Finite Reflection Groups: Invariant functions and functions of the Invariants in finite class of differentiability

TL;DR

This paper investigates the regularity of invariant functions under finite reflection groups, establishing conditions under which such functions can be expressed as functions of invariants with specific differentiability classes.

Contribution

It characterizes the space of invariant functions that can be written as functions of invariants with a given smoothness and identifies when the composition achieves higher differentiability.

Findings

Determines when invariant functions of class C^{hr} can be expressed as functions of invariants of class C^r.

Identifies the subspace of functions F of class C^r for which F∘P is of class C^{hr}.

Provides a detailed analysis of differentiability properties in the context of finite reflection groups.

Abstract

Let $W$ be a finite reflection group. A $W$-invariant function of class~$C^{\infty}$ may be expressed as a functions of class $C^{\infty}$ of the basic invariants. In finite class of differentiability, the situation is not this simple. Let~$h$ be the greatest Coxeter number of the irreducible components of $W$ and $P$ be~the Chevalley mapping, if $f$ is an invariant function of class $C^{hr}$, and $F$ is the function of invariants associated by $f=F\circ P$, then $F$ is of class $C^r$. However if~$F$ is of class $C^r$, in general $f=F\circ P$ is of class $C^r$ and not of class $C^{hr}$. Here we determine the space of $W$-invariant functions that may be written as functions of class $C^r$ of the polynomial invariants and the subspace of functions $F$ of class $C^r$ of the invariants such that the invariant function $f=F\circ P$ is of class $C^{hr}$.

Equations

$$P:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n},\quad P(x)=\big{(}p_{1}(x),...,p_{n}(x)\big{)}.$$

$$C^{hr}({\mathbb{R}}^n{)}^W\ni f⟶F\in C^r({\mathbb{R}}^n)$$

$$W=W^0\times W^1\times \cdots \times W^s$$

$$w(x)=w(x^{0},x^{1},\ldots,x^{s})=\big{(}x^{0},w_{1}(x^{1}),\ldots,w_{s}(x^{s})\big{)}.$$

$$k_1=2\le \cdots \le k_n=h,$$

$${\mathcal{R}}^{\#}=d=\sum _{i=1}^n(k_i-1).$$

$$H_{\tau}=\big{\{}x\in\mathbb{R}^{n}\;;\;\tau(x)=x\big{\}}.$$

$$|J_P|=c\prod _{\tau \in \mathcal{R}}{\lambda }_{\tau }$$

$$A=(a_k{)}_{k\in {\mathbb{N}}^n,|k|\le m}$$

$$A_x:x^{\prime }⟼A_x(x^{\prime })=\sum _{\begin{array}{c}k\in {\mathbb{N}}^n\\ |k|\le m\end{array}}\frac{1}{k!}{a}_k(x)(x^{\prime }-x{)}^k.$$

$$|A|_{K_{s}}^{m}=\sup_{\begin{subarray}{c}x\in K_{s}\\ |k|\leq m\end{subarray}}\big{|}a_{k}(x)\big{|},$$

$$(D^{q}A{)}_x(x^{\prime })=a_q(x)+\sum _{\begin{array}{c}k>q\\ |k|\le m\end{array}}\frac{1}{(k-q)!}{a}_k(x)(x^{\prime }-x{)}^{k-q}.$$

$$(R_{x}A{)}^q(x^{\prime })=(D^{q}A{)}_{x^{\prime }}(x^{\prime })-(D^{q}A{)}_x(x^{\prime }).$$

$$(W_{q}^{r})\hskip 56.9055pt(R_{x}A)^{q}(x^{\prime})=o\big{(}|x^{\prime}-x|^{r-|q|}\big{)},$$

$$\|A\|^{r,m}_{K_{s}}=|A|_{K_{s}}^{m}+\sup_{\begin{subarray}{c}(x,x^{\prime})\in K^{2}_{s}\\ x\neq x^{\prime},\,|k|\leq r\end{subarray}}\Big{(}\frac{|(R_{x}A)^{k}(x^{\prime})|}{|x-x^{\prime}|^{r-|k|}}\Big{)}$$

$$(D^{q}A)_{x}(x^{\prime})=\Big{(}\frac{\partial^{|q|}A}{\partial x^{q}}\Big{)}_{\!x}(x^{\prime})$$

$$P^{*}:\mathcal{E}^{r}\big{(}P(\mathbb{R}^{n})\big{)}\longrightarrow\mathcal{E}^{r}(\mathbb{R}^{n}),\quad\;P^{*}(F)=F\circ P.$$

$$F\big{[}P(x)\big{]}=F\big{[}P(a)\big{]}+\sum_{1\leq|\beta|\leq r}\frac{1}{\beta!}D^{\beta}F\big{[}P(a)\big{]}\big{(}P(x)-P(a)\big{)}^{\beta}+o\big{(}|P(x)-P(a)|^{r}\big{)}.$$

$$f(x)=f(a)+\sum_{1\leq|\alpha|\leq hr}\frac{1}{\alpha!}f_{\alpha}(a)(x-a)^{\alpha}+o\big{(}|P(x)-P(a)|^{r}\big{)}.$$

$$f_x(x^{\prime })=f_0(x)+\sum _{1\le |\alpha |\le h}\frac{1}{\alpha !}{f}_{\alpha }(x)(x_1^{\prime }-x_1{)}^{{\alpha }_1}\cdots (x_{\ell }^{\prime }-x_{\ell }{)}^{{\alpha }_{\ell }}$$

$$f_{x}(x^{\prime})=F_{0}\big{(}P(x)\big{)}+\sum_{1}^{\ell}F_{i}\circ P(x)\Big{(}\sum_{|\alpha|=1}^{k_{i}}\frac{1}{\alpha!}\frac{\partial^{|\alpha|}p_{i}}{\partial x^{\alpha}}(x)(x^{\prime}-x)^{\alpha}\Big{)},$$

$$f_{\alpha }=F_i\circ P\frac{{\partial }^{k_i}{p}_i}{\partial x^{\alpha }}+\sum _{s>i}{F}_s\circ P\frac{{\partial }^{k_i}{p}_s}{\partial x^{\alpha }}\cdot$$

$$p_1(X)=X_1^2+X_2^{2}and{p}_2(X)=\sum _{i=1}^p(X_1\cos 2j\theta +X_2\sin 2j\theta {)}^p$$

$$p_j=\sum _{i=1}^n{x}_i^{2j}(j=1,\dots ,n-1)andq(x)=x_1{x}_2\cdots x_n,$$

$$c_n{F}_n\circ P(x)=\frac{1}{k_n!}{f}_{k_n,0,\dots ,0}(x)with{c}_n\ne 0.$$

$$\frac{{\partial }^{k_i}{p}_i}{\partial x_1^{k_i}}=k_i!c_i$$

$$f_{k_i,0,\dots ,0}=c_i{k}_i!F_i\circ P+\sum _{s>i}{F}_s\circ P\frac{{\partial }^{k_i}{p}_s}{\partial x_1^{k_i}}\cdot$$

$$|F{|}_{P(K)}^r\le C_K|f{|}_K^{hr}.$$

$$(P^{*})^{(-1)}:\mathcal{J}\longrightarrow\mathcal{E}^{r}\big{(}P(\mathbb{R}^{n})\big{)}$$

$$P^{*}:\mathcal{E}^{r}\big{(}P(\mathbb{R}^{n})\big{)}\longrightarrow\mathcal{J}$$

Full text

1. Introduction

Let $W$ be a finite reflection group acting orthogonally on $\mathbb{R}^{n}$. The algebra of invariant polynomials is generated by $n$ algebraically independent $W$-invariant homogeneous polynomials and the degrees of these polynomials are uniquely determined [3], [4]. Let $p_{1},\ldots,p_{n}$ be a set of basic invariants, we define the “Chevalley” mapping

[TABLE]

Glaeser’s theorem [7] shows that $W$-invariant functions of class $C^{\infty}$ may be expressed as functions of class $C^{\infty}$ of the basic invariants, and that the subalgebra $P^{*}(C^{\infty}(\mathbb{R}^{n}))$ of composite mappings of the form ${F\circ P}$ with $F$ of class $C^{\infty}$, is closed in $C^{\infty}(\mathbb{R}^{n})$.

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 $W$. We have:

Proposition 1.1** (see [1]).**

There exists a linear and continuous mapping

[TABLE]

such that $f=F\circ P$.

A general counter-example shows that this result is the best possible.

However if $F$ is of class $C^{r}$, in general $f=F\circ P$ is of class $C^{r}$ and not of class $C^{hr}$. So the following questions naturally arise:

1. What is the space of $W$-invariant functions that may be written as functions of class $C^{r}$ of the polynomial invariants, and

2. What is the subspace of functions $F$ of class $C^{r}$ of the invariants such that the invariant function $f=F\circ P$ is of class $C^{hr}$? To answer this second question we will complete the results given in [1] and determine the subspace of $\mathcal{E}^{r}(P(\mathbb{R}^{n}))$ isomorphic to $C^{hr}(\mathbb{R}^{n})^{W}$.

2. The Chevalley mapping

When the reflection group $W$ is reducible, it is the direct product of its irreducible components, say

[TABLE]

and we may write $\mathbb{R}^{n}=\mathbb{R}^{n_{0}}\oplus\mathbb{R}^{n_{1}}\oplus\cdots\oplus\mathbb{R}^{n_{s}}$ as an orthogonal direct sum. The first component $W^{0}$ is the identity on the $W$-invariant subspace $\mathbb{R}^{n_{0}}$, and $W^{i}$ is an irreducible finite Coxeter group acting on $\mathbb{R}^{n_{i}}$ for $i=1,\ldots,s$. We choose coordinates that fit with this orthogonal direct sum: if $w=w_{0}w_{1}\cdots w_{s}\in W$ with $w_{i}\in W^{i},0\leq i\leq s$, we have for all $x\in\mathbb{R}^{n}$

[TABLE]

The direct product of the identity $P^{0}$ acting on $\mathbb{R}^{n_{0}}$ and Chevalley mappings $P^{i}$ associated with $W^{i}$ acting on $\mathbb{R}^{n_{i}}$, $1\leq i\leq s$, is a Chevalley map associated with $W$ acting on $\mathbb{R}^{n}$.

For an irreducible $W$ (or for an irreducible component) we may assume that the degrees of the polynomials $p_{1},\ldots,p_{n}$ are in increasing order

[TABLE]

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 $P^{0}_{j},1\leq j\leq n_{0}$ are of degree $k_{0}=1$. Then for each $i=1,\ldots,s$ the $P^{i}_{j}$ are of degree $k^{i}_{j}$, $1\leq j\leq n_{i}$. The $k^{i}_{1}$ are equal to $2$, and $k^{i}_{n_{i}}$ is the Coxeter number of $W^{i}$. We denote with $h$ 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

[TABLE]

For each $\tau\in\mathcal{R}$, we consider a linear form $\lambda_{\tau}$, the kernel of which is the reflecting hyperplane

[TABLE]

The jacobian matrix $J_{P}$ of $P$, 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 $|J_{P^{1}}|,\ldots,|J_{P^{n_{s}}}|$, all vanish on $\mathbb{R}^{n_{0}}$. The Jacobian determinant of $P$ is

[TABLE]

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 reflection 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 $\mathbb{R}^{n}$ 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 $P(\mathbb{R}^{n})$ and an homeomorphism that carries the stratification from the fundamental domain $\mkern 2.0mu\overline{\mkern-2.0muC}$ onto $P(\mathbb{R}^{n})$. Finally let us recall that the Chevalley mapping is proper and separates the orbits.

3. Whitney functions and $r$-regular jets of order $m$

A general study of Whitney functions may be found in [9], the notations of which will be used freely.

A jet of order $m\in\mathbb{N}$, on a locally closed set $E\subset\mathbb{R}^{n}$ is a collection

[TABLE]

of real valued functions $a_{k}$ continuous on $E$. At each point $x\in E$ the jet $A$ determines a polynomial of degree $m$, $A_{x}(X)$ and we will sometimes speak of continuous polynomial fields instead of jets. As a function, $A_{x}(X)$ acts on vectors $x^{\prime}-x$ tangent to $\mathbb{R}^{n}$ and we write

[TABLE]

The space $J^{m}(E)$ of jets of order $m$ on $E$ is naturally provided with the Fréchet topology induced by the family of semi-norms

[TABLE]

where $K_{s}$ runs through an exhaustive countable collection of compact sets of $E$.

By formal derivation of $A$ of order $q\in\mathbb{N}^{n},|q|\leq m$, we get jets of the form $(a_{q+k})_{|k|\leq m-|q|}$, inducing polynomials

[TABLE]

For $0\leq|q|\leq r\leq m$, we put

[TABLE]

Definition 3.1**.**

Let $A$ be a jet of order $m$ on $E$. For $r\leq m$, we say that $A$ is $r$-regular on $E$, if for all compact set $K\subset E$, for $(x,x^{\prime})\in K^{2}$, and for all $q\in\mathbb{N}^{n}$ with $|q|\leq r$, it satisfies the Whitney conditions

[TABLE]

Remark 3.2**.**

If $m>r$ there is no need to consider the truncated field $A^{r}=(a_{k})_{k\in\mathbb{N}^{n},|k|\leq r}$ instead of $A$ in the conditions $(W_{q}^{r})$. Actually $(R_{x}A^{r})^{q}(x^{\prime})$ and $(R_{x}A)^{q}(x^{\prime})$ differ by a sum of terms $[\frac{a_{k}(x)}{(k-q)!}](x^{\prime}-x)^{k-q}$, with $a_{k}$ uniformly continuous on $K$ and $|k|-|q|>r-|q|$.

The space of $r$-regular jets of order $m$ on $E$ is naturally provided with the Fréchet topology defined by the family of semi-norms

[TABLE]

when $K_{s},s\in\mathbb{N}$ runs through an exhaustive family of compact sets in $E$. This Fréchet space will be 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 $C^{r}$ on $E$. When $E$ is locally closed we have:

Theorem 3.3** (see [11]).**

The restriction mapping of the space $\mathcal{E}^{r}(\mathbb{R}^{n})$ to the space $\mathcal{E}^{r}(E)$ is surjective. There is a linear section, continuous when the spaces are provided with their natural Fréchet topologies.

When $A\in\mathcal{E}^{r}(E)$ and $|q|\leq r$, the formal derivation $D^{q}A$ and the regular derivative $\partial^{|q|}A/\partial x^{q}$ define the same polynomial

[TABLE]

and they may be identified.

In general the semi-norms $\|.\|^{r}_{K_{s}}$ and $|.|^{r}_{K_{s}}$ are not equivalent on $\mathcal{E}^{r}(E)$. However if $E$ is an open set $\mathcal{O}$, the semi-norms are equivalent. In this case $\mathcal{E}^{r}(\mathcal{O})$ is the space of fields of Taylor polynomials of functions in $C^{r}(\mathcal{O})$ and the two spaces $\mathcal{E}^{r}(\mathcal{O})$ and $C^{r}(\mathcal{O})$ may be identified.

Definition 3.4** (see [10]).**

We say that a compact set $K\subset\mathbb{R}^{n}$ connected by rectifiable arcs, is Whitney $1$-regular if the geodesic distance in K is equivalent to the Euclidian distance. That is, there is a constant $k_{K}>0$ such that for all $(x,x^{\prime})\in K^{2}$, there is a rectifiable arc from $x$ to $x^{\prime}$ in $K$ with length $\ell(x,x^{\prime})\leq k_{K}|x-x^{\prime}|$.

Proposition 3.5** (see [10]).**

If the compact set $K$ is $1$-regular, the norms $\|.\|_{K}^{r}$ and $|.|_{K}^{r}$ are equivalent on $\mathcal{E}^{r}(K)$.

Proposition 3.6** (see [2]).**

The image by the Chevalley mapping $P$ of any closed ball centered at the origin is Whitney $1$-regular.

This entails that any compact in $P(\mathbb{R}^{n})$ is Whitney 1-regular. Hence on $\mathcal{E}^{r}(P(\mathbb{R}^{n}))$ the families of semi-norms $\|.\|_{K}^{r}$ and $|.|_{K}^{r}$ are equivalent.

4. Functions of class $\mathcal{C}^{r}$ of the invariants and invariant functions

We consider the homomorphism induced by $P$:

[TABLE]

For any $(a,x)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$, by the Taylor formula for $F$ between $P(a)$ and $P(x)$, either by using an extension of $F$ to $\mathcal{E}^{r}(\mathbb{R}^{n})$ given by Whitney extension theorem or by using a Taylor integral remainder along an integrable path satisfying the inequality given by the $1$-regularity property of $P(\mathbb{R}^{n})$, we have:

[TABLE]

Expanding $P(x)-P(a)$ by the polynomial Taylor formula, we get a polynomial in $x-a$ of degree $h$. Hence for $f=F\circ P$,

[TABLE]

On a compact $K\subset\mathbb{R}^{n}$ containing $[a,x]$, there exists a constant $C_{K}$ such that $|P(x)-P(a)|^{r}\leq C_{K}|x-a|^{r}$.

When $|\alpha|\leq r$ the $f_{\alpha}$ are continuous derivatives. When $r<|\alpha|\leq hr$ the $f_{\alpha}$ are obtained by the composition process with $|\beta|\leq r$ and they are continuous, so $f$ belongs to $\mathcal{E}^{r,hr}(\mathbb{R}^{n})$.

Lemma 4.1**.**

Let $F$ be of class $C^{r}$ and $f=F\circ P$. For $0\leq|\beta|\leq r,\,F_{\beta}\circ P(x)$ is a linear combination of some $f_{\alpha}(x),\,0\leq|\alpha|\leq hr$.

Proof.

We have $f_{0}(x)=F_{0}(P(x))$. By induction, let us first assume $r=1$. We identify

[TABLE]

with

[TABLE]

where $F_{i}$ stands for $F_{\beta},\beta_{i}=1,\beta_{j}=0$ if $i\neq j$.

For $|\alpha|=k_{i}$, we get

[TABLE]

In particular when $|\alpha|=h$, we have $f_{\alpha}=F_{n}\circ P\;({\partial^{h}p_{n}}/{\partial x^{\alpha})}$ and since $p_{n}$ is of degree $h$ there is a $\partial^{h}p_{n}/\partial x^{\alpha}$ which is not [math].

Hence the result for $F_{n}$. Solving the equations in succession gives the result for the $F_{i}\circ P$, $i=1\ldots n$.

For more explicit computations, observe that if $W$ is reducible, it would be sufficient to study each irreducible component $W_{i}$ in each $\mathbb{R}^{n_{i}}$ and gather the results at the end. For an irreducible component we may use the polynomial invariants given in [8]. Disregarding $D_{n}$ for a while, for the other groups the $k_{i}$ are distinct and there is an invariant set of linear forms $\{L_{1},\ldots,L_{v}\}$ such that their symmetric functions $\sum_{j=1}^{v}L_{j}^{k}$ are $W$-invariant and we may take $p_{i}=\sum_{j=1}^{v}[L_{j}(x)]^{k_{i}}$ with the $k_{i}$s determined in [4]. At least one of the $L_{j}$ 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 two exceptions: $A_{n}$ and $I_{2p}$.

$\scriptstyle\triangleright$ For $I_{2p}$ we may choose

[TABLE]

in which the coefficient of $X_{1}^{p}$ is $\sum_{j=1}^{P}(\cos 2j\theta)^{p}\neq 0$.

$\scriptstyle\triangleright$ For $A_{n}$ we may take $L_{i}=X_{i},i=1,\ldots,n+1$ and there is no possible cancelation either.

Finally for $D_{n}$ if we choose as basic invariant polynomials

[TABLE]

we may use the above method when $1\leq j\leq n-1$, and consider $\partial^{n}q/\partial x_{1}\cdots\partial x_{n}=1$ to get the continuity of $(\partial F/\partial q)\circ P$.

Anyway, one has $\partial^{k_{n}}p_{n}/\partial x_{1}^{k_{n}}=k_{n}!c_{n}$ for some constant $c_{n}\neq 0$, 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 $k_{j}<k_{n}$. The identification shows that

[TABLE]

Assuming that when $s>i$, the $F_{s}\circ P$ are linear combinations of the $f_{\alpha}$, $|\alpha|\leq h$, since $p_{i}(x)$ contains a monomial in $x_{1}^{k_{i}}$, we have

[TABLE]

for some constant $c_{i}\neq 0$, and $\partial^{k_{i}}p_{j}/\partial x_{1}^{k_{i}}=0$ for $j<i$. The identification now gives

[TABLE]

With the induction assumption we have 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 statement of the lemma is true when $r\leq k$. When $|\beta|=k,F_{\beta}\circ P$ is a linear combination of some $f_{\alpha}$s with $|\alpha|\leq hk$. By using the basis step of the induction for the function $g=G\circ P$ with $G=F_{\beta}$, we get that the $G_{i}\circ P$ are linear combinations of $g_{\alpha}$, $|\alpha|\leq h$. The induction assumption for $g$ shows that for $|\gamma|=k+1$, $F_{\gamma}\circ P$ is a linear combination of $f_{\alpha}$, $|\alpha|\leq h(k+1)$. This achieves the induction and the proof of the lemma. ∎

From the lemma we get that for any compact $K\subset\mathbb{R}^{n}$ there is a constant $C_{K}$ depending only on $K$ and $W$ such that

[TABLE]

Any compact in $P(\mathbb{R}^{n})$ being Whitney $1$-regular, there is a constant $C_{K}^{1}$ such that $\|F\|^{r}_{P(K)}\leq C_{K}^{1}|f|^{hr}_{K}$, and of course $\|F\|^{r}_{P(K)}\leq C_{K}^{1}\|f\|^{r,hr}_{K}$.

The linear injective mapping $P^{*}$ is surjective on its image $\mathcal{J}$. This image $\mathcal{J}$ is the space of invariant functions associated with fonctions of class $C^{r}$ of the invariants. By the above inequality, the inverse

[TABLE]

is continuous. Hence by the Banach theorem it is an isomorphism of Fréchet spaces. The image $\mathcal{J}\subset{\mathcal{E}^{r,hr}(\mathbb{R}^{n})}$ of $P^{*}$ is closed in both $\mathcal{E}^{r}(\mathbb{R}^{n})$ and $\mathcal{E}^{r,hr}(\mathbb{R}^{n})$.

The map

[TABLE]

is also an isomorphism of Fréchet spaces and for each compact $K\subset\mathbb{R}^{n}$, there is a constant $C_{K}$ such that $\|f\|^{r,hr}_{K}\leq\|F\|^{r}_{P(K)}$.

Conclusion: the space of invariant functions given by a function of class $C^{r}$ of the invariants is a closed subspace of $\mathcal{E}^{r,hr}(\mathbb{R}^{n})$, isomorphic to the Fréchet space $\mathcal{E}^{r}(P(\mathbb{R}^{n})$ .

5. Invariant functions of class $C^{hr}$ and functions of the invariants

There exists a linear and continuous mapping (see [1]):

[TABLE]

Thanks to the Whitney extension theorem, the target might be $C^{r}(\mathbb{R}^{n})$ as well. The map $L$ is a version of $P^{*(-1)}$, we gave it another name because the source and target are different from those of $P^{*(-1)}$ in the previous section.

Clearly $L$ is injective and surjective on its image.

Let $C$ be a Weyl chamber. Its image $P(C)$ is the interior of $P(\mathbb{R}^{n})$ where F is of class $C^{hr}$. The loss of differentiability happens on the border of $P(\mathbb{R}^{n})$.

Let $S$ be a stratum of $\mkern 2.0mu\overline{\mkern-2.0muC}$, 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 $C^{hr/h_{S}}$, where $h_{S}$ is the highest degree of the $W_{S}$-invariants. Moreover, we have:

Proposition 5.1**.**

If $f=F\circ P$ is of class $C^{hr}$, the derivatives of $F$ of order $\beta=(\beta_{1},\ldots,\beta_{n})$ with $\sum_{i=1}^{n}\beta_{i}k_{i}\leq hr$, are continuous on $P(\mathbb{R}^{n})$.

Proof.

Let us begin with two remarks. First it is sufficient to study the problem for each $W^{i}$ acting on $\mathbb{R}^{n_{i}}$, so we may and will assume $W$ to be irreducible. Then since $P$ is proper if $\partial^{|\beta|}F/\partial x^{\beta}\circ P$ is continuous, so is $\partial^{|\beta|}F/\partial x^{\beta}$.

The first derivatives of $F$ with respect to $p_{j}$ are solutions of the system:

[TABLE]

The solutions are given by:

[TABLE]

where $M_{i,j}$ is the Jacobian of the polynomial mapping:

[TABLE]

This mapping is invariant by the subgroup $W_{i}$ of $W$, that leaves invariant the $i^{th}$ coordinate axis. $W_{i}$ is a reflection group, product of the identity on $\mathbb{R}e_{i}$ and the reflection group generated by the set $\mathcal{R}_{i}\subset\mathcal{R}$ of those of the hyperplanes generating $W$ containing $\mathbb{R}e_{i}$.

Actually $M_{i,j}$ is an homogeneous polynomial of degree $s_{j}=\sum_{i=1,i\neq j}^{n}(k_{i}-1)$ which is $s_{j}-1$ flat at the origin and divisible by $\prod_{\tau\in\mathcal{R}_{i}}\lambda_{\tau}$.

At the origin there is a loss of differentiability of $1+d-s_{j}=k_{j}$ units. In the neighborhood of a stratum $S$ intersection of $p$ reflection hyperplanes, we write

[TABLE]

with $Q$ invertible and $V$ a Chevalley mapping for the isotropy subgroup $W_{S}$ of $S$. We have

[TABLE]

The $V_{l,j}$ and accordingly the $M_{i,j}$ are $(s_{p}^{j}-1)$-flat on $S$, where $s_{p}^{j}$ is the degree of the $V_{l,j}$. Since $p$ is the number of reflections in $W_{S}$, there is a loss of differentiability of $1+p-s_{p}^{j}$ equal to the degree $k^{\prime}_{j}$ of the $j^{\mathrm{th}}$ polynomial of $V$ which is less than or equal to $k_{j}$. So overall the derivation of $F$ with respect to $p_{j}$ brings a loss of differentiability of $k_{j}$ units.

As a consequence a derivation of order $\beta$ brings a loss of differentiability of $\sum_{i=1}^{n}\beta_{i}k_{i}$ units and if this sum is less than $hr$, the derivative will be continuous on $P(\mathbb{R}^{n})$. ∎

In the neighborhood of $S$, the loss of differentiability entailed by a $\beta$-derivation is $\sum_{i=1}^{n}\beta_{i}k^{\prime}_{i}$ where the $k^{\prime}_{i}$ are the degrees of the invariant polynomials of $W_{S}$. Since this sum is less than $\sum_{i=1}^{n}\beta_{i}k_{i}$, more derivatives will be continuous in the neighborhood of $S$.

If $\sum_{i=1}^{n}\beta_{i}k_{i}=hr$, the derivative $\partial^{|\beta|}F/\partial x^{\beta}$ is continuous at the origin. Its derivative with respect to $p_{j}$ is not continuous but its product by $|x|^{k_{j}}$ tends to [math] with x.

When computing $\partial^{|\alpha|}(F\circ P)/\partial x^{\alpha},$ with $|\alpha|\leq hr$ by the Faa di Bruno formula, the derivative $\partial^{|\beta|}F/\partial p^{\beta}\circ P$ is multiplied by (we use the notations of [7]) :

[TABLE]

with

[TABLE]

and

[TABLE]

Remark 5.2**.**

The $p_{j}$ are homogeneous polynomials of degree $k_{j}$. Up to a multiplicative constant, the terms $\prod^{n}_{j=1}\prod_{A_{\alpha}}(\partial^{|q_{i}|}p_{j}/\partial x^{q_{i}})^{\mu_{iq^{i}}}$ are equal to $|x|^{\sum k_{j}\beta_{j}-|\alpha|}$. This exponent is strictly positive when $\sum\beta_{j}k_{j}>hr$ and we have

[TABLE]

In the neighborhood of a stratum $S$, we could show that $\varepsilon_{\beta}(x)$ is equivalent to a power of the distance from $x$ to $S$.

So, when $f\in C^{hr}(\mathbb{R}^{n})^{W}$, the function $L(f)=F$ is in a subspace of $\mathcal{E}^{r}(P(\mathbb{R}^{n}))$, the functions of which are of class $C^{rh}$ on the interior of $P(\mathbb{R}^{n})$, in $\mathcal{E}^{rh/h_{S}}(P(S))$ on the strata $P(S)$ of the border of $P(\mathbb{R}^{n})$. Additionally the behavior of the derivatives of $F$ that are lost on the border is as follows.

When $\sum_{i=1}^{n}\beta_{i}k_{i}>hr$, if the derivative $(\partial^{|\beta|}F/\partial p^{\beta})$ is lost on strata $S=P(A)$ of the border of $P(\mathbb{R}^{n})$, the function equal to $\varepsilon_{\beta}\cdot(\partial^{|\beta|}F/\partial p^{\beta})\circ P$ on the interior $C$ of $P(\mathbb{R}^{n})$ and equal to [math] on $A$ is continuous.

To be more consistent, we put $\varepsilon_{\beta}=1$ when $\sum_{i=1}^{n}\beta_{i}k_{i}\leq hr$ and for any $\beta$ we will consider the functions $\varepsilon_{\beta}\cdot\partial^{|\beta|}F/\partial p^{\beta}\circ P$ that vanish on the strata $A$ when $\partial^{|\beta|}F/\partial p^{\beta}$ does not exist on $P(A)$.

If we define the semi-norms

[TABLE]

[TABLE]

the Faa di Bruno formula shows that $|f|^{hr}_{K_{s}}$ is equivalent to $|F|^{r,hr}_{P(K_{s})}$. By Whitney regularity $\|f\|^{hr}_{K_{s}}$ and $|F|^{r,hr}_{P(K_{s})}$ are also equivalent.

From proposition 1.1, there is a constant $B_{K_{s}}$ such that $\|F\|^{r}_{P(K_{s})}\leq B_{K_{s}}\|f\|^{hr}_{K_{s}}$.

So, we have $|\mkern-2.0mu\|F|\mkern-2.0mu\|^{hr}_{P(K_{s})}\leq B^{1}_{K_{s}}\|f\|^{hr}_{K_{s}}$ for some constant $B^{1}_{Ks}$. Hence the continuity of $L$ when the target has the topology induced by the semi-norms $|\mkern-2.0mu\|.|\mkern-2.0mu\|^{hr}_{P(K_{s})}$.

Since $L$ is a bijection on its image, by the Banach theorem, we have:

Proposition 5.3**.**

The map $L$ is a Fréchet isomorphism of $C^{hr}(\mathbb{R}^{n})^{W}$ onto its image in $\mathcal{E}^{r}(P(\mathbb{R}^{n}))$ provided with the topology induced by the semi-norms $|\mkern-2.0mu\|F|\mkern-2.0mu\|^{hr}_{P(K_{s})}$.