\magnification1200 \font\Bol=cmbx14 \font\smc=cmcsc10 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\smallsmc=cmcsc8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightex=cmex8 \def\multi#1{\vbox{\baselineskip=0pt\halign{\hfil$\scriptstyle\vphantom{(_)}##$\hfil\cr#1\crcr}}} \def\picture #1 by #2 (#3){ \vbox to #2{ \hrule width #1 height 0pt depth 0pt \vfillF \special{picture #3} % this is the low-level interface } } \def\scaledpicture #1 by #2 (#3 scaled #4){{ \dimen0=#1 \dimen1=#2 \divide\dimen0 by 1000 \multiply\dimen0 by #4 \divide\dimen1 by 1000 \multiply\dimen1 by #4 \picture \dimen0 by \dimen1 (#3 scaled #4)} } \def \first {\scaledpicture 2.18in by 2.71in (first scaled 400)} \def \slinkeys {\scaledpicture 12.53in by 5.71in (slinkeys scaled 400)} \def \drop {\scaledpicture 8.49in by 2.71in (drop scaled 400)} \newbox\squ % box character for ends of proofs \def\sqbox{\copy\squ\hskip -.2pt} \font\Ch=msbm10 \def \UA {\hskip -.05in\uparrow} \def\N{\hbox{\Ch N}} \def\Z{\hbox{\Ch Z}} \def\R{\hbox{\Ch R}} \def\C{\hbox{\Ch C}} \def\Q{\hbox{\Ch Q}} \def\S{\hbox{\Ch S}} \def\la{{\lambda}} \def\htq #1{{\widetilde{#1}^q}} \def \CD {{\cal D}} \def \CA {{\cal A}} \def \CH {{\cal H}} \def \CB {{\cal B}} \def \bra { {{\scriptscriptstyle [}a {\scriptscriptstyle ]} }} \def \BR {{\bf R}} \def \BH {{\bf H}} \def \CJ {{\cal J}} \def \CN {{\cal N}} \def \CR {{\cal R}} \def \mBH {{\hskip -.05truein\bf R}} \def \mBR {{\hskip -.05truein\bf R}} \def \DG {\DD_G} \def \bra { {{\scriptscriptstyle [}a {\scriptscriptstyle ]} }} \def \BR {{\bf R}} \def \BH {{\bf H}} \def \CJ {{\cal J}} \def \CN {{\cal N}} \def \CR {{\cal R}} \def \mBH {{\hskip -.05truein\bf R}} \def \mBR {{\hskip -.05truein\bf R}} \def \DG {\DD_G} \def \BGA {{\bf \Gamma}} \def \HBRA {{\bf H}_{[a]}} \def \PBRA {\Phi_{[a]}} \def \DBRA {\DD_{[a]}} \overfullrule=0pt \def\ligne#1{\ifcase#1 \or\hbox{\sqbox} \or\hbox{\sqbox\sqbox} \or\hbox{\sqbox\sqbox\sqbox} \or\hbox{\sqbox\sqbox\sqbox\sqbox} \or\hbox{\sqbox\sqbox\sqbox\sqbox\sqbox} \or\hbox{\sqbox\sqbox\sqbox\sqbox\sqbox\sqbox} \or\hbox{\sqbox\sqbox\sqbox\sqbox\sqbox\sqbox\sqbox} \or\hbox{\sqbox\sqbox\sqbox\sqbox\sqbox\sqbox\sqbox\sqbox}\fi} \def\un#1{\ligne#1} \def\deux#1#2{{\hbox{$\vbox{\offinterlineskip \ligne#2 \vskip -.2pt \ligne#1}$}}} \def\trois#1#2#3{{\hbox{$\vbox{\offinterlineskip \ligne#3 \vskip -.2pt \ligne#2 \vskip -.2pt \ligne#1}$}}} \def\quatre#1#2#3#4{{\hbox{$\vbox{\offinterlineskip \ligne#4 \vskip -.2pt \ligne#3 \vskip -.2pt \ligne#2 \vskip -.2pt \ligne#1}$}}} \def\cinq#1#2#3#4#5{{\hbox{$\vbox{\offinterlineskip \ligne#5 \vskip -.2pt \ligne#4 \vskip -.2pt \ligne#3 \vskip -.2pt \ligne#2 \vskip -.2pt \ligne#1}$}}} \def \QSS {{\cal QI}} \def \SYM {{\cal SYM}} % % Note that you can also say, e.g., % \special{postscript xxx yyy zzz} % to include PostScript graphics in your documents % \overfullrule=0pt \baselineskip 14pt \settabs 8 \columns \+ \hfill&\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & & \hfill \hfill\cr %\voffset=1truein tells how low in the page should we start! %\hoffset=.5truein %tells how far to right should we start! \parindent=.5truein \hfuzz=3.44182pt \hsize 6truein \font\small=cmr6 \font\title=cmbx10 scaled\magstep2 \font\normal=cmr10 \font\small=cmr6 \font\ninerm=cmr9 \font\binerm=cmbx9 \font\ita=cmti10 \font\bol=cmbx10 \def\CF {{\cal F}} \def\CM {{\cal M}} \def\CA {{\cal A}} \def\CB {{\cal B}} \def\CC {{\cal C}} \def\CD {{\cal D}} \def\CE {{\cal E}} \def\CH {{\cal H}} \def\CG {{\cal G}} \def\RFX {Rat[{\cal F};x_1,\ldots ,x_n]} \def \xsig {x_{\sig_1},x_{\sig_2},\ldots ,x_{\sig_n}} \def \PP {P} \def \PS {\Psi} \def \RSYM {Ratsym[\CF;\xon]} \def \part {\vdash} \def \DD {\Delta} \def\del {\partial} \def \con {\subseteq} \def \scong {\ess \cong \ess } \def \brkpg {\supereject} \def\lll{\lambda} \def \La {\Lambda} \def\sig{\sigma} \def \ttt {\theta} \def\dia{\diamondsuit} \def \ggg {\gamma} \def\PI{\Pi} %\def \-> {\rightarrow} \def\LL{\big\langle} \def\RR {\big\rangle} \def\splangle{\langle\ssp } \def\sprangle {\ssp \rangle} \def \UU {{\scriptscriptstyle U} } \def \HHH {{\bf H}} \def\OO {\Omega} \def\DD {\Delta} \def\OM {\Omega} \def\om {\omega} \def\la {\lambda} \def\xon {x_1,x_2,\ldots ,x_n} \def\GA {\Gamma} \def \sas {\vskip .06truein} \def\sa{{\vskip .125truein}} \def\msas{{\vskip -.01truein}} \def\sap{{\vskip .25truein}} \def\sapp {{\vskip .5truein}} \def\sss {\sigma} \def \eee {\epsilon} \def\DDD {\Delta} \def\aaa {\alpha} \def\bbb {\beta} \def\ccc {\gamma} \def\ddd{\delta} \def\ggg {\gamma} \def\aa {\alpha} \def\bb {\beta} \def\cc {\gamma} \def\dd{\delta} \def\gg {\gamma} \def\con {\subseteq} \def \ses {\, = \,} \def \sps {\, + \,} \def \sms {\, - \,} \def \lag {\langle} \def \shhh {{\scriptstyle \cup\hskip -.18em\cup}} \def \rag {\rangle} \def \scs {\ssp , \ssp} \def \ess {\enskip} \def \ssp {\hskip .25em} \def \bigsp {\hskip .5truein} \def \part {\vdash} \def \scos {\ssp :\ssp} \def \DD {\Delta} \normal \def\uul{\underline} \vsize=8truein %\normalbaselineskip=12pt 18pt for space and a half, %\normalbaselines24pt for double space \sap \def\today{\ifcase\month\or January\or February\or March\or April\or may\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def \LRA {{\ssp \longrightarrow \ssp}} \def \RA {{ \rightarrow }} \def \BH {{\bf H}} \def \BM {{\bf M}} \def \CC {{\cal C}} \def \CJ {{\cal J}} \def \CZ {{\cal Z}} \def \CH {{\cal H}} \def \CA {{\cal A}} \def \CS {{\cal S}} \def \CR {{\cal R}} \def \CB {{\cal B}} \def \BV {{\bf H}} \def \BR {{\bf R}} \def \BK {{\bf K}} \def \BQ {{\bf Q}} \def \BM {{\bf M}} \def \om {\omega} \def \BH {{\bf H}} \def \TK {{\tilde K}} \def \TH {{\tilde H}} \def \xon {x_1,\ldots ,x_n} \def \yon {y_1,y_2, \ldots ,y_n} \def \LLL {\langle} \def \RRR {\rangle} \def \BH {{\bf H}} \def \BR {{\bf R}} \def \BQ {{\bf Q}} \def \om {\omega} \def \TK {{\tilde K}} \def \TH {{\tilde H}} \def \xon {x_1,x_2,\ldots ,x_n} \def \yon {y_1,y_2, \ldots ,y_n} \def \LLL {\langle} \def \RRR {\rangle} \def \xom {x_1,\ldots ,x_m} \def \sgmu {\ ^{sg}\BH_\mu} \def \bhmu {\BH_\mu} \def \pmu {\rho_\mu} \def \FP {{\cal FP}[X,Y]} \def \gamu {{\bf \Gamma}_\mu} \def \mgamu {m({\bf \Gamma}_\mu)} \def \RImu {{\cal RI}_\mu} \def \twotabs {\scaledpicture 3.61in by 1.081in (twotabs scaled 800)} \def \interdiag {\scaledpicture 2.61in by 1.581in (interdiag scaled 800)} \def \secinterdiag {\scaledpicture 2.61in by 1.581in (sinterdiag scaled 800)} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} %First page headline in LaTeX for S\'eminaire Lotharingien de Combinatoire %--first part \font\rms=cmr8 \font\its=cmti8 \font\bfs=cmbx8 \headline={\its S\'eminaire Lotharingien de Combinatoire \bfs 50 \rms (2005), Article~B50j\hfill} \footline={\hfil} \def \sij {s_{ij}} \def \oA {\overline A} \def \oB {\overline B} \def \oC {\overline C} \def \BB {{\bf B}} \def \BV {{\bf V}} \def \II {{\rm I}} \def \dij {\delta_{ij} } \def \QSS {{\cal QSS}} \def \SYM {{\cal SYM}} \def \QSS {{\cal QI}} \def \SYM {{\cal SYM}} \def \orb {{[a\, ]_G }} \def \CJ {{\cal J}} \def \ida {\CJ_{\orb}(\BA)} \def \QI {{\cal QI}_m[G]} \def \GG {\Gamma_{{\orb}}} \def \TT {\Theta_\orb} \def \bm {{\bf m}} \def \CL {{\cal L}} \def \SYM {{\cal SYM}} \def \FF {\Phi} \def \PP {\Psi} \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} \def \LA {\Lambda} \def \icon {\supseteq} \def \BA {{\bf A}} {\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 $$ \ggg_Q(x,\del_x)\ses Q(\del_x)+\sum_{|q|d_\BA$ then the fact that $$ 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) $$ \noindent{\bol Proof} Let $\CB^{\le k}=\{ \phi_{i} \}_{i=1}^{m_k} $ so that $\CB^{= k}=\{ \phi_{i} \}_{m_{k-1}_\BA$. That means that if this new basis is $$ \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}<0$. In particular the conditions in (5.14) immediately give us that $$ \del_x^pQ(x)\, \big |_{x=a}\ses \sum_{|\eee_r|x_2>\ldots >x_n$. It will be convenient to denote this order by ``$<_{dl}$''. Recalling that $\mu(\Phi)$ denotes the homogeneous component of least degree in $\Phi$, we will let $ l(\Phi)$ denote the d-lex least monomial in $\mu(\Phi)$. We also set $ l(f)= l(\FF)$ when $f=\mu (\FF)$. \sas \vbox{ \noindent{\bol Remark 6.1} It will be convenient in our further developments to use the symbol ``$\FF\,\big|_{=k}$'' to denote the the homogeneous components of $\FF$ of degree $k$. In the same vein, we let ``$\FF\big|_{_{dl}l(\FF_{i_o})$. Let $l(f)=l(\FF_{i_o})$. Again chose $c$ so that $l(\FF_{i_o})$ does not occur in $f-c\FF_{i_o}$. Here there are two cases. If $ f =c\ssp \mu(\FF_{i_o})$ we are done. If, on the other hand, this difference does not vanish identically, then, since $l(f)=l(\FF_{i_o})$ implies that $degree(\mu(\FF))=degree(\mu(\FF_{i_o}))$, we can use (6.30) with $\FF_1=\FF$ and $\FF_2=-c\, \FF_{i_o}$ and conclude that $$ 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 $k degree \ssp Q$ 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 |_{