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%First page headline in LaTeX for S\'eminaire Lotharingien de Combinatoire
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\font\rms=cmr8 
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\headline={\its S\'eminaire Lotharingien de
Combinatoire \bfs 50 \rms (2005), Article~B50j\hfill}
\footline={\hfil}


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 \null

\vskip -.5truein
\centerline{\Bol Some new Applications}
\vskip5pt
\centerline{\Bol of }
\vskip5pt
\centerline{\Bol   Orbit Harmonics}
\vskip15pt
 \centerline{  by} 
\vskip15pt
 \centerline{\smc  A.M. Garsia and N.R. Wallach }
\vskip20pt  
\def \SS  {\Sigma} 
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{\narrower\smallskip 
\noindent{\smallsmc Abstract} 

\baselineskip10pt
\textfont0=\eightrm
\textfont1=\eighti 
\textfont2=\eightsy 
\eightrm
We prove  a new result in the Theory of Orbit Harmonics and
derive from it a new proof of the  Cohen--Macauliness of the ring ${\cal QI}_m(G)$ of $m$-Quasi-Invariants
of a Coxeter Group $G$. Using the non-degeneracy of the fundamental bilinear form on ${\cal QI}_m(G)$,
this approach yields also a direct and simple proof that the quotient of ${\cal QI}_m(G)$ by the ideal generated by
the homogeneous $G$-invariants affords a graded version of the
left regular representation of $G$. Originally all of these  results were obtained as 
a combination of some deep work of Etingof--Ginzburg [3], Feigin--Veselov [6]
and  Felder--Veselov [5]. The arguments here are quite elementary 
and self contained, except 
those using the non-degeneracy of the fundamental bilinear 
form.   
  
\smallskip}

\vskip15pt

\def \SR {{\cal SR}}


\noindent{\bol Introduction}
\sas

Throughout this paper we let $G$ be a finite reflection group of $n\times n$ matrices,
$\SS(G)$ will denote its class of reflections and for each $s\in\SS(G)$ we choose once and for
all a vector $\aaa_s$  perpendicular to the reflecting hyperplane of $s$. In this manner the 
linear form giving the  equation of this reflecting hyperplane is  given by the scalar product $(x,\aa_s)$. This 
given, a polynomial $P(x)=P(\xon)$ is said to be $G$-$m$-Quasi-Invariant if and only if for
all $s\in\SS(G)$ the polynomial $(1-s)P(x)$ is divisible by $ (x,\aaa_s) ^{2m+1}$.
It easily shown that $G$-$m$-Quasi-Invariants form a finitely generated $G$-invariant graded subalgebra
of the polynomial ring $\BQ[X_n]$,
where $X_n$ is short for $x_1,x_2,\dots,x_n$. 
Denoting this algebra by $\QI$,  we have the proper inclusions
$$
{\cal QI}_1[G]\supset {\cal QI}_2[G]\supset\cdots \supset{\cal QI}_m[G]\supset\cdots .
$$

\eject
%
\headline={\small   Orbit Harmonics  and m-Quasi-Invariants\hfill $\ess\ess\ess\ess\ess$  $\ess\ess\ess$
$\ess\ess\ess$ \folio }%
Clearly,  ${\cal QI}_0[G]=\BQ[X_n]$, 
and ${\cal QI}_\infty[G]$ may be viewed as the
algebra $\LA _G$ of $G$-invariant polynomials.
Our goal here is to derive
new proofs of some of the basic results on $m$-Quasi-Invariants by means of the
Theory of Orbit Harmonics. More generally, using this 
theory, we will prove
the following   basic result.
\sas

\noindent{\bol Theorem I.1}

{\ita Let 
$\bf A$ be a degree-graded $G$-invariant subalgebra of   
$\BQ[X_n]$, and 
suppose that 

\item {\rm (i)} ${\bf A}\icon \LA_G$,
\item {\rm (ii)} 
for some non-trivial homogeneous $G$-invariant $B(x)$ 
we have $B(x)\BQ[X_n]\con {\bf A}$,
\item {\rm (iii)} ${\bf A}$ has a $G$-invariant,  non-degenerate, symmetric  bilinear form $\LL\scs \RR_{\bf A}$ , graded by degree%  
\footnote {$\dag$}{% 
\baselineskip10pt%
\textfont0=\eightrm
\textfont1=\eighti 
\textfont2=\eightsy 
\textfont3=\eightex 
\textfont\bffam\eightbf
\eightrm
Homogeneous elements of $\BA$ of different degrees are orthogonal with respect to $\langle\scs \rangle_{\bf A}$.},
\item {\rm (iv)} 
the orthogonal complement\/ 
${\bf H}_G({\bf A})$, with respect to $\LL\scs \RR_{\bf A}$,  of the ideal ${\cal J}_G(\BA)$ generated
in
$\bf A$ by the homogeneous $G$-invariants has dimension bounded by the order of $G$. 
 
\noindent
Then both 
${\bf H}_G({\bf A})$ and ${\bf A}/{\cal J}_G(\BA)$ afford  the regular representation of $G$ and 
 $\bf A$ is   free over $ \Lambda_G$. 
}
\sa
\def \ux {{\underline x}}

To state a further application of this  Theorem we need to introduce further notation and make some 
preliminary observations. To begin
with, we should note that 
within $G$-$m$-Quasi-Invariants we find $m$-analogues of all the ingredients 
that occur in the relationship between the  polynomial ring 
$\BQ[X_n]$  and the ring of
invariants $\LA_G$. For instance, let us recall that the space $H_G$  of 
``{\ita $G$-Harmonics}'' is defined as the orthogonal complement of the
ideal $\CJ_G$ generated by the homogeneous $G$-invariants. 
Now, it is well known 
that, for a Coxeter  group $G$ of $n\times n$ 
matrices,  
 $\LA_G$ 
is a free polynomial ring on $n$ homogeneous generators $q_1(x),q_2 (x),\ldots q_n (x)$. It follows from this that we have
$$
H_G\ses \big\{P\in\BQ[X_n]\, :\, q_k(\del_x)P(x)=0\, \ess
\hbox{for all}\ess\ess k=1,2,\ldots ,n\big\},
\eqno ({\rm I}.1)
$$
where  for a polynomial $P(x)$ we set $P(\del_x)=P(\del_{x_1},\del_{x_2},\dots ,\del_{x_n})$.
It is also shown in [12] that $H_G$ is the linear span of all the partial derivatives of the discriminant
$$
\Pi_G(x)\ses \prod_{s\in\SS(G)}(x,\aaa_s) \,  . 
\eqno ({\rm I}.2)
$$
In  symbols,
$$
H_G\ses\big\{ Q(\del_x)\Pi_G(x)\, \, Q\in \BQ[X_n]\, \big\}  . 
\eqno ({\rm I}.3)
$$
We have  $m$-analogues of both (I.1) and (I.3). To describe them we need to recall that  
in [2] Chalykh and Veselov show that to each homogeneous $m$-Quasi-Invariant $Q(x)$ of degree $d$ there corresponds a unique
homogeneous differential operator, acting  on $\QI$,  of the form
$$
\ggg_Q(x,\del_x)\ses Q(\del_x)+\sum_{|q|<d }c_q(x)\del_x^q,
\eqno ({\rm I}.4)
$$
where $\del_x^q=\del_{x_1}^{q_1}\del_{x_2}^{q_2}\cdots \del_{x_n}^{q_n}$ and $|q|=q_1+q_2+\cdots +q_n$,
with $c_q(x)$ a rational function in $\xon$ with a denominator which factors into a product of the linear forms $(x,\aaa_s)$.
In fact, there is even an explicit formula for $\ggg_q(x,\del_x)$ which is due to Berest [1]. This is 
$$
\ggg_Q(x,\del_x)\ses{1\over 2^d d!}\sum_{k=0}^d{d\choose k} L_m(G)^{d-k}\underline{Q}  L_m(G)^{k}
\eqno ({\rm I}.5)
$$
where $\underline{Q} $ denotes the operator ``{\ita multiplication  by $Q(x)$}'',
$$
L_m(G)\ses \DD_2\sms 2m \sum_{s\in \SS(G)}{1\over (x,\aaa_s)}\del_{\aaa_s}
\eqno ({\rm I}.6)
$$ 
with $\DD_2$ the ordinary Laplacian and $\del_{\aaa_s}$ the directional derivative corresponding to $\aaa_s$. 
In fact, it develops that the linear extension of the map $Q\rightarrow\gg_Q(x,\del_x)$ defined by (I.5) yields an algebra 
isomorphism of  $\QI$ onto the algebra of operators of the form (I.4) that commute with $L_m(G)$.
In particular for  all $P,Q\in \QI$ we   have
$$
\ggg_{PQ}(x,\del_x)\ses \ggg_{P }(x,\del_x)\ggg_{ Q}(x,\del_x).
\eqno ({\rm I}.7)
$$

This given,  a deep result of Opdam [11] implies that the bilinear form 
defined by setting, for $P,Q\in \QI$
$$
\LL P\scs Q\RR_m\ses \ggg_P(s,\del_x)Q(x)\big|_{x=0}
\eqno ({\rm I}.8)
$$

 \noindent
is 
non-degenerate   on $\QI\times \QI$. 
Now, the space $\BH_G(m)$ of $m$-Harmonics is simply defined
as the orthogonal complement, with respect to $\LL\scs \RR _m$, of the ideal $\CJ_G(m)$ generated in $\QI$ by the homogeneous
$G$ invariants. This given, the $m$-analogue of (I.1) is simply
$$
\BH_G(m)\ses \big\{P\in\BQ[X_n]\, :\, \gg_{ q_k}(x,\del_x)P(x)=0\, \ess
\hbox{for all}\ess\ess k=1,2,\ldots ,n\big\}
\eqno ({\rm I}.9)
$$
It should be mentioned that it follows from this that $\BH_G(m)\con \QI$. This is 
an immediate consequence
of the remarkable property of the operator $L_m(G)$ to the effect that for any two polynomials $P,Q$ 
we have $L_m(G)P=Q$ with $Q\in \QI$ if and only if $P\in \QI$. In 
particular, any polynomial in the kernel
of $L_m(G)$ is necessarily in $\QI$.
\sas 


Now, it develops that a beautiful $m$-analogue of (I.3) was conjectured by Feigin and Veselov in [6] and proved by
Etingov and Ginsburg in [3]. In the present 
notation, this result may be stated as follows.

\sas

\noindent{\bol Theorem I.2} (Theorem 6.20 of [3])

{\ita  
$$
\BH_G(m)\ses\big\{ \ggg_Q(x,\del_x)\Pi_G^{2m+1}(x)\, :\, Q\in \QI\, \big\}  . 
\eqno ({\rm I}.10)
$$
In fact, if $ \CB\subset \QI $ is any basis for the quotient of $\QI/\CJ_G(m)$,
then the collection
$$
\CF\ses \big\{\ggg_b(x,\del_x)\, \Pi_G^{2m+1}(x)\, :\, b\in \CB \big\}
\eqno ({\rm I}.11)
$$
is a basis for $\BH_G(m)$}.
\sa

We shall show here that, by combining the theory of orbit harmonics with 
a Hilbert series result of Felder--Veselov, we can also obtain a rather nice new proof of this result.
\sas

This paper consists of 6 sections. In the first four sections we establish Theorem~I.1 by a sequence of
five steps as follows.  In the first step we introduce, for a given $G$-regular vector $a=(a_1,a_2,\ldots ,a_n)$,
the   ring $\BA_{[a]}$ of the  $G$-orbit of $a$ and show that it affords the regular representation of $G$. 
In the  second  step  we introduce its graded version 
${\bf gr}\, \BA_{[a]}$ and show that it carries a graded version of 
the regular representation of $G$. In the third step, we introduce the space   $H_{[a]}(\BA)$
of orbit $\BA$-harmonics  and use the non-degeneracy of the form $\LL\scs \RR_{\bf A}$
to show that $H_{[a]}(\BA)$ and  
${\bf gr}\, \BA_{[a]}$ are equivalent as graded $G$-modules. 
 In the fourth step we use the dimension bound in (iv) to show that $H_{\bf A}$ and $H_{[a]}(\BA)$
 are one and the same. In the fifth step we use again the   non-degeneracy of the form
to show that $H_{\bf A}$ and the quotient ring ${\bf A}/{\cal I}_G(\BA)$ are equivalent as
graded $G$-modules. In the sixth and final step we combine property~(ii) with a simple
Hilbert series argument and derive that $\BA$ is free over $\LA_G$.
\sas
In the fifth section
we use some basic facts about Coxeter groups and  their corresponding $m$-Quasi-Invariants 
to show that the hypotheses of Theorem~I.1  are satisfied when $\BA=\QI$
and thereby derive that $\QI$ is a Cohen--Macaulay module over the ring of $G$-invariants.
We believe that the resulting proof is simpler,  more elementary and more revealing than the previous proofs.
\sas

In the sixth section we prove Theorem~I.2. This last section has a substantially
different character than the previous ones. It makes crucial use of deep results such as  the symmetry of the Hilbert series
of $m$-Harmonics and properties of the Baker--Akhiezer function of $\QI$  whose proofs, to this date,
are   far from being  elementary.
\sa

Finally, we should mention that some of the methods of the Theory of Orbit Harmonics we use here
were developed and successfully used in [7] in the study of the Garsia--Haiman modules.
This may not be an accident 
since, in a sense, $m$-Quasi-Invariants may be viewed as the  Jack polynomial case 
of the so-called $n!$-conjecture. In particular, since $m$-Quasi-Invariants arose in the study
of operators which commute with $L_m(G)$, by analogy, there must be spaces arising from a study
of operators that commute with the Macdonald operator.  We believe that we have only seen here
the tip of a mathematical iceberg gravid with  combinatorial implications. To dispell any doubts we may have
on this 
score, we urge the reader to view the surprising facts that emerged in [9] in the study of the
simplest possible  cases. Namely when the underlying reflection group reduces to   
 the symmetric group $S_2$ or the dihedral group $D_2$. 
\sap

\noindent{\bol 1. The orbit Ring}
 
In this paper  we adopt the convention that an $n\times n$ matrix $A=\|a_{ij}\|_{1\le i,j\le n }$
  acts  on a point \hbox{$x=(\xon )$} by right multiplication. In this manner the action of $A$ on a polynomial
$P(x) =P(\xon)$ is simply expressed in the form
$$
T_AP(x)=P(xA).
$$
We will work with a fixed finite Coxeter group  $G$  of  $n\times n$ matrices and a general algebra $\BA$
which satisfies the hypotheses of Theorem~I.1. The specialization $\BA=\QI$ will only take place in Section~5.
In the first four sections we will make use of the  invariant $B(x)$ which satisfies  hypothesis in  (ii).
This given,  for our developments it is   necessary that 
we choose once and for all a point $a=(a_1,a_2,\ldots ,a_n)$ which satisfies the following two conditions
$$
\hbox{a)}\ess\ess   B(a)\ne 0 \scs \ess\ess\ess\ess\ess\hbox{and}\ess\ess
\ess\ess\ess\ess 
\hbox{b)}\ess\ess \Pi_G(a)= \prod_{s\in \SS(G)}(a,\aaa_s)\, \ne \, 0  .
\eqno (1.1)
$$
Throughout the paper we denote by $[a]_{G}$ 
the $G$-orbit of $a$.  
In symbols,
$$
\orb\ses \{ aA\, :\, A\in G\}\, .
\eqno (1.2)
$$
Note that because of  (1.1)~b) the point $a$ cannot lie in any of the reflecting 
hyperplanes of $G$.   Since $G$ is a Coxeter group, it follows that  
the stabilizer of $a$, namely the subgroup
$$
G_a\ses \{A\in G\, :\, aA=a \}
$$
reduces to the identity.  This assures that $\orb$ consists of 
$|G|$ distinct points. 

Next we need a polynomial $\phi_a(x)$ such that
$$
\phi_a(x)\ses
\cases{
1 & if $x=a$,\cr\cr
0 & if $x=b$ with $b\in \orb$ and $b\ne a$.
\cr}
\eqno (1.3)
$$
The construction of this polynomial can be carried out in many ways. For the moment
it is immaterial how we pick $\phi_a(x)$. However, to get the best results in 
Section~5, it will be necessary that $\phi_a(x)$ is constructed to have the smallest  possible  degree.
It turns out that we can never do better than the degree of $\Pi_G$. Indeed, suppose (1.3)
holds true and  set
$$
F(x)\ses \sum_{B\in G} 
\det(B) \phi_a(xB)\, .
$$
Now note that from (1.3) it follows that  $F(a)=1\ne 0$. Moreover we also have
$$
F(xA^{-1})\ses 
\det(A)\, F(x)\bigsp (
\hbox{for all }     A\in G)
$$
and this implies that $F(x)$ is a 
$G$-invariant multiple of $\Pi_G(x)$. 
Thus $degree\big(F(x) \big)\ge degree\big(
\Pi_G(x)\big)$ and this can only happen  when
$degree\big(\phi_a(x)) \big)\ge degree\big(
\Pi_G(x)\big)$. 
To show that this minimum can be 
achieved, 
 we use the well known fact that if $\CB=\{ h_1,h_2,\ldots ,h_{|G|}\}$ is a basis for the
$G$-Harmonics then every polynomial $P(x)$ has an expansion of the form
$$
P(x)\ses \sum_{i=1} ^{|G|}h_i(x) A_i(x)\,  \bigsp (\hbox{with $A_i(x)\in \LA_G$}).
\eqno (1.4)
$$
This given, we start by constructing in any manner we please an initial polynomial $P_a(x)$ satisfying  
$$
P_a(x)\ses
\cases{
1 & if $x=a$,\cr\cr
0 & if $x=b$ with $b\in \orb$ and $b\ne a$,
\cr}
$$
then use (1.4) and obtain the expansion
$$
P_a(x)\ses \sum_{i=1} ^{|G|}h_i(x) A_{i,a}(x)\,  \bigsp (\hbox{with $A_{i,a}(x)\in \LA_G$})\, .
$$
This 
done, we claim that we can take
$$
\phi_a(x)\ses \sum_{i=1} ^{|G|}h_i(x) A_{i,a}(a)\ .
\eqno (1.5)
$$
In fact, note that the $G$-invariance of the coefficients $ A_{i,a}(x)$ gives
$$
P_a(b)\sms \phi_a(b)\ses  \sum_{i=1} ^{|G|}h_i(b)\big( A_{i,a}(b)-A_{i,a}(a)\big)\ses 0\bigsp 
\hbox{for all} \ess\ess b\in \orb .
$$
Thus this choice of $\phi_a(x)$ will also satisfy (1.3). Note next that since   (1.5) defines
$\phi_a(x)$ to be a $G$-harmonic polynomial, it follows from (I.3) that
$$
degree\big(\phi_a(x)\big)\le degree\big(\Pi_G(x)\big)\, .
$$
Since we already have the reverse inequality, equality must hold true for this choice of $\phi_a$.
  
\def \QI {\BA}

 
Now for each $b\in \orb$ with $b=aA$ we set 
$$
\phi_b(x)\ses \phi_a(xA^{-1})\ess\ess\ess\hbox{and}\ess\ess\ess  \eee(b)\ses 
\det(A)\, .
$$
Note that from (1.3) it immediately follows that 
$$
\phi_b (x)=\cases{
1 & if $x=b$,\cr\cr
0 & if $x=b'$ with $b'\in \orb$ and $b'\ne b$.
\cr}
\eqno (1.6)
$$
 Next,  we let $\ida   $ denote the ideal generated in $\BA$ by the elements of  $\BA$  that vanish on $\orb$.
In symbols,
$$
\ida \ses\big\{ P(x)\in \QI \, :\, P(b)=0\ess
\hbox{for all} \ess b\in \orb\big\}.
\eqno (1.7)
$$
This given,  it follows that 

\sas

\noindent{\bol Proposition 1.1}
\def \BR {{\bf A}}

{\ita The quotient ring
$$
\BR_{\orb}\ses \QI/\ida
$$
has dimension $|G|$ and affords the left regular representation of $G$}.

\noindent{\bol Proof}


In view of (1.1), we can set
$$
\psi_b(x)\ses \phi_b(x) 
{B(x) \over B(a)  }\, .
\eqno  (1.8)
$$
Note 
that, since $B(x) $ is $G$-invariant, we derive from (1.6) that
$$
\psi_b (x)=\cases{
1 & if $x=b$,\cr\cr
0 & if $x=b'$ with $b'\in \orb$ and $b'\ne b$.
\cr}
\eqno  (1.9)
$$
Moreover, from (ii)  we also derive that
$$
\psi_b (x)\in \QI\ess\ess\ess 
\hbox{for all}\ess\ess b\in \orb\, .
$$
Now note that, for any $P\in\QI$, (1.9)  gives
$$
P(x)\, - \sum_{b\in \orb} P(b)\, \psi_b(x)\in \ida \, .
\eqno  (1.10)
$$
It will be convenient to express this relation by writing
$$
P(x)\, \cong_{[a]_G}\,  \sum_{b\in \orb}  P(b)\, \psi_b(x)\,  .
\eqno  (1.11)
$$
Thus the collection
$$
\big\{\psi_b(x)\big\}_{b\in \orb}
\eqno  (1.12)
$$
is a basis for the quotient ring $\BR_{\orb}$. 

\def \congo {\ssp \cong_{\orb} \ssp }

Since the ideal $\ida$ is $G$-invariant it immediately follows that
$G$ acts on $\BR_{\orb}$. Thus to complete our proof we only need to compute the  
character of this action. 
To this end, note that, for all $\sig$ and $\bb$ in $G$, 
we derive from (1.11) that
$$
\eqalign{
T_\sig\psi_{a\bb }(x)
 \ses 
\psi_a(x\sig\bb^{-1})
&\congo    \sum_{\aa\in G} \psi_a(a\aa \sig\bb^{-1} )\psi_{a\aa}(x)
\cr
&
\congo    \sum_{\aa\in G} \chi\big (\aa \sig\bb^{-1} =id\big )\psi_{a\aa}(x)
\cr
&
\congo    \sum_{\aa\in G} \chi\big (\aa \sig  =\bb \big )\psi_{a\aa}(x),
}
$$
where $\chi({\cal A})=1$ if $\cal A$ is true and  $\chi({\cal A})=0$ otherwise.
Thus the action of $G$ on the basis $\big\{\psi_b(x)\big\}_{b\in \orb}$ is given by the matrix $A(\sig)=\|\chi\big (\aa \sig=\bb \big )\|_{\aa,\bb\in G}$. 
It follows that the character of the $G$ action on  $\BR_{\orb}$ is given by
$$
trace\, A(\sig)\ses \sum_{\aa\in G} \chi\big (\aa \sig =\aa \big )\ses \cases {|G| & if $\sig=id$,\cr\cr
0 & if $\sig\neq id$,}
$$
and this is the character of the left regular representation of 
$G$, precisely as asserted.
\sa

\noindent{\bol Remark 1.1}

The ring $\BR_{\orb}$ is not graded but it has a filtration given by the subspaces 
$$
\CH_{\le k}(\BR_{\orb})\ses \CL_\orb\big [ P \, : \,  P\in \CH_{\le k}\big(\QI\big) \big]
\eqno  (1.13)
$$
where ``$\CL_\orb$'' denotes ``{\ita Linear Span}'' modulo $\ida$ and $\CH_{\le k}\big(\QI\big)$ is the subspace of
$\QI$ spanned by its elements of degree $\le k$. 
Now note that, since by construction
we have
$$
degree \, \psi_b(x)\ses degree\big( B(x)\big) \times    degree (  \phi_a),
\eqno  (1.14)
$$
it immediately follows from the expansion in (1.11) that
$$
\BR_{\orb}\ses \CH_{\le d_{\BA}}\big (\BR_{\orb}\big )
\eqno  (1.15)
$$
where for convenience we have set
$$
 d_{\BA}\ses degree\big( B(x)\big) \times    degree (  \phi_a).
\eqno  (1.16)
$$

It will be convenient  here and after to  adopt the convention that if $V$ is a graded vector  space
then $\CH_{=k}(V)$ denotes the subspace spanned by   homogeneous elements of degree $k$.
Likewise  $\CH_{\le k}(V)$ denotes the subspace spanned by the homogeneous elements of degree $\le k$.



\def \grJ {{\bf gr}\, \ida}
\def \Ra {{ \BR_\orb}}
\def \grRa  {{\bf gr\, \BR}_\orb} 
 

\sap

\noindent{\bol 2. The graded version of $\BR_\orb$ }
\sas

For each polynomial $P$ we shall here and after denote by $h(P)$ the homogeneous component of
highest degree in $P$. This given, we let $\grJ$ be the ideal in $\QI$ generated by the highest degree
components of elements of $\ida$. 
In symbols,
$$
\grJ\ses \big(h(P)\, :\, P\in \ida\, \big)_{\QI}\, .
\eqno (2.1)
$$
 

\noindent
This brings
 us to define the ``{\ita graded version}'' of $\Ra$ as the quotient
$$
\grRa \ses \QI/\grJ \,  .
\eqno (2.2)
$$
Since $\grJ$ is generated by homogeneous polynomials, $\grRa$ is necessarily a graded ring. Note further that
if $P$ is any homogeneous polynomial of degree $>d_\BA$ then the fact that
$$
P(x)\sms \sum_{b\in \orb}P(b)\, \psi_b(x)
\quad  \in\ {\cal J}_\orb({\bf A})
$$
immediately implies that $P\in \grJ$. Thus it follows that $\grRa$ has the direct sum decomposition
$$
\grRa\ses \bigoplus_{k=0}^{ d_\BA}\CH_{=k}(\grRa).
\eqno (2.3)
$$
It will be convenient to choose once and for all, for each $1\le k\le d_\BA$, a  collection  $\CB^{=k} \subset \CH_{=k}
({\bf A}) $ yielding a basis
for the  subspace
$\CH_{=k}(\grRa)$.   
 This given, we have the following useful fact.
\sas

\noindent{\bol Proposition 2.1}

{\ita The collection 
$$
\CB^{\le k}\ses \CB^{=0}\uplus \CB^{=1}\uplus \cdots \uplus \CB^{=k}
$$
is also a basis for the subspace  $\CH_{\le k}(\Ra)$. 
In particular, $\CB^{\le d_\BA} $ is
a basis of $\Ra$}.

\noindent{\bol Proof}

Clearly the result holds true for $k=0$ since $\CB_{=0}$ reduces to the single constant $\bf 1$.
So we may proceed by induction on $k$. Let us then assume that each $P\in \CH_{\le k-1}(\Ra)$ 
has an expansion in terms of $\CB^{\le k-1}$. This given, note that any $P\in \CH_{\le k}(\QI)$,
viewed as a representative of an element of $\grRa$,  may be expanded in terms of the basis $\CB^{\le k}$.
In other words, we will have constants $c_\phi$ such that
$$
P\ses \sum_{\phi\in \CB^{\le k}}c_\phi \, \phi\sps R\ ,
\eqno (2.4)
$$
for a suitable $R\in\grJ$. Now the definition of $\grJ $ yields that there must be elements $f_i\in \ida$ 
and $A_i\in \QI$ giving
$$
R\ses \sum_{i }A_i\,  h(f_i)\ .
\eqno (2.5)
$$
Since $R$ is necessarily of degree $\le k$ we see that there is no loss in assuming that we have
$$
degree \, A_i\, \le\,  k- degree \, h(f_i)\,.
\eqno (2.6)
$$
Using (2.5) we may rewrite (2.4) in the form
$$
P\ses \sum_{\phi\in \CB^{\le k}}c_\phi \, \phi\sps  \sum_{i }A_i\,  f_i\sms  \sum_{i }A_i\,  (f_i-h(f_i))\, ,
\eqno (2.7)
$$
and this implies that 
$$
P\sms \sum_{\phi\in \CB^{\le k}}c_\phi \,\phi\ssp \cong_{\orb}\sms  \sum_{i }A_i\,  (f_i-h(f_i))\, .
\eqno (2.8)
$$
But now (2.6) and the definition of $h(f_i)$ yield that
$$
degree \, \sum_{i } A_i\,  (f_i-h(f_i))\ssp \le\ssp k-1 \ .
\eqno (2.9)
$$
Thus our inductive hypothesis yields that we have constants $d_\phi$ giving
$$
 \sum_{i }A_i\,  (f_i-h(f_i))\ssp \cong_{\orb}  \sum_{\phi\in \CB^{\le k-1}}d_\phi \,\phi\ ,
$$
and this combined with (2.8) completes the induction.  This proves the first assertion.
The second assertion follows from the identity in (1.15).
\sas

Proposition~2.1 has the following remarkable corollary.
\sa

\noindent{\bol Theorem 2.1}

{\ita The ring $\grRa$ yields a graded version of the regular representation. 
In fact, for every 
$0\le k\le d_\BA$,  we have the character relation 
$$
char\ssp \CH_{=k}(\grRa)\ses char\ssp \CH_{\le k}(\Ra) \sms char\ssp \CH_{\le k-1}(\Ra).
\eqno (2.10)
$$
In particular the characters of the subspaces $\CH_{\le k}(\Ra)$ are related to the graded character of  $\grRa$ by the following identity}
$$
\sum_{k= 0}^{d_\BA}q^k char\ssp \CH_{=k}(\grRa)\ses q^{d_\BA} char\ssp \Ra\sps  (1-q)\sum_{k=0}^{d_\BA-1} q^k\, char\ssp \CH_{\le k}(\Ra).
\eqno (2.11)
$$



\noindent{\bol Proof}

Let  $\CB^{\le k}=\{  \phi_{i} \}_{i=1}^{m_k} $ so that $\CB^{= k}=\{  \phi_{i} \}_{m_{k-1}<i\le {m_k}}  $,
  and  let $$A (\sig)\ses \| a_{i,j} (\sig)\|_{i,j=1}^{|G|}$$
be the matrix expressing the action of $G$ on the basis  $\CB^{\le d_\BA}$ as elements of $\Ra$. 
Since $\CH_{\le k}(\Ra)$ is $G$-invariant it follows  that for any  $j\le m_k$ we have the 
expansion 
$$
T_\sig \phi_j \ses\sum_{i=1}^{m_k}\phi_i \, a_{i,j} (\sig)\sps R_{\sig,j}\bigsp \hbox{ (with $R_{\sig,j}\in \ida$ of degree $\le k$).}
\eqno (2.12)
$$
This means that the action of $G$ on the subspace $\CH_{\le k}(\Ra)$ induces a   representation 
given by the matrix
$$
 A^{\le k}(\sig)\ses\big\|  a_{i,j} (\sig) \big\|_{1\le i,j\le m_k}  \ .
\eqno (2.13)
$$
Note further that when $m_{k-1}<j\le m_k$, then $\phi_j$ is homogeneous of degree $k$ and 
equating homogeneous terms of degree $k$ in (2.12) we get
$$\eqalign{
T_\sig \phi_j \ses\sum_{m_{k-1}<i\le m_k} \phi_i \, a_{i,j} (\sig)\sps 
\cases{h(R_{\sig,j}) & if $degree \ssp R=k$,
\cr\cr
0 & if $degree \ssp R_{\sig,j}<k$,
}\hskip2cm\cr
\bigsp\hbox{(with $R_{\sig,j}\in \ida$).}
}
$$
In either case we derive that 
$$
T_\sig  \phi_j \ssp \cong\ssp \sum_{m_{k-1}<i\le m_k} \phi_i \, a_{i,j} (\sig) \bigsp\hbox{(modulo $\grJ$)}.
$$
This shows that the action of $G$ on the subspace $\CH_{=k}\big(\grRa\big)$ induces a representation 
given by the matrix
$$
 A^{=k}(\sig)\ses\big\|  a_{i,j} (\sig) \big\|_{m_{k-1}<i,j\le m_k}  \ .
$$
Thus 
$$
char\ssp \CH_{=k}\big(\grRa\big)\ses trace\ssp A^{=k}\ses trace\ssp   A^{\le k}\sms trace \ssp  A^{\le k-1} \, .
$$ 
This proves (2.10).  This given, multiplying (2.10) by $q^k$ and summing for $0\le k\le d_\BA$,
the identity in (1.15) yields (2.11).                 
\sap

\noindent{\bol 3. The orbit $\bf A$-Harmonics}

Recall that by the hypothesis in (iii) of Theorem~I.1, the algebra $\BA$ has a non-degenerate 
bilinear form $\LL\scs \RR_\BA$.
Using this form we define the $\BA$-harmonics of the orbit $\orb$ to constitute the orthogonal complement
of the ideal $  
{\bf gr} \ida$. In symbols
$$
\BH_{[a]_G}(\BA)\ses \big\{ P\in \QI\, : \LL Q\scs P\RR_\BA\ses 0\ess\ess 
\hbox{for all } Q\in \grJ\big\}\, .
\eqno (3.1)
$$ 
Note that, since the bilinear form  $\LL\scs \RR_\BA$ is also graded, it immediately follows from (3.1) that all the homogeneous components of any $P\in \BH_{[a]}(\BA)$
must be also in 
$\BH_{[a]_G}(\BA)$. Thus $\BH_\orb(\BA)$ is a graded vector space. However we can establish
a considerably stronger result.
\sas

\noindent{\bol Theorem 3.1}

{\ita $\BH_\orb(\BA)$ is a graded $G$-module equivalent to  $\grRa$. In particular it also
carries the  regular representation and thus it has dimension the order of $ G$.} 

\noindent{\bol Proof}

Note that the $G$-invariance of the bilinear form $\LL\scs \RR_\BA$  may be also expressed in the form
$$
 \LL T_\sig P\scs Q \RR_\BA  \ses \LL  P\scs T_\sig^{-1} Q \RR_\BA \bigsp ( 
\hbox{for all }\sig\in G),
\eqno (3.2)
$$
and, since the ideal $
{\bf gr} \ida$ is $G$-invariant, it follows from (3.2) and the definition in  (3.1) that  also $\BH_\orb(\BA)$ is   $G$-invariant. Thus to prove the assertion we need only compute its  
character. To this end, for a given $1\le k\le d_\BA$ let $\{\psi_1,\psi_2,\ldots ,\psi_s\}\con \BA$ be 
a basis for $\CH_{=k}\big(\grJ)$, and let $\{\phi_1,\phi_2,\ldots ,\phi_r\}\con \QI$ be constructed
so that, together with  $\{\psi_1,\psi_2,\ldots ,\psi_s\}$ they yield a basis for $\CH_{=k}\big(\QI  \big)$.
In particular, $\{\phi_1,\phi_2,\ldots ,\phi_r\}\con \QI$ are independent modulo 
$\grJ$, and therefore
the elements of $\grRa$
which they represent are a basis for $\CH_{=k}\big( \grRa \big)$.

Now note that the non-degeneracy of the form $\LL\scs \RR_\BA$ on the subspace $\CH_{=k}\big(\QI  \big)$
assures the  non-singularity of the block matrix
$$
M\ses \left[
\matrix{
\|\LL \phi_i,\phi_j\RR_\BA \|_{1\le i,j\le r} &\|\LL \phi_i,\psi_j\RR_\BA\|_{\multi{1\le i\le r\cr 1\le j\le s}} 
\cr
\|\LL \psi_j,\phi_i\RR_\BA\|_{\multi{1\le i\le r\cr 1\le j\le s}} &\|\LL \psi_i,\psi_j\RR_\BA\|_{1\le i,j\le s}
\cr}\right].
\eqno (3.3)
$$
It then follows that the matrix product
$$
\LL \phi_1,\phi_2,\ldots ,\phi_r,\psi_1,\psi_2,\ldots ,\psi_s\RR M^{-1}
$$
yields a basis for  $\CH_{=k}\big(\QI  \big)$ that is ``{\ita dual}'' to $\LL \phi_1,\phi_2,\ldots ,\phi_r,\psi_1,\psi_2,\ldots ,\psi_s\RR $
 with respect to the bilinear form $<\scs >_\BA$.     That means that if  this new  basis is
$$
\LL \eta_1,\eta_2,\ldots ,\eta_r,\ggg_1,\ggg_2,\ldots ,\ggg_s\RR 
$$
then we shall have
$$
\hbox{a)}\ess\ess \LL \phi_i,\eta_j\RR_\BA\ses \cases {1 & if $i=j$ \cr 0 & if $i\ne j$\cr}\scs \ess\ess \hbox{b)}\ess\ess  
\LL \psi_i,\ggg_j\RR_\BA\ses \cases {1 & if $i=j$ \cr 0 & if $i\ne j$\cr}
\eqno (3.4)
$$            
and
$$
\hbox{a)}\ess\ess  \LL \phi_i,\ggg_j\RR_\BA =0\ess  
\hbox{for all}\ess i,j\scs \ess\ess  
\hbox{b)}\ess\ess  \LL \psi_i,\eta_j\RR_\BA =0\ess  
\hbox{for all}\ess i,j
\eqno (3.5)
$$ 
Note that, since $\psi_1,\psi_2,\ldots ,\psi_s$ span $\CH_{=k} \big(\grJ\big)$,
the relations in (3.5)~b) imply that
$\eta_1,\eta_2,\ldots ,\eta_r$ lie in 
$\CH_{=k} \big({\bf H}_\orb(\BA)\big)$. We claim that $\LL \eta_1,\eta_2,\ldots ,\eta_r\RR$
are in fact a basis of $\CH_{=k} \big( \BH_\orb(\BA)\big)$. 
To show this, note that the duality relations in (3.4) and (3.5) yield
that for any  polynomial $P\in\CH_{=k}\big(\QI \big)$ we have the expansions
$$
P\ses \sum_{i=1}^r \LL P,\eta_i\RR_\BA \phi_i \sps \sum_{j=1}^s \LL P,\ggg_j\RR_\BA \psi_i\ ,
\eqno (3.6)
$$
$$
P\ses \sum_{i=1}^r \LL P,\phi_i\RR_\BA \eta_i \sps \sum_{j=1}^s \LL P,\psi_j\RR_\BA \ggg_i\ .
\eqno (3.7)
$$
In particular, if $P\in \CH_{=k} \big( \BH_\orb(\BA)\big) $, the latter  expansion reduces to
$$
P\ses \sum_{i=1}^r \LL P,\phi_i\RR_\BA \eta_i  \, .
\eqno (3.8)
$$
This shows that  $\LL \eta_1,\eta_2,\ldots ,\eta_r\RR$ span $\CH_{=k} \big( \BH_\orb(\BA)\big)$. Since they are 
independent by construction, it follows that they are a basis as asserted. This given, let $A(\sig)=\|a_{i,j }(\sig)\|_{i,j=1}^r $
be the matrix that expresses the action of $G$ on $\LL \eta_1,\eta_2,\ldots ,\eta_r\RR$. 
That is, for all $\sig\in G$ we have
$$
T_\sig\,  \eta_i\ses \sum_{u=1}^r \, \eta_u\,\, a_{u,i}(\sig)\bigsp\hbox{for $1\le i\le r$} .
\eqno (3.9)
$$
Using the expansion in (3.6) for $P=T_\sig \phi_v$, we get
$$
\eqalign{
T_\sig \phi_v
&\ses  \sum_{i=1}^r \LL T_\sig \phi_v ,\eta_i\RR_\BA \phi_i \sps \sum_{j=1}^s \LL T_\sig \phi_v,\ggg_j\RR_\BA \psi_j
\cr
(\hbox{by (3.2)}) &\ses  \sum_{i=1}^r \LL  \phi_v ,T_\sig^{-1}\eta_i\RR_\BA \phi_i \sps \sum_{j=1}^s \LL T_\sig \phi_v,\ggg_j\RR_\BA \psi_j
\cr
&\cong \ssp  \sum_{i=1}^r \LL  \phi_v ,T_\sig^{-1}\eta_i\RR_\BA \phi_i \bigsp \hbox{(modulo $\grJ$)}.
}
\eqno (3.10)
$$
But, using (3.9), we derive (from (3.4)~a)):
$$
\LL  \phi_v ,T_\sig ^{-1}\eta_i\RR_\BA\ses \sum_{u=1}^r \LL  \phi_v , \eta_u\RR_\BA a_{u,i}( \sig^{-1})\ses a_{v,i}(           \sig^{-1})\ ,
$$
and thus (3.10) reduces  to
$$
T_\sig \phi_v\ssp \cong\ssp   \sum_{i=1}^r    a_{v,i} (  \sig^{-1})\phi_i \bigsp \hbox{(modulo $\grJ$)}.
$$
In other words, the action of $G$ on the basis $\LL \phi_1,\phi_2,\ldots ,\phi_r\RR$ of $\CH_{=k}\big( \grRa \big)$ is given by the matrix
$$
A^ \top  \big( \sig^{-1}\big )\, .
$$
Since
$$
trace\ssp A (\sig)\ses trace \ssp A^\top  \big( \sig^{-1}\big )
$$
we can thereby conclude that
$$
char\ssp \CH_{=k}\big(\BH_\orb(\BA) \big)\ses char\ssp \CH_{=k}\big( \grRa \big)  \big)\, .
$$
This proves that  $  \BH_\orb(\BA) $ and $  \grRa    $ are equivalent as
graded $G$-modules and completes the proof of the theorem.
\sa

\def \CU {{\cal U}}



\noindent{\bol Remark 3.1}

We should note that if $V$ is a  finite dimensional vector space with a non-degenerate, symmetric, bilinear form $\LL\scs \RR_V$,
and $U \subset V$ is a proper 
subspace,  and if we set
$$
U^{\perp }\ses\big\{ P\in V\, :\, \LL P\scs Q\RR_V=0\, \ess  
\hbox{for all} \ess Q\in U\, \big\}\ ,
$$     
then 
$$
{U^{\perp }}^\perp\ses U\, .
\eqno (3.11)
$$
%Proceeding as  we did in the proof of Theorem~3.1, let
%${\cal U}=\{\phi_1,\phi_2,\ldots ,\phi_r\}$ be a basis for $U$ and  ${\cal U'}=\{\psi_1,\psi_2,\ldots ,\psi_{s}\}$
%be constructed so that  $\CU\cup \CU'$ is a   basis for $V$. Let as before 
%$$
%\LL \eta_1,\eta_2,\ldots ,\eta_r\,;\, \ggg_1,\ggg_2,\ldots ,\ggg_{s}\RR 
%$$
%be the dual basis with respect to the form  $\LL\scs \RR_V$. Then the same argument carried out there yields that
%$\LL\ggg_1,\ggg_2,\ldots ,\ggg_{s}\RR$ is a basis for $U^\perp$. This implies that we always have 
%$$
%\dim U^\perp \ses \dim  V \sms \dim U\, .
%$$
%In particular we must also have
%$$
%\eqalign { 
%{\dim U^\perp}^\perp &\ses \dim  V \sms \dim U^\perp
%\cr
% &\ses \dim  V \sms ( \dim  V \sms \dim U)\ses \dim U\, .
%\cr}
%$$
%But then the equality in (3.11) necessarily follows since we trivially have the containment
%$$
%U\con {U^{\perp }}^\perp \, .
%$$
%\sas

Note that, since the form $\LL\scs \RR_\BA$ is graded we can apply this result with
$V=\CH_{=k}(\BA)$ and $U=\CH_{=k}\big( 
{\bf gr} \ida\big)$ and deduce the relations
$$
\CH_{=k}\big(
{\bf H}_\orb(\BA)\big)\ses \Big( \CH_{=k}\big( 
{\bf gr} \ida\big) \Big)^\perp
$$
and
$$
\CH_{=k}\big( 
{\bf gr} \ida\big)  \ses \Big( \CH_{=k}\big(
{\bf H}_\orb(\BA)\big) \Big)^\perp\ .
$$
Similarly we get
$$
{\bf H}_\orb(\BA)\ses \big( 
{\bf gr} \ida\big)^\perp\ess\ess \hbox {and }\ess\ess 
{\bf gr} \ida\ses \big( 
{\bf H}_\orb(\BA) \big)^\perp\ .
\eqno (3.12)
$$
\sap

\noindent{\bol 4. The $\bf A$-Harmonics of $\bf G$}
\sas

In the classical case the space 
$\BH_{G}$ of ``{\ita Harmonics}'' of $G$ is   defined as the orthogonal complement of the 
ideal generated by the homogeneous $G$-invariants.   For $\BA$-harmonics the definition is entirely analogous.
We simply let $\CJ_{G}(\BA)$ be the ideal generated in $\BA$ by the  
homogeneous $G$-invariants 
and set, (using the notation in Remark~3.1):
$$
\BH_{G}(\BA)\ses  \big(\CJ_{G}(\BA)\big)^\perp
\eqno (4.1)
$$
Note that, since the bilinear form $\LL  \scs  \RR_\BA$ is graded, 
$\BH_{G}(\BA)$  will necessarily be a graded 
vector space, as was the case for $\BH_\orb(\BA)$ itself. However,        
the hypothesis in (iv) immediately yields the following remarkable result.

\noindent{\bol Theorem 4.1}

{\ita The two subspaces $\BH_\orb(\BA)$ and $\BH_G(\BA)$ are identical.
In particular,  $\BH_G(\BA)$ carries a graded
version of the left regular representation of $G$. Moreover we also have}
$$
\CJ_{G}(\BA)\ses 
{\bf gr} \ida
\eqno (4.2)
$$
\noindent{\bol Proof}

Note that if $Q$ is a $G$-invariant polynomial then the polynomial $Q(x)-Q(a)$ vanishes at all points of the 
orbit $\orb$. That  is $Q(x)-Q(a)\in \ida $. Thus if $Q$ is also homogeneous of positive degree it necessarily follows  that 
$$
Q(x)\in \grJ\, .
$$
This proves the containment
$$
 \CJ_{G}(\BA)\, \con \,\grJ\, .
$$
Therefore we must also have the reverse containment 
$$
 \big(\CJ_{G}(\BA)\big)^\perp\supseteq \big( \grJ\big)^\perp\ ,
$$
and this can be written as 
$$
\BH_{G}(\BA)\supseteq \BH_\orb(\BA)
\, .
\eqno (4.3)
$$
But now   Theorem~3.1 and  the hypothesis in (iv) give
$$
|G|\ses \dim\, 
\BH_{G}(\BA)\, \geq \dim  \BH_\orb(\BA)\ses |G|\ ,
$$
and (4.3) forces the desired equality  
$$
\BH_{G}(\BA)\ses \BH_\orb(\BA)
\, .
$$
Then it follows that we must also have  
$$
\BH_{G}(\BA)^\perp \ses \BH_\orb(\BA)^\perp
\, ,
$$ 
\vskip -.07truein

\noindent
and the equality in (4.2) follows immediately from Remark~3.1. 
\sa

We should note that  from (4.2) 
we can derive the following result.
\sas

\noindent{\bol Theorem 4.2}
{\ita 

If  $P\in \BA$ is homogeneous then
$$
degree(P)>d_\BA \ess\ess\Longrightarrow\ess\ess P\in \CJ_G(\BA)
\eqno (4.4)
$$
}
\noindent{\bol Proof}  

We have seen that, in terms of the basis elements defined in (1.8), every element  $P\in \BA$ satisfies the identity in (1.10),
that is
$$
P(x)\, - \sum_{b\in \orb} P(b)\, \psi_b(x)\in \ida \, .
\eqno (4.5)
$$
Since all the terms in the sum have degree $\le d_\BA$, if $P$ is homogeneous of degree $>d_\BA$ then (4.5) forces
$$
P(x)\in 
{\bf gr} \ida\ ,
$$
and the equality in (4.2) proves (4.4).
\sas

But the most important consequence of (4.2) is given by the following truly remarkable result.
\sas

\noindent{\bol Theorem 4.3}

{\ita The quotient ring 
$$
\BA/\CJ_G(\BA)
$$
carries the regular representation of $G$ and  therefore has dimension the order of $G$.
}  

\noindent{\bol Proof}

The equality $\CJ_G(\BA)=
{\bf gr} \ida$ forces the equality  
$$
\BA/\CJ_G(\BA)\ses \BA/
{\bf gr} \ida\ ,
$$ 
and therefore the assertion follows from Theorem~2.1.
\sa
\def \CA {{\bf A}}

Before we can complete the proof of Theorem~I.1 we need to recall some   basic facts about 
Cohen--Macaulay algebras.
To begin
with, let us  recall that the Hilbert series of a finitely generated, graded algebra $\CA$ is given by the formal sum
$$
F_\CA(t)\ses \sum_{k\ge 0} t^k \dim \CH_k(\CA)\ ,
\eqno (4.6)
$$
where 
$\CH_k(\CA)$ denotes the subspace spanned by the elements of $\CA$ that are homogeneous of degree $k$.
It is well known that $F_\CA(t)$ is a rational function of the form
$$
F_\CA(t)\ses  {P(t)\over (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})}\ ,
$$
with $d_1,d_2,\ldots ,d_n$ positve integers and 
$P(t)$ a polynomial that does not vanish at $t=1$. The minimum $n$ for which this is possible characterizes the growth of 
$\dim \CH_k(\CA)$ as $k\RA \infty$. This integer is customarily  called the ``{\ita Krull dimension}'' of $\CA$ 
and is denoted  ``$\dim_K\CA$''. 
It is easily shown that we can always find in $\CA$ homogeneous elements $\ttt_1,\ttt_2,\ldots ,\ttt_n$ such
that the quotient of $\CA$ by the ideal generated by  $\ttt_1,\ttt_2,\ldots ,\ttt_n$ is a finite dimensional
vector space. In symbols
$$
\dim\, \CA/(\ttt_1,\ttt_2,\ldots ,\ttt_n)_\CA <\infty
\eqno (4.7)
$$
It is also a fact
that $\dim_K\, \CA$ is also equal to the minimum $n$ for which this is possible.
When (4.7) holds true and $n= \dim_K\, \CA$ then $\{\ttt_1,\ttt_2,\ldots ,\ttt_n\}$ is
called  a ``{homogeneous system of parameters}'', $\cal HSOP$ in brief.

It  follows from (4.7) that if $\eta_1,\eta_2,\ldots ,\eta_N\in \BA $ give a basis for the quotient in (4.7) then
every element of $\CA$ has an expansion of the form
$$
P\ses \sum_{i=1}^N \eta_iP_i(\ttt_1,\ttt_2,\ldots ,\ttt_n)\ ,
\eqno (4.8)
$$
with coefficients $P_i(\ttt_1,\ttt_2,\ldots ,\ttt_n)$  polynomials in their arguments.
The algebra $\CA$ is said to be Cohen--Macaulay, when the coefficients $P_i(\ttt_1,\ttt_2,\ldots ,\ttt_n)$ 
are uniquely determined by $P$. This amounts to the requirement that the collection
$$
\big\{ \eta_i\, \ttt_1 ^{p_1}\ttt_2 ^{p_2}\cdots \ttt_n ^{p_n}\big\}_{i,p}
\eqno (4.9)
$$
is a basis for $\CA$ as a vector space and therefore $\BA$ is a free module over\break $\BQ[\ttt_1,\ttt_2,\ldots ,\ttt_n]$.
 Note that when this happens and
 $ \ttt_1,\ttt_2,\ldots ,\ttt_n;\eta_1,\eta_2,\ldots ,\eta_N $  are homogeneous of  degrees 
$d_1,d_2,\ldots ,d_n; r_1,r_2,\ldots ,r_N$ then   we must necessarily have
$$
F_\CA(t)\ses {\sum_{i=1}^N t^{r_i}\over (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})}\ ,
\eqno (4.10)
$$
from which it follows that $ \dim_K\, \CA=n$.  

This brings us to a useful criterion for assuring the Cohen--Macauliness of a finitely generated graded algebra.
\sa

\def \CL {{\cal L}}
\def \CJ {{\cal J}}
\def \BM {{\cal M}}
\def \CART {{\cal ART}}
\def \QI {{\cal QI}}

\noindent{\bol Proposition 4.1}

{\ita

Let  $ \ttt_1,\ttt_2,\ldots ,\ttt_n $ be an  $\cal HSOP$, and let  $d_i=degree(\ttt_i)$,
Then $\BA$ is free over $\BQ[ \ttt_1,\ttt_2,\ldots ,\ttt_n]$ and therefore Cohen--Macaulay if and only if} 
$$
\lim_{t\RA 
1^-}(1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})F_\CA(t) \ses \dim\, \CA/(\ttt_1,\ttt_2,\ldots ,\ttt_n)_\CA
\eqno (4.11)
$$  
\noindent{\bol Proof}%
\footnote {$\dag$}%
{% 
\baselineskip10pt%
\textfont0=\eightrm
\textfont1=\eighti 
\textfont2=\eightsy 
\textfont3=\eightex 
\textfont\bffam\eightbf
\eightrm
This is a known result but we include a proof 
for the sake of completeness.}

Note first that the necessity of the condition follows immediately from (4.10). To prove the sufficiency,
let $\eta_1,\eta_2,\ldots, \eta_N$ be a homogeneous basis for the quotient $ \CA/(\ttt_1,\ttt_2,\ldots ,\ttt_n)_\CA$
and set $r_i=degree(\eta_i)$. Next let $\BM_i$ denote the subspace spanned by the collection 
$$
\big\{\eta_j\ttt_1^{p_1}\ttt_2^{p_2}\cdots \ttt_n^{p_n}\, :\, 1\le j\le i\, ;\, p_j\ge 0\big\}\, . 
\eqno  (4.12)
$$
It is easily seen that if $\CH_m(\BM_i)$ and $\CH_m(\BM_i/\BM_{i-1})$ denote the subspaces
of $\BM_i$ and $\BM_i/\BM_{i-1}$ spanned by their homogeneous elements of degree $m$ then we must have
$$
\dim\,  \CH_m(\BM_i)\ses \dim\,\CH_m(\BM_i/\BM_{i-1})\sps \dim\,\CH_m( \BM_{i-1})\,.
$$
Multiplying by $t^m$ and summing, we derive the Hilbert series identities 
$$
F_{\BM_i}(t)\ses F_{\BM_i/\BM_{i-1}}(t)\sps F_{ \BM_{i-1}}(t)\bigsp (\hbox{for $1\le i \le N$ with $\BM_0=\{0\}$})\ .
$$
This implies that
$$
F_{\CA}(t)\ses  F_{\BM_1}(t)\sps  F_{\BM_2/\BM_{ 1}}(t)\sps    \cdots \sps  F_{\BM_N/\BM_{N-1}}(t) \ .
\eqno  (4.13)
$$
Now, for a given $1\le i\le N$, let $\phi$ be the map from the polynomial ring\break 
$\BQ[x_1^{d_1},x_2^{d_2},\ldots ,x_n^{d_n}]  $
onto $\BM_i/\BM_{i-1}$ defined by setting for every polynomial\break 
$P( x_1^{d_1},x_2^{d_2},\ldots ,x_n^{d_n})$
$$
\phi P\ses \eta_i \, P( \ttt_1 ,\ttt_2 ,\ldots ,\ttt_n )\ .
$$
Note that, a priori the kernel $\CJ$ of $\phi$ will be an ideal of 
$\BQ[x_1^{d_1},x_2^{d_2},\ldots ,x_n^{d_n}]$, and 
since  $\phi$ preserves degrees, we will have
$$
 F_{\BM_i/\BM_{i-1}}(t)\ses t^{r_i}
F_{\BQ[x_1^{d_1},x_2^{d_2},\ldots ,x_n^{d_n}]/\CJ}(t)\ .
\eqno  (4.14)
$$
Thus, if $\CJ$ happens to be trivial, it follows that
$$
 F_{\BM_i/\BM_{i-1}}(t)\ses {t^{r_i}\over (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})}\, .
\eqno  (4.15)
$$
On the other hand, if $\CJ$ contains a non trivial homogeneous element $P$ of degree $d$, from (4.14) we derive the coefficient-wise inequality
of Hilbert series
$$
 F_{\BM_i/\BM_{i-1}}(t)\, <<\, t^{r_i}  
F_{\BQ[x_1^{d_1},x_2^{d_2},\ldots ,x_n^{d_n}]/(P)}(t)\, ,
\eqno  (4.16)
$$
where the symbol ``$<<$'' is to indicate that the inequality is coefficient-wise.
Since the ring 
$\BQ[x_1^{d_1},x_2^{d_2},\ldots ,x_n^{d_n}]$ has no zero divisors,  we will have
$$
F_{\BQ[x_1^{d_1},x_2^{d_2},\ldots ,x_n^{d_n}]/(P)}(t) \ses   {1-t^d \over (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})}\, ,
$$
and from (4.16) we derive that
$$
 F_{\BM_i/\BM_{i-1}}(t)\, <<\,  t^{r_i}{1-t^d \over (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})}\, .
\eqno  (4.17)
$$
In conclusion, we see that  we will have 
$$
\lim_{t\RA 
1^-} (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n}) F_{\BM_i/\BM_{i-1}}(t)\ses \eee_i\ ,
\eqno  (4.18)
$$
with
$$
\eee_i\ses \cases
{1 & if (4.15) holds true,
\cr\cr
0 & if (4.17) holds true.
}
$$
Thus, passing to the limit as $t\RA 
1^-$ in (4.13) and using (4.18) together with the hypothesis in (4.11), 
we finally obtain   
$$
\eee_1+\eee_2+\cdots +\eee_N\ses \dim\, \CA/(\ttt_1,\ttt_2,\ldots ,\ttt_n)_\CA\ses N\, .
\eqno  (4.19)
$$
This forces all the $\eee_i$ to be equal to one. 
However, this can only hold true when the collection
$$
\big\{\eta_j\ttt_1^{p_1}\ttt_2^{p_2}\cdots \ttt_n^{p_k}\big\}_{j,p}
$$
forms an independent set. Indeed any relation of the form
$$
\sum_{j=1}^N \eta_jP_j(\ttt_1,\ttt_2,\ldots \ttt_n)\ses 0\, .
$$
would force one of the $\eee_i$ to vanish and 
contradict (4.19). This shows that
$$
F_\BA(t)\ses {F_{\BA/(\ttt_1,\ttt_2,\ldots ,\ttt_n)_\BA}(t)\over (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})}
\eqno  (4.20)
$$
and  proves the Cohen--Macauliness of  $\CA$.
\sas



We are now ready to complete  the proof of Theorem~I.1.
\sa

\noindent{\bol Theorem 4.4}

{\ita  The algebra $\BA$ is a free $\LA_G$-module and therefore Cohen--Macaulay.

} 

\noindent{\bol Proof}

 Let  $q_1(x),q_2 (x),\ldots q_n (x)$
be a  fundamental set of homogeneous generators
of $\LA_G$ and suppose that $d_1,d_2,\ldots ,d_n$ are their respective degrees.
It is well known (see [10]) that we must have the equality   
$$
d_1d_2\cdots d_n\ses |G|. 
\eqno (4.21)
$$
From  the hypothesis (ii)  it follows that we have the 
containments
$$
B(x)\BQ[X_n]\subset \BA\subset \BQ[X_n]\, ,
\eqno (4.22)
$$
and since $\BA$ is degree graded, and $B$ is homogeneous, it follows from (4.6) that the Hilbert series $F_\BA(q)$ of $\BA$ will satisfy the 
inequalities
$$
{t^{degree(B)}\over (1-t)^n}<<F_\BA(t)<< {1 \over (1-t)^n}\ .
\eqno (4.23)
$$ 
In particular, this shows that the Krull dimension of $\BA$ is $n$.


This given, multiplying both sides of (4.23) by
$(1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n})$ and passing to the limit as $t\RA 1^-$,  the equality in  (4.21) implies that
$$
lim_{t\RA 1^-}(1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n}) F_\BA(t)\ses |G|\, .
\eqno (4.24)
$$
Moreover, since $q_1(x),q_2 (x),\ldots q_n (x)$ are also generators of the ideal $\CJ_G(\BA)$ we see  that
Theorem~4.3 implies that
$$
\dim \, \BA/\big(q_1 ,q_2 ,\ldots q_n  \big)_\BA\ses |G|\ ,
$$
and Proposition~4.1 then yields that $\BA$ 
is Cohen--Macaulay  over $\LA_G$.  
\sas

\noindent{\bol Remark 4.1}

We should note that (4.20) yields  the Hilbert series identity
$$
F_\BA(t)\ses {F_{\BA/ (q_1 ,q_2 ,\ldots q_n   )_\BA}(t)\over (1-t^{d_1})(1-t^{d_2})\cdots (1-t^{d_n}) }\, .
$$  
In particular it follows  that if $\CB$ is any basis for the quotient 
$$
\BA/ (q_1 ,q_2 ,\ldots q_n   )_\BA
$$
then the collection
$$
\Big\{ b\, q_1^{p_1}q_2^{p_2}\cdots q_n^{p_n}\Big\}_{b\in \CB\, ,\, p_i\ge 0}
$$
is a basis for $\BA$.
\sap

 \def \QI {{\cal QI}_m(G)} 
\def \BR {{\bf R}}
\def \JO {{\CJ_{\orb}(m)}}
\def \OR {{\BR_\orb [m]}}
\def \grJ {{\bf gr}\, \CJ_{\orb}(m)}
\def \OH   {{\bf H}_\orb(m)} 
\def \JG {{\cal J} _G(m)}
\def \GH {{\bf H}_G(m)}
 


\noindent{\bol 5.  $\bf  QI_m(G)$ is a  Cohen--Macaulay algebra. }

In this section we show that $\QI$ satisfies the hypotheses (i)-(iv) of  Theorem~I.1.

Now, (i) is no problem since
we have seen that the definition gives $ \QI\supset {\cal QI}_\infty (G)=\LA_G $ for all $m\ge 0$.  
For property (ii),  we can take
$$
B(x)=\PI_G(x)^{2m}.
\eqno (5.1)
$$
To see this, we need only observe that for any polynomial $P(x)\in \BQ[X_n]$ and
any $s\in \SS(G)$, the $G$-invariance of $\PI_G(x)^{2m}$  gives
$$
(1-s)\PI_G(x)^{2m}P(x)\ses \PI_G(x)^{2m} (1-s)P(x).
$$
Clearly,  $\PI_G(x)^{2m}$  yields the factor $(x,
\aaa_s)^{2m}$ and   another
factor $(x,
\aaa_s)$ comes from \hbox{$(1-s)P(x)$}.
This proves that
$$
 \Pi_G(x)^{2m}\BQ[X_n]\subset \QI\ ,
\eqno (5.2)
$$
as desired.

\noindent
For (iii) we need to make sure that the bilinear form    
$$
\LL P\scs Q\RR_G\ses \ggg_P(x,\del_x) Q(x)\, \Big|_{x=0}
\eqno (5.3)
$$
mentioned in the introduction has the required properties. To begin
with, in the
$S_n$ case a reasonably elementary proof of non-degeneracy can be found in [8],
but in the general case we will have to rely on Opdam's proof [11].
The $G$-invariance as expressed in (3.2)  immediately follows from the  identity
$$
T_\sig\ggg_Q(x,\del_x) T_\sig^{-1} \ses \ggg_{T_\sig Q}(x,\del_x)\bigsp (\, 
\hbox{for all}\ess\ess  \sig\in G\, )
$$
satisfied by all the operators $\ggg_Q$. The latter in turn follows from 
the Berest formula (I.5) and the identity
$$
T_\sig L_m(G)T_\sig^{-1}\ses L_m(G) \bigsp (\, 
\hbox{for all}\ess\ess  \sig\in G\, ),
$$
which can be  easily verified from the definition in (I.6). 
The symmetry of $\LL \scs \RR_m$ is a consequence of the symmetry
of the Baker--Akhiezer function
(see Section~6 for more information on that function) 
of the algebra $\QI$. A reasonably accessible proof of this
result  can be found in [8]. The fact that  $\LL \scs \RR_m$ is graded is an immediate consequence 
of the fact  that for $Q$ homogeneous of degree $d$ the differential operator 
 $\ggg_Q(x,\del_x)$ is also homogeneous of order $d$. Again this can be easily seen from the Berest formula.

We are thus left with the verification of property (iv).
Before we do this, we need to 
fix some notation  and   establish a few auxiliary facts that are of interest
by themselves. To begin
with, we recall that the point $a=(a_1,a_2,\ldots ,a_n)$ has to be  
chosen to have a trivial 
$G$-stabilizer.
In this case we can satisfy all the conditions we need (including (1.1)) 
by requiring that
$$
\Pi_G(a)\ne 0\, 
\eqno (5.4)
$$  
Here, as in Section~1, $\phi_b(x)$ denotes the polynomial defined by (1.6). But it will be 
helpful to assume that
the polynomial $\phi_a(x)$ satifying (1.3) is chosen so as to have  the minimal degree $|\SS(G)|$.
Thus, since $degree \big(\Pi(x_G)\big)= |\SS(G)|$, the definition in (1.8) and (5.1) yield that the polynomial
$\psi_b(x)$ satisfying (1.9) has degree  
$$
d_{m}(G)= (2m+1)|\SS(G)|\, . 
\eqno (5.5)
$$


Before we proceed with our arguments, it will be convenient to adopt a notation that is more adherent to
the present choice $\BA=\QI$. 
To this end, for the rest of this paper we shall assume that
\sas

\vbox{
{\ita

\hsize 5.5truein
\item{\rm(1)}  The ideal  of  $G$-$m$-Quasi-Invariants that vanish in $\orb$ is denoted  $\CJ_{\orb}(m)$.

\item{\rm(2)} The quotient of  $\QI$  by $\CJ_{\orb}(m)$ is denoted $  \BR_\orb [m]$.

\item {\rm(3)} The graded version of $\JO]$ is denoted $  \grJ$.

\item {\rm(4)} The graded version of $\BR_\orb [m]$ is denoted $  {\bf gr }\,  \BR_\orb [m] $
 
\item {\rm(5)} The orthogonal complement of $\grJ$ with respect to the bilinear form in {\rm(5.3)} is denoted $\OH $
and its elements are called  ``{\ita orbit $m$-Harmonics}''. 


\item {\rm(6)} The ideal generated in $\QI$ by the $G$-invariants is denoted $\JG$.

\item {\rm(7)} The orthogonal complement of $\JG$ with respect to the bilinear form in {\rm(5.3)} is denoted $\GH $,
and its elements are called  ``{\ita  $G$-$m$-Harmonics}''. 

}


\hsize6truein

 \vskip -1.8truein
$$
\eqno (5.6)
$$
 \vskip  1.4truein
}

\noindent
This given, since the only place we have used property~(iv) is in the proof of Theorem~4.1, all the results
obtained in Sections~1, 2 and  3  hold true with  $\BA=\QI$. Therefore we can state

\noindent{\bol Theorem 5.1}
{\ita

\item{\rm (a)}  $\OR$ is of dimension $|G|$ and affords the regular representation of $G$.

\item{\rm (b)}  $\OR$ and $\OH $ are equivalent  $G$-modules affording a graded version of the regular representation of $G$.
}

However,  here we have the following two additional results. 
\sas

\noindent{\bol Theorem 5.2}

$$
\BH_{\orb}(m)\ses \big\{Q\in \QI\, :\, \ggg_PQ\ses 0\ess 
\hbox{for all}\ess\ess P\in \grJ\big\}\, .
\eqno (5.7)
$$
\noindent{\bol Proof}

In view of the definition in (5.3), we clearly see that 
the condition $ \ggg_PQ\ses 0$ is stronger than $\LL P\scs Q\RR_m= 0$. Thus we need only establish the containment
$$
\BH_{[a]_G}(m)\con \big\{P\in \QI\, :\, \ggg_PQ\ses 0\ess 
\hbox{for all}\ess\ess P\in \grJ\big\}\, .
\eqno (5.8)
$$
To this end, note that since $\grJ$ is an ideal of $\QI$, the defining condition 
$$
\OH=\big\{Q\in \QI\, :\, \LL P\scs Q\RR_m=0\ess 
\hbox{for all}\ess\ess P\in \grJ\big\}
$$
can be also written in the form
$$
\eqalign{
\OH=\Big\{Q\in \QI\, :\,\LL RP\scs  Q\RR_m\ses 0\hskip3cm\cr
\hskip3cm
\hbox{for all}\ess\ess P\in \grJ \ess\ess\hbox{and}\ess\ess R\in \QI\, \Big\}.
}
$$
On the other hand from (I.7)
 and the definition in (5.3) of the bilinear form $\LL \scs  \RR_m $ we derive that
$$
\LL RQ\scs  P\RR_m\ses \LL R\scs \ggg_QP\RR_m
$$
Thus
$$
\eqalign{
P\in 
\BH_{[a]_G}(m)\ess\ess   
\hbox{implies} \ess\ess \LL R\scs \ggg_QP\RR_m=0\hskip4cm\cr
\hskip3cm
\hbox{for all}\ess\ess Q\in \grJ \ess\ess\hbox{and}\ess\ess R\in \QI\ .
}
$$
But then the non-degeneracy of $\LL \scs  \RR_m $ yields that
$$
P\in 
\BH_{[a]_G}(m) \ess\ess   \Longrightarrow \ess\ess   \ggg_QP=0   \ess\ess  
\hbox{for all}\ess\ess  Q\in \grJ\, .  
$$
This proves (5.8) and completes the proof of the theorem.
\sa

The next result proves the identity in (I.9).

\sas

\noindent{\bol Theorem 5.3}

$$
\BH_G(m)\ses \big\{P\in\BQ[X_n]\, :\, \gg_{ q_k}(x,\del_x)P(x)=0\, \ess
\hbox{for all}\ess\ess k=1,2,\ldots ,n\big\}\ .
\eqno (5.9)
$$
\noindent{\bol Proof}

By definition,
$$
\BH_G(m)\ses  \big\{P\in\BQ[X_n]\, :\, \LL Q\scs P\RR_m=0\, \ess
\hbox{for all}\ess\ess Q\in \JG\big\} \ ,
$$
and, since the  fundamental invariants $q_1,q_2,\ldots ,q_n$ generate the ideal $\JG$, 
this is equivalent to  
$$
\eqalign{
\BH_G(m)\ses  \big\{P\in\BQ[X_n]\, :\, \LL Rq_k\scs P\RR_m=0\hskip4cm\cr
\hskip4cm
\hbox{for all}\ess R\in \QI\ess \hbox{and}\ess k=1,2,\ldots ,n\} 
}
\eqno (5.10)
$$
Now, from 
(5.3) and (I.7) we derive that
$$
\LL Rq_k\scs P\RR_m\ses \LL R\scs \ggg_{q_k} P\RR_m\, .
\eqno (5.11)
$$
But the 
non-degeneracy of $\LL\scs \RR_m$ yields that
$$
 \LL R\scs \ggg_{q_k} P\RR_m=0\ess\ess
\hbox{for all}\ess\ess R\in \QI\ess
\hbox{ implies}\ess\ess \ggg_{q_k}P\ses 0.
\eqno (5.12)
$$
Combining (5.10), (5.11) and (5.12) proves (5.9).
\sas

We are now finally in a position to  establish property (iv) for $\BA=\QI$. 
\sas
\noindent{\bol Theorem 5.4}
We have
$$
 \dim\, \BH_G(m)\ssp\le \ssp|G|.
\eqno (5.13)
$$
\noindent{\bol Proof}

Notice to the reader. The following   argument was provided to us by E-mail by Feigin and Veselov. We feel compelled
to reproduce it here since it is not available in the present literature. Although this result is stated in many places (see [3], [4], [6]),
proofs (when not omitted) give no indication that it could be established in such a simple and elementary manner.
\sa

The idea is to  show that each $m$-Harmonic  $Q\in \GH$ is completely determined by  $|G|$ of its derivatives at the point $a$.  
To do this, we fix once and for all a fundamental  set $q_1,q_2,\ldots ,q_n$ of $G$-invariants and a monomial basis 
$\{x^{\eee _1}, x^{\eee _2},\ldots ,x^{\eee _N} \}$  for the quotient $  \BQ[X_n]/(q_1,q_2,\ldots ,q_n)_{\BQ[X_n]}$. 
Since it is well known that $  \dim\BQ[X_n]/(q_1,q_2,\ldots ,q_n)_{\BQ[X_n]}\ses |G|$, we must necessarily have
$$
N=|G|.
$$
This given, we need only show that every $Q(x)\in \GH$ is completely determined by the $|G|$ derivatives  
$$
\del_x^{\eee_1}Q(x)\, \big|_{x=a}\scs\ess \del_x^{\eee_2}Q(x) \big|_{x=a}\,\scs \ess \ldots  \scs \del_x^{\eee_N}Q(x)\, \big|_{x=a}\, .
$$
That is we must show that
$$
\del_x^{\eee_1}Q(x)\, \big|_{x=a}=0\scs\ess \del_x^{\eee_2}Q(x) \big|_{x=a}=0\,\scs \ess \ldots  \scs \del_x^{\eee_N}Q(x)\, \big|_{x=a}=0 
\eqno (5.14)
$$
forces
$$
Q\equiv 0.
\eqno (5.15)
$$
To do this, the point of departure is the well known fact that every polynomial $P\in \BQ[X_n]$ 
may be given  an expansion of the form
$$
P\ses \sum_{r=1}^{|G|}x^{\eee_r}  A_{P,r}(q_1,q_2,\ldots ,q_n)\ ,
\eqno (5.16)
$$ 
with the coefficients $A_{P,r}$ polynomials in their arguments. Specializing $P$ to the
 monomial $x^p$ (5.16) gives
$$
x^p\ses \sum_{r=1}^{|G|}x^{\eee_r}\ssp  a_{p,r}(x)
\eqno (5.17)
$$ 
with
$$
 a_{p,r}(x)\ses A_{x^p,r}(q_1,q_2,\ldots ,q_n)
\eqno (5.18)
$$
Two important facts should be kept in mind here:

\item  {(1)} We can assume that
$$
degree ( a_{p,r}(x))\ses |p|-|\eee_r|\, .
\eqno (5.19)
$$  

\item  {(2)}  From (5.18), (5.9) and (I.7) it follows that
$$
\ggg_{ a_{p,r}}(x,\del_x)Q(x)\ses A_{x^p,r}(0,0,\ldots ,0)\ssp Q(x)\ess\ess\ess\ess\ess\ess
\hbox{for all}\ess\ess Q\in \GH\, .
\eqno (5.20)
$$

\noindent
Note first that, since it is well known that we must have
$$
\sum_{r=1}^{N}q^{\eee_r}\ses \prod_{i=1}^n(1+q+\cdots +q^{d_i-1})
$$ 
with $d_i=degree(q_i)$, it follows that one of the exponents $\eee_r$ vanishes. Thus one of the conditions
in (5.14) reduces to
$$
Q(a)\ses 0\ssp.
\eqno (5.21)
$$
Our next task is to show that the remaining conditions in  
(5.14) force 
$$
\del_x^pQ(x)\, \big|_{x=a}\ses 0
\eqno (5.22)
$$
for all
$$
  p=(p_1,p_2,\ldots ,p_n)\,.
$$
In view of (5.21), we can proceed by induction on $|p|=p_1+p_2+\cdots +p_n$. We assume (5.22) to be true
for $|p|<d$  and show that it holds true for $|p|=d$. To this end we use (5.17) and write for $|p|=d$,
$$
\del_x^pQ(x)\ses \sum_{|\eee_r|=d}a_{p,r}\ess \del_x^{\eee_r}Q(x)\sps  \sum_{|\eee_r|<d}\del_x^{\eee_r}a_{p,r}(\del_x)Q(x)
$$
Note that  (5.19) says that $a_{p,r}$ in the first sum  reduces to a scalar and in the second sum it must be
a homogeneous polynomial of degree $d-|\eee_r|>0$.  In particular the conditions in (5.14) immediately 
give us that  
$$
\del_x^pQ(x)\, \big |_{x=a}\ses    \sum_{|\eee_r|<d}\del_x^{\eee_r}a_{p,r}(\del_x)Q(x)\big |_{x=a}\, .
\eqno (5.23)
$$
Now, from (I.4) and the fact that $degree(a_{p,r})=d-|\eee_r|$, we obtain an expansion of the form
$$
\ggg_{a_{p,r}}(x,\del_x)\ses a_{p,r}(\del_x)\sps \sum_{|q|<d-|\eee_r|}c_{q,p,r}(x)\del_x^q\ .
$$
Using this in (5.23), we derive that
$$
\eqalign{
\del_x^pQ(x)\, \big |_{x=a}\ses    \sum_{|\eee_r|<d}\del_x^{\eee_r}\ggg_{a_{p,r}}(x,\del_x)Q(x)\big |_{x=a}\hskip3cm\cr
\hskip4cm
\sms  \sum_{|\eee_r|<d} \sum_{|q|<d-|\eee_r|}\del_x^{\eee_r} c_{q,p,r}(x)\del_x^qQ(x)\big |_{x=a}\
\, .
}
$$
But (5.20) and the inductive hypothesis reduces this to
$$
\del_x^pQ(x)\, \big |_{x=a}\ses     
\sms  \sum_{|\eee_r|<d} \sum_{|q|<d-|\eee_r|}\del_x^{\eee_r} c_{q,p,r}(x)\del_x^qQ(x)\big |_{x=a}\
\, .
\eqno (5.24)
$$
Note that this makes perfectly good sense since our assumption in 5.4 assures that the denominators 
that will be produced by the term 
$$
\del_x^{\eee_r} c_{q,p,r}(x)\del_x^qQ(x)
$$
will not vanish at $x=a$. However, (5.24) completes the induction since the derivatives of $Q$ that will be produced by
these terms will necessarily be of order $<d$ and by the inductive hypothesis they will all vanish at $x=a$.. 

This completes the proof of the dimension bound in (5.13).
\sa


\noindent
Theorem~5.3 has a number of immediate corollaries that are worth stating  
explicitly. 
\sa


\noindent{\bol Theorem 5.5}

{\ita The following remarkable equalities hold true for every integer $m\ge 0$:
$$
  \OH\ses \GH \ess\ess \hbox{and}\ess\ess \JG\ses \grJ\,.
\eqno (5.25)
$$
Thus the $m$-Harmonics $\GH$ and the quotient ring
$$
\QI/(q_1,q_2,\ldots , q_n)_{\QI}
\eqno (5.26)
$$
have both dimension $|G|$ and afford  the same graded regular representation of $|G|$ as the
the space of orbit harmonics $\OH$. }
\sa

\noindent{\bol Theorem 5.6}

{\ita For every integer $m\ge 0$ the algebra of $G$-$m$-Quasi-Invariants is a
free module  over the ring of invariants $\La_G$.}
\sas

\noindent{\bol Theorem 5.7}

{\ita Every homogeneous $G$-$m$-Quasi-Invariant $Q\in \QI$ of degree greater than
\hbox{$(2m+1)|\SS(G)|$}
lies  in the ideal $\JG$.
} 
\sa
\def \gaga {{\GG(m)}}
\noindent

Since we have verified that  the algebra  $\BA=\QI$ satisfies conditions  
(i), (ii), (iii), (iv) 
of 
Theorem~I.1, all of these results are simply specializations to $\BA=\QI$ of the 
corresponding results established in Section~4.
\sap

\noindent{\bol 6.  More on the $\bf G$-$\bf  m$-Harmonics}

The goal of this section is to establish Theorem~I.2. The basic tool in this task is a space $\GG(m)$ of
formal power series in $\xon$ which may be viewed as the orthogonal complement  of the ideal
$\JO$. More precisely we set
$$
\GG(m)\ses \Big\{ \FF(x) \, :\, \ggg_Q(x,\del_x)\FF(x)= 0\ess \ess 
\hbox{for all}  \ess Q\in \JO \Big\}
\eqno (6.1)
$$ 
Perhaps a few words are  necessary here to assure that this is a well defined space. To begin
with, we shall
view  each formal power series $\FF(x)$  as the formal   sum
$$
\FF \ses  \FF^{(0)} \sps \FF^{(1)}  \sps \FF^{(2)} \sps \cdots \sps \FF^{(k)} \sps \cdots 
$$
with $\FF^{(k)} $ a  
polynomial in $\xon$ homogeneous of degree $k$. Moreover if $Q\in \JO$
has the decomposition
$$
Q \ses Q_0\sps Q_1   \sps \cdots \sps Q_h  
$$ 
with   $Q_r $  homogeneous of degree $r$, then the  equation
$$
\ggg_Q(x,\del_x)\FF(x)= 0
\eqno (6.2)
$$
simply means that we must have
$$
\sum_{r=0}^h\ggg_{Q_r }(x,\del_x)\FF^{(r+k)}(x)\ses 0 \bigsp  (\, 
\hbox{for all } k\ge 0).
\eqno (6.3)
$$
Thus no infinite sums are involved in checking 
containment in $\gaga$.
\def \SD {{\cal SD}}

To deal with this space,  we need an important ingredient which has been in the background up to this 
moment but which nevertheless is the most significant tool in the Theory of $m$-Quasi-Invariants.
This is a formal power series $\PP_m(x,y)$ referred to as the  ``{\ita Baker--Akhiezer function}'' of $\QI$. 
To use $\PP_m(x,y)$, we shall have to state a number of facts
whose original proofs are in a series of papers scattered over several years. However, a reasonably 
self contained account of this material with detailed proofs of everything we use here (except for the
non-degeneracy of the bilinear form) can be found in [8]. This is a monograph we have put together
for the benefit of future researchers in this area.

To begin
with, it will be good to see how $\PP_m(x,y)$ is defined. Indeed, although this definition will  play  no role here,
the novice in this area has great difficulty locating it in the literature.   
Remarkably, $\PP_m(x,y)$ may be given an explicit (though quite forbidding) construction based on 
a truly remarkable family $\SD_n$ of ``{\ita Shift-Differential}'' operators   that
act  on polynomials by a combination of the ordinary $G$-action followed by differentiation. 
These operators are of the form  
$$
A\ses \sum_{\sig \in G}a_\sig(x,\del_x)\,  T_\sig\ ,
\eqno (6.4)
$$
where 
$$
a_\sig(x,\del_x)\ses \sum_{p}a_p(x)\del_x^p\bigsp \hbox{($\del_x^p=\del_{x_1}^{p_1}\del_{x_2}^{p_2}\cdots \del_{x_n}^{p_n}$)},
\eqno (6.5)
$$
with each $a_p(x)$ a special rational function in the ring  $\CS\CR_n(x)$   generated by the variables $x_i$ together with the
fractions $1/(x,\aaa_s)$ for $s\in \SS(G)$. It it is easily shown that $\SD_n$ is in fact an algebra
since the algebra of differental operators of the form given in (6.5) is invariant under conjugation by elements of $G$.
The building blocks of $\SD_n$ are the $m$-Dunkl operators, which  
can be written in the form 
$$
\nabla_i(m)\ses \del_{x_i}- m\sum_{s\in \SS(G)}  (\aaa_s,e_i) {
{1\over (x,\aaa_s)}}\, (1-s)
\bigsp \hbox { (for $i=1,2,\ldots ,n$)},
  \eqno (6.6)
$$
with $e_i$ the $i^{th}$ coordinate unit vector.  
For fixed $m$, the operators $\{\nabla_i(m)\}_{i=1}^n$ are a commuting set, and
thus it makes perfectly good sense to evaluate a polynomial at $\nabla_1(m),\nabla_2(m),\ldots ,\nabla_n(m)$. 
We adopt the notation
$$
Q[\nabla(m)]\ses Q\big(\nabla_1(m),\nabla_2(m),\ldots ,\nabla_n(m)\big)\bigsp \hbox{(for all $Q\in 
\BQ[X_n]$)}.
$$  
Clearly,  all these  operators belong to the family  $\SD_n$.         
 But once
they are written in the form given in (6.4), we can simply ``{\ita forget}'' the $G$ action
and set
$$
\Gamma  A\ses \sum_{\sig \in G} a_\sig(x,\del_x)\, .
$$
It develops that this seemingly innocent operation  can achieve miracles. For instance one  obtains 
that
 $$
\Gamma  \sum_{i=1}^n\nabla_i(m)^2\ses L_m(G).
\eqno (6.7)
$$
More generally, it can  be shown that for  each $G$-invariant $Q$ we have the beautiful identity
$$
\ggg_Q(x,\del_x)\ses \Gamma Q[\nabla(m)].
\eqno (6.8)
$$ 
This given, let us set
$$
O_m\ses\Gamma \, \Pi_G(\nabla(m)) \Pi_G(\ux)\ ,
$$
where ``$\Pi_G(\ux)$'' denotes the operator multiplication by $\Pi_G(x)$. 
Remarkably, it can be shown that
we have the Opdam commutation relation
$$
L_m(G) \, O_m\ses O_m L_{m-1}(G)\, .
\eqno (6.9)
$$
This given, the Baker--Akhiezer function $\PP_m(x,y)$ is simply defined by setting
$$
\PP_m(x,y)\ses O_m O_{m-1}\cdots O_1 e^{(x,y)}\ .
\eqno (6.10)
$$
Since the definition in (I.6) gives that $L_0(G)=\DD_2$, 
we see that it follows from (6.9) that
$$
L_m(G)\PP_m(x,y)\ses (y,y)\PP_m(x,y)\, .
\eqno (6.11)
$$

\vbox{
\noindent
The properties of $\PP_m(x,y)$ we will use  here  may be stated 
as follows: 
{\ita

\item {\rm(a)} Symmetry: $\PP_m(x,y)=\PP_m(y,x)$.

\item {\rm(b)} $\ggg_Q(x,\del_x)\PP_m(x,y)\ses Q(y) \PP_m(x,y)\ess\ess 
\hbox{for all} \ess\ess Q\in \QI$.\hfill {\rm(6.12)} 

\item {\rm(c)} For all $\sig\in G$ we have   $ \PP_m(x\sig ,y)=\PP_m(x ,y\sig^{-1})$.  

\item 
{\rm(d)}
We have the decomposition  
 $
\PP_m(x,y)= c_m(G)+\sum_{k\ge 1} \PP_m^{(k)}(x,y)
 $ 
with
\itemitem {\rm(i)} $\PP_m(0,0)=\PP_m(0,y)=\PP_m(x,0)=c_m(G)\ne 0$,
  
\itemitem {\rm(ii)} $\PP_m^{(k)}(x,y)$
a polynomial 
  homogeneous of degree $k$  in the $x\, 's$ and $y\, 's$ separately,
\itemitem{\rm(iii)} $\PP_m^{(k)}(x,y)$  in $ \QI$ in $\xon$ and $\yon$. 
}
\sas
}

\noindent
We  should note that (6.12)~{(d)~(ii)} immediately follows from the definition in (6.10) and the fact that the operator  
${\bf \OM}_m=O_mO_{m-1}\cdots O_1$ does not change degrees. 
In fact, from the expansion of the exponential $e^{(x,y)}$
we derive that 
$$
\PP_m^{(k)}(x,y)\ses \sum_{|p|=k}   {y^p\over p!} \ess  {\bf \OM}_mx^p\ .
$$ 
Moreover,  property (6.12)~{(d)~(iii)}  is an immediate consequence of the following remarkable fact which will also 
be needed in the sequel.

\sas
 
 {\ita We have the relation  $L_m(G)P=Q$ with $Q\in \QI$ if and only if $P\in \QI$. } \hfill {\rm(6.13)} 
\sa

\noindent We now  have    all the tools we need  to proceed  with our developments.
We begin with some basic properties of    $  \GG(m)$.
\sas

\noindent{\bol Proposition 6.1}

{\ita  

\itemitem{\rm(1)} All the homogeneous components of every $\FF\in \GG(m)$ are in\break $\QI$.

\itemitem{\rm(2)} The collection $\big\{\PP_m(x,b)\big\}_{b\in \orb}$ is a basis for  $\GG(m)$.

\itemitem{\rm(3)} $ \GG(m)$ is a $G$-module affording the regular representation of $G$.

\itemitem{\rm (4)} Every element $\Phi\in\GG$ may be written in the form
 $$
\ess \ess {\rm a)}\ess\ess  \Phi(x)\ses \ggg_Q \TT (x)\bigsp(\hbox{for some $Q\in \QI$}),
$$
where 
$$
{\rm b)}\ess\ess  \TT (x)\ses \sum_{b\in \orb}\eee_b \Psi_m(x,b)\ses c \Pi(x)^{2m+1}\sps \cdots .
$$}

\noindent{\bol  Proof}

We have seen in (6.7) that $L_m(G)=\ggg_{p_2}(x,\del_x)$ with $p_2(x)=(x,x)$. 
Since $(x,x)$ is $G$-invariant, it follows that the difference $(x,x)-(a,a)$ belongs to 
$\GG(m)$. 
Thus, from the definition in (6.1) it follows that 
$$
L_m(G)\FF\ses (a,a) \FF \bigsp (\, 
\hbox{for all}\ess \FF\in \GG(m)\, ).
\eqno (6.14)
$$ 
But if $\FF^{(i)}$ is the homogeneous component of degree $i$ in $\FF$,  
then (6.14) yields that 
$$
L_m(G)\FF^{(0)}=0\scs L_m(G)\FF^{(1)}=0  \ssp \ess \hbox{and}\ess\ess L_m(G) \FF^{(i)}=(a,a)\FF^{(i-2)}\ess\ess
\hbox{for all} \ess\ess i\ge 2\, .
$$  
Thus property {(1)} follows from (6.13). To show {(2)} we note first that property {(b)}
of the Baker--Akhiezer function implies that for all $Q\in \JO$ we have $\ggg_Q(x,\del_x)\PP_m(x,b)=0$ for
all $b\in \orb$. Thus
$$
\big\{\PP_m(x,b)\big\}_{b\in \orb}\subset \,\, \GG(m)\, .
$$
To show that this collection spans $\GG(m)$, 
we will use the polynomials  $\{\psi_b(x)\}_\orb$ defined 
in (1.8) with the specialization $\BA=\QI$ and
$B(x)=\Pi_G(x)^{2m}$. Since (1.9) immediately gives the identity
$$
1\equiv \sum_{b\in \orb }\psi_b(x) \bigsp (
\hbox{mod}\ess \JO\, ) 
$$
we derive from the definition in (6.1) that, for all $\FF\in \GG(m )$ we have the decomposition
$$
\FF(x)\ses \sum_{b\in \orb } \FF_b(x)\ ,
\eqno (6.15)
$$ 
with
$$
\FF_b(x)\ses \ggg_{\psi_b}(x,\del_x) \FF(x)\,.
\eqno (6.16)
$$
We claim that the latter is none other than a scalar multiple of $\PP_m(x,b)$. 
To prove this, note first that,
for all $Q\in \QI$ 
we have the relation
$$
Q(x)\psi_b(x)\sms Q(b)\psi_b(x)\equiv 0 \bigsp (
\hbox{mod}\ess \JO\, )
\eqno (6.17)
$$
Thus
$$
\eqalign{
\ggg_Q(x,\del_x)\FF_b(x)
&\ses \ggg_Q(x,\del_x)\ggg_{\psi_b}(x,\del_x) \FF(x)
\cr
(\hbox{by (I.7)})&\ses \ggg_{Q\psi_b}(x,\del_x) \FF(x)
\cr
(\hbox{by (6.17)})&\ses Q(b)\ggg_{\psi_b}(x,\del_x) \FF(x)\ses Q(b) \FF_b(x).
\cr}
\eqno (6.18)
$$ 
Using this relation with $Q(x)=\PP^{(k)}_m(x,y)$, we get
$$
\eqalign{
\gg_{\PP^{(k)}_m}\FF_b\, \Big|_{x=0}\ses \PP^{(k)}_m(b,y)\FF_b(0)  .
\cr
}
\eqno (6.19)
$$
On the other hand, since $\gg_{\PP^{(k)}_m}$ decreases degrees by $k$,
denoting by $\FF_b^{(k)}$ the $k^{th }$ homogeneous component of $\FF_b$,
it follows that
$$
\eqalign{
\gg_{\PP^{(k)}_m}\FF_b(x)\, \Big|_{x=0}
&\ses \LL \PP^{(k)}_m\scs \FF_b^{(k)}\RR_m 
\cr
&\ses \LL \FF_b^{(k)}  \scs \PP^{(k)}_m   \RR_m 
\cr
&\ses \ggg_{ \FF_b^{(k)}}    \PP^{(k)}_m(x,y)\, \Big|_{x=0}
\cr
&\ses \ggg_{ \FF_b^{(k)}}    \PP _m(x,y)\, \Big|_{x=0}
\cr
(\hbox{by {(b)} of (6.12) })
&\ses   \FF_b^{(k)}(y)    \PP _m(x,y)\, \Big|_{x=0}
\cr
(\hbox{by {(d) (i)} of (6.12)})
&\ses   \FF_b^{(k)}(y)    c_m(G)
\cr
}
\eqno (6.20)
$$
Combining (6.19) and (6.20) we get that
$$
\PP^{(k)}_m(b,y)\FF_b (0) \ses  \FF_b^{(k)}(y)    c_m(G)\, .
$$
This holding true for all $k$  yields that
$$
\PP _m(b,y)\FF_b (0) \ses  \FF_b (y)    c_m(G)\, .
$$
Solving for $\FF_b (y)$ and using the symmetry of $\PP_m(x,y)$, we now obtain
$$
\FF_b (y)\ses 
{\FF_b (0)\over c_m(G)} \ess\PP _m(y,b) \,,
\eqno (6.21)
$$
as desired. Combining (6.15) with (6.21)  proves that $\big\{\PP_m(x,b)\big\}_{b\in \orb}$ spans\break $\GG(m)$.

To complete the proof of {(2)}, we need to show independence. 
To this end, suppose that for some constants $c_b$ we have
$$
\FF\ses \sum_{b\in \orb}c_b \PP_m(x,b)\, .
\eqno (6.22)
$$
Then the relations in (1.9) and {(b)} of (6.12)  give, for $b'\in \orb$,
$$
\LL\psi_{b'} \scs \FF\RR_m\ses \ggg_{\psi_{b'}}\FF\Big|_{x=0}\ses \sum_{b\in \orb}c_b\, \psi_{b'}(b) \PP_m(x,b)\Big|_{x=0}\ses c_{b'}\,  c_m(G)\, .
\eqno (6.23)
$$
Thus
$$
\FF=0\ess\Longrightarrow\ess c_b=0\ess\ess
\hbox{for all }  b\in\orb\, ,
$$
proving independence. Incidentally, 
(6.23) yields that the expansion  in (6.22) may be   written in the form
$$
\FF\ses \sum_{b\in \orb}  \textstyle{\LL\psi_b \scs \FF\RR_m\over  c_m(G)}\ess \PP_m(x,b)\, .
$$
Finally, note that  property~{(c)} of $\PP_m(x,y)$ gives that
$$
T_\sig \PP_m(x,b)\ses  \PP_m(x,b\sig^{-1})\bigsp (\,  
\hbox{for all } \sig\in G\, ).
$$
Thus the character $\chi$ of the action of $G$  on the basis $\big\{\PP_m(x,b)\big\}_{b\in \orb}$ has the expansion 
$$
\chi(\sig)\ses \sum_{b\in \orb}\PP_m(x,b\sig^{-1})\Big|_{\PP_m(x,b)}\ses \sum_{b\in \orb}\chi(b\sig^{-1}=b)
\ses 
\cases {
|G| & if $\sig=id$,\cr\cr 
 0 &if $\sig\ne id$.}
$$
This gives {(3)}.  

Finally, note that if
$$
\FF(x)\ses \sum_{be\in \orb} c_b \PP_m(x,b)
\eqno (6.24)
$$
and we set
$$
Q\ses \sum_{b'\in \orb}{\textstyle {c_{b'} \over \eee_{b'}}} \, \psi_{b'}(x) 
$$
then clearly $Q\in \QI$, and we also have from  (6.24)
$$
\eqalign{
\ggg_Q\TT
&\ses  
\sum_{b\in \orb}\eee_b\sum_{b'\in \orb}{\textstyle {c_b\over \eee_{b'}}} \, \ggg_{\psi_{b'}}\Psi_m(x,b)
\cr
(\hbox{by (6.12)~{(b)}})&\ses  
\sum_{b\in \orb}\eee_b\sum_{b'\in \orb}{\textstyle {c_b\over \eee_{b'}}} \,  \psi_{b'}(b)\Psi_m(x,b)
\cr
(\hbox{by (1.9)})&\ses  
\sum_{b\in \orb}\eee_b {\textstyle {c_b \over \eee_b}}       \Psi_m(x,b)\ses \Phi(x).
\cr}
$$
This proves {(4)}~(a).

To prove {(4)}~(b), note that from the definition of $\TT(x)$ it follows that for any $\sig\in  G $ we have
$$
T_\sig\TT\ses 
\det(\sig)\, \TT.
$$
This forces all the homogeneous components  of $\TT$ to be $G$-invariant multiples of $\Pi_G(x)^{2m+1}$.
Thus we can write
$$
\TT(x)\ses A(x)\Pi_G(x)^{2m+1}\sps \cdots \bigsp(\hbox{for some $A(x)\in \LA_G$}).
$$
Now, from the definition of $\TT  $ and  (6.12)~{(b)}  we derive that
$$
\ggg_{\Pi ^{2m+1}}\TT  (x)\ses \sum_{b\in[a]_{S_n}}\eee(b)\Pi_G(b)^{2m+1}\Psi_m(x,b)\ses\Pi_G(a)^{2m+1}\sum_{b\in[a]_{S_n}}\Psi_m(x,b)\, .
\eqno (6.25)
$$
Since $\ggg_{\Pi ^{2m+1}}$ decreases degrees by $(2m+1)|\SS(G)|$,
we have
$$
degree\big(\ggg_{\Pi ^{2m+1}}A(x)\Pi_G(x)^{2m+1} \big)\ses degree(A(x))\, .
$$
Thus,
$$
degree\left(\mu\big(\ggg_{\Pi ^{2m+1}}\TT  (x)\big)\right)\ge degree(A(x))\, ,
\eqno (6.26)
$$
where, for each formal power series $\Phi$, we let $\mu(\Phi)$ denote the homogeneous 
component of least degree in $\Phi$.
However, the right hand side of (6.25)  (using (6.12)~{(d)~(i)}) yields that
$$
\mu\big(\ggg_{\Pi ^{2m+1}}\TT  (x)\big) \ses \Pi_G(a)^{2m+1}|G| c_m(G)\ne 0\, . 
\eqno (6.27)
$$
Combining  (6.26) with (6.27) forces  $A(x)$ to be a scalar. 

Note that this argument  also shows that for some constant $C$ we must have
$$
\ggg_{\Pi^{2m+1}} \Pi(x)^{2m+1}=C\ne 0 
\eqno (6.28)
$$
\sa

To introduce our next construct, we need further notation. 
To begin
with, we shall choose once and for all 
a degree lexicographic order of monomials. 
For instance, the one  which corresponds to the total order
$x_1>x_2>\ldots >x_n$. It will be convenient to  denote this order 
by ``$<_{dl}$''. 
Recalling that $\mu(\Phi)$ denotes the homogeneous component of 
least degree in $\Phi$, we will let $ l(\Phi)$
denote the d-lex least monomial in $\mu(\Phi)$. 
We also set $ l(f)= l(\FF)$ when $f=\mu (\FF)$.
\sas

\vbox{
\noindent{\bol Remark 6.1}

It will be convenient in our further developments to use the symbol ``$\FF\,\big|_{=k}$'' to denote the
the homogeneous components of $\FF$ of degree $k$. In the same vein, we let ``$\FF\big|_{<k}$'' and ``$\FF\big|_{\le k}$''
the sum of all the homogeneous components of $\FF$ of degree $<k$ and $\le k$ respectively.
 In our previous notation
$$
\FF\big|_{=k}= \FF^{(k)  }\scs\ess\ess\ess \FF\big|_{<k}=\sum_{i=0}^{k-1} \FF^{(i)}\ess\ess\ess \hbox{and}\ess\ess\ess \FF\big|_{\le k}=\sum_{i=0}^k \FF^{(i)}\ .
\eqno (6.29)
$$
Thus we have 
$$
\mu(\FF)=\FF\big|_{=k}\ess\ess \ess 
\hbox{if and only if}\ess\ess\ess \FF\big|_{<k}=0\ess 
\hbox{and}\ess  \FF\big|_{=k}\ne 0\, .
$$
Thus we clearly see that the map $\FF
\mapsto \mu(\FF)$   is not linear. Nevertheless, if $\mu(\FF_1)$ and 
 $\mu(\FF_2)$ have the same degree then
$$
\mu(\FF_1)+\mu(\FF_2)\neq 0\ess\ess 
\hbox{implies} \ess\ess \mu(\FF_1+\FF_2)= \mu(\FF_1)+\mu(\FF_2).
\eqno (6.30)
$$}

Keeping this in mind,  we let $ \mu(\GG(m)  )$ denote the linear span of the homogeneous components of least degree of elements of $\GG(m) $.
In symbols
$$
 \mu(\GG(m)  )\ses \CL  \big[ \, \mu(\Phi)\, :\, \Phi\in \GG(m) \,  \big]\,.
\eqno (6.31)
$$

The following result shows the intimate relation between $\GG(m) $ and\break
$ \mu(\GG(m) )$.
\sas

\noindent{ \bol Proposition 6.2}

{\ita   We can find in $\GG(m) $ a collection of formal power series 
$$
\FF_1\scs\FF_2\scs\ldots \scs \FF_{|G|} 
\eqno (6.32)
$$
with the property that
$$
l(\FF_1)<_{dl}l(\FF_2)<_{dl}\cdots <_{dl} l(\FF_{|G|})
\eqno (6.33)
$$
and such that each $f\in \mu(\GG(m)  )$ has a unique expansion of the form
$$
f\ses \sum_{l(\FF_i)\, \ge_{dl}\, l(f)} c_i\, \mu(\FF_i)\,.
\eqno (6.34)
$$ 
In particular, the collection 
$$
\mu(\FF_1)\scs \mu(\FF_2)\scs\ldots \scs \mu(\FF_{|G|}) 
\eqno (6.35)
$$
is a basis of $ \mu(\GG(m) )$ and we have
$$
\dim\,  \mu(\GG(m) )\ses |G|\,.
\eqno (6.36)
$$
}
\noindent{ \bol Proof } 

The collection in (6.32) satisfying (6.33)  can be constructed by starting with the basis  
$$
\big\{\PP_\mu(x,b) \big\}_{b\in \orb}\ ,
$$ 
then progressively reducing it to echelon form with respect to the 
degree-lexi\-co\-gra\-phic order of least monomials. This done,    (6.33)  yields that their
 minimum components in (6.35)
are linearly independent. In fact, more than that is true.
Note first that no element $\FF\in \GG(m) $ can have a least monomial that is different from each
of the monomials in (6.33). Indeed, such an element would necessarily be independent of the
$\FF_i$'s, and then $\GG(m) $ would have dimension greater than $|G|$. 
The existence of constants $c_i$ giving (6.34)  may be established by descent induction 
on the degree-lexicographic order of least monomials. To see this let
$$
f=\mu(\FF).
$$
Now the result is immediate if
$l(f)=l(\FF_{|G|})$. In fact, let $c$ be chosen so that $l(\FF_{|G|})$ does not occur 
in  $f-c\, \mu(\FF_{|G|})$.  
Then the difference
 $\FF-c\FF_{|G|}$ must be identically zero, for
otherwise its degree-lexicographically least monomial would necessarily be 
larger than $l (\FF_{|G|})$ and this, as we have seen, is not possible. This
gives 
$$
f\ses c\, \mu(\FF_{|G|}),
$$
and we are done in this case. 
So assume by induction that the expansion in
(6.34)  exists when $l(f)>_{dl}l(\FF_{i_o})$. Let $l(f)=l(\FF_{i_o})$. Again chose $c$ so that 
$l(\FF_{i_o})$ does not occur in $f-c\FF_{i_o}$. Here there are two cases. If $ f =c\ssp \mu(\FF_{i_o})$ we
are done.  If, on the other hand, this difference does not vanish identically, then,
since $l(f)=l(\FF_{i_o})$ implies that $degree(\mu(\FF))=degree(\mu(\FF_{i_o}))$, we can use (6.30) 
with $\FF_1=\FF$ and
$\FF_2=-c\, \FF_{i_o}$ and conclude that 
$$
f\ses \mu(\FF)\ses c\ssp \mu(\FF_{i_o}) \sps \mu(\FF- c\, \FF_{i_o})\ess .
$$
Since now $l(\FF-c\, \FF_{i_o})\ssp >_{dl}\ssp l(\FF_{i_o})$, we can use the induction hypothesis 
and deduce 
that, for some suitable constants $c_i$, we must have
$$
 f \ses c \ssp \mu(\FF_{i_o}) \sps \sum_{i=i_o+1}^{|G|}\ssp c_i\ssp \mu(\FF_i)\ess .
$$
This completes the induction and establishes (4.34). The remaining assertions are 
immediate consequences of (6.34).
\sas

The following remarkable fact provides us with a new tool for studying   $m$-Harmonics.
\sa

\noindent{\bol Theorem 6.1}

{\ita For all finite reflection groups $G$ and all $m\ge 0$ we have}
$$
\mu\big(\GG(m)\big)\ses \BH_G(m) .
\eqno (6.37)
$$
\noindent{\bol Proof}

In view of the fact that from Theorem~5.5 and Proposition~6.2 it follows that
these two spaces have the same dimension, to show the equality in (6.37) it is sufficient to derive the 
containment
$$
\mu\big(\GG(m)\big)\, \subseteq\,  \BH_G(m)   \, .
\eqno (6.38)
$$ 
This in turn immediately follows from  Theorem~5.3 once we verify that
$$
P\in \mu\big(\GG(m)\big)\ess\ess\ess 
\hbox{implies} \ess\ess\ess \ggg_{q_k}P(x)=0\ess\ess\ess (\hbox{for $k=1,2,\ldots ,n$})\, .
\eqno (6.39)
$$
This given, let $Q(x)$ be a homogeneous $G$-invariant  and note that since the difference $Q(x)-Q(a)$ vanishes
throughout 
$\orb$, it follows from the definition in (6.1) that we have
$$
 \ggg_Q(x,\del_x)\FF(x)\ses Q(a) \FF(x)\ess\ess\ess(\, 
\hbox{for all}\ess\ess \FF\in \GG(m)\, ). 
\eqno (6.40)
$$
Thus, if $Q$ is of degree $d$, equating homogeneous components of both sides of (6.40), we derive
that
$$
\ggg_Q(x,\del_x)\FF^{(k)}(x)\ses 
\cases {0 & if $k<d$,
\cr\cr
Q(a) \FF^{(k-d)}(x) &if $k\ge d$.
\cr
}
\eqno (6.41)
$$
Thus, if $\mu(\FF(x))$ has degree $d_o$, then it follows from this that
$$
\ggg_Q(x,\del_x)\FF^{(k)}(x)\ses 0\ess\ess\ess\ess 
\hbox{for all} \ess\ess   k-d<d_o \, .
\eqno (6.42)
$$
In particular, we must have
$$
\ggg_Q(x,\del_x)\mu\big(\FF (x)\big)\ses 0\, ,
$$
and (6.39) then immediately follows from the definition in (6.31). This establishes (6.38) and completes our proof.
\sa

We now have all the tools we need for the proof of Theorem~I.2. The argument is based on the following two basic observations
\sa

\noindent{\bol Proposition 6.3}

{\ita Let $d_{mn}=maxdegree\big(\BH_G(m)\big)$. 
Then
  

\itemitem{$\bf Ob_1:$} To show that an element $\FF\in \GG(m)$ vanishes, it is sufficient to check that all  
its homogeneous components of degree $\le d_{mn}$ vanish.


\itemitem{$\bf Ob_2:$} To show that a polynomial $Q(x)\in \QI$ belongs to $\JO$, 
 it is sufficent to show that $\ggg_Q(x,\del_x)$  kills
the  element $\TT(m)$ defined in {\rm(4)} of Proposition~{\rm6.1}.

} 
\noindent{\bol Proof}

Suppose that $\FF\in \GG$ satisfies
$$
 \FF^{(k)}=0\ess\ess\ess (\hbox{for  $k\le d_{mn}$}),
\eqno (6.43)
$$
and suppose if possible that  $\FF\ne 0$. Now we have shown that that for any $\FF\in \GG$ we have $\mu(\FF)\in \BH_G(m)$.
Thus (6.43) gives $\mu(\FF)=0$. But this is absurd since by definition  $\FF\ne 0$
implies $\mu(\FF)\ne 0 $.
This proves $\bf Ob_1$. Note next that the equation $\ggg_Q\TT=0$ together with (6.12)~{(b) } yields the identity
$$
\sum_{b\in\orb }\eee(b) Q(b)\PP_m(x,b)\ses 0\, .
$$
However, the independence of $\big\{\PP_m(x,b) \big\})_{b\in\orb}$ then forces $Q_(b)=0$ for all $b\in \orb$, But
that is $Q \in \JO$. This proves $\bf Ob_2$. 
\sa

We 
shall now establish Theorem~I.2 by proving a bit more.
\sa

\vbox{
\noindent{\bol Theorem 6.2}

{\ita Let $\CB\in \QI$ yield  a homogeneous basis for the quotient $\QI/\CJ_G(m)$,
and let $\CB_k$ be the subset of elements of degree $k$ in
$\cal B$. Then the collection
$$
\big\{ \ggg_b(x,\del_x)\Pi_G(x)^{2m+1}\big\}_{b\in \CB_k}
$$
is a basis of  $\CH_{=k}\big(\BH_G(m)\big)$.
}}

\noindent{\bol Proof}


We shall have to use two 
facts, namely that  
$$
 d_{mn}= degree\big(\Pi_G(x)^{2m+1}\big)
\eqno (6.44)
$$
and  
that the Hilbert series $F_{\BH_G(m)}(t)$ is palindromic. 
That is, we have
$$
dim\big(\CH_k(\BH_\orb\big)\ses dim\big(\CH_{ d_{mn}-k}(\BH_\orb\big)\, .
\eqno (6.45)
$$

We should note that (6.44) immediately follows from the 
definition of the polynomial $\psi_b(x)$ given in (1.8),
since we proved there 
that the maximum degree of the ordinary $G$-harmonics is equal 
to the degree of the
discriminant $\Pi_G(x)$. 
 
As for (6.45) we have to refer to the paper of
Felder--Veselov [5] for a proof.

This given, 
we shall  prove our result by induction on $k$. More  precisely 
we shall assume that 
\sas

\hsize=5.8truein
\itemitem {$1_k:$}{\ita  $P(\del_x)\PBRA)\ssp |_{<d_{mn}-k}\ses  0$ implies that
$P$ is congruent $
\hbox{mod}\ssp \JO$ to a polynomial $Q$ of degree $\leq k$;}

\noindent
and

\itemitem {$2_k:$} {\ita  $\big\{ \ggg_b(x,\del_x)\Pi_G(x)^{2m+1}\big\}_{b\in \CB_{ k}}$
is a basis for $\CH_{=d_{mn}-k}\big (\BH_G(m)\big)$; }
\sa 

\hsize=6truein
\def \Obo {{\bf Ob_1}}
\def \Obt {{\bf Ob_2}}
\def \Ob  {{\bf Ob_2}}
 
\noindent
hold true for all $k< k_o$ 
and complete the induction by showing that
 $1_{k_o}$ and $2_{k_o}$ must hold as well.

\def \PBRA {{\TT}}

\def \DBRA   {\Pi_G(x)^{2m+1}} 

\noindent  
To this end, note first that, since  $dim\big(\CH_0(\BH_\orb\big)=1$,
it follows from (6.39) that we also have
$$
dim\big(\CH_{d_{mn}}\big(\BH_\orb\big)\big)=1\, .
\eqno (6.46)
$$
Since the polynomial $\Pi_G(x)^{2m+1}$ is in $\QI$ and is clearly killed by all $G$-invariant
differential 
operators, it follows that it lies in  $\CH_{d_{mn}}\big(\BH_\orb\big)$.
But then (6.44) yields that every element of $\CH_{d_{mn}}\big(\BH_\orb\big)$ is necessarily a multiple of\break
$\Pi_G(x)^{2m+1}$. This proves $2_0$.  
To start, we need also check  the validity of  $1_0$. Note that
$1_0$ says that any $P$ such that 
$$
\ggg_P(x,\del_x)\PBRA \ssp |_{< d_{mn}}\ses 0
\eqno (6.47)
$$
must be congruent to a constant $
\hbox{mod}\ssp \JO$. Note that if 
$\ggg_P(x,\del_x)\PBRA\ssp |_{=d_{mn}}$ also vanishes then $\Obo$ gives that $\ggg_P(x,\del_x)$ kills $\PBRA$, and
then $\Obt$ yields that $P$ is congruent to zero  $
\hbox{mod}\ssp \JO$. On the other hand, if
$\ggg_P(x,\del_x)\PBRA\ssp |_{ =d_{mn}}$ 
does not vanish, then, since the homogeneous elements of
degree $d_{mn}$ in $\BH_\bra$ are all multiples of $\DBRA$, from {(4)~b)} of Proposition~6.1
we derive that
$$
\ggg_P(x,\del_x)\PBRA\ssp |_{=d_{mn}}\ses c\ssp \mu(\TT)
$$
for a suitable constant $c$. But then all the homogeneous components of degree $d_{mn}$
or less in 
$$
(\ggg_P(x,\del_x)-c)\ssp \PBRA
$$ 
must vanish, and $\Obo$ and $\Obt$ again yield that 
$P-c\in \JO$. This gives $1_0$. 

We are thus in a position to proceed with our induction, and we shall assume that
$1_k$ and $2_k$ hold for all $k<k_o$. We start by proving $2_{k_o}$.
To this end, note that by (6.45) for $k=k_o$ we need only show that  
$\big\{\ggg_b(x,\del_x) \DBRA \big\}_{b\in {\cal B}_{k_o}}\ssp $ is an independent set. 
So, let there 
be constants $c_b$ such that
$$
\sum_{b\in {\cal B}_{k_o}}\ssp c_b\ssp \ggg_b(x,\del_x) \DBRA \ses 0\,,
\eqno (6.48)
$$
and set
$$
P(x) \ses \sum_{b\in {\cal B}_{k_o}}\ssp c_b\ssp b(x)\,.
\eqno (6.49)
$$
Now, (4.48) implies that 
$$
\ggg_P(x,\del_x)\PBRA \ssp |_{\leq d_{mn}-k_o}\ses 0\ess .
$$
However, this brings us into $1_{k_o-1}$ and by induction we can find a polynomial $Q$
of degree $< k_o$ congruent to $P$ modulo $\JO$. But now, since 
$degree\ssp P=k_o> degree \ssp Q$ and $P$ is homogeneous, we deduce that
$P\in 
{\bf gr}\ssp \JO$. Now we have seen in (5.25) that $ 
{\bf gr}\ssp \JO=\CJ_G(m)$,
so it follows that $P\in  \JG$. But this together with (6.49) 
contradicts the
independence of $\CB$ $
\hbox{mod} \, \JG$. This forces the vanishing of all the coefficients 
$c_b$ in (6.49). Thus $2_{k_o}$ must hold true as desired.

Next we show $1_{k_o}$. So  let
$$
\ggg_P(x,\del_x )\PBRA \ssp |_{<d_{mn}-k_o}\ses 0\ess .
\eqno (6.50)
$$
If $\ggg_P(x,\del_x )\PBRA =0$, then by $\Ob $ we must have $P\in \JO$ and we are done.
If $\ggg_P(x,\del_x )\PBRA$ 
does not vanish, then (6.50) gives that
$$
degree\ssp \mu\big(\ggg_P(x,\del_x)\PBRA \big)\ses d_{mn}-k_1 \bigsp(\ssp{\rm with}\ssp k_1\leq k_o\ssp )\ess .
$$
By $2_{k_1}$, which is now available up to and including $k_o$, we can find a homogeneous
polynomial $Q_1$ of degree $k_1$ such that
$$
\mu\big( \ggg_P(x,\del_x)\PBRA \big)\ses \ggg_{Q_1}(x,\del_x)\ssp \DBRA\ess .
\eqno 
(6.51)
$$ 
This in turn implies that for a suitable constant $c$
$$
\ggg_{P-cQ_1}(x,\del_x ) \PBRA\ssp |_{\leq d_{mn}-k_1}\ses 0\ess .
$$
However, since $k_1\leq k_o$, this brings us down into the domain of $1_{k_o-1}$, so we can
use the induction hypothesis and conclude that $P-cQ_1$ is congruent $
\hbox{mod}\ssp \JO$   
to a polynomial $Q_2$ of degree at most $k_o-1$. 
In other words, we have shown that
$P$ is congruent $
\hbox{mod}\ssp \JO$ to the polynomial $Q=cQ_1+Q_2$ which is
of degree at most $k_o$, which is precisely what we needed to show. 
This completes the induction and our proof.
\sa



\sapp
\centerline {\bol REFERENCES}
\sa 

\item{ [1]} Yu. Berest, {\ita Huygens principle and the byspectral problem,}
CRM Proc.\ and Lecture Notes~{\bol 13}, Amer.\ Math.\ Soc., R.I., 1998, 
pp.~11--30.
\sa 


\item{[2]}  O. A. Chalykh and A. P. Veselov, {\ita Commutative Rings of Partial Differential Operators and Lie Algebras},
 Comm.\ Math.\ Phys.\ {\bf 126} (1990), 597--611.
\sa 

\item{[3]}  P. Etingof and V. Ginzburg, {\ita Om $m$-quasi-invariants of a Coxeter group},
 Mosc.\ Math.\ J. {\bol 2}  (2002), 555--566.
\sa 

\item{[4]} P. Etingof and E. Strickland, {\ita Lectures on quasi-invariants of Coxeter groups and the Cherednik algebra},
 Enseign.\ Math.\ (2) {\bol 49}  (2003), 35--65.



\sa 
\item{[5]} G. Felder and A. P. Veselov, {\ita Action of Coxeter Groups on $m$-harmonic polynomials and Knizhnik--Zamolodchikov equations},
 Mosc.\ Math.\ J.  {\bol 3}  (2003), 1269--1291.
\sa

\item{[6]} M. Feigin and A. P. Veselov, {\ita Quasi-invariants of Coxeter groups and $m$-harmonmic polynomials},
 Int.\ Math.\ Res.\ Not.\  2002,  no.~10, 521--545.
\sa 


\item{[7]} A. Garsia and M. Haiman, {\ita Some Natural bigraded Modules and the 
$q,t$-Kostka coefficients}, The Foata Festschrift. Electronic J. Combin.\
{\bol 3} (1996), Research Paper~24, 60 pp. 
\sa 

\item{[8]} A. Garsia and N. Wallach,  {\ita Shift Differential Operators and the Theory of $m$-Quasi-Invariants},
UCSD lecture notes, 2004. 
\sa

\item{[9]} A. Garsia and N. Wallach, 
{\ita Combinatorial aspects of the Baker--Akhiezer function for $S_2$},   
to appear in the European Journal of Combinatorics.

\sa
\item{[10]} J. Humphreys. {\ita Reflection Groups and Coxeter Groups}, 
Cambridge Studies in Advanced Mathematics {\bol 29}, 
Camb.\ Univ.\ Press, 1990, p.~62.

\sa
\item {[11]} E. M. Opdam, {\ita Some applications of Shift Operators}, 
Invent.\ Math.\ {\bol 98} (1989), 1--18.

\sa
\item {[12]} R. Steinberg, {\ita Differential Equations Invariant under finite reflection Groups},
Trans.\ Amer.\ Math.\ Soc.\ {\bol 112} (1964), 392--400.

\vskip1cm
{\smallsmc\baselineskip10pt
Department of Mathematics,
University of California, San Diego,
9500 Gilman Drive, Dept 0112,
La Jolla, CA 92093-0112 USA.

}

 
\end





 
