Invariant derivations and differential forms\\ for reflection groups
Victor\ Reiner
Anne V.\ Shepler
Classical invariant theory of a complex reflection group $W$ highlights three beautiful structures: \begin{itemize em[ ] \hspace{-3ex$\bullet$ the $W$-invariant polynomials constitute a polynomial algebra, over which em[ ] \hspace{-3ex$\bullet$ the $W$-invariant differential forms with polynomial coefficients constitute an exterior algebra, and %over the $W$-invariant polynomials, em[ ] \hspace{-3ex$\bullet$ the relative invariants of any $W$-representation constitute a free module. \end{itemize When $W$ is a {duality (or {well-generated) group, we give an explicit description of the isotypic component within the differential forms of the irreducible reflection representation. This resolves a conjecture of Armstrong, Rhoades and the first author, and relates to Lie-theoretic conjectures and results of Bazlov, Broer, Joseph, Reeder, and Stembridge, and also Deconcini, Papi, and Procesi. We establish this result by examining the space of $W$-invariant differential derivations; these are derivations whose coefficients are not just polynomials, but differential forms with polynomial coefficients. For every complex reflection group $W$, we show that the space of invariant differential derivations is finitely generated as a module over the invariant differential forms by the basic derivations together with their exterior derivatives. When $W$ is a duality group, we show that the space of invariant differential derivations is free as a module over the exterior subalgebra of $W$-invariant forms generated by all but the top-degree exterior generator. (The basic invariant of highest degree is omitted.) Our arguments for duality groups are case-free, i.e., they do not rely on any reflection group classification.
1612.01031.tex
%  Vic Reiner and Anne Shepler, December 2016
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\title[Invariant derivations and differential forms]
{Invariant derivations and differential forms\\ for reflection groups}

\dedicatory{To Peter Orlik and Hiroaki Terao on their 80th and 65th birthdays, respectively.}

\author{Victor\ Reiner}
\author{Anne V.\ Shepler}
\thanks{Research supported in part by NSF grants DMS-1601961,DMS-1101177,   
and Simons Foundation Grant \#429539.}
\email{reiner@math.umn.edu, ashepler@unt.edu}
%\address{School of Mathematics\\University of Minnesota\\Minneapolis, MN 55455, USA}

\keywords{Reflection groups.  Invariant theory. Weyl groups. Coxeter groups.}

\begin{abstract}
Classical invariant theory of a complex reflection group $W$ 
highlights three beautiful structures:
\begin{itemize}
\item[ ] \hspace{-3ex}$\bullet$ 
the $W$-invariant polynomials constitute a polynomial algebra, over which
\item[ ] \hspace{-3ex}$\bullet$ 
the $W$-invariant differential forms with polynomial coefficients
constitute an exterior algebra, and %over the $W$-invariant polynomials,
\item[ ] \hspace{-3ex}$\bullet$ 
the relative invariants of any $W$-representation
constitute a free module.
\end{itemize}
When $W$ is a {\it duality} (or {\it well-generated}) group,
we give an explicit description 
of the isotypic component within the differential forms 
of the irreducible reflection representation.
This resolves a conjecture of Armstrong, Rhoades
 and the first author, 
and relates to Lie-theoretic conjectures and
results of 
Bazlov, Broer, Joseph, Reeder, and Stembridge,
and also Deconcini, Papi, and Procesi.
We establish 
this result by examining the space of 
$W$-invariant differential derivations; these are derivations whose coefficients
are not just polynomials, 
but differential forms with polynomial coefficients.

For every complex reflection group $W$,
we show that the space of invariant differential derivations
is finitely generated as a module over
the invariant differential forms by
the basic derivations together
with their exterior derivatives.
When $W$ is a duality group,
we show that the space of invariant differential
derivations is free as a module over the exterior subalgebra
of $W$-invariant forms generated by all 
but the top-degree exterior generator.
(The basic invariant of highest degree is omitted.)

Our arguments for duality groups are case-free, i.e., 
they do not rely on any reflection group classification.
\end{abstract}

\maketitle

\vspace{-2ex}

\section{Introduction}
A celebrated result of Solomon~\cite{Solomon}
exhibits the set of differential forms invariant
under the action of a complex reflection group
as an {\em exterior algebra}.  A similar result holds when
we consider derivations instead of differential forms, 
i.e., elements of
$S(V^*)\ot V$ instead of $S(V^*)\ot \wedge V^*$, for a reflection representation $V$
with symmetric algebra $S(V^*)$.
The polynomial degrees of generators of these sets of invariants
are positioned into various combinatorial identities
expressing the geometry, topology, and representation theory
of reflection groups.
%
Recently, a theory of
Catalan combinatorics for reflection groups (e.g., see~\cite{ArmstrongRRhoades}) 
has prompted questions about the structure of invariant forms for other representations of
a reflection group.  Of particular interest are the {\em differential derivations},
elements of $S(V^*)\ot \wedge V^*\ot V$.
The first author together with Armstrong and Rhoades
conjectured a 
formula~\cite[Conj.~11.5${}^\prime$]{ArmstrongRRhoades} 
for the Poincar\'e polynomial
of the invariant differential derivations when the
reflection group $W$ is real, i.e., a finite Coxeter group.  
We verify this conjecture and show that
the set of invariant differential derivations, 
$$(S(V^*)\ot \wedge V^*\ot V)^W\, ,$$
is a free module over an exterior algebra
constructed from exterior derivatives $df_i$ of 
all but one of the basic invariants $f_i$;
the last basic invariant of highest polynomial degree is
omitted.
In fact, we give the 
explicit structure of the invariant differential derivations
for all complex reflection groups $W$ that are duality groups.
We also give a basis for nonduality groups.
We explain these two main results next.

\subsection*{Invariant theory of reflection groups}
Recall that a {\it reflection} on a finite dimensional
vector space $V=\CC^\ell$ is a nonidentity general linear transformation
that fixes a hyperplane in $V$ pointwise. A {\it complex reflection group} $W$
is a subgroup of $\text{GL}(V)$
generated by {\it reflections}; we assume all reflection groups
are finite.
We fix a $\CC$-basis
$x_1,\ldots,x_\ell$ of $V^*$ with dual $\CC$-basis
$y_1, \ldots, y_\ell$ of $V$ 
and identify the symmetric algebra $\SSS:=\Sym(V^*)$ 
with the polynomial ring
$\CC[x_1,\ldots,x_\ell]$,
which carries a $W$-action by linear substitutions.
Shephard and Todd~\cite{ShephardTodd} and Chevalley~\cite{Chevalley}
showed that the $W$-invariant subalgebra $\SSS^W$ is again {\it polynomial}:
$$
\SSS^W=\CC[f_1,\ldots,f_\ell]
$$ 
for certain algebraically independent polynomials
$f_1, \ldots, f_\ell$ called {\em basic invariants}.  One can
choose $f_1,\ldots,f_\ell$ homogeneous;
we assume 
$\deg(f_1) \leq \cdots \leq \deg(f_\ell)$ after re-indexing.
It follows that 
for any $W$-representation $U$,
the space of {\it relative invariants} $(\SSS \otimes U)^W$
forms a free $\SSS^W$-module of rank $\dim_\CC(U)$
(see, e.g., Hochster and Eagon~\cite[Prop.~16]{HochsterEagon}).
Note that 
we take all tensor products and exterior algebras over $\CC$
unless otherwise indicated.  We also assume all representations
are complex and finite dimensional.

\subsection*{Differential forms and derivations}
Two particular cases have received much attention.
First, when 
$U=\wedge V^*$, 
one may identify $\SSS\ot U=\SSS \otimes \wedge V^*$ 
with the $\SSS$-module of {\it differential forms with polynomial coefficients}
generated by $dx_1, \ldots, dx_\ell$.
Solomon's theorem~\cite{ShephardTodd, Solomon} 
asserts that $(\SSS \otimes \wedge V^*)^W$ is not
just a free $\SSS^W$-module, but, in fact, an exterior algebra
over $\SSS^W$ on exterior generators $df_1,\ldots,df_\ell$,
where $df_j:=\sum_{i=1}^\ell \frac{\partial f_j}{\partial x_i} \otimes x_i$
has degree $e_i:=\deg(f_i)-1$:
$$
(\SSS \otimes \wedge V^*)^W = \bigwedge_{S^W}\{ df_1, \ldots, df_{\ell}\}
\, .
$$
Second, when $U=V$, one
may identify $\SSS\ot U=\SSS \otimes V$ 
with the set of {\it derivations} $S \rightarrow S$ on $V$
generated by the partial derivatives 
$\del/\del x_1, \ldots, \del/\del x_n$.
Here, a derivation 
$\theta=\sum_{i=1}^\ell \theta^{(i)} \otimes y_i$
maps $f$ in $S$ 
to 
$\theta(f)=\sum_{i=1}^\ell 
\theta^{(i)} \sfrac{\partial}{\partial x_i}(f)$.
One may choose a homogeneous
basis $\theta_1,\ldots,\theta_\ell$
for the free $\SSS^W$-module $(\SSS \otimes V)^W$, i.e., 
$$(\SSS \otimes V)^W = S^W\theta_1 \oplus \ldots\oplus S^W \theta_\ell\, $$
for some {\em basic derivations}
$\theta_j=\sum_{i=1}^\ell \theta_i^{(j)} \otimes y_i$
with each $\theta_i^{(j)}$ in $\SSS$  homogeneous of
fixed degree, say $e_j^*$.


\subsection*{Duality groups}
Our main results combine these contexts, with special results
for duality groups.
An irreducible complex reflection group $W$ is a {\it duality group}
if its {\it coexponents} $e_1^* \geq \cdots \geq e_\ell^*$
and {\it exponents} $e_1 \leq \cdots \leq e_\ell$ determine each other 
via the relation
$
e_i + e_i^* = h,
$
where $h:= \deg(f_\ell)=e_\ell+1$ is the largest degree of a basic invariant, 
called the {\it Coxeter number} for the duality group $W$.
%

Duality groups include all irreducible 
{\it real} reflection groups (i.e., finite Coxeter groups) 
as well as symmetry groups
of {\it regular complex polytopes}~\cite{Coxeter}. 
It was observed by Orlik and Solomon~\cite{OrlikSolomon} 
in a case-by-case fashion 
(using the 
classification~\cite{ShephardTodd})
that an irreducible 
complex reflection group $W$ is a duality group if and only if
it is {\it well-generated}, that is, generated by $\ell=\dim(V)$ reflections,
but we will not need this fact in the sequel.

\subsection*{Main theorems}
We consider the set $M$ of mixed forms, called {\em differential derivations},
$$
M:=\SSS \otimes \wedge V^* \otimes V.
$$
We view $M$
as an $(\SSS \otimes \wedge V^*)$-module
via multiplication in the first two tensor positions.
Its $W$-invariant subspace $M^W$
is then a $(\SSS \otimes \wedge V^*)^W$-module.
In general, $M^W$ will not
be a {\it free} module.  Nevertheless, 
our first main result asserts that for duality groups, 
$M^W$  {\it is} a free $R$-module,
where $R$ is the subalgebra of 
the invariant forms generated by all $df_i$ but the last,
$$
R:=\bigwedge_{\SSS^W}
\{df_1,\ldots,df_{\ell-1}\}\, ,
$$
with only $df_{\ell}$ omitted.  To give an $R$-basis, 
we extend the usual exterior derivative operator $d$ on $\SSS$ to
a function on $\SSS\ot \wedge V^* \ot V$ defining
$
d(f\ot \omega \ot y) = \sum_{1\leq i\leq \ell} 
\sfrac{\del f}{\del x_i} \ot (x_i\wedge \omega) \ \ot y.
$
We apply $d$ to derivations, 
after identifying them with their images under the
inclusion 
$$
\begin{array}{rcl}
S \ot V &\hookrightarrow &S \ot \wedge V^* \ot V\\
f \otimes y\ &\longmapsto &f \otimes\ \, 1\ \otimes \ \, y.
\end{array}
$$
Our first main result is shown with case-free arguments
(it does not depend on any classification).
%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
\label{main-result}
For $W$ a duality (well-generated) complex reflection group,
$(\SSS \otimes \wedge V^* \otimes V)^W$ forms a
free $R$-module on the $2\ell$ basis elements
$
\{\theta_1,\ldots,\theta_\ell, \,\, d\theta_1, \ldots, d \theta_\ell\}\, .
$ 
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%


For arbitrary complex reflection groups, we have no uniform statement like
Theorem~\ref{main-result}.  However,
we give an explicit free
$S^W$-basis for $(\SSS \otimes \wedge V^* \otimes V)^W$
in the remaining (non-duality) cases, showing this:

%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
\label{generation-result}
For $W$ any complex reflection group,
$(\SSS \otimes \wedge V^* \otimes V)^W$ is generated as a
module over the exterior algebra 
$(\SSS \otimes \wedge V^*)^W=\bigwedge_{\SSS^W}\{df_1,\ldots,df_{\ell}\}$ by the $2\ell$ generators
$
\{\theta_1,\ldots,\theta_\ell, \,\, d\theta_1, \ldots, d \theta_\ell\}.
$ 
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%

To be clear:
$(\SSS \otimes \wedge V^* \otimes V)^W$ is 
not {\it freely} generated as a 
module over $\bigwedge_{\SSS^W}\{df_1,\ldots,df_{\ell}\}$
by $\{ \theta_i, d\theta_i \}_{i=1}^\ell$.

\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}
\label{rank-one-example}
A rank $\ell=1$ reflection group
$W \subset \text{GL}(V)
=\text{GL}_1(\CC)=\CC^\times$ is a {\it cyclic group}
$
W=\langle \zeta \rangle \cong \ZZ/h\ZZ
$
for some $\zeta=e^{\frac{2\pi i}{h}}$ in $\CC$.
Let $V=\CC x$ and $V^*=\CC y$ with $y$ dual to $x$.
Under the generator of $W$,
$x \mapsto \zeta^{-1}x$ and $y \mapsto \zeta y$.
Then
$$
\SSS=\CC[x],\quad
\SSS^W=\CC[f_1] \quad\text{ where }f_1=f_\ell=x^h \text{ has degree }h.
$$
The $\SSS^W$-module of invariant forms,
$(\SSS \otimes \wedge V^*)^W$, is an exterior algebra over
$\SSS^W$ generated by $df_1=h x^{h-1} \otimes x$ 
of degree $e_1=h-1$, that is,
$$
\begin{array}{rccccl}
(\SSS \otimes \wedge V^*)^W
 &=&(\SSS \otimes \wedge^0 V^*)^W & \oplus & (\SSS \otimes \wedge^1 V^*)^W\\
 &=&\underbrace{\SSS^W (1 \otimes 1)}_{:=R} 
                           & \oplus & \SSS^W (x^{h-1} \otimes x).
\end{array}
$$
On the other hand, the $\SSS^W$-module
of invariant derivations is
$$(\SSS \otimes V)^W=\SSS^W (x\otimes y),
$$ where
$\theta_1= x \otimes y$ has degree $e_1^*=1$.
In particular, $W$ is a duality group, since $e_1^*+e_1=1+(h-1)=\deg(f_\ell)$.
Now consider the invariant differential derivations:
An easy check confirms that
$M^W=(\SSS \otimes \wedge V^* \otimes V)^W$ is a free module
over $R=\SSS^W$ with basis 
$
\{ \theta_1, d \theta_1\}=\{ x \otimes 1 \otimes y, 1 \otimes x \otimes y\}
$,
that is, 
$$
(\SSS \otimes \wedge V^* \otimes V)^W 
= \SSS^W(x \otimes 1 \otimes y) \oplus \SSS^W( 1 \otimes x \otimes y)\, .$$
\end{example}

\medskip

\subsection*{Outline}
Section~\ref{further-context-section} provides further context and implications of
Theorem~\ref{main-result} while Section~\ref{Lie-theory-section}
gives its relation to some theorems and conjectures in Lie theory.
We collect some tools
for establishing helpful reflection group numerology
(like Molien's theorem and a lemma by Opdam)
in Section~\ref{Molien-strategy-section}
and reap that numerology in Section~\ref{Numerology-section}.
Section~\ref{Saito-criterion-section} is a slight
digression deriving basis conditions reminiscent of Saito's Criterion for
free arrangements.
A linear independence condition is given in
Section~\ref{independence-section}. 
We complete the proof of Theorem~\ref{main-result} 
in Section~\ref{well-generated-section}
after showing that duality groups exhibit auspicious numerology
in Section~\ref{numerology-works-section}.

The remainder is about nonduality groups and 
Theorem~\ref{generation-result}.
Section~\ref{2-dimensional-groups-section}
addresses rank two reflection groups, while
Section~\ref{G31-section} considers the 
Shephard and Todd group $G_{31}$ --- the only irreducible non-duality
group that is neither of rank two, nor within the Shephard and Todd infinite
family of monomial groups $G(r,p,n)$.
Section~\ref{poorly-generated-section} assembles the
proof of Theorem~\ref{generation-result},
relegating the case of $G(r,p,n)$ to
Appendix~\ref{appendix-section}.  When $p=1$ or $p=r$,
these are duality groups and covered by
Theorem~\ref{main-result}; when $1 < p <r$,
they are nonduality groups and we give
an alternate basis for the
invariant differential derivations in this appendix.

We consider some further questions in Section~\ref{questions-section}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Implications of Theorem~\ref{main-result}}
\label{further-context-section}

To provide context for Theorem~\ref{main-result},
we first note a few of its consequences and
special cases.

%%
\subsection*{The case of exterior degree zero}

Theorem~\ref{main-result}
implies something that we already knew about 
$$
(\SSS \ot \wedge^0 V^* \ot V)^W=(\SSS \ot V)^W,
$$
namely, that it is a free
$\SSS^W$-module on the basis $\{\theta_j \}_{j \in \{1,\ldots,\ell\}}$;
this is true even when $W$ is not a duality group.
We will end up using this fact in the proof of the theorem.


%%
\subsection*{The case of top exterior degree}
At the opposite extreme, Theorem~\ref{main-result} asserts
that 
$$(\SSS \ot \wedge^\ell V^* \ot V)^W
\cong (S\ot \det \ot V)^W$$ 
is free
as an $\SSS^W$-module on the 
basis $\{df_1 \cdots df_{\ell-1}\, d\theta_k \}_{k \in \{1,\ldots, \ell\}}$.
This agrees 
with the polynomial degrees of a basis
found
in~\cite{Shepler} for any reflection group $W$,
as we may view $(S\ot \det \ot V)^W$ as
the space of invariant derivations
for the ``twisted reflection representation''
$\det \ot V$, where $\det: W\mapsto \CC^*$ is the determinant
character of $W$ acting on $V$.
Indeed, if $J=\det\left( \frac{\partial f_i}{\partial x_j}\right)$ is the Jacobian
determinant of the basic invariants $f_1,\ldots,f_\ell$ 
(see Section~\ref{J-and-Q-subsection}), then 
the forms $df_1\cdots df_{\ell-1}\, d\theta_k$ have degrees
$$e_1+\ldots+ e_{\ell-1}+ e_k^*-1
=(e_1+\ldots+ e_{\ell})- (e_{\ell}+1) + e_k^*
=\deg J - (h - e_k^*)
= \deg J - e_{k}
$$
when $G$ is a duality group 
(see~\cite[Cor.~13(e)]{Shepler}).




%%
\subsection*{The Hilbert series consequence}
Theorem~\ref{main-result} has implications for 
{\it Hilbert series} 
analogous to those given by the 
Shephard-Todd-Chevalley and Solomon theorems
with $S=\oplus_{i \geq 0}\, S_i$ graded by polynomial degree.
Just as these classical results immediately imply that
$$
\begin{aligned}
\hspace{14ex}
\label{hilb-of-polynomial-invariants}
\Hilb(\SSS^W; q)
 &:=& &\sum_{i \geq 0} \quad\ \  \  q^i \dim \SSS^W_i 
 &=& \ \ \prod_{i=1}^\ell (1-q^{e_i+1})^{-1},& \\ 
\label{hilb-of-derivations}
\Hilb\big( (\SSS \otimes V)^W; q \big) 
 &:=& &\sum_{i \geq 0} \quad\ \ q^i \dim \left( S_i \otimes V \right)^W 
 &=& \ \ \Hilb(\SSS^W; q) \cdot \sum_{i=1}^\ell\ q^{e_i^*} \ ,&
\hfill\text{and}
\\
\label{hilb-of-exterior-invariants}
\Hilb \big( (\SSS \otimes \wedge V^*)^W; q,t \big) 
 &:=& &\sum_{i,j \geq 0}\, q^i\, t^j\,
 \dim ( S_i \otimes \wedge^j V^* )^W 
 &=& \ \ \Hilb(\SSS^W; q)  \cdot \prod_{i=1}^\ell \left( 1+q^{e_i}t \right) ,&
\end{aligned}
$$
Theorem~\ref{main-result} analogously
immediately implies that
\begin{equation}
\label{Hilbert-series-consequence}
\begin{array}{lll}
\Hilb\left( \left(\SSS \otimes \wedge V^* \otimes V \right)^W; q,t\right)
&:=\ \ \displaystyle\sum_{i,j \geq 0} \, q^i \, t^j\, 
    \dim \left( S_i \otimes \wedge^j V^* \otimes V\right)^W \\
&\ =\ \ \Hilb(S^W; q,t) \cdot  \displaystyle\sum_{i=1}^\ell\ (q^{e_i^*}+q^{e_i^*-1}t) \\
&\ =\ \ \Hilb(S^W;q) \cdot (q+t)\cdot 
       \left( \displaystyle\prod_{i=1}^{\ell-1}\ 1+q^{e_i}t \right)
       \left( \displaystyle\sum_{i=1}^\ell\ q^{e_i^*-1} \right) \, .
\end{array}
\end{equation}
Our original motivation, in fact, was
the special case of~\eqref{Hilbert-series-consequence}
for real reflection groups $W$, which appeared 
as~\cite[Conj.~11.5${}^\prime$]{ArmstrongRRhoades},
based on Coxeter-Catalan combinatorics and computer experimentation.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Lie theory connection}
\label{Lie-theory-section}
We now explain how the case of Theorem~\ref{main-result} when $W$ is a {\it Weyl group},
i.e, a finite crystallographic real reflection group,
relates to Lie-theoretic results and 
work of Bazlov, Broer, 
Joseph, Reeder, and Stembridge, and also DeConcini, Papi, and Procesi. 
Let $G$ be a simply-connected, compact simple Lie group 
%connected compact Lie group, 
with a choice of maximal torus $T$.  Denote by $\lieg$ and $\lieh$ the
complexification of their corresponding Lie algebras,
and let $W:=N_G(T)/T$ be the associated Weyl group
acting on a real vector space $V$.  
Then $G$ acts on $\wedge\lieg^*$, while $W$ acts on $\SSS:=S(\lieh^*)$ and on its
{\it coinvariant algebra}
$$
\SSS/\SSS^W_+ \cong H^*(G/T)
$$
where the last isomorphism to cohomology, due to Borel, is grade-doubling,
and where $S^W_+$ is the ideal generated by invariant polynomials of positive
degree.
Classical results (see~\cite{Reeder1}) give isomorphisms
\begin{equation}
\label{Borel-Cartan-Leray-result}
\left(\wedge\lieg^*\right)^G 
\quad \cong \quad H^*(G) 
\quad \cong \quad \left( H^*(G/T) \otimes H^*(T) \right)^W
\quad \cong \quad 
\left( S/\SSS^W_+ \otimes \wedge\lieh^*\right)^W
\end{equation}
exhibiting both of these rings as (isomorphic) exterior algebras,
with exterior generators $P_1,P_2,\ldots,P_\ell$ where $P_i$ lies in
$\left(\wedge^{2e_i+1}\lieg^*\right)^G$.  The isomorphism
is again homogeneous after doubling the grading in $S/S_+^W$.

Reeder~\cite{Reeder2} conjectured a similar relation between $G$-invariants and $W$-invariants,
relating two Hilbert series associated with a finite-dimensional $G$-representation $M$:
$$
\begin{aligned}
P_G(M;t)
  &:= \sum_{j \geq 0} \, t^j \,
       \dim (\wedge^j \lieg^* \otimes M)^G \, ,
\\
P_W(M^T;q,t)
  &:= \sum_{i,j \geq 0} \,q^i\, t^j\, 
       \dim \left( 
                   (S/S^W_+)_i
                    \otimes \wedge^j\lieh^* \otimes M^T 
                \right)^W \, .
\end{aligned}
$$
\begin{conjecture}\cite[Conj.~7.1]{Reeder2}
\label{Reeder-conjecture}
If $M$ is {\bf small}, meaning its weight space 
$M_{2\alpha}=0$ for all roots $\alpha$, one has
$$
P_G(M;t)=P_W(M^T;q,t) |_{q=t^2}.
$$
\end{conjecture}
\noindent
Various special cases of Conjecture~\ref{Reeder-conjecture} were known at the time that it was formulated.  For example, when $M$ is the trivial $G$-representation it follows
from \eqref{Borel-Cartan-Leray-result} above. 
Reeder~\cite[Cor.~4.2]{Reeder2} proved the $t=1$
specialization of Conjecture~\ref{Reeder-conjecture} and
credited it also to Kostant:   
For $M$ small, 
\begin{equation}
\label{Kostant-Reeder-ungraded-theorem}
\dim (\wedge \lieg^* \otimes M)^G 
=P_G(M;t)|_{t=1}
=P_W(M^T;q,t) |_{q=t=1}
=\dim \left( S/S^W_+ \otimes \wedge\lieh^* \otimes M^T \right)^W.
\end{equation}
The type $A$ special case was also known to follow from the ``first-layer'' formulas of Stembridge \cite{Stembridge}.  The type $B$ special case was 
recently\footnote{DeConcini and Papi, personal communication, 2016} 
verified in work of DeConcini and Papi, and 
Stembridge
independently\footnote{Stembridge, personal communication, 2016.}   
verified 
Conjecture~\ref{Reeder-conjecture} case-by-case
for all types.


A further bit of motivation comes from a generalization of
Chevalley's restriction theorem due to Broer~\cite{Broer}.  Chevalley's result asserts
that restriction $\lieg^* \rightarrow \lieh^*$ induces an isomorphism of polynomial rings
\begin{equation}
\label{Chevalley-restriction-theorem}
S(\lieg^*)^G \rightarrow \SSS^W,
\end{equation}
while Broer~\cite{Broer} showed more generally that, for any small $G$-module $M$, restriction
also induces an isomorphism 
(of modules over the polynomial rings in~\eqref{Chevalley-restriction-theorem})
$$
(S(\lieg^*) \otimes M)^G \rightarrow (\SSS \otimes M^T)^W.
$$

This last result in particular suggests the following enhanced
version of Reeder's Conjecture~\ref{Reeder-conjecture}.

\begin{conjecture}
\label{extended-Reeder-conjecture}
{(Enhanced Reeder conjecture)}
For small $G$-representations $M$, one has an isomorphism 
$$
(\wedge \lieg^* \otimes M)^G 
\cong
\left(  S/S^W_+ \otimes \wedge\lieh^* \otimes M^T \right)^W
$$
as modules over the exterior algebra in~\eqref{Borel-Cartan-Leray-result}
which is degree-preserving after doubling the grading in $S/S^W_+$.
\end{conjecture}

There are two particularly important cases of small $G$-representations 
where DeConcini and Papi
have recently\footnote{DeConcini and Papi, personal communication, 2016} 
proven this Enhanced Reeder Conjecture~\ref{extended-Reeder-conjecture} uniformly,
namely for the {\it adjoint representation}\footnote{It is exactly in the case of the adjoint representation
that Bazlov~\cite{Bazlov} proved Conjecture~\ref{Reeder-conjecture}, and he credits this special case of
the conjecture to Joseph~\cite{Joseph}}  $M=\lieg$ and the {\it little adjoint representation},
which are the $\lieg$-irreducibles whose highest weights are the highest root and highest short root,
respectively.   The adjoint representation case of Conjecture~\ref{extended-Reeder-conjecture} connects our work to
the following result of DeConcini, Papi, and Procesi.

\begin{theorem}{\cite[Thm~1.1]{DPP}}
\label{DPP-theorem}
Regard
$
(\wedge \lieg^* \otimes \lieg)^G
$
as a module over the exterior subalgebra $R$ 
of $(\wedge \lieg^*)^G$ generated by $P_1,P_2,\ldots,P_{\ell-1}$,
via multiplication in the first tensor factor.
Then $
(\wedge \lieg^* \otimes \lieg)^G
$
is free as an $R$-module, with 
basis elements $\{f_i,u_i\}_{i=1}^\ell$ of degrees
$\deg(f_i)=2e_i, \deg(u_i)=2e_i-1$.
\end{theorem}

\noindent
An alternate proof of Theorem~\ref{DPP-theorem} follows from
our Theorem~\ref{main-result} using
the adjoint special case of Conjecture~\ref{extended-Reeder-conjecture} 
after modding out by $S^W_+$
and bearing in mind that 
$\{e^*_i\}_{i=1}^\ell=\{e_i\}_{i=1}^\ell$ for Weyl groups $W$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Degree sums and Opdam's Lemma}
\label{Molien-strategy-section}

Before determining bases for the 
invariant differential derivations, we first establish tools for investigating the relevant numerology,
most notably a lemma by Opdam useful for finding
the sum of degrees in a basis.


\subsection*{Degree sum}
Let $A$ be a graded $k$-algebra and integral domain for $k$ a field. 
Consider a free graded $A$-module 
$M \cong A^p$ of finite rank, say
with homogeneous basis $m_1,\ldots, m_p$.
The (unordered) list of degrees 
$\deg(m_1),\ldots,\deg(m_p)$ 
are uniquely determined by
the quotient of Hilbert series
$$
\sum_{i=1}^p q^{\deg(m_i)}=
\Hilb(M,q)/\Hilb(A,q).
$$
Thus we may assign to any such $M$ the {\em degree sum}
$$
\Delta_A(M):=\sum_{i=1}^p \deg(m_i) = 
\left[ \sfrac{\partial}{\partial q} 
\frac{\Hilb(M,q)}{\Hilb(A,q)} \right]_{q=1}.
$$
If one knows this degree sum {\em a priori},
then one may determine an explicit $A$-basis for $M$
by just checking independence over the 
fraction field:

\begin{lemma}
\label{commutative-algebra-lemma}
Let $A$ be a graded $k$-algebra and integral domain
and let $M \cong A^p$ a free graded $A$-module.
A set of homogeneous
elements $\{n_1, \ldots, n_p\}$ in $M$ with 
$\sum_{i=1}^p \deg(n_i)=\Delta_A(M)$ 
forms an $A$-basis
for $M$ if and only if it is linearly independent over the fraction field $K=\Frac(A)$.
\end{lemma}
\begin{proof}
The forward implication is clear.  For the reverse implication, note that
linear independence is equivalent to
nonsingularity of the matrix $B$ in $A^{p \times p}$ with $\nnn = B \mmm$,
for  $\mathbf{n}=[n_1,\ldots,n_p]^T$ and $\mathbf{m}=[m_1,\ldots,m_p]^T$.
Since $\det(B) \neq 0$, its expansion contains a nonzero term 
indexed by a permutation $\pi$ 
with $b_{ i , \pi(i) }$ nonzero for each $i=1,2,...,p$;
after re-indexing, 
one may assume $\pi$ is the identity permutation.
Hence $\deg(n_i) =\deg( b_{ i , i} ) + \deg(m_i)$ 
for $i=1,\dots,p$.
Since $\sum_i \deg(n_i)=\Delta_A(M)=\sum_i \deg(m_i)$,
one concludes that $\deg(n_i)=\deg(m_i)$ for each $i$.   Thus after re-ordering the
rows and columns of $B$ in increasing order of degree, $B$ will be block
upper triangular, with each diagonal block an invertible matrix 
with entries in $k$.
Therefore $B$ represents an $A$-module automorphism of $M$  sending the
$A$-basis $\mmm$ to $\nnn$.
\end{proof}

\subsection*{Modules over the Invariant Ring}
As mentioned in the Introduction, 
a result of Hochster and Eagon
implies that for any 
representation $U$ of a complex reflection group $W$,
the set of all {\it relative invariants} $M=(\SSS \ot U)^W$ is 
a free module of finite rank $p=\dim_\CC U$
over the graded $k$-algebra $A=S^W$.
We introduce an abbreviation for the above degree sum:
\begin{definition}
\label{Saito-degree-def}
Let $W$ be a complex reflection group.
For any $W$-representation $U$, set
$$
\Delta(U):=\Delta_{S^W}\big( (S\ot U)^W\big)\, 
=\sum_{1\leq i \leq p}\deg\psi_i 
$$
for any 
$\SSS^W$-basis $\{\psi_i\}_{i=1}^p$ of $(\SSS\ot U)^W$.
\end{definition}


\subsection*{Local Data}
An {\it a priori} calculation 
for the degree sum $\Delta(U)$
appears in a preprint of Opdam~\cite[\S2]{Opdam}. 
For completeness, and 
since his proof skips a few mildly tricky steps, 
we review the statement and proof.

The formula for $\Delta(U)$ is expressed in what is sometimes called the {\it local data}
for $U$ at each reflecting hyperplane $H$ of $W$.
The pointwise stabilizer subgroup $W_H$ in $W$ of $H$ is cyclic,
say of order $e_H$; 
note that $e_H$ is the maximal order of a reflection in $W$ 
fixing $H$ pointwise.  The $W_H$-irreducible representations 
are the powers $\{\det^j\}_{j=0}^{e_H-1}$
of the $1$-dimensional (linear) 
character $\det:= \det \downarrow_{W_H}^{W}$ 
restricted from $W$ to $W_H$ acting on $V$.  
It is convenient to introduce
the {\it representation ring} 
$$R(W_H):=\ZZ[v]/(v^{e_H}-1),$$ where $v^j$ represents
the class of the
$1$-dimensional representation $\det^j$,
and define a $\ZZ$-linear
functional  
$$D_H: R(W_H) \rightarrow \ZZ,
\quad v^j  \mapsto  j\, .$$
Then for any $W$-representation $U$, 
the functional $\Hderiv$ on the restricted representation
$U\downarrow_{W_H}^{W}$
can be expressed in terms of the inner products
$
\mu_{H,j}:= \langle U\downarrow_{W_H}^{W}, {\det}^j \rangle_{W_H}$ as
$$
\Hderiv \left(U\downarrow_{W_H}^{W} \right)=\sum_{j=0}^{e_H-1} j \cdot \mu_{H,j}\, .
$$

\subsection*{Molien's Theorem}
We use a standard variant of Molien's theorem (e.g.,
see~\cite{OrlikTerao}) with $\Tr$ indicating trace:
\begin{lemma} For any $W$-representation $U$,
\begin{equation}
\label{standard-Molien-variant}
\Hilb\left((S \ot U)^W,q \right)
=\tfrac{1}{|W|\rule{0ex}{1.5ex}} \sum_{w \in W} \frac{\Tr_U(w^{-1})}{\det(1-qw)}\ .
\end{equation}
\end{lemma}
Molien's formula for a complex reflection group $W$ 
immediately implies (take $U$ 
to be the trivial representation)
the well-known fact that
$$
\prod_{i=1}^\ell \frac{1}{1-q^{e_i+1}}
=\Hilb(S^W,q)
=\tfrac{1}{|W|} \sum_{w \in W} \frac{1}{\det(1-qw)}\ 
$$
as $S^W=\CC[f_1,\ldots,f_\ell]$ with
$\deg(f_i)=e_i+1$.
As noted by Shephard and Todd \cite[\S8]{ShephardTodd}, 
comparing the first two coefficients in the
Laurent expansions about $q=1$ on the left and right immediately
gives these facts:
\begin{eqnarray}
\label{Shephard-Todd-product-fact}
|W|&=&\prod_{i=1}^\ell (e_i+1),\\
\label{Shephard-Todd-sum-of-exponents-fact}
N:= \#\{\text{reflections in }W\} &=&
\sum_{i=1}^\ell e_i\, .
\end{eqnarray}


\subsection*{Opdam's lemma}
We now record
Opdam's lemma~\cite[Lemma~2.1]{Opdam}.
\begin{lemma}
\label{Opdam-lemma}
Let $U$ be a representation
of a complex reflection group $W$.
Then
$
\Delta(U) = \sum_H \Hderiv \left(U\downarrow_{W_H}^{W}\right),
$ 
where the sum runs over all reflecting hyperplanes $H$ for $W$.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{Opdam-lemma}.]
We rewrite $
\Delta(U)=\left[ 
                  \frac{\partial}{\partial q} 
                  \frac{\Hilb\left((S \ot U)^W,\ q \right)}
                       {\Hilb\left(S^W,\ q \right)\rule{0ex}{1.5ex}}
                 \right]_{q=1}
$
as
\begin{equation}
\label{linearity-consequence}
\Delta(U)
=
\sum_{w \in W} \Tr_U(w^{-1})\
  \tfrac{1}{|W|\rule{0ex}{1.5ex}} \left[\sfrac{\del}{\del q}
  \frac{\prod_{i=1}^\ell (1-q^{e_i+1})}{\det(1-qw)}
\right]_{q=1}\, 
\end{equation}
using Molien's formula \eqref{standard-Molien-variant}
and the fact that
$1/\Hilb(S^W,q) = \prod_{i=1}^\ell (1-q^{e_i+1})$.

Note that $\prod_{i=1}^\ell (1-q^{e_i+1})$ and $\det(1-qw)$ have zeroes at $q=1$ of 
orders $\ell$ and $\dim(V^w)$, respectively.  Hence their quotient has a zero at $q=1$ of order
$\ell-\dim(V^w)$.  
Thus after applying $\frac{\del}{\del q}$ and setting $q=1$, 
the quotient will vanish
unless $\ell-\dim(V^w) \leq 1$, that is, unless $w$ is either the identity 
$I$ of $W$ or a reflection.

When $w=I$, one has 
$$
\tfrac{1}{|W|\rule{0ex}{1.5ex}} \left[\sfrac{\del}{\del q}
  \frac{\prod_{i=1}^\ell (1-q^{e_i+1})}{\det(1-qw)}
  \right]_{q=1}
  =
\tfrac{1}{|W|\rule{0ex}{1.5ex}} \left[\sfrac{\del}{\del q}
  \prod_{i=1}^\ell \frac{(1-q^{e_i+1})}{(1-q)}
  \right]_{q=1}
=  \tfrac{1}{|W|\rule{0ex}{1.5ex}} \left[\sfrac{\del}{\del q}
  \prod_{i=1}^\ell [e_i+1]_q
  \right]_{q=1}
  = \ \tfrac{1}{2} \sum_{i=1}^\ell e_i
$$
using the abbreviation $[n]_q:=1+q+q^2+\cdots+q^{n-1}$.
For the second-to-last equality, note that  
the $q$-symmetry of the
polynomial $f(q):=\prod_{i=1}^\ell [e_i+1]_q$
implies that
$[\frac{\del}{\del q}f(q)]_{q=1}=\tfrac{1}{2}\cdot (\deg f)\cdot f(1)$
with $\deg f = \sum_{i=1}^\ell e_i$ and 
$f(1)= \prod_{i=1}^\ell (e_i+1)=|W|$
by Equation~(\ref{Shephard-Todd-product-fact}).

When $w$ is a reflection, say with $\lambda:=\det(w)$, one has 
$$
\tfrac{1}{|W|\rule{0ex}{1.5ex}} \left[\sfrac{\del}{\del q}
  \frac{\prod_{i=1}^\ell (1-q^{e_i+1})}{\det(1-qw)}
  \right]_{q=1}
 =\ \tfrac{1}{|W|\rule{0ex}{1.5ex}} \left[\sfrac{\del}{\del q}
  \frac{\prod_{i=1}^\ell (1-q^{e_i+1})}{(1-q)^{\ell-1}\, (1-\lambda q)}
  \right]_{q=1}\, .
  $$
The right side can be rewritten via Leibniz rule
and miraculously simplified, giving
$$ 
\begin{aligned}
\tfrac{1}{|W|\rule{0ex}{1.5ex}} 
& \left[ 
\frac{ \prod_{i=1}^\ell (1-q^{e_i+1}) }{ (1-q)^{\ell-1} } 
\cdot  \frac{\sfrac{\del}{\del q} 
\prod_{i=1}^\ell (1-q^{e_i+1})}{(1-q)^{\ell-1}} 
\cdot \frac{1}{(1-\lambda q)}
\right. \\ 
& \left.\qquad\qquad \,\, 
+ \,\,
    \frac{ \prod_{i=1}^\ell (1-q^{e_i+1}) }{ (1-q)^{\ell} } 
\cdot \frac{(\ell-1)}{(1-\lambda q)}
 \,\, + \,\,
  \frac{ \prod_{i=1}^\ell (1-q^{e_i+1}) }{ (1-q)^{\ell-1} } \cdot \frac{\lambda}{(1-\lambda q)^2}
  \right]_{q=1}\\
& 
= 
  \tfrac{1}{|W|\rule{0ex}{1.5ex}} 
 \left(
  - \sum_{j=1}^\ell (e_j+1) 
\prod_{\substack{i=1\\ i \neq j}}^\ell 
(e_i+1) \cdot \frac{1}{(1-\lambda)}
  \,\, + \,\,
    \prod_{i=1}^\ell (e_i+1)  \cdot \frac{(\ell-1)}{(1-\lambda)}
  \,\, + \,\,
  0 \cdot \frac{\lambda}{(1-\lambda)^2}
  \right) 
\ \ \\
&=\ \ 
  \frac{-\ell + (\ell-1)}{1-\lambda}
\ \  =\ \ 
\frac{1}{\lambda-1} \, .
\end{aligned} 
$$
We substitute back into Equation~\eqref{linearity-consequence}, noting 
that $\Tr_U(I)=\dim U$, and obtain
\begin{equation}
\label{second-linearity-consequence}
\Delta(U)
=\frac{\dim U }{2} \sum_{i=1}^\ell e_i
  +\sum_{w}
    \frac{\Tr_U(w^{-1})}{\det(w)-1}\, , 
\end{equation}
with the summation over all reflections $w$ in $W$.  
We evaluate the rightmost sum here by 
grouping together
reflections $w$ about a common fixed reflecting hyperplane
$H$:
\begin{equation}
\label{third-linearity-consequence}
\begin{aligned}
\sum_{\substack{\text{reflections}\\w \in W_H\rule{0ex}{1.5ex}}} 
    \frac{\Tr_U(w^{-1})}{\det(w)-1} 
 &=\left[
     \frac{\Tr_U(I)}{1-q} - 
   \sum_{w \in W_H} \frac{\Tr_U(w^{-1})}{1-\det(w)q} 
     \right]_{q=1} 
 =\scalebox{1.3}{\Bigg[}
     \frac{\dim U}{1-q} - 
       \sum_{w \in W_H} 
        \frac{\displaystyle\sum_{j=0}^{e_H-1} \mu_{H,j}\cdot \det(w)^{-j}}{1-\det(w)q} 
\ \scalebox{1.3}{\Bigg]}_{q=1} .
\end{aligned}
\end{equation}
But
Molien's formula~\eqref{standard-Molien-variant} applied
to the $W_H$-representation $\det^j$ implies that
$$
 \sum_{w \in W_H} \frac{\det(w)^{-j}}{1-\det(w)q}
= \Hilb\left( \left(\CC[x] \otimes {\det}^{j} \right)^{W_H} , q \right)
= \frac{q^j}{1-q^{e_H}}\ 
$$
since the $G_H$-invariant polynomials
in $\CC[x]$ are generated by $x^{e_H}$
(see Example~\ref{rank-one-example})
and every ($\det^{-j}$)-invariant polynomial
for $G_H$ 
is an invariant polynomial times $x^j$.
We substitute back into~\eqref{third-linearity-consequence}:
\begin{equation}
\label{fourth-linearity-consequence}
\begin{aligned}
\Delta(U)
&=
\frac{\dim(U)}{2} \sum_{i=1}^\ell e_i
  +\sum_H
\scalebox{1.3}{\Bigg[}
     \frac{\dim U}{1-q} - 
        \frac{\displaystyle \sum_{j=0}^{e_H-1} \mu_{H,j}\, q^j}{1-q^{e_H}}
     \ \scalebox{1.3}{\Bigg]}_{q=1}\\
&=\frac{\dim(U)}{2} \sum_{i=1}^\ell e_i
  +\sum_H
    \left[
\frac{1}{1-q^{e_H}} \left(
       [e_H]_q \cdot \dim U - e_H \cdot \sum_{j=0}^{e_H-1} \mu_{H,j}\ q^j \right)
     \right]_{q=1}.
\end{aligned}
\end{equation}
L'H\^opital's rule then gives
\begin{equation}
\label{post-L'Hopital-formula}
\begin{aligned}
\Delta(U)
&=
\frac{\dim(U)}{2} \sum_{i=1}^\ell e_i
+ \sum_H \frac{1}{-e_H} 
   \left(
      \sbinom{e_H}{2} \dim U - e_H  \cdot \sum_{j=0}^{e_H-1} j \cdot \mu_{H,j}
   \right) \\
&=
\frac{\dim(U)}{2} \sum_{i=1}^\ell e_i
+ \sum_H 
    \Hderiv(U\downarrow_{W_H}^{W}) - \frac{\dim U}{2} (e_H-1)\\
&=
\frac{\dim(U)}{2} \left( \sum_{i=1}^\ell e_i - \sum_H (e_H-1) \right)
+ \sum_H \Hderiv(U\downarrow_{W_H}^{W})\, .
\end{aligned}
\end{equation}
However, $\sum_H (e_H-1)
=\sum_H (|G_H|-1)
=N=\sum_{i=1}^\ell e_i$ by\footnote{On the other hand, 
applying \eqref{post-L'Hopital-formula} 
to the trivial $W$-representation $U$, which
has $\Delta(U)=0$ and  each $\Hderiv(U\downarrow_{W_H}^{W})=0$,
gives an alternate derivation of Equation~\eqref{Shephard-Todd-sum-of-exponents-fact}.} Equation~\eqref{Shephard-Todd-sum-of-exponents-fact}, 
and the lemma follows.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerology from Opdam's lemma}
\label{Numerology-section}
Again, let $W$ be a complex reflection group.
This section harvests numerology implied by Opdam's lemma.


\subsection*{Number of Reflecting Hyperplanes}
We first see  that 
Lemma~\ref{Opdam-lemma} implies that the coexponents $e^*_i$
sum to the number $N^*$ of reflecting hyperplanes for $W$.

\begin{example}
Let 
$U=V$, the reflection representation.  Each reflecting hyperplane $H$ has
$\mu_{H,j}(V)=0$ for $j \neq 0,1$, with $\mu_{H,0}=\ell-1$ and $\mu_{H,1}=1$,
and hence $\Hderiv(V\downarrow_{W_H}^{W})=1$.
Thus in this case, Lemma~\ref{Opdam-lemma} implies that the coexponents $e^*_i:=\deg(\theta_i)$ for the 
$S^W$-basis $\{\theta_i\}_{i=1}^\ell$ of $(S \ot V)^W$ satisfy the well-known formula (e.g., see~\cite[p.~130]{BMR})
\begin{equation}
\label{sum-of-coexponents-is-N*}
N^*:=
\#\{\text{reflecting hyperplanes in }W\}
= \sum_H 1 
=\Delta(V) 
=
\sum_{i=1}^\ell e^*_i\, .
\end{equation}
\end{example}


\subsection*{Graded representations}
In order to apply Lemma~\ref{Opdam-lemma} to {\it graded} 
$W$-representations, 
we consider the {\it graded} representation ring
$R(W_H)[[t]]$,
that is, the ring of graded virtual $W_H$-characters. 
We also extend $\Hderiv$ to a map
$R(W_H)[[t]] \rightarrow \ZZ$ coefficientwise, defining
$\Hderiv \left( \sum_{k \geq 0} t^k \chi \right) := \sum_{k \geq 0} t^k \Hderiv(\chi)$.
The sum in the following corollary is over all reflecting hyperplanes $H$
of $W$.

\begin{corollary}
\label{general-exterior-tensor-degree-sum-corollary}
For any $W$-representation $U$,
$$
\sum_{m=0}^\ell \Delta( \wedge^m V^* \otimes U)\, t^m =
(1+t)^{\ell-1} \sum_H \Hderiv \left( (1+v^{e_H-1} t) \sum_{j=0}^{e_H-1} \mu_{H,j}\, v^j \right).
$$
\end{corollary}
\begin{proof}
Recall that each $U\downarrow_{W_H}^{W} = \sum_{j=0}^{e_H-1} \mu_{H,j}\, v^j$ in $R(W_H)$.
The restriction $V^*\downarrow^W_{W_H}$  is a sum of $\ell-1$ copies
of the trivial representation and one copy of $\det^{e_H-1}$.  
Hence 
the restriction 
$\wedge V^* \downarrow^W_{W_H}$ will be represented by
$(1+t)^{\ell-1} (1+v^{e_H-1}t)$, and 
$(\wedge V^* \otimes U) \downarrow^W_{W_H}$ will be represented by
$
(1+t)^{\ell-1} (1+v^{e_H-1}t) 
\sum_{j=0}^{e_H-1} \mu_{H,j}\, v^j
$
in $R(W_H)[[t]]$.
The result then follows from Lemma~\ref{Opdam-lemma}.
\end{proof}

\subsection*{Solomon's theorem}
We illustrate in the next example how
Corollary~\ref{general-exterior-tensor-degree-sum-corollary} 
gives
Solomon's result~\cite{Solomon}
that the space of $W$-invariant differential forms 
is generated by $df_1,\ldots, df_\ell$
as an exterior algebra\footnote{For the sketch of a geometric alternative proof of Solomon's result, see Berest, Etingof, and Ginzburg~\cite[Remark~1.17]{BEG}.}.
\begin{example}
We show that $(S \otimes \wedge^m V^*)^W$ has $S^W$-basis 
$\{ df_I \}_{I \in \binom{[\ell]}{m}}$
where $\binom{[\ell]}{m}$ denotes the collection of all $m$-element subsets
$I=\{i_1,\ldots,i_\ell\}$ of the set $[\ell]:=\{1,2,\ldots,\ell\}$,
with $1\leq i_1<\ldots<i_m\leq \ell$, and where
$$
df_I:=
df_{i_1} \wedge\cdots \wedge df_{i_m}.
$$
Apply Corollary~\ref{general-exterior-tensor-degree-sum-corollary}  
to the trivial representation $U$,
obtaining
$$
\begin{aligned}
\sum_{m=0}^\ell \Delta( \wedge^m V^*) t^m 
=(1+t)^{\ell-1} \sum_H \Hderiv (1+v^{e_H-1} t) 
=(1+t)^{\ell-1} \sum_H (e_H-1)\, t 
=Nt(1+t)^{\ell-1} 
\end{aligned}
$$
where the last equality uses Equation~\eqref{Shephard-Todd-sum-of-exponents-fact}.  
Therefore
$
 \Delta( \wedge^m V^*) = 
  \binom{\ell-1}{m-1} N.
$
Note that the sum of degrees of the elements in the alleged basis  $\{ df_I \}_{I \in \binom{[\ell]}{m}}$
matches this:
 $$
 \sum_{I \in \binom{[\ell]}{m}} \sum_{i \in I} e_i
 =\sum_{i=1}^\ell e_i\, \#\left\{I \in \sbinom{[\ell]}{m}: i \in I\right\}
 =N \sbinom{\ell-1}{m-1}
 =\Delta(\wedge^m V^*).
$$
Hence by 
Lemma~\ref{commutative-algebra-lemma}, 
it suffices only to check the linear independence
of $df_I$ over 
$K=\Frac(S)$.  As observed by Solomon, 
the elements $df_1,\ldots,df_\ell$ in $K \ot \wedge^m V^*$ have their wedges $\{df_I\}$ forming 
a $K$-basis for  $K \ot \wedge V^*$ if and only their top wedge is nonvanishing:
$$
df_1\wedge \cdots\wedge  df_\ell 
= \det\left(\frac{\partial f_i}{\partial x_j}\right) dx_1\wedge \cdots\wedge dx_\ell \neq 0.
$$
But this follows immediately
from the Jacobi Criterion~\cite{Jacobi}:
The algebraic independence of $f_1,\ldots,f_\ell$ 
implies
that the matrix of coefficients
$(\frac{\partial f_i}{\partial x_j})$
of $df_1, \ldots, df_\ell$
is nonsingular.
\end{example}

\subsection*{Numerology of differential derivations}
Consider the $W$-representation 
$U:=\wedge^m V^* \ot V$
of dimension 
$$
p:= \ell \sbinom{\ell}{m}
=\dim_\CC (\wedge^m V^* \ot V) \, .$$
We show that $\Delta(\wedge^m V^*\ot V)$
depends only on $N$, $N^*$, and $\ell:=\dim_\CC V$.
\begin{prop}
\label{Molien-derivative-prop}
For any complex reflection group $W$
and $1\leq m\leq \ell$,
$$
\Saitodegree(\wedge^m V^*\ot V)=
(\ell-1)\sbinom{\ell-1}{m-1}N + \sbinom{\ell-1}{m}N^*\, .
$$
\end{prop}

\begin{proof}
We take $U=V$ in 
Corollary~\ref{general-exterior-tensor-degree-sum-corollary}:
$$
\begin{aligned}
\sum_{m=0}^\ell \Delta( \wedge^m V^* \otimes V) t^m 
&=(1+t)^{\ell-1} \sum_H \Hderiv \left( (1+v^{e_H-1} t) ( \ell-1 + v ) \right) \\
&=(1+t)^{\ell-1} \sum_H \Hderiv \left( \ell-1 + (\ell-1)tv^{e_H-1}+ v + t \right) \\
&=(1+t)^{\ell-1} \sum_H \big( (\ell-1)t(e_H-1) + 1 \big)
= (1+t)^{\ell-1}\big( (\ell-1)Nt+N^* \big),
\end{aligned}
$$
using Equations~\eqref{Shephard-Todd-sum-of-exponents-fact} and~
\eqref{sum-of-coexponents-is-N*}.  
The proposition now follows  from the binomial theorem.
\end{proof}

Proposition~\ref{Molien-derivative-prop} and
Lemma~\ref{commutative-algebra-lemma} imply a corollary used repeatedly in the proofs of Theorems~\ref{main-result} and \ref{generation-result}.

\begin{corollary}
\label{key-numerology-corollary}
Any set of homogeneous elements $\{\psi_i\}_{i=1,2,\ldots,\ell\binom{\ell}{m}}$
in
$(S \ot \wedge^m V^* \ot V)^W$ with
$$\sum_i \deg(\psi_i)=
\Saitodegree(\wedge^m V^*\ot V)=
(\ell-1)\sbinom{\ell-1}{m-1}N + \sbinom{\ell-1}{m}N^*
$$
forms an $S^W$-basis for $(S \ot \wedge^m V^* \ot V)^W$ if and only
if they are linearly independent over $K=\Frac(S)$.
\end{corollary}












%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Digression: A Saito Criterion}
\label{Saito-criterion-section}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This section, although not needed for the sequel, extends Corollary~\ref{key-numerology-corollary} to a condition similar to {\it Saito's criterion} for 
free hyperplane arrangements~\cite[\S4.2]{OrlikTerao}
for any complex reflection group $W$.
We first
recall some facts about the coefficient matrices
for the differential forms $df_i$ and
basic derivations $\theta_i$.

\subsection*{Jacobian matrix and product of hyperplanes}
\label{J-and-Q-subsection}

Recall 
the defining polynomial $Q$ and the Jacobian polynomial $J$
for a complex reflection group $W$:
$$
\begin{aligned}
Q&:=\prod_{H} l_H,& \quad\text{and}\quad
J&:=\prod_{H} l_H^{\, e_H-1}\, .
\end{aligned}
$$
Here, the product is taken over all reflecting hyperplanes $H=\ker l_H$ in $V$
for some choice of 
linear forms $l_H\in V^*$ with $e_H=|\text{Stab}_W(H)|$.
Note that $Q$ and $J$ are only well-defined
up to nonzero scalars.
Let $\Jac(f)$ and $M(\theta)$, respectively, be the matrices in 
$S^{\ell \times \ell}$ that express
$\{df_i\}_{i\in [\ell]}$ and $\{\theta_i\}_{i\in [\ell]}$
in the $\SSS$-bases $\{dx_i\}_{i \in [\ell]}$ and $\{y_i\}_{i \in [\ell]}$
for $\SSS \otimes V^*$ and $\SSS \otimes V$, respectively.
Steinberg~\cite{Steinberg} and 
Orlik and Solomon~\cite[\S2]{OrlikSolomon}, respectively, showed that
$$
J=\det(\Jac(f))\quad\quad\text{and}\quad\quad
Q=\det(M(\theta))\, ,
$$
so that
\begin{equation}
\begin{aligned}
\label{N-and-N*-definition}
N 
  &= \sum_{i=1}^\ell {e_i}=\deg(J)
=\ \# \{\text{reflections in }W\}\, \quad\text{and}\\
N^* &=
    \sum_{i=1}^\ell {e^*_i}=\deg(Q)
=\  \#\{\text{reflecting hyperplanes for }W\}
\, .
\end{aligned}
\end{equation}
Note that Terao~\cite{Terao} also
showed that invariant derivations
$\theta_i$ also give an $\SSS$-basis for the {\it module of derivations}
of the reflection hyperplane arrangement of $W$; see 
also~\cite[\S4.1,~\S6.3]{OrlikTerao}.


When $W$ is a duality group, observe that (by definition)
\begin{equation}
\label{h-expressed-in-N-and-N*}
h:=\deg(f_\ell)=\frac{1}{\ell}\sum_{i=1}^\ell (e_i+e^*_i) = \frac{N+N^*}{\ell}
\, .
\end{equation}

\subsection*{Matrix of Coefficients}
We capture the coefficients
of any set of invariant differential derivations
in a matrix of coefficients.
Consider the
obvious free $\SSS$-basis for
$\SSS\ot \wedge^m V^* \ot V$
given by
\begin{equation}
\label{obvious-basis}
\{dx_{I} \ot y_j: 1\leq i,j\leq \ell\}
\qquad\qquad\text{with}\qquad
dx_{I}:=1\ot x_{i_1}\wedge \cdots\wedge x_{i_m}
\end{equation}
for $m$-subsets $I=\{i_1 < \cdots < i_m\}$ of $[\ell]$
and $[\ell]:=\{1,\ldots, \ell\}$.
Given a collection $\BBB\subset
(S \ot \wedge^m V^* \ot V)^W$,
let  $\Coef(\BBB)$ denote its coefficient matrix in $\SSS^{p \times p}$
with respect to the $\SSS$-basis in \eqref{obvious-basis}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
\begin{lemma}
\label{mandatory-divisibility-lemma}
For any $\BBB=\{\psi_i\}_{i=1}^p \subset 
(\SSS\ot \wedge^m V^* \ot V)^W$, 
the product
$J^{(\ell-1)\binom{\ell-1}{m-1}} Q^{\binom{\ell-1}{m}}$ divides
$\det\Coef(\BBB)$.
\end{lemma}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Fix a reflecting hyperplane $H$ in $V$ for $W$ and 
a reflection $s$ in $W$ of maximal order $e_H$ fixing $H$.
Choose coordinates
$x_1, \ldots, x_\ell$ of $V^*$ so that
$l_H=x_1$ and $s$
acts diagonally
with nonidentity eigenvalue $\xi$: 
$$
s(y_i)=\begin{cases} 
\xi^{-1} y_1 &\text{ if }i=1, \\
y_i &\text{ if }i \neq 1,
\end{cases}
\qquad \text{ and } \qquad
s(x_i)=\begin{cases} 
\xi x_1 &\text{ if }i=1, \\
x_i &\text{ if }i \neq 1\, .
\end{cases}
$$
Each row of $\Coef(\BBB)$ 
lists the coefficients $f_{I,j}$ of some invariant 
differential derivation
$\psi_i=\sum%_{I \in \bbinom{[\ell]}{m},\, j \in [\ell]} 
f_{I,j}\, dx_I\otimes y_j$
in $(\SSS\ot \wedge^m V^* \ot V)$,
while each column of
$\Coef(\BBB)$ is indexed by a pair $(I,j)$
for $I$ an $m$-subset of $[\ell]$
and $j\in[\ell]$.
Note two observations:
\begin{itemize}
\item
For each pair $(I,j)$ with $j \neq 1$ but $1\in I$,
the polynomial $f_{I,j}$ is divisible by $x_1^{e_H-1}$ since
$$
s(dx_I \otimes y_j) = \xi(dx_I\otimes y_j)
\quad\text{implies that}\quad
s(f_{I,j})=\xi^{-1} f_{I,j}\, ;
$$
thus $(\ell-1)\binom{\ell-1}{m-1}$ different columns 
of the matrix
$\Coef(\BBB)$ are divisible by $x_1^{e_H-1}$.
\item
For each pair $(I,j)$ with $j = 1$ but $1\in I$,
the polynomial $f_{I,j}$ is divisible by $x_1$,
since
$$
s(dx_I\otimes y_j) = \xi^{-1}(dx_I\otimes y_j)
\quad\text{implies that}\quad
s(f_{I,j})=\xi f_{I,j}\, ; 
$$
thus $\binom{\ell-1}{m}$ different columns of the matrix
$\Coef(\BBB)$ are divisible by $x_1$.
\end{itemize}
Hence $\ell_H=x_1$ when raised to the power
${(e_H-1)(\ell-1)\binom{\ell-1}{m-1}+\binom{\ell-1}{m}}$ divides
$\det\Coef(\BBB)$.  
This holds for each reflecting hyperplane $H$,
hence unique factorization implies that 
$J^{(\ell-1)\binom{\ell-1}{m-1}} Q^{\binom{\ell-1}{m}}$ divides $\det\Coef(\BBB)$.
\end{proof}

\subsection*{Saito-like Criterion}
Again, let $K=\CC(x_1,\ldots,x_\ell)$
denote the {\it fraction field} of $S=\CC[x_1,\ldots,x_\ell]$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}
\label{Saito-criterion}
For a homogeneous subset $\BBB=\{\psi_i\}_{i=1}^p$ of
$(\SSS\ot \wedge^m V^* \ot V)^W$, 
the following are equivalent:
\begin{enumerate}
\item[(a)]
$\BBB$ forms an $\SSS^W$-basis for 
$(\SSS\ot \wedge^m V^* \ot V)^W$.
\rule{0ex}{3ex}
\item[(b)] $\det\Coef(\BBB)$ is nonzero of degree
$(\ell-1)\binom{\ell-1}{m-1}N+\binom{\ell-1}{m}N^*$.
\rule{0ex}{3ex}
\item[(c)] 
$\det\Coef(\BBB)=
c\cdot J^{(\ell-1)\binom{\ell-1}{m-1}} Q^{\binom{\ell-1}{m}}
$
for some nonzero scalar $c$ in $\CC$.
\rule{0ex}{3ex}
\rule{0ex}{3ex}
\item[(d)]
$
\sum_{i=1}^p \deg(\psi_i) = 
(\ell-1)\binom{\ell-1}{m-1}N+\binom{\ell-1}{m}N^*$
and $\BBB$ is $K$-linearly independent in $K \ot \wedge^m V^* \ot V$.
\rule{0ex}{3ex}
\end{enumerate}
\end{corollary}
%
\begin{proof}
Equivalence of (a), (d) is Corollary~\ref{key-numerology-corollary},
of  (d),(b) is linear algebra, and of (b), (c) 
is Lemma~\ref{mandatory-divisibility-lemma}.
\end{proof}

\begin{remark}
An argument with Cramer's rule shows that 
Corollary~\ref{Saito-criterion}~(c) implies~(a)
directly
without using Corollary~\ref{key-numerology-corollary},
or appealing to any Hilbert series argument. 
Indeed,~(c) implies that
$\BBB$ {\it spans}
$(\SSS \otimes \wedge^m V^* \otimes V)^W$ over $\SSS^W$,
as we explain next.
Label the $\SSS$-basis elements 
$dx_I\otimes y_j$ of 
$\SSS \otimes \wedge^m V^* \otimes V$ as $z_1,\ldots,z_p$ 
for convenience.
Then the matrix  $\Coef(\BBB)$ in $\SSS^{p \times p}$
expresses the elements 
$\BBB=\{\psi_i\}_{i=1}^p$ in the $\SSS$-basis $\{z_i\}_{i=1}^p$:
\begin{equation}
\label{change-of-basis-matrix-equation}
\psi_j = \sum_{i=1}^p \Coef(\BBB)_{ij} \cdot z_i\, .
\end{equation}
To show that a typical element 
$\sum_{i=1}^p s_i z_i$ in $(\SSS \otimes \wedge^m V^* \otimes V)^W$
lies in the $\SSS^W$-span of $\BBB$,
find $k_i$ in the fraction field $K$ of $\SSS$ 
(using  $\det\Coef(\BBB) \neq 0$)
with
\begin{equation}
\label{fraction-field-expression}
\sum_{i=1}^p s_i z_i = \sum_{j=1}^p k_j \psi_j\, .
\end{equation}
We may assume each $k_j$ lies in $K^W$, else apply 
the symmetrizer $\frac{1}{|W|} \sum_{g \in W} g(-)$ to
\eqref{fraction-field-expression}
and use the $W$-invariance of $\sum_i s_i z_i$ and of each $\psi_j$.
To show the $k_j$ actually lie in $\SSS^W$,  
substitute~\eqref{change-of-basis-matrix-equation}
into~\eqref{fraction-field-expression}, giving a matrix
equation relating the column vectors 
$\sss=\left[s_1,\ldots,s_p\right]^t$ and 
$\kk=\left[k_1,\ldots,k_p\right]^t$ in $K^p$:
$$
\sss=\Coef(\BBB) \cdot \kk.
$$
Cramer's rule then implies that
\begin{equation}
\label{Cramer-consequence}
k_i = \frac{\ \ \det \Coef(\BBB_{(i)})} 
                      {\det\Coef(\BBB)}
\end{equation}
where the numerator matrix $\Coef(\BBB_{(i)})$ 
is obtained from $\Coef(\BBB)$ 
by replacing its $i^{th}$ column with $\sss$.  
Then since $\Coef(\BBB_{(i)})$ expresses the
elements $\psi_1,\ldots,\psi_{i-1},\sum_i s_i z_i,\psi_{i+1},\ldots,\psi_p$ 
of
$(\SSS \otimes \wedge^m V^* \otimes V)^W$ in terms of the basis $\{z_i\}$,
Lemma~\ref{mandatory-divisibility-lemma} implies that 
its determinant is divisible by the nonzero polynomial
$\det\Coef(\BBB)$.  
Thus the right side of \eqref{Cramer-consequence} lies in $\SSS$, 
so that its left side $k_i$ lies in 
$K^W \cap \SSS = \SSS^W$, as desired.  
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Independence over the fraction field}
\label{independence-section}

In this section, we use Springer's theory of regular elements
to investigate differential derivations
with coefficients in the fraction field
$K=\CC(x_1,\ldots,x_\ell)$
of $S=\CC[x_1,\ldots,x_\ell]$.
We will later show that 
Theorem~\ref{main-result}
(for duality groups) follows from
a more general statement
established in this section for arbitrary
reflection groups, Theorem~\ref{linear-independence-theorem},
describing a $K$-vector space basis 
for $(K \ot \wedge^m V^*  \ot V)^W$.

We first give a definition and a lemma.
Recall the notation $V^{\reg}$ 
for the complement within $V$ of the union of 
all reflecting hyperplanes for $W$, that is, the subset of vectors
in $V$ having regular $W$-orbit.
Recall the notation 
$
df_I:=df_{i_1}\wedge \cdots\wedge df_{i_m}\in \SSS\ot \wedge^m V^*
$
for subsets 
$I=\{i_1 < \ldots < i_m\} \subset [\ell]:=\{1,2,\ldots,\ell\}$,
and the notation  $\binom{[\ell]}{m}$ for the collection of all 
$m$-subsets $I \subset [\ell]$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{definition}
\label{Euler-map-definitions}
Define a $K$-linear map
$$
K \otimes V^*  \overset{E}{\longrightarrow} K
\quad\quad\text{by}\quad E(dx_i)=E(1\ot x_i)=x_i\, .$$ 
One also has a $K$-linear map
$
K \otimes V^* \otimes V  \xrightarrow{\ E \otimes \one_V }
K \otimes V\, .
$
\end{definition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
Note that by {\it Euler's identity}, for any homogeneous $f$ in $\SSS$, one has 
\begin{equation}
\label{Euler-and-d-on-polynomial}
E(df) =\deg(f) \cdot f.
\end{equation}
Similarly, when $\theta=\sum_{j=1}^\ell \theta^{(j)}\otimes y_j$ 
in $\SSS \otimes V$ with each $\theta^{(j)}$ homogeneous 
of the same degree $\deg(\theta)$, one has
\begin{equation}
\label{Euler-and-d-on-derivation}
(E\ot {\bf 1}_V )(d \theta) = \deg(\theta) \cdot \theta.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%

We first give a 
$K$-basis for differential derivations with coefficients in $K$.
\begin{lemma}
\label{more-obvious-basis-lemma}
For each $0 \leq m \leq \ell$, 
the following set gives a $K$-basis for 
$K \ot \wedge^m V^*  \ot V$:
\begin{equation}
\label{more-obvious-basis}
\widetilde{\BBB}^{(m)} 
:= \big\{ df_{I} \, \theta_k \big\}_{ 
I \in \tbinom{[\ell]}{m}, \ k \in [\ell]}\ .
\end{equation}
%
Furthermore, elements of $\SSS \ot V^* \ot V$
can be expressed in the $K$-basis $\widetilde{\BBB}^{(1)}$ 
with coefficients in $(JQ)^{-\ell}\rule{0ex}{1.5ex}S$.
\end{lemma}

\begin{proof}
The matrix that expresses 
$\widetilde{\BBB}^{(m)}$
in the usual $\SSS$-basis
$
\{ dx_{I}\ot y_j : I \in \tbinom{[\ell]}{m}, \ j \in [\ell] \}
$
of $\SSS \ot \wedge^m V^*  \ot V$
is the tensor product of the matrices $\wedge^m(\Jac(f)) \otimes M(\theta)$, where $\wedge^m(\Jac(f))$ is
the $m^{th}$ exterior power of $\Jac(f)$.  The invertibility of $\Jac(f)$ 
and functoriality of $\wedge^m(-)$ imply the invertibility of
$\wedge^m(\Jac(f))$. 
Then since $M(\theta)$ is also invertible, so is the tensor product
$\wedge^m(\Jac(f)) \otimes M(\theta)$,
and hence $\widetilde{\BBB}^{(m)}$ is another $K$-basis.
The last assertion of the proposition then follows, since
in the $m=1$ case, 
$$
\det( \Jac(f) \otimes M(\theta) )
= \det( \Jac(f) )^\ell \cdot \det( M(\theta) )^\ell
= J^\ell Q^\ell. \qedhere
$$
\end{proof}

We will show that
Theorem~\ref{main-result} follows from the next theorem.
\begin{theorem}
\label{linear-independence-theorem} 
Let $W$ be a complex reflection group
with homogeneous basic invariants $f_1,\ldots,f_\ell$,
and an index $i_0$ in $1,2, \ldots, \ell$ that satisfies
\begin{equation}
\label{regular-vector-assumption}
V^\reg \cap \bigcap_{i \neq i_0} f_i^{-1}\{0\} \neq \varnothing.
\end{equation}
Then for each $m=0,1,\ldots,\ell$, the following set gives a
$K$-vector space basis for $K \ot \wedge^m V^*  \ot V$:
\begin{equation}
\label{more-general-putative-basis}
\BBB^{(m)} := 
 \Big\{ df_{I}\, d\theta_k\Big\}_{I \in \sbinom{[\ell] \setminus \{i_0\}}{m-1},
                              \, k \in [\ell]} 
\quad \sqcup \quad
 \Big\{ df_{I}\, \theta_k \Big\}_{I \in \sbinom{[\ell] \setminus \{i_0\}}{m},
                              \, k \in [\ell]}.
\end{equation}
\end{theorem}
%
\begin{proof}[Proof of Theorem~\ref{linear-independence-theorem}.]
There is nothing to prove in the case $m=0$.
We consider first the extreme case  $m=1$, then
the opposite extreme case $m = \ell$, and finally the intermediate cases with $2 \leq  m \leq \ell-1$.

\vskip.1in
\noindent
{\sf The case $m=1$.}
Note that the set $\BBB^{(1)}$ 
that we want to show is a $K$-basis for $K \ot V^* \ot V$,
$$
\BBB^{(1)} = \{ df_i\, \theta_k: i \in [\ell] \setminus \{i_0\}, k \in [\ell]\} 
                      \quad \sqcup \quad  \{ d\theta_k : k \in [\ell]\},
$$
has substantial overlap with the known $K$-basis 
$\widetilde{\BBB}^{(1)}$ for $K \ot V^* \ot V$ given in 
Lemma~\ref{more-obvious-basis-lemma},
$$
\widetilde{\BBB}^{(1)} 
  = \{ df_i \, \theta_k : i,k \in [\ell] \} 
  = \{ df_i \, \theta_k : i \in [\ell] \setminus \{i_0\}, k \in [\ell]\}
               \quad \sqcup \quad \{ df_{i_0} \theta_k : k \in [\ell]\}.
$$
Thus we need only show that when working in the quotient
of $K \ot V^* \ot V$ by the $K$-subspace spanned by
$$
\BBB^{(1)} \cap \widetilde{\BBB}^{(1)} 
= \{ df_i \, \theta_k: i \in [\ell] \setminus \{i_0\}, k \in [\ell] \}\, ,
$$
a {\it nonsingular} matrix in $K^{\ell \times \ell}$ expresses the images of the 
elements 
$$
\{ d\theta_k : k \in [\ell] \}=
\BBB^{(1)} \setminus \BBB^{(1)} \cap \widetilde{\BBB}^{(1)}
$$
uniquely in terms of the images of the elements
$$
\{ df_{i_0} \, \theta_k : k \in [\ell] \}=
\widetilde{\BBB}^{(1)}  \setminus \BBB^{(1)} \cap \widetilde{\BBB}^{(1)}\, .
$$
Here is how one produces this $\ell \times \ell$ matrix.  First
use Lemma~\ref{more-obvious-basis-lemma} to uniquely write
\begin{equation}
\label{d-thetas-in-terms-of-more-obvious-basis}
d \theta_k = \sum_{i,j \in [\ell]} r_{i,j,k}\ df_i \, \theta_j
\quad\text{ for each } k\in [\ell]\, ,
\end{equation}
with $r_{i,j,k}$ in $(JQ)^{-\ell}S$.
%
Then the matrix in $K^{\ell \times \ell}$ that we wish to show is 
nonsingular is $(r_{i_0,j,k})_{j,k \in [\ell]}$. 

To this end, apply to \eqref{d-thetas-in-terms-of-more-obvious-basis} the map $E \otimes \one_V$ from 
Definition~\ref{Euler-map-definitions}, giving
a system of equations in $K \otimes V$:
$$
e^*_k \ \theta_k = \sum_{i,j \in [\ell]} r_{i,j,k}\ 
\deg(f_i)\ f_i \, \theta_j
\quad\text{ for each } k\in [\ell].
$$
Since $\{ \theta_j \}_{j \in [\ell]}$ forms a $K$-basis for $K \otimes V$,
this gives a linear system in $K$: 
\begin{equation}
\label{K-linear-system}
e^*_k\ \delta_{j,k} = \sum_{i \in [\ell]} r_{i,j,k}\ \deg(f_i)\ f_i
\quad\text{ for each } j,k\in[\ell]\, ,
\end{equation}
where
$\delta_{j,k}$ denotes the Kronecker delta function.

To show that
$(r_{i_0,j,k})_{j,k \in [\ell]}$ in $K^{\ell \times \ell}$ is nonsingular,
we will evaluate each of its entries at a carefully chosen vector $v$.
By the hypothesis \eqref{regular-vector-assumption}, one can choose a 
vector $v$ in $V^\reg$ with the property that $f_i(v)=0$ 
for $i \neq i_0$. Since the coefficients $r_{i,j,k}$ lie in 
${(JQ)^{-\ell}} S$, and since $J, Q$ vanish nowhere on $V^\reg$,
one may evaluate the linear system
\eqref{K-linear-system} at $v$ to obtain a linear system over $\CC$:
\begin{equation}
\label{specialized-matrix-equation}
e^*_k\  \delta_{j,k} = r_{i_0,j,k}(v)\ \deg(f_{i_0})\ f_{i_0}(v)
\quad\text{ for each } j,k\in[\ell]\, .
\end{equation}
We claim $f_{i_0}(v) \neq 0$:  otherwise 
$f_i(v)=0$ for every $i$ in $[\ell]$, meaning $v$ is in
the common zero locus within $V$ of the homogeneous system of
parameters $f_1, \ldots, f_{\ell}$ in $\SSS$, forcing the
contradiction $v=0$.
As the coexponents $e^*_k$ are also nonzero, \eqref{specialized-matrix-equation}
shows that the specialized matrix $(r_{i_0,j,k}(v))_{j,k \in [\ell]}$ 
in $\CC^{\ell \times \ell}$ is diagonal with nonzero determinant.
Hence it is nonsingular, and so is the unspecialized matrix 
$(r_{i_0,j,k})_{j,k \in [\ell]}$ in $K^{\ell \times \ell}$, as desired.


\vskip.1in
\noindent
{\sf The case $m=\ell$.}
To show that
$
\BBB^{(\ell)} = \{ df_{[\ell]\setminus\{i_0\}}\, d\theta_k \}_{k \in [\ell]}
$
is $K$-linearly independent,
consider a dependence
$$
0 = \sum_{k \in [\ell]} c_k\ df_{[\ell]\setminus\{i_0\}}\, d\theta_k.
$$
Substitute the expressions for $d\theta_k$ 
from Equation~\eqref{d-thetas-in-terms-of-more-obvious-basis} to obtain
$$
0 = \sum_{i,j,k \in [\ell]} 
     c_k\ r_{i,j,k}\ df_{[\ell]\setminus\{i_0\}}\, df_i \, \theta_j 
= \sum_{j,k \in [\ell]} 
     c_k\ (-1)^{i_0}\ r_{i_0,j,k}\ df_{[\ell]}\, \theta_j\, .
$$
But $\{df_{[\ell]}\, \theta_j\}_{j \in [\ell]}$ is a $K$-basis
for $K \ot \wedge^{\ell}(V^*) \ot V$
by  Lemma~\ref{more-obvious-basis-lemma}, 
hence, 
$$
0 = \sum_{k \in [\ell]} 
     c_k\ (-1)^{i_0}\ r_{i_0,j,k}
\quad\text{ for each } j \in [\ell]\, .
$$
The matrix $(r_{i_0,j,k})_{j,k \in [\ell]}$ was already
shown nonsingular in the  $m=1$ case, and hence
$c_k=0$ for each $k$.


\vskip.1in
\noindent
{\sf The intermediate cases $2 \leq  m \leq  \ell-1$.}
To show that $\BBB^{(m)}$ is $K$-linearly independent, consider a dependence
\begin{equation}
\label{intermediate-m-dependence}
0 = \sum_{\substack{I \in \bbinom{[\ell] \setminus \{i_0\}}{m-1} \\
                     k \in [\ell]\rule{0ex}{2ex} }} 
c_{I,k}\ df_I \, d\theta_k
\quad + \quad
  \sum_{\substack{I \in \bbinom{[\ell] \setminus \{i_0\}}{m} \\
                     k \in [\ell]\rule{0ex}{2ex} }} 
c_{I,k}\ df_{I}\, \theta_k\, .
\end{equation}
It suffices to show all coefficients $c_{I,k}$ in the first sum vanish:
If so, then the second sum gives a dependence among a subset of the $K$-basis $\widetilde{\BBB}^{(m)}$ from 
Lemma~\ref{more-obvious-basis-lemma}, and hence
its coefficients $c_{I,k}$ must also vanish.
To this end, fix a subset $I_0 \in \sbinom{[\ell] \setminus \{i_0\}}{m-1}$
and consider its complementary subset within $[\ell] \setminus \{i_0\}$, 
namely,
$$
I_0^c:=[\ell] \setminus \{i_0\} \setminus I.
$$ 
Since 
$
|I_0^c|=(\ell-1)-(m-1)=\ell-m,
$
we note that
\begin{itemize}
\item
$I_0^c \cap I \neq \varnothing$ for each 
$I \subset [\ell] \setminus \{i_0\}$ with $|I|=m$, and
\item
$I_0^c \cap I \neq \varnothing$ for each
$I \subset [\ell] \setminus \{i_0\}$ with $|I|=m-1$ and $I \neq I_0$.
\end{itemize}
Consequently, multiplying both sides of 
Equation~\eqref{intermediate-m-dependence} by $df_{I_0^c}$ causes all terms in the second sum to vanish, 
as well as most of the terms in the first sum, leaving only
$$
0 = \sum_{k \in [\ell] } 
\pm c_{I_0,k}\ df_{[\ell] \setminus \{i_0\}}\, d\theta_k\, ,$$
with sign corresponding to that in 
$df_{I_0^c}\wedge df_{I_0}= \pm df_{[\ell] \setminus \{i_0\}}$.
But then by the case $m=\ell$ already proven, the
coefficients $c_{I_0,k}=0$ all vanish, and thus the
coefficients in the first sum of 
Equation~\eqref{intermediate-m-dependence} vanish, as desired.
This completes the proof of Theorem~\ref{linear-independence-theorem}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerology of duality groups}
\label{numerology-works-section}

We now fix our focus on duality groups
and the candidate basis for $(\SSS\ot \wedge^m V^* \ot V)^W$
given in Theorem~\ref{main-result}. We check in this section
that these sets, composing the putative basis,
have appropriate degree sum.  Let
\begin{equation}
\label{specific-candidate-basis}
\BBB^{(m)} = \Big\{ df_I \,\theta_k \Big\}_{_I \in \bbinom{[\ell-1]}{m},\, k \in [\ell]}
\ \sqcup \
\Big\{ df_I\, d\theta_k \Big\}_{I \in \bbinom{[\ell-1]}{m-1},\, k \in [\ell]} \ \quad\text{for}\quad 0 \leq m \leq \ell\, . 
\end{equation}
Here one interprets the second set as empty when $m=0$ and
interprets the first set as empty when $m=\ell$.
\begin{lemma}
\label{rightdegree}
For a duality group $W$, 
the sum of the degrees of elements in $\BBB^{(m)}$ above is
$$
\Delta(\wedge^m V^*\ot V)=(\ell-1)\sbinom{\ell-1}{m-1}N + \sbinom{\ell-1}{m}N^*.
$$
\end{lemma}


\begin{proof}
Using the shorthand
notation $e_I:=\sum_{i \in I} e_i$
for subsets $I \subset [\ell]$,
note that 
$$
\deg (df_I\, \theta_k) = e_I+e^*_k
\quad\quad\text{and}\quad\quad
\deg (df_I \, d\theta_k) = e_I+e^*_k-1,
$$
and therefore the sum of degrees for $\BBB^{(m)}$ is
$$
 \sum_{\substack{I \in \bbinom{[\ell-1]}{m},\\ k \in [\ell]
\rule{0ex}{2ex}}}
 (e_I + e^*_k)
 +
\sum_{\substack{I \in \bbinom{[\ell-1]}{m-1},\\k \in [\ell]
\rule{0ex}{2ex}}}
  (e_I + e^*_k-1) 
=
\sum_{k \in [\ell]}
  \left( \sum_{I \in  \bbinom{[\ell-1]}{m}} e^*_k
         +  \sum_{I \in  \bbinom{[\ell-1]}{m-1}} (e^*_k-1) \right)
+\sum_{k\in [\ell]} \left(
            \sum_{I \in  \bbinom{[\ell-1]}{m}} e_I
           + \sum_{I \in  \bbinom{[\ell-1]}{m-1}} e_I
        \right).
$$
Now the first sum over $k$ can be rewritten as 
$$ 
\sbinom{\ell-1}{m} N^* + \sbinom{\ell-1}{m-1}(N^*- \ell)
=\sbinom{\ell}{m} N^* - \ell\sbinom{\ell-1}{m-1},
$$
while the second sum over $k$ can be rewritten as
$$
\ell \left( 
               \sum_{i \in [\ell-1]} \sbinom{\ell-2}{m-1} e_i +
               \sum_{i \in [\ell-1]} \sbinom{\ell-2}{m-2} e_i
             \right) 
= \ell\sbinom{\ell-1}{m-1} ( N - (h-1) )
$$
since 
$\sum_{i \in [\ell-1]} e_i 
 = (\sum_{i \in [\ell]} e_i ) - e_\ell
 = N- (h-1)$
by Equation~\eqref{Shephard-Todd-sum-of-exponents-fact},
as $e_\ell+1=\deg(f_\ell)=h$ by definition.
Hence the degree sum is 
$$
\sbinom{\ell}{m} N^*  + \ell N\sbinom{\ell-1}{m-1} 
-\ell h \sbinom{\ell-1}{m-1}
=
 \sbinom{\ell-1}{m} N^*  + (\ell-1)\sbinom{\ell-1}{m-1}N
=\Delta(\wedge^m V^*\ot V),
$$
where the first equality used the duality group equation
$N + N^* = h \ell$ (see~(\ref{h-expressed-in-N-and-N*})).
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\section{Duality groups and 
proof of Theorem~\ref{main-result}}
\label{well-generated-section}
We now investigate differential derivations
invariant under duality groups
and prove Theorem~\ref{main-result}.
We combine the linear independence results from Section~\ref{independence-section} with the numerology of the last section.
We first check that 
the hypothesis~\eqref{regular-vector-assumption} in 
Theorem~\ref{linear-independence-theorem} holds for all duality groups when one
chooses the index $i_0=\ell$.
We emphasize that although both
Lemma~\ref{well-generated-implies-unique-lowest-coexponent}
and its consequence 
Corollary~\ref{well-generated-implies-unique-highest-degree}
below could easily be checked case-by-case, we give case-free proofs
so that Theorem~\ref{main-result} relies on no classification
of reflection groups.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
\label{well-generated-implies-unique-lowest-coexponent}
An irreducible complex reflection group $W$ has exactly one
coexponent equal to $1$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\begin{proof} 
The number of coexponents of $W$ equal to $1$ is
the number of $\CC$-linearly independent $W$-invariant derivations 
$\theta=\sum_{i=1}^\ell \theta^{(i)} \otimes y_i$ 
with each $\deg \theta^{(i)}=1$.  We recall three facts about
such derivations $\theta$:
\begin{itemize}
\item The image of $\theta$ regarded as a map 
$V^* \overset{\theta}{\rightarrow} V^*$ is the 
$W$-stable subspace 
$\CC \theta^{(1)} + \cdots +\CC \theta^{(\ell)}$
of $V^*$.

\item 
Choosing coordinates $x_1,\ldots,x_\ell$
for $V^*$ so that the hyperplane $x_1=0$ is fixed by some
reflection $s$ in $W$ guarantees that $\theta^{(1)}= c_\theta x_1$ 
for some nonzero scalar $c_\theta$
(as in the proof of 
Lemma~\ref{mandatory-divisibility-lemma}).


\item The {\it Euler derivation}
$\theta_E:=x_1 y_1 + \cdots + x_\ell y_\ell$
is an example of such a derivation.
\end{itemize}

\noindent
We claim this forces $\theta=c_\theta \theta_E$
for any such $\theta$: Since
$c_\theta \theta_E-\theta$ 
is another such derivation with vanishing $y_1$ term,
its coefficients
span a proper $W$-stable subspace of 
the irreducible module $V^*$ and hence vanish.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{remark}
The hypotheses that $W$ is irreducible
and $W$ is a reflection group are both required in the last lemma.
For example, the derivation
$\theta = x_2 \otimes y_1+ x_1\otimes y_2 \ (\not\in \CC\theta_E)$
is invariant under the reducible reflection group generated by the
transposition $y_1 \leftrightarrow y_2$ acting on $\CC^2$, 
while the derivation
$\theta = x_2\otimes y_1 - x_1 \otimes y_2\ \ (\not\in \CC\theta_E)$
is invariant under the irreducible group acting on $\CC^2$
generated by $
\left(\begin{smallmatrix}
0 & -1\\
1 & \ 0
\end{smallmatrix}\right),
$
which is not a reflection group.
\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}
\label{well-generated-implies-unique-highest-degree}
For any duality group $W$, 
there is a unique highest exponent $e_\ell$ and 
accompanying unique highest degree 
$h=\deg(f_\ell)=e_\ell+1$.
\end{corollary}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} 
\label{well-generated-implies-h-regular}
Duality groups $W$ satisfy 
hypothesis \eqref{regular-vector-assumption} 
in Theorem~\ref{linear-independence-theorem}
with $i_0=\ell$, that is,
$$
V^\reg \ \cap\ \bigcap_{i=1}^{\ell-1} f_i^{-1}\{0\} \neq \varnothing.
$$
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
(cf. \cite[p.~4]{BessisR})
Springer~\cite[Prop.~3.2(i)]{Springer} showed that if $W$
is a complex reflection group  with 
basic invariants $f_1,\ldots,f_\ell$ and $\zeta$ is any primitive
$d^{th}$ root of unity in $\CC$, then
\begin{equation}
\label{Springer-fact}
\bigcap_{\substack{i=1,\ldots,\ell:\\ d \, \nmid\,  \deg(f_i)}}
f_i^{-1}\{0\} 
= \bigcup_{g \in W} \ker(\zeta \one_V - g).
\end{equation}
On the other hand, Lehrer and Michel~\cite[Thm.~1.2]{LehrerMichel} 
showed existence of $g$ in $W$ with
$\ker(\zeta \one_V - g) \cap V^\reg \neq \varnothing$
if and only if $d$ divides as many 
degrees $\deg(f_i)=e_i+1$ as codegrees $e^*_i-1$.
For a duality group $W$, the equations $e_i + e^*_i=h=\deg(f_\ell)$ 
imply $h$ 
divides as many degrees as codegrees.  Also,
Corollary~\ref{well-generated-implies-unique-highest-degree}
implies that $f_\ell$ is the {\it only} basic invariant of
degree $h$, so that \eqref{Springer-fact} gives the result.
\end{proof}


We can now deduce the two equivalent statements of our main result.
Recall that $R$ is the exterior subalgebra of 
the $W$-invariant forms 
$(\SSS \otimes \wedge V^*)^W
=\bigwedge_{\SSS^W}\{df_1,\ldots,df_{\ell}\}$
generated by all $df_i$ except for the last one $df_\ell$, that is,
$
R:=\bigwedge_{\SSS^W}
\{df_1,\ldots,df_{\ell-1}\}.
$

\vskip.1in
\noindent
{\bf Theorem~\ref{main-result}.}
{\it
For $W$ a duality (well-generated) complex reflection group,
$(\SSS \otimes \wedge V^* \otimes V)^W$ forms a
free $R$-module on $R$-basis 
$
\{\theta_1,\ldots,\theta_\ell, \,\, d\theta_1, \ldots, d \theta_\ell\}.
$ 
Equivalently, $(\SSS\ot \wedge^m V^* \ot V)^W$ for $0 \leq m \leq \ell$
has $\SSS^W$-basis
\begin{equation*}
\BBB^{(m)} = \Big\{ df_I \theta_k \Big\}_{I \in \bbinom{[\ell-1]}{m},\, k \in [\ell]}
\ \sqcup \
\Big\{ df_I d\theta_k \Big\}_{I \in \bbinom{[\ell-1]}{m-1},\, k \in [\ell]}\, .
\end{equation*}
}
\vskip.1in

\begin{proof}[Proof of Theorem~\ref{main-result}]
Lemma~\ref{well-generated-implies-h-regular} and
Theorem~\ref{linear-independence-theorem} imply
that $\BBB^{(m)}$ has nonsingular coefficient matrix.
Indeed, when $i_0=\ell$,
the set $\BBB^{(m)}$ in \eqref{more-general-putative-basis}
agrees with that in~\eqref{specific-candidate-basis}.
Note that this set has cardinality 
$$
\sbinom{\ell-1}{m-1} \ell + \sbinom{\ell-1}{m} \ell
= \sbinom{\ell}{m} \ell=\dim_K \left( K \ot \wedge^m V^*  \ot V \right)
=\rank_{\SSS^W}(\SSS\ot\wedge^mV^*\ot V)^W\, .
$$
Lemma~\ref{rightdegree} shows that  their degree sum
is appropriate, and the theorem then follows 
from Corollary~\ref{key-numerology-corollary}.
\end{proof}



%%%%%%%%%%%%%
\begin{remark}
Theorem~\ref{main-result} has an amusingly compact rephrasing:  defining
by convention $f_0:=1$,  the last assertion of the 
theorem is equivalent to the assertion
that $(\SSS\ot \wedge^m V^* \ot V)^W$ is a 
free $\SSS^W$-module on basis
\begin{equation}
\label{cute-rephrased-basis}
\{  \
df_{i_m} \cdots df_{i_2} \cdot d( f_{i_1} \theta_k ) 
:\ 
0 \leq i_1 < i_2 < \cdots <i_m \leq \ell-1
\text{ and } k \in [\ell]
\,  \}\, .
\end{equation}
The reason is that  the elements in \eqref{cute-rephrased-basis} with $i_1=0$
coincide with the basis elements $\{df_I d\theta_k\}$ in the second part of 
$\BBB^{(m)}$, while the elements in \eqref{cute-rephrased-basis} with $i_1 \geq 1$
are almost the same as the basis elements $\{df_I  \theta_k\}$ in the first part of 
$\BBB^{(m)}$, but differ from them by $f_{i_1}$ times an element in the second part
of $\BBB^{(m)}$.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two dimensional reflection groups}
\label{2-dimensional-groups-section}

We now consider reflection groups
acting on $2$-dimensional complex space,
i.e., the case when $\ell=2$.
We found in Theorem~\ref{main-result}
an $\SSS^W$-basis for $(\SSS \ot \wedge V^* \ot V)^W$
when $W$ is a duality group.
Here, we find a 
{\it different} choice of basis
that works for {\it any} 
rank $2$ complex reflection group $W$, 
duality or not.  
Let us assume that
we have named 
the basic derivations in $(\SSS \otimes V)^W$
\begin{equation}
\label{rank-two-derivations}
\begin{aligned}
\theta_1&:=x_1 \otimes y_1 + x_2 \otimes y_2\ (=\theta_E)
\, , \qquad\text{and}\\
\theta_2&:=a \otimes y_1 + b \otimes y_2\, ,\\
\end{aligned}
\end{equation}
for some $a,b$ in $\CC[x_1,x_2]$.
With this indexing,  
$e_1^*=1$ and $e_2^*=\deg(a)=\deg(b)$.

\begin{theorem}
\label{rank-two-theorem}
For a complex reflection group $W$ acting on $\CC^2$,
and $0 \leq m \leq 2$, the following sets $\BBB^{(m)}$
give free $S^W$-bases for
$(\SSS \ot \wedge^m V^*  \ot V)^W$:
\begin{equation}
\begin{aligned}
\BBB^{(0)}&:=\{\theta_1, \theta_2 \} \\
\BBB^{(1)}&:=\{df_1\, \theta_1, df_2\, \theta_1, 
d\theta_1, d\theta_2 \} \\
\BBB^{(2)}&:=\{df_1\, d\theta_1, df_2\, d\theta_1 \}.
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
The $m=0$ case is immediate.
For $m=1,2$, with basic derivations 
as in~\eqref{rank-two-derivations},
one has
$$
\Coef(\BBB^{(2)})
=\bordermatrix{~ 
& \text{\tiny $df_1 d\theta_1$} & 
\text{\tiny $df_2 d\theta_1$} 
\cr & & \cr
\text{\tiny $dx_1 dx_2 \otimes y_1$} & 
-\frac{\del f_1}{\del x_2} &
-\frac{\del f_2}{\del x_2} \cr
  & &  \cr
\text{\tiny $dx_1 dx_2 \otimes y_2$} & 
\frac{\del f_1}{\del x_1} & \frac{\del f_2}{\del x_1}\cr
  & & },
 \quad
\Coef(\BBB^{(1)})
= \bordermatrix{~ & 
\text{\tiny $df_1 \theta_1$} & 
\text{\tiny $df_2 \theta_1$} & 
\text{\tiny $d\theta_1$} & 
\text{\tiny $d\theta_2$} \cr
 & & & & \cr
\text{\tiny $dx_1 \otimes y_1$} & x_1\frac{\del f_1}{\del x_1} & x_1\frac{\del f_2}{\del x_1} & 1         & \frac{\del a}{\del x_1} \cr
 & & & & \cr
\text{\tiny $dx_2 \otimes y_2$} &x_2\frac{\del f_1}{\del x_2} & x_2\frac{\del f_2}{\del x_2} & 1 & \frac{\del b}{\del x_2} \cr
 & & & & \cr
\text{\tiny $dx_1 \otimes y_2$} & x_2\frac{\del f_1}{\del x_1} & x_2\frac{\del f_2}{\del x_1} & 0 & \frac{\del b}{\del x_1} \cr
 & & & & \cr
\text{\tiny $dx_2 \otimes y_1$} & x_1\frac{\del f_1}{\del x_2} & x_1\frac{\del f_2}{\del x_2} & 0 & \frac{\del a}{\del x_2} \cr
  & & & & \cr}.
$$
One now computes that $\BBB^{(1)},  \BBB^{(2)}$ have the right degree sums,
and satisfy the hypotheses of Corollary~\ref{key-numerology-corollary}:
$$
\begin{aligned}
\det \Coef(\BBB^{(2)})&= \tfrac{\del f_1}{\del x_1} 
\tfrac{\del f_2}{\del x_2}
-\tfrac{\del f_1}{\del x_2} \tfrac{\del f_2}{\del x_1} 
=\det \Jac(f_1,f_2)
=J
=J^{(\ell-1)\binom{\ell-1}{m-1}} Q^{\binom{\ell-1}{m}} \text{ for }\ell=2=m,\\
\det \Coef(\BBB^{(1)})
&= \left(
\tfrac{\del f_1}{\del x_1} \tfrac{\del f_2}{\del x_2}
-\tfrac{\del f_1}{\del x_2} \tfrac{\del f_2}{\del x_1}
\right)
\left( 
x_1 \left( x_1\tfrac{\del b}{\del x_1} + x_2 \tfrac{\del b}{\del x_2}\right)
- x_2 \left( x_2 \tfrac{\del a}{\del x_2} + x_1 \tfrac{\del a}{\del x_1}\right)
\right)
\rule{0ex}{5ex} \\
&= J\, (x_1 e_2^* b - x_2 e_2^* a) 
= e_2^*\, J \det(M(\theta_1,\theta_2))  
= e_2^*\, JQ
=e_2^*\, J^{(\ell-1)\binom{\ell-1}{m-1}} 
Q^{\binom{\ell-1}{m}} \text{ for }\ell=2, m=1.
\rule{0ex}{3ex} 
\qedhere
\end{aligned}
$$
\end{proof}

This gives an immediate Hilbert series corollary when $\ell=2$.
\begin{corollary}
For a complex reflection group $W$ acting on $\CC^2$,
$$
\frac{\Hilb( (\SSS \ot \wedge V^*  \ot V)^W; q,t)} 
{\Hilb( \SSS^W;q)}= 
 t^0(q+q^{e_2^*})
      +t^1(1+q^{e_2^*-1}+q^{e_1+1}+q^{e_2+1})
      +t^2(q^{e_1}+q^{e_2}).  
$$
\end{corollary}

In the case of a duality group $W$ with $\ell=2$, one can
check that this agrees with 
description \eqref{Hilbert-series-consequence},
bearing in mind that $e_1^*=1$ and $e_2^*+e_1=h=e_2+1$ 
with the above conventions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The reflection group $G_{31}$}
\label{G31-section}

The group $W=G_{31}$ is an irreducible complex reflection group 
of rank $4$ containing $60$ reflections, each of order $2$
(so $N^*=N=60$), although it is not the complexification of
a Coxeter group.  It is not a duality group; the exponents are $(7,11,19,23)$
and the coexponents are $(1,13,17,29)$.  
Using a computer to complete
a Molien-style summation as
in Lemma~\ref{standard-Molien-variant}
for $W=G_{31}$ (taking $U=V=\CC^4$),  one obtains
\begin{equation}
\label{G31-hilbert-series}
\frac{\Hilb( (\SSS \ot \wedge V^*  \ot V)^W; q,t)} 
{\Hilb( \SSS^W;q)}
=(1 + q^7 t)(1 + q^{11} t)(q+t)(1+q^{12})(1 + q^{19} t + q^{16} + q^{23} t)\, .
\end{equation}
It is not hard to see that this is inconsistent with a description of
$(\SSS \ot \wedge^m V^*  \ot V)^W$ exactly as in Theorem~\ref{main-result}.
However, rewriting 
the right side of \eqref{G31-hilbert-series} as
$$(1 + q^7 t)(1 + q^{11} t) \cdot
   (q+t)\left[(1 + q^{12} + q^{16} + q^{28}) + 
           (q^{19}t + q^{23} t+ q^{19+12}+q^{23+12}t) \right]
$$
suggests a modified statement.  
Let 
$$R':=\bigwedge_{\SSS^W}\{df_1,df_2\}$$
as a subalgebra of 
$(\SSS \ot \wedge V^*)^W=\bigwedge_{\SSS^W}\{df_1,df_2,df_3,df_4\}$.

\begin{theorem}
\label{G31-theorem}
For $W=G_{31}$, the $R'$-module 
$(\SSS \ot \wedge V^* \ot V)^W$ 
is free with $R'$-basis
$$
\big\{ \theta_i \,\, , d\theta_i \big\}_{i=1,2,3,4} 
\ \sqcup \ 
\Bigg\{ 
\begin{aligned}
df_3\, \theta_1,&  &df_4\, \theta_1,&  &df_3\, \theta_2,&   & df_4\, \theta_2,  \\
df_3\, d\theta_1,& &df_4\, d\theta_1,& &df_3\, d\theta_2,& &df_4\, d\theta_2 
\end{aligned}
\Bigg\}
\, .
$$
\end{theorem}
%
\begin{proof}
One can check that the elements listed above that lie in $\SSS\ot\wedge^m V^*\ot V$ 
have degrees adding to
$$
\Saitodegree(\wedge^m V^*\ot V)
=60\left( 3\binom{3}{m-1} + \binom{3}{m} \right), \quad\text{for}\quad
0\leq m\leq 4.
$$
Thus the theorem follows from Corollary~\ref{key-numerology-corollary}
after one checks that each matrix of coefficients 
for $0\leq m\leq 4$ is nonsingular.  
We did this in {\tt Mathematica}, using explicit choices of  
basic invariant polynomials $f_1,f_2,f_3,f_4$ of degrees $8,12,20,24$
and basic derivations $\theta_1,\theta_2,\theta_3,\theta_4$
of degrees $1,13,17,29$, constructed as prescribed by 
Orlik and Terao
(using Maschke's~\cite{Maschke} 
invariants $F_8, F_{12}, F_{20}$, 
with $F_{24}=\det \mathrm{Hessian}(F_8)$; 
see also Dimca and Sticlaru~\cite{DimcaSticlaru} and 
also~\cite[p.~285]{OrlikTerao}).
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem~\ref{generation-result}}
\label{poorly-generated-section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We first recall the statement of the theorem.
\vskip.1in

%%%%%%%%%%%%%%%%%%%%
\noindent
{\bf Theorem~\ref{generation-result}.}
{\it
For $W$ any complex reflection group,
$(\SSS \otimes \wedge V^* \otimes V)^W$ is generated as a
module over the exterior algebra 
$(\SSS \otimes \wedge V^*)^W=\bigwedge_{\SSS^W}\{df_1,\ldots,df_{\ell}\}$ by the $2\ell$ generators
$
\{\theta_1,\ldots,\theta_\ell, \,\, d\theta_1, \ldots, d \theta_\ell\}.
$ 
}
%%%%%%%%%%%%%%%%%%%%%%%

\begin{proof}[Proof]
The general statement follows from the case where $W$ is irreducible.
For irreducible $W$, we proceed case-by-case, taking
advantage of the fact that the irreducible non-duality 
(that is, not well-generated)
reflection groups fall into three camps:
\begin{itemize}
\item The 2-dimensional groups ($\ell=2$).
\item The exceptional group $G_{31}$ (with $\ell=4$).
\item The infinite family of monomial groups $G(r,p,\ell)$ for $1<p<r$.
\end{itemize}
Reflection groups of dimension $2$
were considered in Section~\ref{2-dimensional-groups-section}; Theorem~\ref{rank-two-theorem} gives a basis.
The group $G_{31}$ was considered in Section~\ref{G31-section};
Theorem~\ref{G31-theorem} gives a basis.
The groups $G(r,p,\ell)$ are considered in the appendix,
as some direct computation is required to prove
the pattern in this general case; 
Theorem~\ref{monomial-group-theorem} gives a basis.
In each case, we provided an explicit $S^W$-module basis
for $(\SSS \otimes \wedge V^m \otimes V)^W$ 
whose elements all have
either the form $df_I \theta_k$ or $df_I d\theta_k$ for various subsets $I \subset [\ell]$
and $k$ in $[\ell]$.
%

\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\section{Remarks and questions}
\label{questions-section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\subsection*{What about $U=\wedge^k V$?}

One might wonder whether for complex reflection groups $W$,
or even just duality groups, one can
factor the Hilbert series more generally for 
$
\left( \SSS \otimes \wedge V^* \otimes \wedge^k V \right)^W
$
when $k$ takes values besides $k=0,1$.
One can manipulate Molien-style computations using 
this consequence of Lemma~\ref{standard-Molien-variant}:
$$
\begin{aligned}
\Hilb\left( \left( \SSS \otimes 
\wedge V^* \otimes \wedge V\right)^W; 
\ q, t, u \right)
&:= \sum_{i,j,k} \left( 
                 \dim \SSS_i \otimes \wedge^j V^* \otimes \wedge^k V
              \right)^W q^i\, t^j\, u^k\\
&=\tfrac{1}{|W|\rule{0ex}{1.5ex}} \sum_{w \in W} \frac{\det(1+uw^{-1})\det(1+tw)}{\det(1-qw)}\ .
\end{aligned}
$$
Things appear not to factor so nicely unless $k$ lies in 
$\{0,1,\ell-1,\ell\}$, but at least we
have a reciprocity:
\begin{prop}
\label{perfectpairing}
Let $W$ be a complex reflection group and set 
$$
\tau(q,t,u) := \Hilb \big( (\SSS \otimes \wedge V^* \ot \wedge V)^W; 
q,t, u\big)\, .
$$
Then $\tau$ satisfies the reciprocity
$$
\tau(q,t,u) =  t^{\ell}\, u^{\ell}\, \tau(q,t^{-1}, u^{-1})\, .
$$
\end{prop}
\begin{proof}
Let $\CC_{\det}$ be a $1$-dimensional $W$-module carrying the
determinant character of $W$ acting on $V$,
and likewise for $\CC_{\det^{-1}}$.
The $W$-equivariant perfect pairings
$$
\wedge^j\, V^*\otimes \wedge^{\ell-m}\, V^*
\longrightarrow \wedge^{\ell}\, V^*
\cong \CC_{\det^{-1}}
\quad\text{and}\quad
\wedge^k\, V\otimes \wedge^{\ell-m}\, V
\longrightarrow \wedge^{\ell}\, V
\cong \CC_{\det}
$$
imply that
$$
\SSS\ot \wedge^j \, V^* \ot \CC_{\det^{-1}}
\cong
\SSS\ot \wedge^{\ell-j}\, V\, 
\quad\text{and}\quad
\wedge^k\,  V \ot \CC_{\det}
\cong
\wedge^{\ell-k}\, V^*\, 
$$
as $W$-modules
(see~\cite{Shepler}, proof of Corollary 4),
since $V\cong V^{**}$ as $W$-modules.
The result then follows from the isomorphisms
of $W$-modules
$$
\SSS\ot \wedge^j\, V^* \ot \wedge^k\, V
\cong
(\SSS\ot \wedge^j \, V^* \ot \CC_{\det^{-1}})
\ot
(\wedge^k\,  V \ot \CC_{\det})
\cong
(\SSS\ot \wedge^{\ell-j}\, V)
\ot (\wedge^{\ell-k}\, V^*)\, .
$$
\end{proof}

A similar argument confirms the following.
\begin{prop}
Let $W$ be a complex reflection group and
set 
$$\tau(\chi, q,t,u)
:=
\Hilb \big( 
(\SSS \otimes \wedge V^* \ot \wedge V\ot \CC_\chi)^W; q,t, u \big) 
=\tfrac{1}{|W|\rule{0ex}{1.5ex}} 
\sum_{w \in W}\chi^{-1}(w)\, \frac{\det(1+uw^{-1})\det(1+tw)}{\det(1-qw)}\, 
$$
for any character $\chi: W \rightarrow \CC^*$
afforded by a $1$-dimensional $W$-module $\CC_\chi$.
Then $\tau$ satisfies the reciprocity
$$
\tau(\chi; q, t, u) =
t^{\ell}\, u^{\ell}\, \tau(\chi; q, t^{-1},u^{-1})\, .
$$
\end{prop}

\begin{remark}
The last two results generalize an observation 
for real reflection groups
from~\cite[Eqn.~(1.24)]{GNS}.
\end{remark}

\begin{example}
For the Weyl group $W=W(F_4)$, with exponents $(1,5,7,11)$,  and $V^* \cong V$, 
a computation in {\tt Mathematica} gives
$$
\begin{aligned}
&\hspace{-5ex}\
\frac{ \Hilb\left( \left( 
\SSS \otimes \wedge V \otimes \wedge V \right)^W; q, t, u \right) }
{\Hilb(\SSS^W,q)}
\hfill \\
&   = u^0 (1 + q t)(1 + q^5 t)(1 + q^7 t)(1 + q^{11} t)  \\
 &\quad + u^1 (q+t)(1 + q^4 + q^6+q^{10})
      (1 + qt)(1 + q^5 t)(1 + q^7 t) \\
 &\quad + u^2 (q + t)(1 + q t) (1+q^4)
\Big( (q^5+q^7-q^9+q^{11}+q^{13})(1+t^2)
+ ( 1+q^6+q^8+q^{10}+q^{12}+q^{18})t \Big) \\
&\quad + u^3 (1+qt)(1 + q^4+ q^6+q^{10})(q+t)(q^5 + t)(q^7 + t) \\
&\quad + u^4 (q + t)(q^5 + t)(q^7 + t)(q^{11} + t) \, .
\end{aligned}
$$
The coefficient of $u^2$ does not seem to factor further,
but Proposition~\ref{perfectpairing} explains 
the duality between the coefficients of 
$u^k$ and of $u^{\ell-k}$.
\end{example}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendix: The case of $G(r,p,\ell)$}
\label{appendix-section}
The Shephard and Todd
infinite family of reflection groups
$G(r,p,\ell)$ includes the Weyl groups 
of types $B_\ell$,  $D_\ell$, the dihedral groups, and symmetric groups.
To define these groups, fix an integer $r\geq 1$.  Then $G(r,1,\ell)$
is the set of $\ell \times \ell$
monomial matrices (i.e., matrices with a single 
nonzero entry in each row and column) whose nonzero
entries are complex $r$-th roots of unity.
As an abstract group, $G(r,1,\ell)$ is the direct product
of the symmetric group of order $\ell !$ and a cyclic group:
$$G(r,1,\ell)\cong \text{Sym}_\ell\times(\mathbb{Z}/r\mathbb{Z})^\ell
\cong \text{Sym}_\ell \ltimes \mathbb{Z}/r\mathbb{Z}\, .$$
Each group $G(r,1,\ell)$ acts on $V=\CC^\ell$ as a 
reflection group generated
by complex reflections of order $2$ and order $r$.
In fact, the group $G(r,1,\ell)$ is the symmetry group of
the complex cross-polytope in $\CC^\ell$, a {\it regular complex polytope} as studied
by Shephard~\cite{Shephard} and Coxeter~\cite{Coxeter}.

For integers $p \geq 1$ dividing $r$,
the group $G(r,p,\ell)$ consists of those matrices in $G(r,1,\ell)$ 
whose product of nonzero entries is an $(r/p)$-th root-of-unity.
Both $G(r,1,\ell)$ and $G(r,r,\ell)$ are duality groups,
and hence covered by Theorem~\ref{main-result}.
When $1<p<r$, the group $G(r,p,\ell)$ is a nonduality-group.

We record here a convenient choice of basic invariant polynomials and derivations for $G(r,p,\ell)$.

\begin{proposition}
Let $W=G(r,p,\ell)$ with $1\leq p < r$ and $p$ dividing $r$.
One may choose 
basic $W$-invariant polynomials $\{f_i\}_{i=1}^\ell$ and 
derivations $\{\theta_i\}_{i=1}^\ell$ as follows:
$$
\begin{aligned}
f_k&=x_1^{rk}+\cdots+x_\ell^{rk} \text{ for }k=1,2,\ldots,\ell-1, \text{ and }
f_\ell=(x_1 \cdots x_\ell)^{\frac{r}{p}},
 & \\
\theta_k &= x_1^{(k-1)r+1} \otimes y_1 + \cdots + x_\ell^{(k-1)r+1} \otimes y_\ell
\text{ for }k=1,2,\ldots,\ell\, .
\end{aligned}
$$
In particular, $W$ has 
$$
\begin{aligned}
&\text{exponents}\quad&
&(e_1,\ldots,e_{\ell-1},e_\ell)
=(r-1,2r-1,\ldots,(\ell-1)r-1,\tfrac{\ell r}{p}-1), \\
&\text{coexponents}\quad&
&(e_1^*,\ldots,e_\ell^*)
=(1,r+1,2r+1,\ldots,(\ell-1)r+1),\\
&\text{number of reflections}\quad& 
&N
=\sbinom{\ell}{2}r+\ell\left(\tfrac{r}{p}-1\right),
\quad\text{and}\\
&\text{number of hyperplanes}\quad& 
&N^*
=\sbinom{\ell}{2}r+\ell\, .
\end{aligned}
$$
\end{proposition}
\begin{proof}
The $W$-invariant derivations $\{\theta_i\}_{i=1}^\ell$ above are the 
usual choice, for example, as in Orlik and Terao~\cite[Prop.~6.77]{OrlikTerao}.
(Or use the $m=0$ case of
Corollary~\ref{key-numerology-corollary} to verify the $\theta_i$ are basic derivations, 
since their associated coefficient matrix is an easy variant of a
Vandermonde matrix.)  
The $W$-invariant polynomials $\{f_i\}_{i=1}^\ell$ above are closely related to a more usual choice $\{f_i'\}_{i=1}^\ell$ of basic $W$-invariants 
$$
e_1(x_1^r,\ldots,x_\ell^r),
e_2(x_1^r,\ldots,x_\ell^r),
\cdots,
e_{\ell-1}(x_1^r,\ldots,x_\ell^r),
(x_1\cdots x_\ell)^{\frac{r}{p}},
$$
in which $e_k(x_1,\ldots,x_\ell)$ is the $k^{th}$ elementary symmetric polynomial in $x_1,\ldots,x_\ell$,
the sum of all square-free monomials of degree $k$;
e.g., see Smith~\cite[\S7.4, Ex.~1]{Smith}.
  However, 
$\{f_i\}_{i=1}^{\ell-1}$ and $\{f'_i\}_{i=1}^{\ell-1}$ 
generate the same subalgebra of polynomials
when working over a field of characteristic zero 
because the collection of power sums and 
elementary symmetric functions
can be expressed as polynomials in each other; 
e.g., see~\cite[Thm.~7.4.4, Cor.~7.7.2, Prop.~7.7.6]{Stanley}.
\end{proof}

\begin{theorem}
\label{monomial-group-theorem} 
Let $W=G(r,p,\ell)$ with $1\leq p<r$ and $p$ dividing $r$.  
Then for $1 \leq m \leq \ell$,
the set
$$
\begin{aligned}
\BBB^{(\ell,m)}
&=\left\{df_I\, \theta_k\right\}_{ 
I \in \bbinom{[\ell]}{m}, \ k \in [\ell-m]} \
\ \  \sqcup\ \
\left\{df_{I}\, d\theta_k\right\}_{ 
I \in \bbinom{[\ell]}{m-1},\ k \in [\ell-m+1]}
\end{aligned}
$$
gives a free basis for the $\SSS^W$-module of invariants
$\big(\SSS\ot \wedge^m V^* \ot V)^W$.
\end{theorem}

\begin{proof}
We will apply Corollary~\ref{key-numerology-corollary}
to $\BBB^{(\ell,m)}$.  There are several steps.
\vskip.1in
\noindent
{\sf Step 1.}
First note that $\BBB^{(\ell,m)}$ has the correct cardinality:
$$
|\BBB^{(\ell,m)}|
=(\ell-m)\sbinom{\ell}{m} + (\ell-m+1)\sbinom{\ell}{m-1}
=(\ell-m)\sbinom{\ell}{m} + m\sbinom{\ell}{m}
=\ell\sbinom{\ell}{m}
=\rank_{\SSS^W}(\SSS\ot\wedge V^*\wedge V)^W\, .
$$

\vskip.1in
\noindent
{\sf Step 2.}
We check that the sum of the degrees of the elements in $\BBB^{(\ell,m)}$ is $\Delta(\wedge^mV^*\ot V)$,
which is straightforward albeit tedious.
Again using the shorthand notation $e_I:=\sum_{i \in I} e_i$ for $I \subset [\ell]$,
this sum is
$$
\sum_{ \substack{I \in \bbinom{[\ell]}{m} \\ 
k\in[\ell-m]\rule{0ex}{2ex}}} 
(e_I+e_k^*)
 + \sum_{ \substack{I\in \bbinom{[\ell]}{m-1} \\ 
k\in[\ell-m+1]\rule{0ex}{2ex}}} (e_{I}+e_k^*-1)\, ,
 $$
 which one can rewrite as
$$
\begin{aligned}
&\sbinom{\ell}{m} \sum_{k \in [\ell-m]}e_k^* \ + 
     \sbinom{\ell}{m-1} \sum_{k \in [\ell-m+1]}(e_k^*-1)\ + \\
& \qquad \qquad       (\ell-m)\left(  \sum_{I \in \bbinom{[\ell-1]}{m}} e_I 
                                      +  \sum_{I \in \bbinom{[\ell-1]}{m-1}} \left( e_I + e_\ell\right) \right) +
          (\ell-m+1)\left(  \sum_{I \in \bbinom{[\ell-1]}{m-1}} e_{I} 
                                      +  \sum_{I \in \bbinom{[\ell-1]}{m-2}} \left( e_{I} + e_\ell\right) \right)\, .
\end{aligned}
$$
Bearing in mind that $e_i=ir-1$ for $1 \leq i \leq \ell-1$, 
we employ a shorthand notation,
\begin{equation}
\label{sum-shorthand}
g(n,m):=\sum_{
\substack{I \in \bbinom{[k]}{m} \\ i \in I\rule{0ex}{1.5ex} 
}}e_i 
=\sum_{\substack{I \in \bbinom{[k]}{m} \\ i \in I\rule{0ex}{1.5ex} 
} } (ir-1) 
=\sbinom{k-1}{m-1} \sum_{i \in I} (ir-1) 
=\sbinom{k-1}{m-1} \left(r \sbinom{k+1}{2}-k \right),
\end{equation}
to rewrite the degree sum
as
$$
\begin{aligned}
&\sbinom{\ell}{m} \left( r \sbinom{\ell-m}{2} + \ell-m \right) +
    \sbinom{\ell}{m-1} \left( r \sbinom{\ell-m+1}{2} \right) +\\
& \qquad \qquad     (\ell-m) \left( g(\ell-1,m) + g(\ell-1,m-1) + \sbinom{\ell-1}{m-1}e_\ell \right) + \\
 &    \qquad \qquad\qquad    (\ell-m+1) \left( g(\ell-1,m-1) + g(\ell-1,m-2) + \sbinom{\ell-1}{m-2}e_\ell \right) \, .
\end{aligned}
$$
(Here, we use the fact that  $e_i^*=(i-1)r+1$ for all $i$.)
Finally, substituting in the right side 
of~\eqref{sum-shorthand} for all $g(n,m)$,
and $\frac{\ell r}{p}-1$ for $e_{\ell}$, we obtain
$$
\begin{aligned}
(\ell-1)\sbinom{\ell-1}{m-1}
&
\left( \sbinom{\ell}{2}r+\ell \left(\frac{\ell r}{p}-1\right) \right)
+ \sbinom{\ell-1}{m}\left( \sbinom{\ell}{2}r+\ell\right)\\
&=(\ell-1)\sbinom{\ell-1}{m-1} N+ \sbinom{\ell-1}{m}N^* 
=\Delta(\wedge^m V \ot V^*). 
\end{aligned}
$$
The first equality here was checked by hand and 
corroborated with computer algebra packages.
\vskip.1in
\noindent
{\sf Step 3.}
At this stage, to apply Corollary~\ref{key-numerology-corollary}, 
we need only show that the
set $\BBB^{(\ell,m)}$ is $K$-linearly independent in $K \otimes \wedge^m V^* \otimes V$.
We will use this to reduce to the case where $p=1$, that is, $W=G(r,1,\ell)$.

Note that the formulas for $\theta_k, d\theta_k, df_k$ in $W=G(r,p,\ell)$ depend on $p$ in only one place, namely $df_\ell$:
\begin{equation}
\label{differential-formulas}
\begin{aligned}
df_k&=
\begin{cases}
kr \sum_{j=1}^\ell x_j^{kr-1} dx_j &\text{ for }1 \leq k \leq \ell-1\, ,\\
\frac{r}{p} (x_1 \cdots x_\ell)^{\frac{r}{p}}  \sum_{j=1}^\ell x_i^{-1} dx_i 
&\text{ for }k=\ell\, ,\rule{0ex}{4ex}
\end{cases}\\
\theta_k&=\sum_{j=1}^\ell x_j^{(k-1)r+1} \otimes y_j\, ,\\
d\theta_k&=((k-1)r+1) \sum_{j=1}^\ell x_j^{(k-1)r} dx_j \otimes y_j\, .
\end{aligned}
\end{equation}

In checking whether the elements of 
$\BBB^{(\ell,m)}$ are $K$-linearly independent, we
are free to scale them by elements of the (rational function) field 
$K=\CC(x_1,\ldots,x_\ell)$. 
Hence, in \eqref{differential-formulas}, 
we may rescale each element $df_k$
by $kr$, for $1 \leq k \leq \ell-1$ and
also divide $df_\ell$ by 
$\frac{r}{p} (x_1 \cdots x_\ell)^{\frac{r}{p}}$.  
Hence Theorem~\ref{monomial-group-theorem} is equivalent to 
asserting $K$-linearly independence of the set 
$$
\BBB^{(\ell,m)}
=\left\{df_I \, \theta_k\right\}_{I \in \bbinom{[\ell]}{m},\ k \in [\ell-m]}
   \quad \sqcup \quad
 \left\{df_{I} \, d\theta_k\right\}_{I \in \bbinom{[\ell]}{m-1},\ k \in [\ell-m+1]}
$$
where one keeps $\theta_k, d\theta_k$ as in \eqref{differential-formulas} 
but re-defines $df_k$ more uniformly as follows:
$$
df_k:= \sum_{j=1}^\ell x_j^{(k-1)r-1} dx_j \text{ for }
1 \leq k \leq \ell\, .
$$
Note that we have also employed a cyclic shift of the indexing, that is,
the old $df_\ell$ has been replaced by the new $df_1$, the old
$df_1$ by the new $df_2$, etc.

This new $K$-linear independence assertion does not involve the parameter $p$.   
Therefore Theorem~\ref{monomial-group-theorem} 
for $W=G(r,p,\ell)$ with $1 \leq p < r$ follows once one 
proves it for $W=G(r,1,\ell)$, that is, with $p=1$.


\vskip.1in
\noindent
{\sf Step 4.}
We rescale the $K$-basis elements
$\{dx_I \otimes y_k\}_{I \in \bbinom{\ell}{m}, k \in [\ell]}$ in $K \otimes \wedge^* V^* \otimes V$
as follows:
$$
dx_I \otimes y_k \longmapsto
      x_k^{-1} x_I \, dx_I \otimes y_k
$$
(recall that 
$dx_I\ot y_k
=1\ot x_{i_1}\wedge\cdots\wedge x_{i_m}\ot y_k$
for $I=\{i_1 < \cdots< i_m\}$).
Then Theorem~\ref{monomial-group-theorem} 
is equivalent to the assertion that $\BBB^{(\ell,m)}$ is $K$-linearly independent
after redefining 
\begin{equation}
\label{last-redefinition-of-invariants}
\begin{array}{rlll}
df_k &:=\sum_{j=1}^\ell x_j^{(k-1)r} dx_j &=\sum_{j=1}^\ell z_j^{k-1} dx_j, 
\\
\theta_k&:=\sum_{j=1}^\ell x_j^{(k-1)r}\otimes  y_j &=\sum_{j=1}^\ell z_j^{k-1} \otimes y_j, 
\rule{0ex}{3.5ex}\\
d\theta_k&:=\sum_{j=1}^\ell x_j^{(k-1)r} dx_j \otimes y_j &=\sum_{j=1}^\ell z_j^{k-1} dx_j \otimes y_j,
\rule{0ex}{3.5ex}\\
\end{array}
\end{equation}
for $k=1,2,\ldots,\ell$,
where we have set $z_j:=x_j^r$ in $K$.


\vskip.1in
\noindent
{\sf Step 5.}
Consider the matrix $B^{(\ell,m)}$ with entries in $\CC(z_1,\ldots,z_\ell)$
whose columns express each element of $\BBB^{(\ell,m)}$,
defined via \eqref{last-redefinition-of-invariants},
in terms of $\{ dx_I \otimes y_k\}$ for
$I \in \sbinom{[\ell]}{m}$ and $k \in [\ell]$.
Since each $d\theta_k$ is a $K$-linear combination of
terms of the form $dx_j \otimes y_j$, the
expansion of each $df_{I}\, d\theta_k$ in $\BBB^{(\ell,m)}$
has nonzero coefficient of $dx_I \otimes y_k$ only when $k \in I$.
This leads to a block upper-triangular decomposition:
$$
B^{(\ell,m)} \quad =\quad
\bordermatrix{~      
& \{\text{\tiny $df_{I}\, d\theta_k$}\}  
&  \{\text{\tiny $df_{I}\, \theta_k$}\}  \cr
& & \cr
 \{\text{\tiny $dx_I \otimes y_k$}\}_{k \in I} & C^{(\ell,m)} & * \cr
 & & \cr
 \{\text{\tiny $dx_I \otimes y_k$}\}_{k \not\in I} & 0 & D^{(\ell,m)} \cr
 }.
$$
By convention here, $C^{(\ell,0)}, D^{(\ell,\ell)}$ are $0 \times 0$ matrices.

Thus it remains to show $C^{(\ell,m)}, D^{(\ell,m)}$
are invertible.  
We reduce this to showing invertibility of only the $D^{(\ell,m)}$,
since we claim that for $m=1,2,\ldots,\ell$, the
matrices $C^{(\ell,m)}$ and $D^{(\ell,m-1)}$ differ only by row-scalings.
To justify this claim, consider a pair $(J,j_0)$ with 
$J=\{j_1 <\cdots<j_{m-1}\}$ for a $(m-1)$-subset of $[\ell]$
and $j \in [\ell-m+1]$.  Then  $(J,j_0)$
indexes both a column in $C^{(\ell,m)}$, 
that lists the expansion coefficients in 
\begin{equation}
\label{C-expansion}
\begin{aligned}
df_{J}\, d\theta_{j_0}
&=\left( \sum_i z_i^{j_1-1} dx_i\right)
\cdots
\left( \sum_i z_i^{j_{m-1}-1} dx_i \right)
\left( \sum_i z_i^{j_0-1} dx_i \otimes y_i\right)\\
&=\sum_{(i_1,\ldots,i_{m-1},i_0)} 
     z_{i_1}^{j_1-1} \cdots  z_{i_{m-1}}^{j_{m-1}-1}  z_{i_0}^{j_0-1} \cdot
       dx_{i_1} \cdots dx_{i_{m-1}} dx_{i_0} \otimes y_{i_0}\, ,
\end{aligned}
\end{equation}
and a column in $D^{(\ell,m-1)}$, that lists the expansion coefficients in 
\begin{equation}
\label{D-expansion}
\begin{aligned}
df_{J} \, \theta_{j_0}
&=\left( \sum_i z_i^{j_1-1} dx_i\right)
\cdots
\left( \sum_i z_i^{j_{m-1}-1} dx_i \right)
\left( \sum_i z_i^{j_0-1} \otimes y_i\right)\\
&=\sum_{(i_1,\ldots,i_{m-1},i_0)} 
     z_{i_1}^{j_1-1} \cdots  z_{i_{m-1}}^{j_{m-1}-1}\,  
    z_{i_0}^{j_0-1}\, 
       dx_{i_1} \cdots dx_{i_{m-1}} \otimes y_{i_0}\, .
\end{aligned}
\end{equation}
On the other hand, a pair $(I,k)$ where $I$ is an $m$-subset of $[\ell]$ and $k \in I$
will index both a row for $dx_I \otimes y_k$ in $C^{(\ell,m)}$
and a row for $dx_{I \setminus \{k\}} \otimes y_k$ in  $D^{(\ell,m-1)}$.
If one assumes that $I=\{i_0,i_1,\ldots,i_{m-1}\}$ in the two expansions \eqref{C-expansion},
\eqref{D-expansion} above, then one can see that these two rows will differ by
a sign;  this sign is the product of the signs of two permutations,
namely those permutations that sort the ordered sequences
$(i_1,\ldots,i_{m-1},i_0)$ and $(i_1,\ldots,i_{m-1})$ into the 
usual integer orders on $I$ and $I \setminus \{k\}$, respectively.



\vskip.1in
\noindent
{\sf Step 6.}
It remains to show invertibility for $0 \leq m \leq \ell$ of the square matrix $D^{(\ell,m)}$ whose
columns give the coefficients in $\CC(z_1,\ldots,z_\ell)$ that expand each element of
 $\{df_I\, \theta_k\}_{I \in \bbinom{[\ell]}{m},\, k \in [\ell-m]}$
in terms of the elements $\{ dx_I \otimes y_k \}_{I \in \bbinom{[\ell]}{m}, k \not\in I}$,
ignoring coefficients on all other $dx_I \otimes y_k$.

In fact, we will show that $\det D^{(\ell,m)}$ has coefficient $\pm 1$ on its {\it lexicographically-largest} 
monomial, that is, the monomial
$
z_\ell^{a_\ell} z_{\ell-1}^{a_{\ell-1}} \cdots z_2^{a_2} z_1^{a_1}
$  
that achieves the maximum exponent $a_\ell$, and 
among all such monomials with maximum $a_\ell$,
also maximizes $a_{\ell-1}$, and so on.

The argument will proceed by induction on $\ell$, 
via consideration of another block decomposition:
$$
D^{(\ell,m)} \quad =\quad
\bordermatrix{~      
   & \text{\small $\{df_I d\theta_k\}_{\ell \in I}$}  
&  \text{\small $\{df_I \theta_{\ell-m}\} $} 
& \text{\small $\{df_I \theta_k\}_{\ell \not\in I, k \in [\ell-1-m]}$} \cr
 \text{\small$\{ dx_I \otimes y_k \}_{\ell \in I}$} &
 \alpha & * & * \cr
\text{\small $\{ dx_I \otimes y_\ell \}$}   
         &  *  & 
 \beta & * \cr
 \text{\small$\{dx_I \otimes y_k\}_{\ell \not\in I \sqcup\{k\}}$} 
& * & * & \gamma \cr }.
$$
Since the row indices $dx_I \otimes y_k$ always have $k \not\in I$,
the highest powers of $x_\ell$ in entries of $\det D^{(\ell,m)}$ are
\begin{itemize}
\item at most $z_\ell^{\ell-1}$ in the top block of rows $(\alpha,*,*)$,
and $z_\ell^{\ell-1}$ occurs only in the left block $\alpha$,
\item at most $z_\ell^{\ell-m-1}$ in the middle block of rows $(*,\beta,*)$,
and $z_\ell^{\ell-m-1}$ occurs only in the middle block $\beta$,
\item only $z_\ell^0$  in the bottom block of rows $(*,*,\gamma)$, that is, 
no $z_\ell$'s occur at all there.
\end{itemize}
Consequently, the highest power of $z_\ell$ in $\det D^{(\ell,m)}$
can only occur in a term from the permutation expansion that is
also a term from the product $\det \alpha \cdot \det \beta \cdot \det \gamma$.
Furthermore, it must be a term that always picks entries of $\alpha$ 
divisible by $z_\ell^{\ell-1}$, and entries of $\beta$ divisible by
$z_\ell^{\ell-m-1}$.

Upon examining $\alpha, \beta, \gamma$ more closely, 
one finds that 
\begin{equation}
\label{closer-examination-of-alpha-beta-gamma}
\begin{aligned}
\gamma&=D^{(\ell-1,m)},\\
\beta&=z_\ell^{\ell-m-1} \cdot \wedge^m\, {\VDM}^{(\ell-1)} ,\\
\alpha&=z_\ell^{\ell-1} D^{(\ell-1,m-1)} +O(z_\ell^{\ell-2})
\end{aligned}
\end{equation}
where $\wedge^m A$ is the $m^{th}$ {\it exterior power} of the
matrix $A$, and ${\VDM}^{(n)}:=[ z_i^{j-1} ]_{i,j=1,2,\ldots,n}$
is an $n \times n$ {\it Vandermonde matrix}.
For example, when $\ell=4,m=2$, the matrix $\beta$ looks like this:
$$
\bordermatrix{~ & 
\text{\tiny $df_3 df_2 \theta_2$} & 
\text{\tiny $df_3 df_1 \theta_2$} & 
\text{\tiny $ df_2 df_1 \theta_2$} \cr
 & & & \cr
\text{\tiny $dx_3 dx_2 \otimes y_4$} 
& (z_3^2 z_2^1-z_2^2 z_3^1)z_4^1 
& (z_3^2 z_2^0-z_2^2 z_3^0)z_4^1 & (z_3^1 z_2^0-z_2^1 z_3^0)z_4^1  \cr
 & & & \cr
\text{\tiny $dx_3 dx_1 \otimes y_4$} 
& (z_3^2 z_1^1-z_1^2 z_3^1)z_4^1 
& (z_3^2 z_1^0-z_1^2 z_3^0)z_4^1 & (z_3^1 z_1^0-z_1^1 z_3^0)z_4^1 \cr
 & & & \cr
\text{\tiny $dx_2 dx_1 \otimes y_4$} 
& (z_2^2 z_1^1-z_1^2 z_2^1)z_4^1 & (z_2^2 z_1^0-z_1^2 z_2^0)z_4^1 & (z_2^1 z_1^0-z_1^1 z_2^0)z_4^1
}
=z_4^{4-2-1} \cdot \wedge^2\, {\VDM}^{(3)}.
$$
Meanwhile, the matrix $\alpha$ looks like this, underlining the 
terms contributing the highest power $z_4^3=z_\ell^{\ell-1}$:

\begin{tiny}
$$
\bordermatrix{~ & df_4 df_3 \theta_2 & df_4 df_2 \theta_2 
                  &  df_4 df_1 \theta_2 &  df_4 df_3 \theta_1 
                      &  df_4 df_2 \theta_1 &  df_4 df_1 \theta_1 \cr
 & & & & & & \cr
dx_4 dx_3\otimes y_2 & (\underline{z_4^3 z_3^2} - z_4^2 z_3^3) z_2^1  &
                       (\underline{z_4^3 z_3^1} - z_4^1 z_3^3) z_2^1 &
                        (\underline{z_4^3 z_3^0} - z_4^0 z_3^3) z_2^1 &
                         (\underline{z_4^3 z_3^2} - z_4^2 z_3^3) z_2^0 &
                          (\underline{z_4^3 z_3^1} - z_4^1 z_3^3) z_2^0 &
                            (\underline{z_4^3 z_3^0} - z_4^0 z_3^3) z_2^0 \cr
 & & & & & & \cr
dx_4 dx_3\otimes y_1 & (\underline{z_4^3 z_3^2} - z_4^2 z_3^3) z_1^1  &
                       (\underline{z_4^3 z_3^1} - z_4^1 z_3^3) z_1^1 &
                        (\underline{z_4^3 z_3^0} - z_4^0 z_3^3) z_1^1 &
                         (\underline{z_4^3 z_3^2} - z_4^2 z_3^3) z_1^0 &
                          (\underline{z_4^3 z_3^1} - z_4^1 z_3^3) z_1^0 &
                            (\underline{z_4^3 z_3^0} - z_4^0 z_3^3) z_1^0 \cr
 & & & & & & \cr
dx_4 dx_2\otimes y_3 & (\underline{z_4^3 z_2^2} - z_4^2 z_2^3) z_3^1  &
                       (\underline{z_4^3 z_2^1} - z_4^1 z_2^3) z_3^1 &
                        (\underline{z_4^3 z_2^0} - z_4^0 z_2^3) z_3^1 &
                         (\underline{z_4^3 z_2^2} - z_4^2 z_2^3) z_3^0 &
                          (\underline{z_4^3 z_2^1} - z_4^1 z_2^3) z_3^0 &
                            (\underline{z_4^3 z_2^0} - z_4^0 z_2^3) z_3^0 \cr
 & & & & & & \cr
dx_4 dx_2\otimes y_1 & (\underline{z_4^3 z_2^2} - z_4^2 z_2^3) z_1^1  &
                       (\underline{z_4^3 z_2^1} - z_4^1 z_2^3) z_1^1 &
                        (\underline{z_4^3 z_2^0} - z_4^0 z_2^3) z_1^1 &
                         (\underline{z_4^3 z_2^2} - z_4^2 z_2^3) z_1^0 &
                          (\underline{z_4^3 z_2^1} - z_4^1 z_2^3) z_1^0 &
                            (\underline{z_4^3 z_2^0} - z_4^0 z_2^3) z_1^0 \cr
 & & & & & & \cr
dx_4 dx_1\otimes y_3 & (\underline{z_4^3 z_1^2} - z_4^2 z_1^3) z_3^1  &
                       (\underline{z_4^3 z_1^1} - z_4^1 z_1^3) z_3^1 &
                        (\underline{z_4^3 z_1^0} - z_4^0 z_1^3) z_3^1 &
                         (\underline{z_4^3 z_1^2} - z_4^2 z_1^3) z_3^0 &
                          (\underline{z_4^3 z_1^1} - z_4^1 z_1^3) z_3^0 &
                            (\underline{z_4^3 z_1^0} - z_4^0 z_1^3) z_3^0 \cr
 & & & & & & \cr
dx_4 dx_1\otimes y_2 & (\underline{z_4^3 z_1^2} - z_4^2 z_1^3) z_2^1  &
                       (\underline{z_4^3 z_1^1} - z_4^1 z_1^3) z_2^1 &
                        (\underline{z_4^3 z_1^0} - z_4^0 z_1^3) z_2^1 &
                         (\underline{z_4^3 z_1^2} - z_4^2 z_1^3) z_2^0 &
                          (\underline{z_4^3 z_1^1} - z_4^1 z_1^3) z_2^0 &
                            (\underline{z_4^3 z_1^0} - z_4^0 z_1^3) z_2^0 
}
$$
\end{tiny}
with determinant $z_4^{4-1} \cdot D^{(3,1)} + O(z_4^{4-2})$.

\normalsize
As the lex-largest monomial in $\det D^{(\ell,m)}$
has the same coefficient as in $\det \alpha \cdot \det \beta \cdot \det \gamma$,
the descriptions in \eqref{closer-examination-of-alpha-beta-gamma}
imply that this monomial is a power of $z_\ell$ 
times the product of the lex-largest monomials in 
$$ 
\det D^{(\ell-1,m)}, 
\,\, \det D^{(\ell-1,m-1)}, 
\,\, \det \wedge^m\, {\VDM}^{(\ell-1)}. 
$$
By induction on $\ell$, the coefficient on the lex-largest monomials
in $\det D^{(\ell-1,m)}, \det D^{(\ell-1,m-1)}$
are both $\pm 1$.  Regarding
$\det \wedge^m\,  {\VDM}^{(\ell-1)}$, the Sylvester-Franke Theorem
asserts that $\det \wedge^m A = \left( \det A \right)^{\binom{n-1}{m-1}}$,
and since one has a coefficient of $\pm 1$ on
the lex-largest monomial 
$z_{\ell-1}^{\ell-2} \cdots z_2^1 z_1^0$ in $\det{\VDM}^{(\ell-1)}$,
the same is also true for $\det \wedge^m\, {\VDM}^{(\ell-1)}$.
Thus the same holds for $\det D^{(\ell,m)}$, completing the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section*{Acknowledgments}
The authors thank Corrado DeConcini, Paolo Papi, Mark Reeder and John Stembridge for helpful conversations and references, and making them aware of their unpublished work.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\begin{thebibliography}{}
\bibitem{ArmstrongRRhoades}
D.\ Armstrong, V.\ Reiner and B.\ Rhoades,
Parking spaces.
{\it Adv.\ Math.} {\bf 269} (2015), 647--706. 

\bibitem{Bazlov}
Y.\ Bazlov,
Graded multiplicities in the exterior algebra. 
{\it Adv.\ Math.} {\bf 158} (2001), 129--153. 

\bibitem{BEG}
Y.\ Berest, P.\ Etingof, and V.\ Ginzburg,
Finite-dimensional representations of rational Cherednik algebras. 
{\it Int.\ Math.\ Res.\ Not.}  {\bf 19} (2003), 1053--1088.
 
%\bibitem{Bessis}
%D.\ Bessis,
%Finite complex reflection arrangements are $K(\pi,1)$,
%{\it Ann.\ of Math.} {\bf 181} (2015), 809--904. 

\bibitem{BessisR}
D.\ Bessis and V.\ Reiner, 
Cyclic sieving of noncrossing partitions 
for complex reflection groups.
{\it Ann.\ Comb.} {\bf 15} (2011), 197--222. 

\bibitem{Broer}
A.\ Broer, 
The sum of generalized exponents and 
Chevalley's restriction theorem for modules of covariants.
{\it Indag.\ Math.\ (N.S.)} {\bf 6} (1995), 385--396. 

\bibitem{BMR}
M.\ Brou\'e, G.\ Malle, and R.\ Rouquier,
Complex reflection groups, braid groups, and Hecke algebras.
{\it J.\ Reiner.\ Angew.\ Math.} {\bf 500} (1998), 127-190.

\bibitem{Chevalley}
C.\ Chevalley, 
Invariants of finite groups generated by reflections.
{\it Amer.\ J.\ Math.} {\bf 77} (1955), 778--782. 

\bibitem{Coxeter}
H.S.M.\ Coxeter, 
Regular complex polytopes, 2nd edition. 
Cambridge University Press, Cambridge, 1991. 

\bibitem{DPP}
C.\ De Concini, P.\ Papi, and C.\ Procesi,
The adjoint representation inside the exterior
algebra of a simple Lie algebra
{\it Adv.\ Math.} {\bf 280} (2015) 21--46.

\bibitem{DimcaSticlaru}
A.\ Dimca and G.\ Sticlaru,
 On the Milnor monodromy of the exceptional reflection arrangement of type $G_{31}$. Alexandru
{\tt arXiv:1606.06615}

\bibitem{GNS}
A.\ Gyoja, K.\ Nishiyama, and H.\ Shimura,
Invariants for representations of Weyl groups and two sided cells,
{\it J.\ Math.\ Soc.\ Japan} {\bf 51} (1999), 1--34.

\bibitem{HochsterEagon}
M.\ Hochster and J.A.~Eagon,
Cohen-Macaulay rings, invariant theory, and the 
generic perfection of determinantal loci.
{\it Amer.\ J.\ Math.} {\bf 93} (1971), 1020--1058. 

\bibitem{Jacobi}
C.G.J.\ Jacobi, 
``De determinantibus functionalibus'', 
{\it J.\ Reine Angew.\ Math.} {\bf 22} (1841),
no.\ 4, 319--359.

\bibitem{Joseph}
Joseph, Anthony,
Sur l'annulateur d'un module de Verma. 
NATO Adv.\ Sci.\ Inst.\ Ser.\ C Math.\ Phys.\ Sci., 514, 
Representation theories and algebraic geometry 
(Montreal, PQ, 1997), 237--300, Kluwer Acad.\ Publ., Dordrecht, 1998. 

\bibitem{LehrerMichel}
G.I.\ Lehrer and J.\ Michel, 
Invariant theory and eigenspaces for unitary reflection groups. 
{\it C.\ R.\ Math.\ Acad.\ Sci.\ Paris} {\bf 336} (2003), 795--800.

\bibitem{Maschke}
H.\ Maschke, 
Ueber die quatern\"{a}re, endliche, lineare Substitutionsgruppe 
der Borchardt'schen Moduln, 
{\it Math.\ Ann.} {\bf 30} (1887), 496--515.

\bibitem{Opdam}
E.M.\ Opdam,
Complex reflection groups and fake Degrees. 
{\it Technical report of the Mathematics Dept., Univ.\ of Leiden, W98-17}
(1998); {\tt arXiv:math/9808026}

\bibitem{OrlikSolomon} 
P.\ Orlik and L.\ Solomon, 
Unitary reflection groups and cohomology, 
{\it Invent.\ Math.} {\bf 59} (1980),
77--94.

\bibitem{OrlikTerao}
P.\ Orlik and H.\ Terao,
Arrangements of hyperplanes.
{\it Grundlehren der Mathematischen Wissenschaften},
{\bf 300}. Springer-Verlag, Berlin, 1992. 

\bibitem{Reeder1}
M.\ Reeder,
On the cohomology of compact Lie groups. 
{\it Enseign.\ Math.} {\bf (2) 41} (1995), 181--200. 

\bibitem{Reeder2}
M.\ Reeder,
Exterior powers of the adjoint representation. 
{\it Canad.\ J.\ Math.} {\bf 49} (1997), 133--159. 

\bibitem{Shephard}
G.C.\ Shephard, 
Regular complex polytopes.
{\it Proc.\ London Math.\ Soc.} {\bf 2} (1952), 82--97.

\bibitem{ShephardTodd}
G.C.\ Shephard and J.\ A.\ Todd,
Finite unitary reflection groups,
{\it Canad.\ J.\ Math.} {\bf 6} (1954), 274--304.

\bibitem{Shepler}
A.V.\ Shepler,
Generalized exponents and forms,
{\it J.\ of Algebraic Combin.}, 
{\bf 22} (2005), no.\ 1, 115--132. 

\bibitem{Smith}
L.\ Smith,
Polynomial invariants of finite groups,
{\it Research Notes in Mathematics} {\bf 6}.
A K Peters, Ltd., Wellesley, MA, 1995. 

\bibitem{Solomon}
L.\ Solomon,
Invariants of finite reflection groups,
{\it Nagoya Math J.} {\bf 22} (1963), 57--64.

\bibitem{Springer}
T.A.\ Springer, 
Regular elements of finite reflection groups.
{\it Invent.\ Math.\ } {\bf 25} (1974), 159--198. 

\bibitem{Stanley}
R.P.\ Stanley,
Enumerative combinatorics, Vol. 2.
{\it Cambridge Studies in Advanced Mathematics} {\bf 62}. 
Cambridge University Press, Cambridge, 1999.

\bibitem{Steinberg}
R.\ Steinberg,
Invariants of finite reflection groups,
{\it Canad.\ J.\ Math.} {\bf 12} (1960), 616--618. 

\bibitem{Stembridge}
J.\ Stembridge, 
First layer formulas for characters of $SL(n,\CC)$, 
{\it Trans.\ Amer.\ Math.\ Soc.}  {\bf 299} (1987), 319--350.

\bibitem{Terao}
H.\ Terao, 
Free arrangements of hyperplanes and unitary reflection groups.
{\it Proc.\ Japan Acad.\ Ser.\ A Math.\ Sci.} {\bf 56} (1980), no.\ 8, 389--392. 
\end{thebibliography}
\vspace{-2ex}
\end{document}


1612.01031.extracted_bib
{ArmstrongRRhoades D.\ Armstrong, V.\ Reiner and B.\ Rhoades, Parking spaces. {Adv.\ Math. {269 (2015), 647--706.
1612.01031.extracted_bib
{Bazlov Y.\ Bazlov, Graded multiplicities in the exterior algebra. {Adv.\ Math. {158 (2001), 129--153.
1612.01031.extracted_bib
{BEG Y.\ Berest, P.\ Etingof, and V.\ Ginzburg, Finite-dimensional representations of rational Cherednik algebras. {Int.\ Math.\ Res.\ Not. {19 (2003), 1053--1088. %
1612.01031.extracted_bib
{Bessis %D.\ Bessis, %Finite complex reflection arrangements are $K(\pi,1)$, %{Ann.\ of Math. {181 (2015), 809--904.
1612.01031.extracted_bib
{BessisR D.\ Bessis and V.\ Reiner, Cyclic sieving of noncrossing partitions for complex reflection groups. {Ann.\ Comb. {15 (2011), 197--222.
1612.01031.extracted_bib
{Broer A.\ Broer, The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants. {Indag.\ Math.\ (N.S.) {6 (1995), 385--396.
1612.01031.extracted_bib
{BMR M.\ Brou\'e, G.\ Malle, and R.\ Rouquier, Complex reflection groups, braid groups, and Hecke algebras. {J.\ Reiner.\ Angew.\ Math. {500 (1998), 127-190.
1612.01031.extracted_bib
{Chevalley C.\ Chevalley, Invariants of finite groups generated by reflections. {Amer.\ J.\ Math. {77 (1955), 778--782.
1612.01031.extracted_bib
{Coxeter H.S.M.\ Coxeter, Regular complex polytopes, 2nd edition. Cambridge University Press, Cambridge, 1991.
1612.01031.extracted_bib
{DPP C.\ De Concini, P.\ Papi, and C.\ Procesi, The adjoint representation inside the exterior algebra of a simple Lie algebra {Adv.\ Math. {280 (2015) 21--46.
1612.01031.extracted_bib
{DimcaSticlaru A.\ Dimca and G.\ Sticlaru, On the Milnor monodromy of the exceptional reflection arrangement of type $G_{31$. Alexandru {\tt arXiv:1606.06615
1612.01031.extracted_bib
{GNS A.\ Gyoja, K.\ Nishiyama, and H.\ Shimura, Invariants for representations of Weyl groups and two sided cells, {J.\ Math.\ Soc.\ Japan {51 (1999), 1--34.
1612.01031.extracted_bib
{HochsterEagon M.\ Hochster and J.A.~Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. {Amer.\ J.\ Math. {93 (1971), 1020--1058.
1612.01031.extracted_bib
{Jacobi C.G.J.\ Jacobi, ``De determinantibus functionalibus'', {J.\ Reine Angew.\ Math. {22 (1841), no.\ 4, 319--359.
1612.01031.extracted_bib
{Joseph Joseph, Anthony, Sur l'annulateur d'un module de Verma. NATO Adv.\ Sci.\ Inst.\ Ser.\ C Math.\ Phys.\ Sci., 514, Representation theories and algebraic geometry (Montreal, PQ, 1997), 237--300, Kluwer Acad.\ Publ., Dordrecht, 1998.
1612.01031.extracted_bib
{LehrerMichel G.I.\ Lehrer and J.\ Michel, Invariant theory and eigenspaces for unitary reflection groups. {C.\ R.\ Math.\ Acad.\ Sci.\ Paris {336 (2003), 795--800.
1612.01031.extracted_bib
{Maschke H.\ Maschke, Ueber die quatern\"{are, endliche, lineare Substitutionsgruppe der Borchardt'schen Moduln, {Math.\ Ann. {30 (1887), 496--515.
1612.01031.extracted_bib
{Opdam E.M.\ Opdam, Complex reflection groups and fake Degrees. {Technical report of the Mathematics Dept., Univ.\ of Leiden, W98-17 (1998); {\tt arXiv:math/9808026
1612.01031.extracted_bib
{OrlikSolomon P.\ Orlik and L.\ Solomon, Unitary reflection groups and cohomology, {Invent.\ Math. {59 (1980), 77--94.
1612.01031.extracted_bib
{OrlikTerao P.\ Orlik and H.\ Terao, Arrangements of hyperplanes. {Grundlehren der Mathematischen Wissenschaften, {300. Springer-Verlag, Berlin, 1992.
1612.01031.extracted_bib
{Reeder1 M.\ Reeder, On the cohomology of compact Lie groups. {Enseign.\ Math. {(2) 41 (1995), 181--200.
1612.01031.extracted_bib
{Reeder2 M.\ Reeder, Exterior powers of the adjoint representation. {Canad.\ J.\ Math. {49 (1997), 133--159.
1612.01031.extracted_bib
{Shephard G.C.\ Shephard, Regular complex polytopes. {Proc.\ London Math.\ Soc. {2 (1952), 82--97.
1612.01031.extracted_bib
{ShephardTodd G.C.\ Shephard and J.\ A.\ Todd, Finite unitary reflection groups, {Canad.\ J.\ Math. {6 (1954), 274--304.
1612.01031.extracted_bib
{Shepler A.V.\ Shepler, Generalized exponents and forms, {J.\ of Algebraic Combin., {22 (2005), no.\ 1, 115--132.
1612.01031.extracted_bib
{Smith L.\ Smith, Polynomial invariants of finite groups, {Research Notes in Mathematics {6. A K Peters, Ltd., Wellesley, MA, 1995.
1612.01031.extracted_bib
{Solomon L.\ Solomon, Invariants of finite reflection groups, {Nagoya Math J. {22 (1963), 57--64.
1612.01031.extracted_bib
{Springer T.A.\ Springer, Regular elements of finite reflection groups. {Invent.\ Math.\ {25 (1974), 159--198.
1612.01031.extracted_bib
{Stanley R.P.\ Stanley, Enumerative combinatorics, Vol. 2. {Cambridge Studies in Advanced Mathematics {62. Cambridge University Press, Cambridge, 1999.
1612.01031.extracted_bib
{Steinberg R.\ Steinberg, Invariants of finite reflection groups, {Canad.\ J.\ Math. {12 (1960), 616--618.
1612.01031.extracted_bib
{Stembridge J.\ Stembridge, First layer formulas for characters of $SL(n,\CC)$, {Trans.\ Amer.\ Math.\ Soc. {299 (1987), 319--350.
1612.01031.extracted_bib
{Terao H.\ Terao, Free arrangements of hyperplanes and unitary reflection groups. {Proc.\ Japan Acad.\ Ser.\ A Math.\ Sci. {56 (1980), no.\ 8, 389--392.