*(\mathcal{E}^r(P(\mathbb{R}^n))$ is closed in the Fr\'echet space $\mathcal{E}^{r,rh}(\mathbb{R}^n)$ and then
a Fr\'echet space itself. Then the Banach theorem shows that $P^*$ is a topological isomorphism onto its image.
\begin{rem}From ~\eqref{eqn2}, ~\eqref{eqn3}, we see that a `truncated' Faa di Bruno formula gives
$\abs{f}_{hr}^K\le C^2_K\abs{F}_r^{P(K)}$ for some constant $C^2_K$ that depends on the compact $K$, \emph{a fortiori}
$\abs{f}_{hr}^K \le C^2_K\norm{F}_r^{P(K)}$. This shows the continuity of the linear mapping $P^*$ directly and
without using the Whitney 1-regularity. \end{rem}
\begin{prop} $P^*(\mathcal{E}^r(P(\mathbb{R}^n))$ is closed in $\mathcal{E}^{r,rh}(\mathbb{R}^n)$ if and only if $P(\mathbb{R}^n)$
is Whitney 1-regular.\end{prop}
By theorem \ref{thm5}, the 1-regularity is sufficient for $P^*(\mathcal{E}^r(P(\mathbb{R}^n))$ to be closed. Conversely,
let us assume that $P^*(\mathcal{E}^r(P(\mathbb{R}^n))$ is closed and consider a sequence $(F_n)_{n\in N}$ in
$\mathcal{E}^1(P(\mathbb{R}^n)$, which is Cauchy for the topology induced by the semi-norms $\vert .\vert_1^{P(K)}$.
The sequence $(f_n=F_n\circ P)_{n\in N}$ in $P^*(\mathcal{E}^1(P(\mathbb{R}^n))\subset \mathcal{E}^{1,h}(\mathbb{R}^n)$
which is also Cauchy by Faa di Bruno's formula, converges to an $f$ in $P^*(\mathcal{E}^1(P(\mathbb{R}^n))$ which
is closed by assumption. As a consequence, the limit $F$ of $(F_n)_{n\in N}$ with $f=F\circ P$ is in
$\mathcal{E}^1(P(\mathbb{R}^n)$. Thus $\mathcal{E}^1(P(\mathbb{R}^n))$ is complete for the topology induced by the semi-norms
$\vert.\vert_1^{P(K)}$ and the Banach isomorphism theorem shows that this topology is equivalent to the topology induced by
the semi-norms $\Vert.\Vert_1^{P(K)}$. Glaeser's proposition \ref{prop2} then shows that $P(\mathbb{R}^n)$ is Whitney 1-regular.
\begin{rem} Finally, one might wish to prove the 1-regularity of $P(\mathbb{R}^n)$ by using Glaeser's proposition \ref{prop2}.
By Banach theorem, we just have to prove that $\mathcal{E}^1(P(\mathbb{R}^n))$ is complete for the topology induced by the
semi-norms $\abs{ .}_1^{P(K)}$. So we consider a sequence $(F_n)_{n\in N}\subset \mathcal{E}^1(P(\mathbb{R}^n)$ which is Cauchy
for the topology induced by these semi-norms. The sequence $(f_n=F_n\circ P)_{n\in N}$ is Cauchy in $\mathcal{E}^{1,h}(\mathbb{R}^n)$
since as we already noticed $\abs{ f}_{h}^K \le C^2_K\abs {F}_1^{P(K)}$, hence $(f_n)$ converges to an $f\in \mathcal{E}^{1,h}(\mathbb{R}^n)$,
which is of the form $F\circ P$ with $F=\lim_n F_n$.
The function $f$ induces a formally holomorphic jet still denoted by ${f} \in \mathcal{H}^{1,h}(\mathbb{R}^n)$ (see ~\cite{14} and ~\cite{4}).
If there was an extension $\tilde{f}$ to $\mathcal{H}^{1,h}(P^{-1}(\mathbb{R}^n))$ of the form $\tilde{f}=\tilde{F}\circ P$,
$\tilde{F}$ identified with $F$ on $P(\mathbb{R}^n)$, would be of class $\mathcal{C}^1$ on the regular image of $P$
in $\mathbb{R}^n$, and $1$-continuous everywhere by \ref{lem4}. Since the critical image is contained in the null
set of the discriminant polynomial, $\tilde{F}$ would be of class $\mathcal{C}^1$ everywhere in $\mathbb{R}^n$, and in particular
$F\in \mathcal{E}^1(P(\mathbb{R}^n))$. Unfortunately, no such extension is available. As mentioned above, Whitney's extension does
not take into account the part of the jet $f$ beyond the degree $r$. It is possible to provide a linear and continuous version of
{\L}ojasiewicz extension operator (see ~\cite{14}) that would give an $\tilde{f}\in \mathcal{H}^{1, h}(P^{-1}(\mathbb{R}^n))$,
but it is not an algebra homomorphism and $\tilde{f}$ would not be of the form $\tilde{F}\circ P$ with an $\tilde{F}$ of degree
$1$ and lemma \ref{lem4} does not apply.
\end{rem}
\appendix
\section{Whitney 1-Regularity of $P(R^n)$.}
The Whitney 1-regularity of the image $P(\mathbb{R}^n)$ of the Chevalley mapping associated with any finite reflection group
is a result which is of interest by itself.
In ~\cite{7}, theorem \ref{thm1} below was stated but lemma \ref{lem1} was not
proved for all Coxeter groups.
Since the 1-regularity is not altered by diffeomorphism, it does not depend on
the choice of coordinates. It does not depend on the choice of the set of basic invariants either,
since a change of basic invariants is an invertible polynomial map on the
target.
When $W$ is reducible we choose coordinates such that the Chevalley map $P$
is the product of the Chevalley maps $P^i$s associated with the irreducible components
$W^i$s of $W$ acting on the subspaces $\mathbb{R}^{n_i}$ of $\mathbb{R}^{n}$.
If for each $i$, $P^i(\mathbb{R}^{n_i})$ is 1-regular, so is
$P(\mathbb{R}^{n})$. As a consequence it is sufficient to prove the regularity
when $W$ is irreducible and from now on, we will assume $W$ to be
a finite Coxeter group.
\subsection{Strata of $P(\mathbb{R}^n)$ and minors of the jacobian.}
Let $C$ be a Weyl Chamber. The walls of $\bar{C}$ are contained in $n$ hyperplanes
$(H_{\omega})_{\omega \in \Omega}$, where $\Omega$ is a subset of
cardinal $n$ in $\mathcal {R}$.
A stratum $S$ of dimension $k$ in $\bar{C}$ is the intersection of $(n-k)$ of the $H_{\omega}$.
The $\lambda_{\tau}$ that are linear combinations of the $\lambda_{\omega}$ vanishing on $S$ will
also vanish there, so that $p \ge n-k$ hyperplanes $H_{\tau}, \tau\in \mathcal {R}$ will intersect along $S$.
The points in $S$ have the same isotropy subgroup $W_{S}$, generated by the reflections in the $p$ hyperplanes
$H_{\tau}$ containing $S$. In a neighborhood of $S$, since $P$ is $W_S$ invariant we can write $P=Q\circ V$, where
$Q$ is invertible and $V$ is a Chevalley mapping for $W_S$.
\noindent We write $W_S= W^0\times W^1 \times \cdots \times W^\ell $ where $W^0$ is the identity on $S$ and the $W^i$s
are the irreducible components of $W_S$. If we choose coordinates fitted to the orthogonal direct sum
$\mathbb {R}^n=\mathbb {R}^{k}\oplus \mathbb {R}^{n_1}\oplus \cdots \oplus \mathbb {R}^{n_\ell}$, and we have
$V={\rm Id}_k\times V_1\times \cdots\times V_\ell$.
\noindent The equation of an $H_{\tau}$ containing $S$ depends of the $x_{k+1},x_{k+2},\ldots, x_n$ but perhaps not
all of them. There is a partition among the $x_i$s, $k+1\le i\le n$ and a corresponding one among the $\lambda_\tau$.
Let $r*~~k$, one of the $(k+1)\times (k+1)$ minors does not vanish and
there may not be critical points.
Let us show that the critical points of $p_{k+1}$ on $P^{-1}(m^k)$ are non degenerate. At a critical point
\[{\partial p_{k+1}\over \partial x_i}-\lambda_k {\partial p_{k}\over \partial x_i}-\ldots
-\lambda_1{\partial p_{1}\over \partial x_i}=\sum_{j}\Big({\partial q_{k+1}\over \partial v_j}
-\lambda_k {\partial q_{k}\over \partial v_j}-\ldots -\lambda_1{\partial q_{1}\over \partial v_j}
\Big)\Big({\partial v_j \over\partial x_i}\Big)=0.\]
When $1\le i\le k,\; \partial v_{j}/\partial x_i =1$ if $i=j$, and $=0$ otherwise.
The only terms remaining in the sum are the
$({\partial q_{k+1}/ \partial v_i}-\lambda_k {\partial q_{k}/ \partial v_i}-\ldots -\lambda_1{\partial q_{1}/ \partial v_i})$
and these are 0.
\noindent When $k+1\le i\le n,\; \partial v_{j}/\partial x_i =0$ either because $v_j$ does not depend on $x_i$
or is an homogeneous polynomial of degree $\ge 2$ that vanishes with its derivatives on $S$, but the
$({\partial q_{k+1}/ \partial v_i}-\lambda_k {\partial q_{k}/ \partial v_i}-\ldots
-\lambda_1{\partial q_{1}/ \partial v_i})$ are not all 0, since $Q$ is invertible.
Actually the vectors $(\partial P/\partial x_1,\ldots,\partial P/\partial x_k)= (\partial Q/\partial x_1,
\ldots,\partial Q/\partial x_k)$ define the tangent plane to $P(S)$ of dimension $k$.
The $\partial P/\partial x_s, k+1\le s\le n$ are linear combinations of them, hence the vanishing of the bordering minors.
The $\partial Q/\partial x_s$, $k+1\le s\le n$, however, span the complement of the tangent space, and they are linearly
independent. As a consequence the minors of order $\ge k+1$ in the jacobian of $Q$ do not vanish, and for
$i\ge k+1,\;{\partial q_{k+1}/ \partial v_i}-\lambda_k {\partial q_{k}/ \partial v_i}-\ldots -\lambda_1{\partial q_{1}/ \partial v_i}\neq 0$.
In restriction to the kernel of the first differential, which is the orthogonal of $S$, in the quadratic differential
\[{\partial^2 p_{k+1}\over \partial x_i\partial x_j}-\lambda_k {\partial^2 p_{k}\over \partial x_i\partial x_j}
-\ldots -\lambda_1{\partial^2 p_{1}\over \partial x_i\partial x_j}\] \[=\sum_{r,s\ge k+1}({\partial^2 q_{k+1}\over
\partial v_r\partial v_s}-\lambda_k {\partial^2 q_{k}\over \partial v_r\partial v_s}-\ldots -\lambda_1
{\partial^2 q_{1}\over \partial v_r\partial v_s})({\partial v_r \over\partial x_i})({\partial v_s \over\partial x_j}) \]
\[+\sum_{r\ge k+1}({\partial q_{k+1}\over \partial v_r}-\lambda_k {\partial q_{k}\over \partial v_r}-\ldots -\lambda_1
{\partial q_{1}\over \partial v_r})({\partial^2v_r\over \partial x_i\partial x_j})\]
the mixed derivatives are all 0 either because $v_j$ does not depend on $x_i$ or $x_j$ or is an homogeneous polynomial
of degree $> 2$ or a sum of squares of $x_i$s. For the same reasons many terms in the pure derivatives also vanish, but some do not.
In each $V_s:\mathbb{R}^{n_s}\rightarrow \mathbb{R}^{n_s}$, let $v_s^1$ be the first $W^s$-invariant which is the sum of the squares
of the $x_i$ in $\mathbb{R}^{n_s}$. By the above remark, ${\partial q_{k+1}/ \partial v_s^1}-\lambda_k
{\partial q_{k}/ \partial v_s^1}-\ldots -\lambda_1{\partial q_{1}/ \partial v_s^1}$ does not vanish, and for each
$x_i\in \mathbb{R}^{n_s}$ of which $v_s^1$ actually depends, we have ${\partial^2v_s^1/ \partial x_i^2}=2$, so that:
\[{\partial^2 p_{k+1}\over \partial x_i^2}-\lambda_k {\partial^2 p_{k}\over \partial x_i^2}-\ldots -\lambda_1
{\partial^2 p_{1}\over \partial x_i^2}= 2({\partial q_{k+1}\over \partial v_s^1}-\lambda_k
{\partial q_{k}\over \partial v_s^1}-\ldots -\lambda_1{\partial q_{1}\over \partial v_s^1})\neq 0\]
Accordingly $p_{k+1}=q_{k+1}\circ V$ is a Morse function on $P_k^{-1}(m^k)$.
\noindent Observe that for each irreducible component, the quadratic differential is definite with the sign of
${\partial q_{k+1}/ \partial v_s^1}-\lambda_k {\partial q_{k}/ \partial v_s^1}-\ldots -\lambda_1
{\partial q_{1}/ \partial v_s^1}$, but the sign may be different for different irreducible components.
\noindent By the equivariant Morse lemma ~\cite{2}, in the neighborhood of a critical point at the intersection of
$S$ and $P_k^{-1}(m^k)$, $p_{k+1}$ is $W_S$-locally equivalent to a $W_S$-invariant quadratic form which is the
direct sum of definite quadratic forms in each $\mathbb{R}^{n_i}$.
$\square$
\begin{lem}\label{lem2}{\rm ~\cite{7}} The reconstruction of the topology of a level set of a function on
$\mathbb{R}_{+}^a\oplus \mathbb{R}_{+}^b$ in the neighborhood of the critical point $0\oplus 0$ with the quadratic differential
$Q_{+}\oplus Q_{-}$ is trivial if $a, b>0$ and consists of the birth (death) of a simplex otherwise.\end{lem}
\noindent Theorem \ref{thm1} may now be proved by induction on $k$, see ~\cite{7}.\medskip%
In particular, for almost all $m^k\in \Pi_k,\; P_k^{-1}(m^k)\cap \bar{C}$ is connected.
\begin{cor}\label{cor1} Every variety $P_k^{-1}(m^k) \cap \bar{C},\; k=1,\ldots, n$, is connected or empty. \end{cor}
\noindent The corollary may be derived from \ref{thm1}, exactly as in ~\cite{11}. Basically if $m^k$ is not in the regular image,
we have to consider two cases. First if for some $j,\;10$, the $x_i$s and as a consequence the $\partial p_{k+1}/ \partial p_j,\; j=1,\ldots, k $ are
bounded on $\overline {S}\cap K$.
\noindent $P$ is a homeomorphism of $\bar{S}$ onto its image $P(\bar{S})$, and so is $P_k$ (on any compact it
is continuous and one to one). Hence $p_{k+1}$ which is continuous with respect to the variables $(x_1,\ldots,x_k)$
on $\bar{S}$, is also continuous in the variables $(p_1,\ldots,p_k)$ on $P_k(\bar{S})$ and moreover by the previous paragraph it is Lipschitz.
The border of $\Pi_{k+1}$ is contained in the images $P_{k+1}(\bar{S})$ of closed strata of dimension $k$.
These images are graphs of functions $p_{k+1}$ on $P_k(\bar{S})$. By \ref{cor2}, the graph of one of the $p_{k+1}$,
say $p_{k+1}^{\rm max}$, is above the others and another, say the graph of $p_{k+1}^{\rm min}$, is below the others.
In $\Pi_k$, the images of the closure of strata of dimension $k$ will intersect along the images of strata of lesser
dimension. Above such points of intersection the mapping $p_{k+1}^{\rm max}$ (resp. $p_{k+1}^{\rm min}$) may and will
change but globally $p_{k+1}^{\rm max}$ (resp. $p_{k+1}^{\rm min}$) will still be continuous and Lipschitz since the
functions above the two strata are glued by their common value above the stratum of lesser dimension along which they intersect.
Now, it would be easy to get for any two points $A$ and $B$ in a compact $K\subset P(\mathbb{R}^n)$ a continuous arc
$AB\subset K$ of length $\ell(AB) \leq k_K \abs{AB}$, following the method in ~\cite{11} and already in ~\cite{3}.
This justifies the statement in ~\cite{1}: \emph {the prism between graphs of Lipschitz functions over a Whitney
1-regular domain is Whitney 1-regular.}
$\Pi_1$ is Whitney 1-regular. By induction, assuming $\Pi_{k}$ to be Whitney 1-regular, the prism $\Pi_{k+1}$ over it
and between the graphs of the Lipschitz functions $p_{k+1}^{\rm min}$ and $p_{k+1}^{\rm max}$ will also be Whitney 1-regular.
Hence all the $\Pi_k$ and in particular $\Pi_n=P(\mathbb{R}^n)$ are Whitney 1-regular and we can state:
\begin{thm} The image of the Chevalley mapping $P(\mathbb{R}^n)$ is Whitney 1-regular.
\end{thm}
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