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\newcommand\R{\mathbb{R}}
\newcommand\orb{\mathcal{O}}
\newcommand\bq{\mathbf{q}}
\newcommand\br[1]{\langle #1\rangle}
\newcommand\tup[1]{\textup{(}#1\textup{)}}
\newcommand\mycase[2]{\smallbreak Case #1: \emph{#2}.}
\newcommand\mcase[1]{\smallbreak Case #1.}
\newcommand\sle{\leqslant}
\newcommand\sge{\geqslant}
\newcommand\vpo{^{\vphantom1}}
\newcommand\vph{^{\vphantom{\vee}}}
\newcommand\isom{\cong}
\newcommand\mtset{\varnothing}
\newcommand\xtal{crystallographic}
\newcommand\type{first fault}
\newcommand\pseries{Poincar\'e series}
\newcommand\eps{\varepsilon}
\newcommand\dcup{\ \dot\cup\ }
\newcommand\fold{\Phi^\vee\!/\sigma}
\newcommand\Efold{(\fold)_0}
\newcommand\Dfold{\Sigma}
\DeclareMathOperator{\Span}{Span}%
\DeclareMathOperator{\sgn}{sgn}%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Auteur
\author{\firstname{John} \middlename{R.} \lastname{Stembridge}}
\address{
Dept.~of Mathematics\\
University of Michigan\\
Ann Arbor\\
Michigan 48109--1043, USA}
\email{jrs@umich.edu}
%%%%% Sujet
\keywords{Weyl group, root system, Poincar\'e series, inversion}
%% Mathematical classification (2010)
\subjclass{05E15, 05A19, 20F55}
%%%%% Gestion
\DOI{10.5802/alco.62}
\datereceived{2018-07-24}
\daterevised{2019-01-17}
\dateaccepted{2019-01-21}
%%%%% Titre et résumé
\title[Odd inversions in Weyl groups]{Sign-twisted Poincar\'e series and odd inversions in Weyl groups}
\begin{abstract}
Following recent work of Brenti and Carnevale, we investigate a
sign-twisted Poincar\'e series for finite Weyl groups $W$ that tracks
``odd inversions''; i.e.~the number of odd-height positive roots transformed
into negative roots by each member of~$W$. We prove that the series
is divisible by the corresponding series for any parabolic
subgroup $W_J$, and provide sufficient conditions for when the
quotient of the two series equals the restriction of the first
series to coset representatives for $W/W_J$. We also show that the
series has an explicit factorization involving the degrees of the
free generators of the polynomial invariants of a canonically
associated reflection group.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction}
\subsection{Overview}
Let $\Phi$ be a finite \xtal{} root system with Weyl group $W$.
In~\cite{BC2}, Brenti and Carnevale consider the sign-twisted \pseries
\[
F(\Phi;q):=\sum_{w\in W}\sgn(w)q^{\ell_1(w)},
\]
where $\ell_1(w)$ denotes the number of ``odd inversions'' of $w$;
\ie the number of positive roots of odd height mapped to negative
roots by $w$. In particular, they give an explicit product formula for
the above series in all cases except $E_8$.
For the root systems $B_n$ and~$C_n$, they also have
multivariate refinements that keep track of more detailed
information about the odd inversions, and in earlier work for
types $A$ and $B$~\cite{BC1}, they consider variations that include
restricting the series to the distinguished coset representatives
for any parabolic quotient $W/W_J$.
Among their motivations, we should mention work of Klopsch and Voll~\cite{KV}.
In the course of enumerating flags of subspaces over a finite field
that are non-degenerate with respect to various classes of bilinear forms,
they conjectured a formula (proved in~\cite{BC1}) that amounted to an
explicit evaluation of the above series restricted to any parabolic
quotient in type~$A$. A similar explicit evaluation
for parabolic quotients in type~$B$ arose conjecturally in the work
of Stasinski and Voll~\cite{SV2} on the zeta functions for certain group
schemes. (This conjecture was proved independently in~\cite{BC1}
and~\cite{L}.)
The main goal of the present article is to provide further insights
into these sign-twisted series, with an emphasis on unified,
classification-free results.
For example, it seems natural from the perspective of general root
systems to exploit the fact that the reflections corresponding to the
even roots permute the odd roots. Although this diverges somewhat from
the point of view taken in~\cite{BC2} (see Remark~\ref{R6.2} below),
it leads to multivariate refinements that share the common feature that
odd roots in the same ``even orbit'' are given the same weight. Thus the
multivariate series we consider here involve 2 variables for $A_n$,
$B_n$, $F_4$ and $G_2$, and 4 variables for $C_n$.
Our main results are as follows:
\begin{enumerate}[\arabic*.]
\item %1.
A combinatorial quotient theorem proving
that for every parabolic subsystem~$\Phi_J$, the corresponding series
$F(\Phi_J;q)$ is a divisor of $F(\Phi;q)$ (see Theorem~\ref{T5.1} and
its corollaries). This makes it clear a priori (\ie without
use of the classification of finite root systems) that the series
$F(\Phi;q)$ necessarily has many irreducible factors.
\item %2.
An explicit product formula for the multivariate series for each
irreducible root system (Theorem~\ref{T6.1}). The proof is by induction with
respect to rank, using Theorem~\ref{T5.1}. Each inductive step requires
the evaluation of a sign-twisted sum over a small number of right coset
representatives. In the classical cases, there are only 2 or 4 terms in
each sum.
\item %3.
A general result that identifies, in a classification-free way,
sufficient conditions for the multivariate series for a parabolic
quotient $W/W_J$ to be the ratio of the corresponding series for $W$
and $W_J$ (Theorem~\ref{T7.1}).
Conjecturally, the conditions are also necessary.
\item %4.
A uniform presentation of the univariate product formula,
revealing that
\[
F(\Phi;q)=(1-q^{a_1})\cdots(1-q^{a_k}),
\]
where $a_1,\dots,a_k$ are the degrees of the free generators of the
polynomial invariants of a certain reflection group canonically associated
to $\Phi$ (Theorem~\ref{T8.5}). Although our only proof of this formula is
case-by-case, it does suggest a possible framework for a
classification-free approach to the factorization of $F(\Phi;q)$.
\end{enumerate}
\subsection{Preliminaries}
Throughout, $V$ shall denote a real Euclidean space with inner
product $\br{\cdot\,,\cdot}$
and $\Phi\subset V$ a finite \xtal{} root system with
co-root system $\Phi^\vee=\{\beta^\vee:\beta\in\Phi\}$,
positive roots $\Phi^+$, and Weyl group $W=W(\Phi)$.
We use a common index set $I$ for the simple
roots $\{\alpha_i:i\in I\}\subset\Phi$
and simple reflections $\{s_i:i\in I\}\subset W$.
We will embed $\Phi^\vee$ in $V$ by
taking $\beta^\vee=2\beta/\br{\beta,\beta}$.
If $L$ is a sublattice of the root lattice $\Z\Phi$, it is easy to
show that $\Phi\cap L$ is again a root system in $V$,
possibly of lower rank. The main lattice of interest here is
\begin{equation}\label{eq1.0}
L_0:=\bigl\{\textstyle{\sum a_i\alpha_i}:a_i\in\Z,
\ \textstyle{\sum a_i}\text{ even}\bigr\}.
\end{equation}
This has index 2 in $\Z\Phi$, and thus partitions $\Phi$ into
``even'' and ``odd'' roots; \ie
\[
\Phi=\Phi_0\cup\Phi_1,
\]
where $\Phi_0=\Phi\cap L_0$ is the root subsystem consisting of all roots
of even height, and $\Phi_1$ is the set of roots with odd height.
Note that the even Weyl group $W(\Phi_0)$ necessarily permutes
$\Phi_0$ and $\Phi_1$.
Recall that the length function on $W$ counts inversions; \ie
$\ell(w)=|\Phi(w)|$, where
\[
\Phi(w)=\{\beta\in\Phi^+:-w\beta\in\Phi^+\}.
\]
We may thus refine length by separating the contributions of
even and odd roots; \ie
\[
\ell(w)=\ell_0(w)+\ell_1(w),
\]
where $\ell_i(w):=|\Phi_i(w)|$ and $\Phi_i(w):=\Phi_i\cap\Phi(w)$.
As discussed earlier, our central object of interest is
the series
\begin{equation}\label{eq1.1}
F(\Phi;q):=\sum_{w\in W}\sgn(w)q^{\ell_1(w)}.
\end{equation}
Note that since every nontrivial $w\in W$ inverts at least one simple
(and therefore odd) root, it follows that $F(\Phi;q)$ has constant
term~1. Likewise, since there is a unique element that inverts all of
the simple roots (the longest element), it is therefore also the unique
element that inverts all of the odd roots. Thus the degree
of $F(\Phi;q)$ is the number of odd positive roots.
It would be reasonable to consider a generalization of~\eqref{eq1.1} in which
the pair $(\Phi_0,\Phi_1)$ is replaced with $(\Phi',\,\Phi\setminus\Phi')$
for \emph{any} root subsystem $\Phi'\subset\Phi$. However, it should be
noted that the corresponding series will vanish unless $\Phi'$ contains
none of the simple roots of~$\Phi$. Indeed, the simple reflection $s_i$
has only one inversion: the simple root $\alpha_i$. If this simple root
belongs to $\Phi'$, then $s_i$ must necessarily permute
$\Phi^+\setminus\Phi'$. Thus multiplication by $s_i$ would change the
sign while preserving the number of inversions not in $\Phi'$.
That is, the contributions from pairs $(w,ws_i)$ would cancel out.
We remark that each equal rank real form of a complex semisimple Lie
group with root system $\Phi$ induces a parity function on $\Phi$
(more precisely, a homomorphism $\Z\Phi\to\Z/2\Z$) defined up to the
action of $W$. The even roots in this setting correspond
to the roots of the maximal compact subgroup.
Thus the parity function we are concerned with here is associated to the
quasisplit equal rank real form; \ie the case where there is a positive
system in which all simple roots have odd parity.
(See for example~\cite{Kn}, especially Chapter~VI and the discussion
surrounding~(6.99).)
For each $J\subseteq I$, we let $\Phi_J$ denote the parabolic subsystem
of $\Phi$ generated by $\{\alpha_j:j\in J\}$,
and $W_J=W(\Phi_J)=\br{s_j:j\in J}$ the associated Weyl group.
It is well known that
\begin{align*}
W^J&:=\{w\in W:\ell(ws_j)>\ell(w)\text{ for all $j\in J$}\},\\
{}^JW&:=\{w\in W:\ell(s_jw)>\ell(w)\text{ for all $j\in J$}\},
\end{align*}
are the unique representatives of minimum length for the left and right cosets
of $W_J$ in $W$, respectively. Moreover, one has $\ell(xy)=\ell(x)+\ell(y)$
for all $x\in W^J$, $y\in W_J$ and all $x\in W_J$, $y\in{}^JW$.
The following key feature of inversion sets will be used repeatedly.
\begin{lemma}\label{L1.1}
For all $x,y\in W$ such that $\ell(xy)=\ell(x)+\ell(y)$, we have
\[
\Phi(xy)=\Phi(y)\dcup y^{-1}\Phi(x)\quad\text{\tup{disjoint union}}.
\]
\end{lemma}
\begin{proof}
It is clear that if $\beta\in\Phi^+$ is an inversion of $xy$,
then either $y\beta<0$, or $y\beta>0$ and $xy\beta<0$; \ie
$\beta\in\Phi(y)$ or $y\beta\in\Phi(x)$.
Thus $\Phi(xy)\subseteq\Phi(y)\cup y^{-1}\Phi(x)$. If this union failed
to be disjoint, or the inclusion was proper, then we would have
\[
\ell(xy)=|\Phi(xy)|<|\Phi(x)|+|\Phi(y)|=\ell(x)+\ell(y),
\]
a contradiction.
\end{proof}
\section{Orbits of odd roots}\label{S2}
%Recall that the even Weyl group $W(\Phi_0)$ permutes the odd roots;
%the structure of these orbits will play a key role in what follows.
%The $W$-orbit structure is well-known: two roots (whether or not they
%are odd) belong to the same $W$-orbit if and only if they have the
%same length and belong to the same irreducible component of $\Phi$.
%The latter means that there is a path between the two
%roots along edges that connect non-orthogonal roots.
%Moreover, at most two root lengths (long and short) occur within any
%irreducible component.
Recall that the even Weyl group $W(\Phi_0)$ permutes the odd roots;
the structure of these orbits will play a key role in what follows.
The $W$-orbit structure is well-known: two roots (whether or not they
are odd) belong to the same $W$-orbit if and only if they belong to the
same irreducible component of $\Phi$ and have the same length.
Moreover, at most two root lengths (long and short) occur within any
irreducible component.
\begin{lemma}\label{L2.1}
Let $\gamma_1,\gamma_2$ be distinct odd roots such
that $\br{\gamma_2\vph,\gamma_1^\vee}>0$.
\begin{enumerate}[(\alph*)]
\item\label{lemma2.1_a} If $\gamma_1$ and $\gamma_2$ have the same length,
then they belong to the same $W(\Phi_0)$-orbit.
\item\label{lemma2.1_b} If $\gamma_1$ is long and $\gamma_2$ is short,
then $\gamma_1=\gamma_2+\beta$, where $\beta\in\Phi_0$
and $\br{\beta,\gamma_1^\vee}=1$.
\end{enumerate}
\end{lemma}
\begin{proof}
In either case, we must have $\br{\gamma_2\vph,\gamma_1^\vee}=1$
(\eg see~\cite[VI.1.3]{B}).
Hence the reflection corresponding to $\gamma_1$ maps $\gamma_2$ to
$-\beta$ where $\beta=\gamma_1-\gamma_2$ is a necessarily even root of
the same length as $\gamma_2$. Furthermore,
$\br{\beta,\gamma_1^\vee}=\br{\gamma_1\vph-\gamma_2\vph,\gamma_1^\vee}=1$,
so the conclusions in~\ref{lemma2.1_b} hold for either case. In~\ref{lemma2.1_a}, the root $\beta$
must also have the same length as $\gamma_1$,
so $\br{\gamma_1,\beta^\vee}=\br{\beta,\gamma_1^\vee}=1$ and the
(even) reflection corresponding to $\beta$ maps $\gamma_1$ to
$\gamma_1-\beta=\gamma_2$.
\end{proof}
\begin{lemma}\label{L2.2}
Let $\orb_1,\orb_2$ be $W(\Phi_0)$-orbits generated by odd roots
of the same length. If $\orb_1-\orb_2\subset\Span\Phi_0$,
then $\orb_1=\pm\orb_2$ or $\orb_1,\orb_2\subset\Span\Phi_0$.
\end{lemma}
\begin{proof}
Let $V_0=\Span\Phi_0$ and assume $\orb_1\ne\pm\orb_2$.
If we select arbitrary representatives $\gamma_i$ from $\orb_i$,
then $\gamma_1\ne\pm\gamma_2$ and Lemma~\ref{L2.1}\ref{lemma2.1_a} (applied to
both $\gamma_2$ and $-\gamma_2$) shows that $\gamma_1$ and $\gamma_2$
must be orthogonal.
Hence $\Span\orb_1$ and $\Span\orb_2$ are orthogonal as well.
On the other hand, the $W(\Phi_0)$-orbit generated by any
vector $\gamma$ is spanned by $\gamma$ and the irreducible components
of $\Phi_0$ for which the orthogonal projection of $\gamma$ is nonzero.
In particular, the span of such an orbit contains the orthogonal projection
of $\gamma$ onto $V_0$, and hence the projection onto $V_0^\perp$ as well.
However, we are given that $\gamma_1-\gamma_2\in V_0$,
so $\gamma_1$ and $\gamma_2$ have the same orthogonal projection
onto $V_0^\perp$. Thus the projections cannot be orthogonal unless they
are~0; \ie $\gamma_1,\gamma_2\in V_0$.
\end{proof}
\begin{lemma}\label{L2.3}
If $\Phi$ and $\Phi_0$ have the same rank and $\Phi$ is irreducible,
then any two odd roots of the same length are in the same $W(\Phi_0)$-orbit.
\end{lemma}
\begin{proof}
Since $\Phi$ and $\Phi_0$ are assumed to have the same rank, every
root in $\Phi$ must be in the linear span of $\Phi_0$. Thus given any odd
root $\gamma$ that is $\Phi_0$-dominant, we may consider its
coordinates with respect to the simple roots of $\Phi_0$, say
$\gamma=\sum c_i\beta_i$. One knows that dominance forces $c_i\sge0$
and the support (\ie $\{i:c_i>0\}$) must coincide with the union of
one or more irreducible components of $\Phi_0$. (This amounts to the
fact that the inverse of any irreducible Cartan matrix has strictly
positive entries.)
Now suppose that $\gamma_1,\gamma_2,\dots$ are the $\Phi_0$-dominant
members of the distinct $W(\Phi_0)$-orbits of odd roots.
If $\gamma_1$ and $\gamma_2$ have the same length and overlapping support,
the dominance forces $\br{\gamma_2\vph,\gamma_1^\vee}>0$ and we contradict
Lemma~\ref{L2.1}\ref{lemma2.1_a}. In other words, distinct orbits generated by odd
roots of the same length have disjoint, orthogonal support.
Alternatively, suppose that $\gamma_1$ is long and $\gamma_2$ is short.
If they have overlapping support, we still
have $\br{\gamma_2\vph,\gamma_1^\vee}>0$ and Lemma~\ref{L2.1}\ref{lemma2.1_b} implies
that $\gamma_1=\gamma_2+\beta$ where $\beta$ is an even root such that
$\br{\beta,\gamma_1^\vee}=1$. The latter forces $\beta$ to be positive,
since $\gamma_1$ is dominant. Thus the support of $\gamma_1$ contains
the support of~$\gamma_2$. However, an even root such as $\beta$ must
have support that is confined to a single irreducible component
of $\Phi_0$, so if there is a gap between the supports of $\gamma_1$
and $\gamma_2$, it is this one irreducible component, and $\beta$ is
the orthogonal projection of $\gamma_1$ onto this component.
Returning to the hypothesis that $\gamma_1$ and $\gamma_2$ have
the same length, consider that our task is to show that this
leads to a contradiction. Since $\gamma_1$ and $\gamma_2$ must
have disjoint orthogonal support and $\Phi$ is irreducible, this is
possible only if another orbit representative, say $\gamma_3$, has
support overlapping $\gamma_1$ or $\gamma_2$ or both. As we have seen,
this is possible only if $\gamma_3$ is long and $\gamma_1$ and $\gamma_2$
are short. Since this eliminates the possibility of $\gamma_1$ and
$\gamma_2$ both being long, this means that $\gamma_3$ generates the
only long odd orbit, and irreducibility forces its support to include
the supports of both $\gamma_1$ and~$\gamma_2$. Furthermore, since the
supports of $\gamma_1$ and $\gamma_2$ are disjoint, the inclusions must
be proper, and as we have shown, each ``support gap'' must be a single
irreducible component of $\Phi_0$. This forces the support of $\gamma_3$
to consist of exactly two irreducible components, and the
orthogonal projections of $\gamma_3$ onto these two components must be
even roots, contradicting the fact that $\gamma_3$ is odd.
\end{proof}
\begin{prop}\label{P2.4}
{\hskip 10pt} %kluge so that enumerate starts on a fresh line
\begin{enumerate}[(\alph*)]
\item\label{prop2.4_a} Every $W(\Phi_0)$-orbit of odd roots includes a
simple or a negative simple root.
\item\label{prop2.4_b} If the nodes of the Dynkin diagram of $\Phi$ are 2-colored
so that each edge is incident to a black vertex and a white vertex,
then the black simple roots and the negatives of the white simple roots
in the same $W$-orbit belong to the same $W(\Phi_0)$-orbit.
\item\label{prop2.4_c} If $\Phi$ is irreducible of rank $n$, then either
\begin{enumerate}[(i)]
\item\label{C1} $\Phi_0$ has rank $n-1$, all simple roots of $\Phi_0$
have height~2 in $\Phi$, and every opposite pair of odd
roots $\pm\gamma$ belong to distinct $W(\Phi_0)$-orbits, or
\item\label{C2} $\Phi_0$ has rank $n$, one simple root of $\,\Phi_0$
has height~4 in $\Phi$, and all odd roots of the same length are in
the same $W(\Phi_0)$-orbit.
\end{enumerate}
Furthermore, case~\ref{C1} occurs if and only if $\Phi\isom A_n$ or $C_n$.
\end{enumerate}
\end{prop}
\begin{proof}
Choose a black-white coloring of the Dynkin diagram as in~\ref{prop2.4_b},
and assume that $\Phi$ is irreducible of rank $n$. Since the diagram is
a tree, there are exactly $n-1$ roots $\beta_1,\dots,\beta_{n-1}$ of
height~2 in $\Phi$; each is of the form $\alpha_i+\alpha_j$ where $i$
is a black node and $j$ is an adjacent white node. If the edge between
$i$ and $j$ is simple (\ie the parabolic subsystem generated
by $\alpha_i$ and $\alpha_j$ is isomorphic to $A_2$), then the
reflection corresponding to $\alpha_i+\alpha_j$ interchanges
$\alpha_i$ with $-\alpha_j$ and $-\alpha_i$ with $\alpha_j$.
It follows that for any nodes $i$ and $j$ connected by a path of
simple edges, the $W(\Phi_0)$-orbit of $\alpha_i$ contains
$\alpha_j$ if $i$ and $j$ have the same color, and the orbit contains
$-\alpha_j$ if $i$ and $j$ have different colors. On the other hand,
it is well-known that two simple (or negative simple) roots belong to
the same $W$-orbit if and only if they are in the same connected
component of the Dynkin diagram after all non-simple edges have been
deleted. Thus~\ref{prop2.4_b} follows.
Now consider that as a member of the even root system $\Phi_0$,
each root $\beta_i$ is clearly indecomposable (\ie
not expressible as a sum of two positive even roots), and thus is
necessarily a simple root of $\Phi_0$. On the other hand, as a root
subsystem of a rank $n$ root system, $\Phi_0$ necessarily has rank $\sle n$,
and thus it can have at most one additional indecomposable root.
\begin{enonce*}[remark]{Case~1: $\Phi_0$ has rank $n-1$}
%\mycase{1}{$\Phi_0$ has rank $n-1$}
Define an integral co-weight $\omega$ for $\Phi$ by setting
$\br{\alpha_i,\omega}=1$ (if $i$ is black) and $-1$ (if $i$ is white).
We therefore have $\br{\beta_i,\omega}=0$ for all $i$ and hence
$W(\Phi_0)$ fixes $\omega$, since $\beta_1,\dots,\beta_{n-1}$ forms
a base for $\Phi_0$ in this case. Thus $\br{\gamma,\omega}$ is a
constant as $\gamma$ varies over any $W(\Phi_0)$-orbit in $\Phi$.
For odd $\gamma$, this constant is the sum of an odd number of $\pm1$'s,
and hence nonzero. Consequently, the $W(\Phi_0)$-orbits of $\gamma$
and $-\gamma$ are distinct for every odd root $\gamma$, proving~\ref{C1}.
We claim that $\br{\gamma,\omega}$ is limited to the values $0,\pm1$
for all $\gamma\in\Phi$ (\ie $\omega$ is minuscule).
Indeed, even roots belong to the hyperplane orthogonal to $\omega$,
and if there were an odd root $\gamma$ such that $\br{\gamma,\omega}\sge3$,
then there would be an even root of the form $\beta=\gamma\pm\alpha_i$
for some~$i$. In that case, we would have $\br{\beta,\omega}\sge2$,
a contradiction.
Thus every $W(\Phi_0)$-orbit of odd roots belongs to
the (affine) hyperplane $\br{\,\cdot,\omega}=1$ or its negative,
and both hyperplanes contain at least one simple root or negative
simple root from each $W$-orbit. To prove~\ref{prop2.4_a} for this case,
it therefore suffices to show that all roots of a given length
within one of these hyperplanes belong to the same $W(\Phi_0)$-orbit.
If there were two such orbits, say $\orb_1$ and $\orb_2$,
then we cannot have $\orb_1=\pm\orb_2$ (the orbit opposite to
$\orb_1$ is in the opposite hyperplane), and yet we do have
$\br{\gamma_1-\gamma_2,\omega}=0$ for all $\gamma_i\in\orb_i$,
so Lemma~\ref{L2.2} implies $\orb_i\subset\Span\Phi_0$, a contradiction.
To see that $\Phi=A_n$ and $C_n$ belong to Case~1,
it suffices to show that the even roots are confined
to a central hyperplane in these cases.
For this, let $\eps_1,\dots,\eps_n$ be the standard orthornomal basis
for $\R^n$, and realize $C_n$ using the simple
roots $\alpha_1=2\eps_1$ and $\alpha_i=\eps_i-\eps_{i-1}$ ($1*\ell(w)$ in this case. The remaining possibility is that
the first and last terms are deleted; \ie $s_iws_j=w$, or equivalently,
$w^{-1}\alpha_i=\pm\alpha_j$. However, this contradicts the fact that
$w^{-1}\alpha_i$ is even.
For the second claim, consider that for all indices $k\in I$, we have
\[
(s_iw)^{-1}\alpha_k = w^{-1}s_i\alpha_k=w^{-1}\alpha_k-aw^{-1}\alpha_i
\]
where $a$ is the integer $\br{\alpha_k\vph,\alpha_i^\vee}$.
Thus if $i$ is the \type{} of $w$, it follows
that $(s_iw)^{-1}\alpha_k$ is even if and only if $w^{-1}\alpha_k$ is even.
In particular, $i$ must be the \type{} of $s_iw$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{P4.1}]
Lemma~\ref{L4.2} implies that the map $w\mapsto s_iw$ (with $i$ being the
\type{} of $w$) defines a sign-reversing involution on the members
of $W^J$ not in~$N(\Phi_0)$. Exchanging $w$ with $s_iw$ if necessary,
we may assume that $\ell(s_iw)>\ell(w)$. In that case,
\[
\Phi(s_iw)=\Phi(w)\dcup\{w^{-1}\alpha_i\}.
\]
Thus the inversion sets of $w$ and $s_iw$ are identical aside from
the presence of a single even root, so their contributions to the
sum cancel out, leaving only the contributions from $N(\Phi_0)\cap W^J$.
\end{proof}
\begin{rema}\label{R4.3}
We have previously noted that the longest element $w_0$
is the unique member of $W$ that inverts all odd roots.
Since $\Phi(w)\subseteq\Phi^+\setminus\Phi_J$
for all $w\in W^J$, we thus have the degree bound
\[
\deg F_J(\Phi;q)\ \sle\ |\Phi_1^+\setminus\Phi_J|
\ =\ \deg F(\Phi;q)-\deg F(\Phi_J;q).
\]
Furthermore, there may be multiple members of $W^J$ that invert all of
$\Phi_1^+\setminus\Phi_J$, resulting in the possibility of cancellation
and a degree that does not achieve this bound.
For example, when $(\Phi,\Phi_J)\isom(D_4,A_1^{\oplus 3})$,
there are 5 odd positive roots not in $\Phi_J$, but it is easily
checked that $F_J(D_4;q)=(1-q^2)(1+3q^2)$. This also shows that
the irreducible factors of $F_J(\Phi;q)$ need not be cyclotomic.
\end{rema}
From now on, we will focus on three specializations
of the variables $q_\beta$; namely,
\begin{enumerate}[(\roman*)]
\item\label{Sect4_i} The univariate grading; \ie $q_\beta$ is independent of $\beta$.
\item\label{Sect4_ii} The coarse grading; \ie $q_\beta$ depends only on the
$W$-orbit of $\beta$.
\item\label{Sect4_iii} The fine grading; \ie $q_\beta$ depends only on the
$W(\Phi_0)$-orbit of $\beta$.
\end{enumerate}
By Proposition~\ref{P2.4}, we know that in most irreducible cases,
each $W$-orbit of roots contains only one $W(\Phi_0)$-orbit of odd
roots, and hence, the coarse and fine gradings coincide.
The exceptions are $A_n$ (which has two $W(\Phi_0)$-orbits
of odd roots) and $C_n$ (which has four; two long and two short).
Among these exceptions, $A_1$ and $C_2$ are somewhat
degenerate in having an odd $W(\Phi_0)$-orbit that consists only
of negative roots. (Recall Remark~\ref{R2.5}\ref{rema2.5_a}.) The corresponding
series $F(\Phi;\bq)$ under the fine grading therefore depends on
only 1 or 3 variables, rather than 2 or 4, respectively.
\section{Perfect cosets and divisibility}
Fix $J\subseteq I$ and consider a minimal right coset representative
$y\in {}^JW$. Recall that this means $\ell(s_jy)>\ell(y)$ for all $j\in J$,
or equivalently $y^{-1}\alpha_j>0$ for such $j$.
We say that $y$ is \emph{perfect} if, in addition, $y^{-1}\alpha_j$ is
odd for all $j\in J$. (Thus in the terminology of the previous section,
the perfect members of ${}^JW$ have no faults in $J$.) We let
\[
{}^JW_1:=\{y\in W:y^{-1}\alpha_j\in\Phi_1^+\text{ for all $j\in J$}\}
\]
denote the set of minimal coset representatives that are perfect, and
\[
G_J(\Phi;\bq):=\sum_{y\in{}^JW_1}\sgn(y)
\prod_{\beta\in\Phi_1(y)}q_\beta
\]
the corresponding sign-twisted series.
\begin{theorem}\label{T5.1}
For all $J\subseteq I$, we have
\[
F(\Phi;\bq)=\sum_{y\in{}^JW_1}
\Bigl(\sgn(y)\!\prod_{\beta\in\Phi_1(y)}q_\beta\Bigr)
F(\Phi_J;y^{-1}\bq)
\]
under the fine\footnotemark grading for $\Phi$, where $y^{-1}\bq$ denotes
the change of variable $q_\beta\to q_{y^{-1}\beta}$ for
odd $\beta\in\Phi_J^+$. In particular,
if $F(\Phi_J;y^{-1}\bq)=F(\Phi_J;\bq)$ for all $y\in{}^JW_1$, then
\[
F(\Phi;\bq)=G_J(\Phi;\bq)F(\Phi_J;\bq)
\]
under the fine grading for $\Phi$.
\end{theorem}
\footnotetext{Note that the fine grading for $\Phi$ may force the
identification of some variables appearing in $F(\Phi_J;\bq)$ that
would otherwise be distinct in the fine grading for $\Phi_J$.}
Note that if $y$ is perfect and $\beta\in\Phi_J$ is odd,
then $y^{-1}\beta$ is clearly another odd root in the same $W$-orbit.
Hence, for the coarse specialization of the variables $q_\beta$, we have
\begin{coro}\label{C5.2}
In all cases, $F(\Phi;\bq)=G_J(\Phi;\bq)F(\Phi_J;\bq)$
under the coarse grading.
\end{coro}
Thus it is clear a priori (\ie without use of the classification
of finite root systems, or explicit evaluation) that $F(\Phi;\bq)$
has many divisors under the coarse grading, and therefore is likely to have
a nice factorization.
For the finer gradings available with $A_n$ and $C_n$, it need not
be the case that $F(\Phi;\bq)$ is divisible by $F(\Phi_J;\bq)$.
For example, a posteriori (see Theorem~\ref{T6.1} below) $F(A_n;\bq)$
is divisible by $F(A_k;\bq)$ under the fine grading only if $k$ is even,
or $n$ is odd and $k+1$ divides $n+1$. On the other hand, the following
result shows that $A_n$ and $C_n$ have parabolic subsystems of co-rank
at most~2 where divisibility is clear a priori.
\begin{coro}\label{C5.3}
If $\Phi_J\isom A_{2n}$, then $F(\Phi;\bq)=G_J(\Phi;\bq)F(\Phi_J;\bq)$
in the fine grading.
\end{coro}
\begin{proof}
For perfect $y\in{}^JW$, the map $\beta\mapsto y^{-1}\beta$ acts as
a parity-preserving isomorphism between two copies of $A_{2n}$
embedded in $\Phi$ (namely, $\Phi_J$ and $y^{-1}\Phi_J$). Although
it is possible that $\beta$ and $y^{-1}\beta$ are in distinct
$W(\Phi_0)$-orbits for odd $\beta\in\Phi_J$,
there is a diagram automorphism of $A_{2n}$ that interchanges the black
and white vertices of the 2-colored Dynkin diagram.
By Proposition~\ref{P2.4}, it follows that this automorphism interchanges
the two orbits of odd roots, and thus $F(A_{2n};\bq)$ is invariant under
interchanging the variables $q_\beta$ in these two orbits. Thus either way,
we have $F(\Phi_J;y^{-1}\bq)=F(\Phi_J;\bq)$ for all $y$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T5.1}]
Suppose that $y\in{}^JW$ is not perfect, so that there is some $j\in J$
such that $\beta=y^{-1}\alpha_j$ is an even (necessarily positive) root.
Since $\ell(xy)=\ell(x)+\ell(y)$ for all $x\in W_J$, Lemma~\ref{L1.1} implies
\begin{equation}\label{eq5.1}
\Phi(xy)=\Phi(y)\dcup y^{-1}\Phi(x).
\end{equation}
Replacing $x$ with $xs_j$ if necessary so that
$\ell(xs_j)>\ell(x)$, we also have
\[
\Phi(xs_jy)\,=\,\Phi(y)\dcup\{y^{-1}\alpha_j\}\dcup y^{-1}s_j\Phi(x)
\,=\,\Phi(y)\dcup\{\beta\}\dcup ty^{-1}\Phi(x),
\]
where $t=y^{-1}s_jy$ is the reflection corresponding to $\beta$.
Since $\beta$ is even, $t$ belongs to $W(\Phi_0)$, so for each
$W(\Phi_0)$-orbit of odd roots, the number of such roots in
$y^{-1}\Phi(x)$ and $ty^{-1}\Phi(x)$ is the same. It follows that
the contributions of $xy$ and $xs_jy$ to the series $F(\Phi;\bq)$
cancel out, and thus the net contribution of the coset $W_Jy$ is zero.
For perfect $y$, we still have~\eqref{eq5.1}, whence
\[
\sum_{x\in W_J}\sgn(xy)\prod_{\beta\in\Phi_1(xy)}q_\beta
\ =\ \sgn(y)F(\Phi_J;y^{-1}\bq)\prod_{\beta\in\Phi_1(y)}q_\beta
\]
and the result follows.
\end{proof}
Let $\rho^\vee$ be the unique co-weight such that
$\br{\alpha_i,\rho^\vee}=1$ for all $i\in I$.
(It is well-known that $2\rho^\vee$ is also the sum of
all positive co-roots.)
We have found that the most convenient way to explicitly compute
$G_J(\Phi;\bq)$ under the coarse grading is as follows.
\begin{prop}\label{P5.4}
Let $\Phi=\orb_1\dcup\orb_2\dcup\cdots$ be the partition of $\Phi$
into $W$-orbits, and choose variables $q_i$ so that $q_i=q_\beta$ for
all $\beta\in\orb_i\cap\Phi_1^+$.
\begin{enumerate}[(\alph*)]
\item\label{prop5.4_a} The map $w\mapsto w\rho^\vee$ defines a bijection
between the set ${}^JW_1$ of perfect coset representatives and
\[
(W\rho^\vee)_J:=\{\theta\in W\rho^\vee:
\br{\alpha_j,\theta}\text{ is odd and positive for all $j\in J$}\}.
\]
\item\label{prop5.4_b} Under the coarse grading, we have
\[
G_J(\Phi;\bq)=\sum_{\theta\in(W\rho^\vee)_J}(-1)^{N(\theta)}_{\vphantom{1}}
q_1^{N_1(\theta)}q_2^{N_2(\theta)}\cdots,
\]
where $N(\theta):=|\{\beta\in\Phi^+:\br{\beta,\theta}<0\}|$ and
\[
N_i(\theta):=|\{\beta\in\orb_i\cap\Phi^+:\br{\beta,\theta}
\text{ is odd and negative}\}|.
\]
\end{enumerate}
\end{prop}
Note that the roots $\beta$ contributing to $N_i(\theta)$ may be
odd or even.
\begin{proof}
\ref{prop5.4_a} Recall that $w\in{}^JW_1$ if and only if $w^{-1}\alpha_j$ is
odd and positive for all $j\in J$. On the other hand,
the height of a root $\beta$ is $\br{\beta,\rho^\vee}$,
so $\br{\alpha_j,w\rho^\vee}=\br{w^{-1}\alpha_j,\rho^\vee}$
is odd and positive if and only if $w^{-1}\alpha_j$ is odd and positive.
Since $\rho^\vee$ is regular, the map $w\mapsto w\rho^\vee$ is
injective, and the claim follows.
\ref{prop5.4_b} Given $\theta=w\rho^\vee$ and a root $\beta$,
we have $\br{\beta,\theta}=\br{w^{-1}\beta,\rho^\vee}$, so
\[
\Phi(w)=\{-w^{-1}\beta:\beta\in\Phi^+,\,\br{\beta,\theta}<0\}
\]
and $N(\theta)$ is the total number of inversions for $w$;
\ie $N(\theta)=\ell(w)$. Since $\beta$ and $-w^{-1}\beta$ clearly
belong to the same $W$-orbit, and $-w^{-1}\beta$ is odd if and only if
$\br{\beta,\theta}$ is odd, we obtain that $N_i(\theta)$ is the number
of odd inversions for $w$ in the $W$-orbit $\orb_i$.
\end{proof}
\begin{remas}\label{R5.5}\hskip0pt
\begin{enumerate}[(\alph*)]
\item\label{rema5.5_a} %(a)
Determining the points in the $W$-orbit of $\rho^\vee$ that correspond
to the perfect cosets for $W_J\backslash W$ does not require generating
the entire orbit. Indeed, we claim that it is possible to efficiently
generate the much smaller set ${}^JW\rho^\vee$ consisting
of members of the orbit that are $\Phi_J$-dominant, and select from
these the ones that are perfect. Indeed, we claim that ${}^JW\rho^\vee$
is the smallest set $X$ of co-weights that
\begin{enumerate}[(i)]
\item\label{rema5.5_a_i} contains $\rho^\vee$, and
\item\label{rema5.5_a_ii} contains $\theta-\beta^\vee$ if $\theta\in X$,
$\beta\in\Phi^+\setminus\Phi_J$, $\br{\beta,\theta}=1$,
and $\theta-\beta^\vee$ is $\Phi_J$-dominant.
\end{enumerate}
Given that $\theta$ and $\beta$ satisfy the hypotheses for~\ref{rema5.5_a_ii},
it follows that $\theta-\beta^\vee$ is the image of $\theta$ under
the reflection corresponding to $\beta$. Thus the fact that ${}^JW\rho^\vee$
contains $X$ follows by induction with respect to height. Conversely,
given $w\in{}^JW$, either $w=1$ and $w\rho^\vee=\rho^\vee$, or we
have $\ell(ws_i)<\ell(w)$ and $\beta=-w\alpha_i>0$ for some $i$.
We must also have $\beta\notin\Phi_J$, otherwise $ws_iw^{-1}\in W_J$
and $\ell(ws_i)=\ell(ws_iw^{-1}\cdot w)>\ell(w)$, a contradiction.
Furthermore, it must be the case that $ws_i\in{}^JW$. Otherwise,
for some $j\in J$ we would have
\[
\ell(s_jws_i)<\ell(ws_i)<\ell(w)<\ell(s_jw),
\]
contradicting the fact that $s_jws_i$ and $s_jw$ differ in
length by~1. Thus by induction with respect to length,
we may assume $ws_i\rho^\vee\in X$.
However, $ws_i\rho^\vee=w(\rho^\vee-\alpha_i^\vee)=w\rho^\vee+\beta^\vee$
and $\br{\beta,ws_i\rho^\vee}=\br{\alpha_i,\rho^\vee}=1$,
whence $ws_i\rho^\vee-\beta^\vee=w\rho^\vee\in X$ by~\ref{rema5.5_a_ii}.
\item\label{rema5.5_b} %(b)
For example, if $(\Phi,\Phi_J)\isom(E_8,E_7)$, then there are
only 240 cosets for $W_J\backslash W$ (one for each root), and it is
a fast calculation to generate the 240 vectors in ${}^JW\rho^\vee$
by the above algorithm.
It turns out that 32 of the 240 vectors correspond to cosets whose
minimal representatives are perfect.
In Table~\ref{T2}, we list the co-weight coordinates of these 32 vectors;
\ie for each $w\in{}^JW_1$, we list $\br{w\rho^\vee,\alpha_i}$
for $1\sle i\sle 8$, along with $\ell(w)=N(w\rho^\vee)$
and $\ell_1(w)=N_1(w\rho^\vee)$ (cf.~the formulas in
Proposition~\ref{P5.4}\ref{prop5.4_b}).
Note that we have indexed the simple roots
of $E_8$ as in this Dynkin diagram:
\[
1{-}3{-}\vbox{\offinterlineskip\hbox{$2$}\vglue 2pt\hbox{\hskip 2.5pt\vrule height7pt}\vglue 2pt\hbox{$4$}}{-}5{-}6{-}7{-}8.
\]
The fact that the first 7 coordinates are odd and positive in each
of the 32 cases confirms that these correspond to minimal coset
representatives that are perfect.
%(b) For example, if $(\Phi,\Phi_J)\isom(E_8,E_7)$, then there are
%only 240 cosets for $W_J\backslash W$ (one for each root), and it is
%a fast calculation to generate the 240 members of ${}^JW\rho^\vee$
%by the above algorithm. One discovers that 32 of the 240 cosets
%are perfect.
\end{enumerate}
\end{remas}
\begin{table}[htb]\centering
%\begin{center}
%\def\spacing{\omit\vrule height2pt&&&&&&&&&&&&&\cr}
%\mbox{
%\vbox{\offinterlineskip
%\halign{\vrule\strut#&\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ %\hfil$#$\ &\ \hfil$#$\ &\vrule#&\ \hfil$#$\hfil\ &\vrule#&\ \hfil$#$\hfil\ &\vrule#\cr
%\noalign{\hrule}\spacing\spacing
%&\rlap{$\br{w\rho^\vee,\alpha_i},i=1,\dots,8$}&&&&&&&&&\ell(w)&&\ell_1(w)&\cr
%\spacing\spacing\noalign{\hrule}\spacing\spacing
\renewcommand{\arraystretch}{1.1}
\renewcommand{\tabcolsep}{3.5pt}
\begin{tabular}{|cccccccr|c|c|}
\hline
\multicolumn{8}{|c|}{$\br{w\rho^\vee,\alpha_i},i=1,\dots,8$}&$\ell(w)$&$\ell_1(w)$\\
\hline
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0& 0\\
1 & 1 & 3 & 1 & 1 & 1 & 1 &$-6$ & 6& 3\\
1 & 1 & 1 & 3 & 1 & 1 & 1 &$-10$ & 10& 5\\
7 & 1 & 1 & 1 & 1 & 1 & 1 &$-11$ & 12& 6\\
1 & 1 & 3 & 1 & 1 & 3 & 1 &$-14$ & 14& 7\\
1 & 1 & 5 & 1 & 1 & 1 & 1 &$-16$ & 18& 9\\
3 & 1 & 1 & 1 & 1 & 5 & 1 &$-19$ & 20& 10\\
1 & 1 & 1 & 3 & 1 & 3 & 1 &$-21$ & 24& 12\\
1 & 1 & 1 & 3 & 1 & 1 & 5 &$-25$ & 28& 14\\
1 & 1 & 1 & 1 & 5 & 1 & 1 &$-23$ & 28& 14\\
1 & 5 & 1 & 1 & 1 & 3 & 1 &$-23$ & 28& 14\\
3 & 1 & 1 & 1 & 1 & 1 & 9 &$-27$ & 28& 14\\
1 & 1 & 3 & 1 & 1 & 1 & 7 &$-26$ & 28& 15\\
1 & 3 & 1 & 1 & 3 & 1 & 3 &$-24$ & 28& 15\\
1 & 1 & 1 & 1 & 1 & 1 &11 &$-28$ & 28& 15\\
1 & 7 & 1 & 1 & 1 & 1 & 1 &$-22$ & 28& 15\\
%\noalign{\hrule}}}
\hline
\end{tabular}
\qquad
%\vbox{\offinterlineskip
%\halign{\vrule\strut#&\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ \hfil$#$\ &\ %\hfil$#$\ &\ \hfil$#$\ &\vrule#&\ \hfil$#$\hfil\ &\vrule#&\ \hfil$#$\hfil\ &\vrule#\cr
%\noalign{\hrule}\spacing\spacing
%&\rlap{$\br{w\rho^\vee,\alpha_i},i=1,\dots,8$}&&&&&&&&&\ell(w)&&\ell_1(w)&\cr
%\spacing\spacing\noalign{\hrule}\spacing\spacing
\begin{tabular}{|cccccccr|c|c|}
\hline
\multicolumn{8}{|c|}{$\br{w\rho^\vee,\alpha_i},i=1,\dots,8$}&$\ell(w)$&$\ell_1(w)$\\
\hline
1 & 1 & 3 & 1 & 1 & 1 & 7 &$-27$ & 29& 14\\
1 & 3 & 1 & 1 & 3 & 1 & 3 &$-25$ & 29& 14\\
1 & 1 & 1 & 1 & 1 & 1 &11 &$-29$ & 29& 14\\
1 & 7 & 1 & 1 & 1 & 1 & 1 &$-23$ & 29& 14\\
1 & 1 & 1 & 3 & 1 & 1 & 5 &$-26$ & 29& 15\\
1 & 1 & 1 & 1 & 5 & 1 & 1 &$-24$ & 29& 15\\
1 & 5 & 1 & 1 & 1 & 3 & 1 &$-24$ & 29& 15\\
3 & 1 & 1 & 1 & 1 & 1 & 9 &$-28$ & 29& 15\\
1 & 1 & 1 & 3 & 1 & 3 & 1 &$-26$ & 33& 17\\
3 & 1 & 1 & 1 & 1 & 5 & 1 &$-28$ & 37& 19\\
1 & 1 & 5 & 1 & 1 & 1 & 1 &$-27$ & 39& 20\\
1 & 1 & 3 & 1 & 1 & 3 & 1 &$-29$ & 43& 22\\
7 & 1 & 1 & 1 & 1 & 1 & 1 &$-28$ & 45& 23\\
1 & 1 & 1 & 3 & 1 & 1 & 1 &$-29$ & 47& 24\\
1 & 1 & 3 & 1 & 1 & 1 & 1 &$-29$ & 51& 26\\
1 & 1 & 1 & 1 & 1 & 1 & 1 &$-28$ & 57& 29\\
\hline
\end{tabular}
%\noalign{\hrule}}}
%}
%\end{center}
%\smallskip
\vspace{2mm} %%%%L'"\abovecaptionskip" ne fonctionne pas. HervÃ©
\caption{Perfect coset representatives for $W(E_7)\backslash W(E_8)$.}
\label{T2}
\end{table}
\section{Factoring the fine graded series}
Following the road map laid out in the previous section, we are now in
a position to explicitly evaluate the fine or coarse graded series for
each irreducible root system. To set a convention for naming the
variables for the coarse grading, let
\begin{itemize}
\item $q_\beta=q$ if $\Phi$ is of type $A$, $D$, or $E$,
\item $q_\beta=p$ (if $\beta$ is short) or $q$ (if $\beta$ is long)
for $B_n$, $F_4$, and $G_2$, and
\item $q_\beta=q$ (if $\beta$ is short) or $r$ (if $\beta$ is long)
for $C_n$
\end{itemize}
for all odd positive roots $\beta$. In this way, the variable attached
to an odd root in $A_{n-1}$ is unchanged when $A_{n-1}$ is embedded as a
parabolic subsystem of $B_n$ or $C_n$.
In types $A$ and $C$, where each $W$-orbit of roots contains two
$W(\Phi_0)$-orbits of odd roots (Proposition~\ref{P2.4}),
we will use the variables $q_\pm$ and $r_\pm$ for the fine grading.
For $C_n$, the $W(\Phi_0)$-orbit containing the long simple root
will be weighted $r_+$, and the short simple root that is adjacent
to it in the Dynkin diagram will be weighted $q_+$. For $A_{2n-1}$,
the odd orbit containing the simple roots at both ends of
the Dynkin diagram will be weighted $q_+$. For $A_{2n}$, we have
noted previously that the diagram automorphism interchanges the two
$W(\Phi_0)$-orbits of odd roots, so the fine series is unaffected by
the choice of which one is weighted $q_+$; either choice will suffice
despite being non-canonical.
For the fine series, it will be convenient to let
$\bq\to\bar\bq$ denote the transformation of the variables
in which the value of $q_\beta$ on a $W(\Phi_0)$-orbit $\orb$ is
replaced by its value on the opposite orbit $-\orb$. For $A_n$
and $C_n$, this amounts to interchanging $q_+$ with $q_-$ and $r_+$
with~$r_-$. In particular, as noted above,
\begin{equation}\label{eq6.1}
F(A_{2n};\bq)=F(A_{2n};\bar\bq).
\end{equation}
Of course in the other irreducible cases, we know that every
$W(\Phi_0)$-orbit is centrally symmetric ($\orb=-\orb$)
and this transformation has no effect (Proposition~\ref{P2.4}).
In the following, we let $(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$.
\begin{theorem}\label{T6.1}
For the finely graded series, we have
\begin{align*}
F(A_{2n-1};\bq)&=(1-q_+^n)(q_+q_-;q_+q_-)_{n-1},\\
F(A_{2n};\bq) &=(q_+q_-;q_+q_-)_n,\\
F(B_{2n};\bq) &=(q^2;q^2)_n(p;q^2)_n,\\
F(B_{2n+1};\bq)&=(q^2;q^2)_n(p;q^2)_{n+1},\\
F(C_{2n};\bq) &=(1-q_+^n)(q_+q_-;q_+q_-)_{n-1}(r_+r_-;q_+q_-)_n,\\
F(C_{2n+1};\bq)&=(1-r_+\vpo
q_-^n)(q_+q_-;q_+q_-)_n(r_+r_-;q_+q_-)_n,\\
F(D_{2n};\bq) &=(1-q^n)^2(q^2;q^2)^2_{n-1},\\
F(D_{2n+1};\bq)&=(q^2;q^2)^2_n,\\
F(E_6;\bq)&=(q^2;q^2)_4,\\
F(E_7;\bq)&=(q^2;q)_7,\\
F(E_8;\bq)&=(1-q^8)(q^2;q^2)_7,\\
F(F_4;\bq)&=(1-p^2)(1-q^2)(1-p^2q^2)(1-p^2q^4),\\
F(G_2;\bq)&=(1-p^2)(1-q^2).
\end{align*}
\end{theorem}
\begin{proof}
Proceed by induction with respect to rank.
In most cases, the claimed identity may be proved by using
Proposition~\ref{P5.4} to compute the series $G_J(\Phi;\bq)$ for a suitable
parabolic subsystem $\Phi_J$ of co-rank~1, and then appealing to
Corollary~\ref{C5.2} and the induction hypothesis. However, for $A_n$
and~$C_n$, our induction will directly rely on Theorem~\ref{T5.1}.
\begin{enonce*}[remark]{Case~1: $\Phi=A_n$}
%\mycase{1}{$\Phi=A_n$}
With coordinates chosen so that $\alpha_i=\eps_{i+1}-\eps_i$ ($1\sle i\sle n$),
the co-weight $\rho^\vee$ is the orthogonal projection of
$(0,1,\dots,n)$ into the hyperplane normal to $(1,\dots,1)$.
The $W$-orbit of $\rho^\vee$ may thus be identified with the
permutations of $\{0,\dots,n\}$. If $\Phi_J\isom A_{n-1}$ omits the first
simple root, then the members of this orbit that are $\Phi_J$-dominant
are of the form $(a_0,\dots,a_n)$ with $a_1<\dots0$ and $y\beta<0$) are also
equivalent to having $-y\beta$ be an inversion for $y^{-1}$. Moreover,
$\beta$ is odd if and only if $-y\beta$ is odd, and the two roots
necessarily belong to opposite $W(\Phi_0)$-orbits (since $y\in W(\Phi_0)$).
Thus the above sum is identical to the result of substituting
$\bq\to\bar\bq$ in Proposition~\ref{P4.1}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T7.1}]
By definition, we have
\begin{equation}\label{eq7.2}
F(\Phi;\bq)=\sum_{x\in W^{\!J}}\,\sum_{y\in W_J}
\sgn(xy)\prod_{\beta\in\Phi_1\!(xy)}q_\beta.
\end{equation}
By Proposition~\ref{P4.1}, this sum may be restricted to those
pairs $(x,y)$ with $xy\in N(\Phi_0)$.
Applying Lemma~\ref{L4.2} to the pair $(\Phi_J,\Phi_\mtset)$, we know
that the map $y\mapsto s_jy$ (with $j$ being the \type{} of $y$ relative
to $\Phi_J$) is a sign-reversing involution on the members of $W_J$ not
in $N(\Phi_{J,0})$. Replacing $y$ with $s_jy$ if necessary, we may
assume $\ell(s_jy)>\ell(y)$. Since $\ell(xy)=\ell(x)+\ell(y)$ for
all $x\in W^J$ and $y\in W_J$, Lemma~\ref{L1.1} implies
\begin{align*}
\Phi(xy)&=\Phi(y)\dcup y^{-1}\Phi(x),\\
\Phi(xs_jy)&=\Phi(y)\dcup\{y^{-1}\alpha_j\}\dcup y^{-1}s_j\Phi(x)\\
&=\Phi(y)\dcup\{y^{-1}\alpha_j\}\dcup ty^{-1}\Phi(x),
\end{align*}
where $t=y^{-1}s_jy$ is the reflection corresponding to the root
$y^{-1}\alpha_j\in\Phi_J$. However, $j$ is the \type{} of $y$ relative to
$\Phi_J$, so this is an even root and $t$ belongs to
$W(\Phi_{J,0})\subset W(\Phi_0)$. It follows that for each
$W(\Phi_0)$-orbit of odd roots, the number of such
roots is the same in $y^{-1}\Phi(x)$ and $ty^{-1}\Phi(x)$.
Furthermore, $N(\Phi_0)$ includes $xs_jy=xyt$ if and only if it also
includes $xy$. Thus the contributions of $xy$ and $xs_jy$ to~\eqref{eq7.2}
cancel out under the fine grading, and we can require $y\in N(\Phi_{J,0})$
in addition to the restriction $xy\in N(\Phi_0)$.
Given the hypothesis $N(\Phi_{J,0})\subset N(\Phi_0)$,
our restrictions on $x$ and $y$ can be separated into $x\in N(\Phi_0)$,
$y\in N(\Phi_{J,0})$. Since
$\Phi_1(xy)=\Phi_1(y)\dcup y^{-1}\Phi_1(x)$, we obtain
\begin{align*}
F(\Phi;\bq)&=\sum_{x\in N(\Phi_0)\cap W^J}
\ \sum_{y\in N(\Phi_{J,0})}
\sgn(xy)\prod_{\beta\in\Phi_1\!(xy)}q_\beta\\
&=\sum_{y\in N(\Phi_{J,0})}\Bigl(\sgn(y)\!
\prod_{\beta\in\Phi_1\!(y)}q_\beta\Bigr)
F_J(\Phi;y^{-1}\bq)
\end{align*}
under the fine grading. It therefore suffices to show that
\begin{equation}\label{eq7.3}
F_J(\Phi;\bq)=F_J(\Phi;y^{-1}\bq)
\ \text{for all $y\in N(\Phi_{J,0})$}.
\end{equation}
Indeed, in that case we could factor $F_J(\Phi;\bq)$ out of the
above sum, and Proposition~\ref{P4.1} would clearly identify the remaining
sum as $F(\Phi_J;\bq)$.
This invariance is immediate whenever $y$ belongs to $W(\Phi_{J,0})$
(a subgroup of $W(\Phi_0)$), so we may assume
$y\in\Gamma(\Phi_{J,0})\subseteq\Gamma(\Phi_0)$ and that $y$ acts
nontrivially on the $W(\Phi_0)$-orbits of odd roots.
We may also assume $\Phi_J$ is proper (otherwise $W^J$ is trivial)
and that $y\in\Gamma(\Phi_{K,0})$ for some irreducible
component $\Phi_K$ of~$\Phi_J$, since $\Gamma(\Phi_{J,0})$ is generated
by such groups. In that case, $y$ may act non-trivially on the orbits
of odd roots in at most one irreducible component of $\Phi$ (the one
containing $\Phi_K$), so we may further assume that $\Phi$ is irreducible.
By Proposition~\ref{P3.2}, this is possible
only if $\Phi\isom A_{2n-1}$ or~$C_n$.
In these cases, there is only one nontrivial element $y\in\Gamma(\Phi_0)$,
and Proposition~\ref{P3.2} provides an explicit description of it.
For $A_{2n-1}$ and $C_{2n-1}$, it involves the simple reflections at
both ends of the Dynkin diagram, and thus cannot belong to any proper
parabolic subgroup of $W$ that is irreducible. For $C_{2n}$,
it is also the nontrivial element of $\Gamma(\Phi_{J,0})$,
where $\Phi_J$ is the parabolic subsystem spanned by the
short simple roots. Thus the conditions of the previous paragraph may
be met only if $(\Phi,\Phi_J)\isom(C_{2n},A_{2n-1})$.
Since $\Gamma(\Phi_0)$ acts on the $W(\Phi_0)$-orbits by interchanging
opposite pairs (Proposition~\ref{P3.2} again), proving
\begin{equation}\label{eq7.4}
F_J(\Phi;\bq)=F_J(\Phi;\bar\bq)
\end{equation}
in this one case completes the proof of~\eqref{eq7.3}.
Our strategy for this case is to show that the conditions
of Lemma~\ref{L7.2} are met; \ie that $W(\Phi_0)$ contains
all of the perfect coset representatives. The key point is that the
long roots of $C_{2n}$ are all odd. Thus every $y\in{}^JW_1$ has the
property that $y^{-1}\alpha_i$ is odd, whether $\alpha_i$ is short
(\ie $i\in J$) or $\alpha_i$ is long (\ie $i\notin J$). In other words,
${}^JW_1\subset N(\Phi_0)$, and the only question is whether ${}^JW_1$
(or the entirety of ${}^JW$)
contains any elements in the nontrivial coset of $N(\Phi_0)/W(\Phi_0)$.
For this, consider any $y\in{}^JW$. Using the coordinates for $C_{2n}$
we introduced in Section~\ref{S2}, we claim that either
$y\eps_{2n}=\eps_{2n}$ or $y\eps_{2n}=-\eps_1$. Otherwise,
either $y\eps_{2n}=\eps_i$ for some $i<2n$ or $y\eps_{2n}=-\eps_i$
for some $i>1$. In the former case, we have
$y^{-1}(\eps_{i+1}-\eps_i)=\pm\eps_j-\eps_{2n}<0$
for some $j$, and in the latter case, we have
$y^{-1}(\eps_i-\eps_{i-1})=-\eps_{2n}\pm\eps_j<0$ for some $j$.
Either way, we contradict the fact that $y\in{}^JW$ and the claim follows.
Thus $y$ maps the (odd) root $2\eps_{2n}$ to either $2\eps_{2n}$
or $-2\eps_1$. Both roots belong to the same $W(\Phi_0)$-orbit,
so this confirms that every $y\in{}^JW_1$ belongs to $W(\Phi_0)$.
Thus Lemma~\ref{L7.2} implies
\[
F_J(\Phi;\bar\bq)=F(\Phi;\bq)/F(\Phi_J;\bq)=F(C_{2n};\bq)/F(A_{2n-1};\bq)
\]
under the fine grading. However, one can see from Theorem~\ref{T6.1}
that the right hand side is invariant under $\bq\to\bar\bq$, so the same
is true for $F_J(\Phi;\bq)$ and~\eqref{eq7.4} follows.
\end{proof}
\goodbreak
\begin{remas}\label{R7.3}\hskip0pt
\begin{enumerate}[(\alph*)]
\item\label{rema7.3_a} %(a)
The irreducible root systems for which $\Gamma(\Phi_0)$ is
trivial are $A_{2n}$, $E_6$, $E_8$, $F_4$, and $G_2$ (see Table~\ref{T1}).
Thus the conditions of Theorem~\ref{T7.1} are easily met whenever the
irreducible components of $\Phi_J$ are on this list.
Conversely, if $\Phi$ is on this list, then this is the only way to meet
the conditions.
\item\label{rema7.3_b} %(b)
It follows in particular that the parabolic subsystems
of $A_{2n}$ whose irreducible components all have even rank are coherent.
Conversely, it is easy to see from Theorem~\ref{T6.1} that there are no
other parabolic subsystems of $A_{2n}$ with this property. Indeed, these
are the only cases for which $F(\Phi_J;\bq)$ is a divisor of $F(\Phi;\bq)$.
\item\label{rema7.3_c} %(c)
In the case $\Phi=B_n$, the nontrivial element in $\Gamma(\Phi_0)$ is
the short simple reflection. Thus Theorem~\ref{T7.1} in this case implies
that a parabolic subsystem of $B_n$ is coherent if each of its irreducible
components contain either the short simple root or an even number of
long simple roots.
\item\label{rema7.3_d} %(d)
It seems likely that the converse of Theorem~\ref{T7.1} is true; \ie
$\Phi_J$ is coherent if and only
if $\Gamma(\Phi_{J,0})\subseteq\Gamma(\Phi_0)$.
We have confirmed this for all $\Phi$ of rank at most~8, and the
argument in~\ref{rema7.3_b} confirms it for $A_{2n}$.
\end{enumerate}
\end{remas}
It appears that there is potential for a very general factorization
theorem involving nearly all of the parabolic series $F_J(\Phi;\bq)$.
For example, Brenti and Carnevale have proved explicit
product formulas for all of the univariate parabolic series $F_J(A_n;q)$
and $F_J(B_n;q)$ (see Corollary~4.3 and Theorem~5.4 in~\cite{BC1});
the latter was also obtained independently by Landesman~\cite{L}.
These formulas are clearly expressible as products of
cyclotomic polynomials, so they support the following:
\begin{conj}\label{C7.4}
If $\Phi$ is irreducible, then the univariate series $F_J(\Phi;q)$
is a product of cyclotomic polynomials unless
\begin{enumerate}[(\alph*)]
\item\label{conj7.4_a} $\Phi\isom D_{2n}$ and $I\supsetneq J\supseteq K$,
where $\Phi_K\isom A_1^{\oplus n+1}$, or
\item\label{conj7.4_b} $\Phi\isom E_7$ and $I\supsetneq J\supseteq K$,
where $\Phi_K\isom A_1\oplus D_4$, or
\item\label{conj7.4_c} $\Phi\isom E_8$ and $I\supsetneq J\supseteq K$,
where $\Phi_K\isom D_6$.
\end{enumerate}
\end{conj}
Note that in each of~\ref{conj7.4_a}--\ref{conj7.4_c}, the subset $K\subset I$ is uniquely
determined.
We have confirmed the above conjecture for all root systems of
rank at most 8, so the only open cases involve $C_n$ and $D_n$.
Note that Brenti and Carnevale have explicit evaluations
of the series $F_J(D_n;q)$ for the cases with $|J|=1$, two cases
with $|J|=2$, and some conjectured evaluations in a few cases
with $|J|\sle 3$ (see Sections~5--6 of~\cite{BC3});
all are consistent with the above conjecture.
\begin{remas}\label{R7.5}\hskip0pt
\begin{enumerate}[(\alph*)]
\item\label{rema7.5_a} %(a)
One way to prove that $F_J(\Phi;q)$ factors into cyclotomic polynomials
would be to show that it is a divisor of $F(\Phi;q)$.
(And empirically, it does not seem to matter whether we use the fine,
coarse, or univariate gradings: for ranks up through 8, all three gradings
share the same divisibility properties.) We will not attempt to formulate
a comprehensive conjecture, but it appears that
\begin{enumerate}[(\roman*)]
\item\label{rema7.5_a_i} %(i)
$F_J(\Phi;\bq)$ is
always a divisor of $F(\Phi;\bq)$ under the fine grading
for $A_{2n}$ or $B_n$, and
\item\label{rema7.5_a_ii} %(ii)
among the classical cases, a sufficient
condition for divisibility is that $\Gamma(\Phi_0)\cap W_J$ is trivial.
Unfortunately, this condition is not sufficient for the exceptional cases.
\end{enumerate}
\item\label{rema7.5_b} %(b)
An even stronger divisibility condition would be for
\[
H_J(\Phi;\bq):=\frac{F(\Phi;\bq)}{F_J(\Phi;\bq)F(\Phi_J;\bq)}
\]
to be a polynomial. If we assume $\Phi$ is irreducible and restrict to
the coarse grading, so that $F(\Phi_J;\bq)$ is assured to be a divisor
of $F(\Phi;\bq)$ (Corollary~\ref{C5.2}),
the only examples we have found where $F_J(\Phi;\bq)$
divides $F(\Phi;\bq)$ but $H_J(\Phi;\bq)$ fails to be a polynomial
involve $D_n$ for $n\sge7$ and violate the condition in~\ref{rema7.5_a};
\ie $\Gamma(\Phi_0)\cap W_J$ is nontrivial.
\end{enumerate}
\end{remas}
\begin{rema}\label{R7.6}
It would be natural to investigate the analogue of $F_J(\Phi;\bq)$ in
which the sum over $W^J$ is replaced with a sum over ${}^JW$. However,
these series do not appear to offer as much potential for
factorization theorems. For example, when $\Phi=F_4$ or~$E_6$, the
univariate series factors into a product of cyclotomic polynomials
only in the trivial case (\ie $J=I$) and in the case $J=\mtset$
(which coincides with the full series $F(\Phi;q)$). Out of the
64 choices for $J$ available with $\Phi=C_6$ and 128 with $D_7$,
only 6 in each case yield series that factor into cyclotomic polynomials.
\end{rema}
\section{A unified presentation of the univariate factorization}
There is a uniform way to present the product formula for
the univariate series $F(\Phi;q)$ that involves data
attached to a root system canonically associated to $\Phi$.
To construct this second root system, we will need to work
on the dual side with the co-root system~$\Phi^\vee$.
Recall that the longest element $w_0$ of $W$ induces a diagram
automorphism of $\Phi$ and $\Phi^\vee$ via $\beta\mapsto -w_0\beta$.
Let $\sigma$ denote the induced map on the index set $I$ for the simple
roots and co-roots, and $I/\sigma$ the set of $\sigma$-orbits on $I$.
For each $\sigma$-orbit $J$, define
\[
\alpha_J^\vee:=\sum_{j\in J}\alpha_j^\vee=
\begin{cases}
\hskip 15pt\alpha_j^\vee&\text{if $j\in J$ and $\sigma(j)=j$},\\
\alpha_{j\vpo}^\vee+\alpha_{\sigma(j)}^\vee
&\text{if $j\in J$ and $\sigma(j)\ne j$},
\end{cases}
\]
and let $s_J$ denote the longest element of $W_J$. It is easy to
check that either
\begin{enumerate}[(\roman*)]
\item\label{Sect8_i} $s_J$ is a simple reflection
and $\alpha^\vee_J\in\Phi^\vee$ is simple, or
\item\label{Sect8_ii} $s_J$ is a product of two commuting simple reflections
and $\alpha_J^\vee\notin\Phi^\vee$, or
\item\label{Sect8_iii} $s_J$ is the non-simple reflection in a copy of $A_2$
and $\alpha^\vee_J\in\Phi^\vee$ has height~2.
\end{enumerate}
Note that~\ref{Sect8_iii} occurs only if $\Phi^\vee$ has an irreducible component
of type $A$ with even rank. Indeed, this is the only circumstance in
which $\sigma$ reverses an edge of the diagram.
Since conjugation by $w_0$ acts as a diagram automorphism of any
$\sigma$-stable parabolic subgroup $W_J$, it follows that the longest
element of any such $W_J$ belongs to $Z(w_0)$, the centralizer of $w_0$.
In particular, for each $\sigma$-orbit $J$, we have $s_J\in Z(w_0)$.
\begin{lemma}\label{L8.1}
The centralizer $Z(w_0)$ is generated by $\{s_J:J\in I/\sigma\}$.
\end{lemma}
\begin{proof}
Consider that the $-w_0$-orbits $\{\beta^\vee,-w_0\beta^\vee\}$
in $\Phi^\vee$ are permuted by $Z(w_0)$ and that each orbit consists of
all positive or all negative co-roots. Therefore
if $w\alpha_j^\vee<0$ for some $w\in Z(w_0)$, then the same is true for
every index in the $\sigma$-orbit $J$ of $j$, and thus $w$ must be the
longest element in its $W_J$-coset. Hence $\ell(ws_J)=\ell(w)-\ell(s_J)$
and it follows by induction that every member of $Z(w_0)$ has a
length-additive expression as a product of elements
chosen from $\{s_J:J\in I/\sigma\}$.
\end{proof}
Let $V_{\pm1}$ denote the $\pm1$-eigenspaces for the action of $w_0$
on the ambient space $V$ that contains $\Phi$ and $\Phi^\vee$.
Since $w_0$ is an involution, $V$ is the direct sum of
these eigenspaces. Moreover, it is easy to see
that $\{\alpha_J^\vee:J\in I/\sigma\}$ is a basis for $V_{-1}$; the
parameter
\[
k:=|I/\sigma|=\dim V_{-1}
\]
will be important in what follows.
Given that some of the generators $s_J$ may be products of two commuting
reflections as in~\ref{Sect8_ii}, it is clear that $s_J$ need not be a reflection,
and $Z(w_0)$ need not be a reflection subgroup of $W$. However
as a centralizer, $Z(w_0)$ acts naturally on the eigenspaces of~$w_0$,
including $V_{-1}$.
\begin{lemma}\label{L8.2}
The centralizer $Z(w_0)$ acts faithfully on $V_{-1}$.
Moreover, the generator $s_J$ acts on $V_{-1}$ as a reflection
through the hyperplane orthogonal to $\alpha_J^\vee$.
\end{lemma}
\begin{proof}
In case the $\sigma$-orbit $J$ is of type~\ref{Sect8_i} or~\ref{Sect8_iii},
$s_J$ acts on $V$ as a reflection through the hyperplane orthogonal
to $\alpha_J^\vee$. Since $\alpha_J^\vee\in V_{-1}$, it restricts
to a hyperplane reflection on $V_{-1}$ as well.
If $J$ is of type~\ref{Sect8_ii}, so that $s_J$ is a product of two commuting simple
reflections, say $s_is_j$, then the $-1$-eigenspace of $s_J$ on $V$ is
spanned by $\alpha_i^\vee$ and $\alpha_j^\vee$, but the intersection of
this space with $V_{-1}$ is spanned
by $\alpha_J^\vee=\alpha_i^\vee+\alpha_j^\vee$, so again it acts as a
hyperplane reflection on $V_{-1}$. If some $w\in Z(w_0)$ acts trivially
on $V_{-1}$, it cannot have any inversions (as a member of $W$);
otherwise, the proof of Lemma~\ref{L8.1} shows that it would have to invert
the simple co-roots $\alpha_j^\vee$ for all $j$ in some
$\sigma$-orbit $J$, and thus it would have to act nontrivially on
$\alpha_J^\vee\in V_{-1}$. Hence the action is faithful.
\end{proof}
Define $\fold$ to be the set of $-w_0$-orbit
sums in $\Phi^\vee$; \ie
\begin{equation}\label{eq8.1}
\fold:=\{\beta^\vee:w_0\beta^\vee=-\beta^\vee\}
\ \cup\ \{\beta^\vee-w_0\beta^\vee:w_0\beta^\vee\ne-\beta^\vee\}.
\end{equation}
It is clear that $\fold$ is a union of $Z(w_0)$-orbits
and that it contains each $\alpha_J^\vee$.
The following result is well-known; we include a proof for the
sake of completeness.
\begin{prop}\label{P8.3}
The orbit sums $\fold$ form a \xtal{} root system in $V_{-1}$
of \tup{full} rank $k$ with base $\{\alpha_J^\vee:J\in I/\sigma\}$.
Hence $Z(w_0)$ is isomorphic to the Weyl
group $W(\fold)$ via a map in which $s_J$ acts as the
reflection corresponding to~$\alpha_J^\vee$.
\end{prop}
\begin{proof}
Let $\Dfold=\{\alpha_J^\vee:J\in I/\sigma\}$.
By Lemma~\ref{L8.2}, it suffices to confirm that
\begin{enumerate}[(\alph*)]
\item\label{proof_of_prop8.3_a} every member of $\fold$ is in the nonnegative or nonpositive
$\Z$-span of $\Dfold$,
\item\label{proof_of_prop8.3_b} every $Z(w_0)$-orbit in $\fold$ meets $\Dfold$, and
\item\label{proof_of_prop8.3_c} no nontrivial rescaling of $\fold$ intersects $\fold$
(\ie $\fold$ is reduced).
\end{enumerate}
Given a $-w_0$-orbit $\orb=\{\beta^\vee,-w_0\beta^\vee\}$,
we may exchange $\beta^\vee$ and $\orb$ with $w_0\beta^\vee$
and $-\orb$ if necessary so that $\beta^\vee>0$.
In that case, $\beta^\vee$ and $-w_0\beta^\vee$ must have
nonnegative integer coordinates with respect to the simple roots
of $\Phi^\vee$, so the same is true for the $\Dfold$-coordinates of
the orbit sum, proving~\ref{proof_of_prop8.3_a}.
Since $w_0\in Z(w_0)$ inverts all co-roots, there must be an
element $w\in Z(w_0)$ and a generator $s_J$ such that $w\orb$ is a
positive orbit and $s_Jw\orb$ is negative. However, $s_J$ inverts
only the roots in the subsystem $\Phi_J^\vee$; in most cases, this
forces $w\orb$ to be $\{\alpha_j^\vee:j\in J\}$ and the corresponding
orbit sum to be $\alpha_J^\vee$. The one exception occurs when
the $\sigma$-orbit $J$ is of type~\ref{Sect8_iii}, in which cases there are
two positive $-w_0$-orbits in $\Phi^\vee_J$: one singleton
and one doubleton. However they both have sum $\alpha_J^\vee$,
so~\ref{proof_of_prop8.3_b} follows either way. (This also shows that the union
in~\eqref{eq8.1} need not be disjoint.)
Finally, suppose $\fold$ and $k\fold$ had a nontrivial intersection for
some scalar $k>1$. In that case, \ref{proof_of_prop8.3_b}~would force some $Z(w_0)$-orbit
in $\fold$ to have $\Dfold$-coordinates in $k\Z$, contradicting the
fact that the orbit must include some $\alpha_J^\vee$.
\end{proof}
Now let $L_0^\vee$ denote the kernel of the parity
morphism $\Z\Phi^\vee\to\Z/2\Z$ in which every
simple co-root is odd; this is the co-root counterpart to the
lattice $L_0$ we introduced in~\eqref{eq1.0}.
Having constructed a root system $\fold$
contained in $\Z\Phi^\vee$, it follows that
\[
\Efold\,:=\,L_0^\vee\ \cap\ (\Phi^\vee/\sigma)
\]
is a root subsystem of $\fold$. Note that if $w_0$ acts as $-1$
on $\Phi^\vee$, the involution $\sigma$ is trivial,
$\fold=\Phi^\vee$, and $\Efold=L_0^\vee\cap\Phi^\vee$ is the
even-height root subsystem of $\Phi^\vee$.
It is important to note that in general, some of the simple
roots of $\fold$ may be ``even'' (\ie members of $L_0^\vee$) and
thus part of $\Efold$. That is, the parity that $L_0^\vee$ induces
on $\fold$ need not be the one that declares
all of its simple roots to be odd.
\begin{exam}\label{E8.4}
Consider $\Phi=A_n$, with the diagram numbered $1{-}2{-}\cdots{-}n$.
Here, the action of $\sigma$ is such that $\sigma(i)=n+1-i$.
If $n=2r$, the simple roots of $\fold$ are
\[
\alpha^\vee_r+\alpha^\vee_{r+1},
\ \ \alpha^\vee_{r-1}+\alpha^\vee_{r+2},
\dotsc,\ \ \alpha^\vee_1+\alpha^\vee_{2r}.
\]
The first one is of type~\ref{Sect8_iii} while the others are of type~\ref{Sect8_ii}
and thus longer, and it is not hard to check that the root system
they generate is isomorphic to $B_r$. Moreover, all of these roots
are in $L_0^\vee$, so we have $\fold=\Efold\isom B_r$.
On the other hand, if $n=2r-1$, the simple roots of $\fold$ are
\[
\alpha^\vee_{r},
\ \ \alpha^\vee_{r-1}+\alpha^\vee_{r+1},
\dotsc,\ \ \alpha^\vee_1+\alpha^\vee_{2r-1}.
\]
The first one is again short, but it is of type~\ref{Sect8_i} and has odd
height in $\Phi^\vee$. Thus $\fold\isom B_r$,
but in this case $\Efold\isom D_r$ contains only the long roots.
(In the somewhat degenerate case $\Phi=A_1$, one should
recognize that $\Efold\isom D_1$ is an empty root system.)
\end{exam}
\begin{table}[htb]\centering
\renewcommand{\arraystretch}{1.1}
\renewcommand{\arraycolsep}{5pt}
$
%\hbox{\vbox{\offinterlineskip
%\def\spacing{\omit\vrule height3pt&&&&&&&&&&\cr}
%\halign{\vrule\strut#&&\ \,\hfil$#$\hfil\ \,&\vrule#\cr
%\noalign{\hrule}\spacing
\begin{array}{|c|c|c|c|c|}
\hline
\Phi &\Phi_0 &\fold &\Efold &k \\
\hline
A_{2n-1} &A_{n-1}\oplus A_{n-1} &B_n &D_n &n\\
A_{2n} &A_n\oplus A_{n-1} &B_n &B_n &n\\
B_{2n} &B_n\oplus D_n &C_{2n} &A_{2n-1} &2n\\
B_{2n+1} &B_n\oplus D_{n+1} &C_{2n+1} &A_{2n} &2n+1\\
C_{2n} &A_{2n-1} &B_{2n} &B_n\oplus D_n &2n\\
C_{2n+1} &A_{2n} &B_{2n+1} &B_n\oplus D_{n+1} &2n+1\\
D_{2n} &D_n\oplus D_n &D_{2n} &D_n\oplus D_n &2n\\
D_{2n+1} &D_n\oplus D_{n+1} &C_{2n} &C_n\oplus C_n &2n\\
E_6 &A_5\oplus A_1 &F_4 &B_4 &4\\
E_7 &A_7 &E_7 &A_7 &7\\
E_8 &D_8 &E_8 &D_8 &8\\
F_4 &C_3\oplus A_1 &F_4 &C_3\oplus A_1 &4\\
G_2 &A_1\oplus A_1 &G_2 &A_1\oplus A_1 &2\\
\hline
\end{array}
$
\vspace*{2mm}
\caption{Listings of $\Phi_0$, $\fold$, $\Efold$, and $k=\dim V_{-1}$.}
\label{T3}
\end{table}
In Table~\ref{T3}, we list the isomorphism classes of $\Phi_0$,
$\fold$, and $\Efold$ as well as the parameter $k=\dim V_{-1}$
(the rank of $\fold$) for each irreducible $\Phi$.
\medbreak
To place the following result in its proper context, recall that
for any finite reflection group action, such as $W$ acting on $V$,
there is a canonically associated multiset of positive
integers $d_1,\dotsc,d_m$ (where $m=\dim V$).
These are the degrees of a homogeneous set of free generators for
the $W$-invariants in the symmetric algebra $S(V)$.
(The existence of such generators is Chevalley's Theorem;
\eg see~\cite[\S3.5]{H}.)
\begin{theorem}\label{T8.5}
If $a_1,\dots,a_k$ are the degrees of the free generating
invariants for the action of $W(\Efold)$ on $S(V_{-1})$, then
\[
F(\Phi;q)\,=\,\prod_{i=1}^k(1-q^{a_i}).
\]
\end{theorem}
\begin{proof}
It suffices to prove this when $\Phi$ is irreducible. In such cases,
we may obtain the univariate series $F(\Phi;q)$ by specializing
Theorem~\ref{T6.1}. On the product side, one may use Table~\ref{T3}
to determine the degrees $a_1,\dots,a_k$. Since the components of
$\Efold$ are always of classical type, one only needs to know that
the degrees of the generating invariants for the action on the
span of such a component are as follows:
\begin{align*}
A_n:&\ \ 2, 3, \dots, n+1,\\
B_n,C_n:&\ \ 2, 4, 6,\dots, 2n,\\
D_n:&\ \ n, 2, 4, 6,\dots, 2n-2.
\end{align*}
It should be noted that the rank $r$ of $\Efold$ may be less than $k$,
in which case the action of $W(\Efold)$ on the span of $\Efold$ accounts
for only $r$ of the $k$ generating invariants in $S(V_{-1})$.
In such cases, the remaining $k-r$ generators all have degree~1.
(This happens when $\Phi\isom A_1$, $C_2$, or $B_n$.)
\end{proof}
\begin{remas}\label{R8.6}\hskip0pt
\begin{enumerate}[(\alph*)]
\item %(a)
Let $G=W(\Efold)$. The Hilbert series for the $G$-invariants is
\[
H(S(V_{-1})^G;q)=\prod_{i=1}^k\frac{1}{1-q^{a_i}},
\]
so an equivalent formulation of Theorem~\ref{T8.5} is the identity
\[
F(\Phi;q)\,H(S(V_{-1})^G;q)=1.
\]
\item %(b)
The degrees of the generating invariants also appear in Chevalley's
formula for the (untwisted) \pseries{} $P(W;q)=\sum_{w\in W}q^{\ell(w)}$
for any Weyl group $W$ (\eg see~\cite[\S3.15]{H}).
Thus another way to formulate Theorem~\ref{T8.5} is the identity
\begin{equation}\label{eq8.2}
\frac{F(\Phi;q)}{(1-q)^k}\,=\,P(W(\Efold);q).
\end{equation}
It should be stressed that the length function here is the one
intrinsic to the Weyl group of the root system $\Efold$. It is not the
same as the length functions for the Weyl groups of $\fold$ or $\Phi$
(although the proof of Lemma~\ref{L8.1} shows that the latter two are
related). Note also that the factor $(1-q)^{-k}$ and the right hand side
of~\eqref{eq8.2} may be interpreted as the Hilbert series for $S(V_{-1})$
and its associated co-invariant algebra, respectively.
\item %(c)
It would be interesting to develop a multivariate refinement of
Theorem~\ref{T8.5} for either the coarse or fine gradings of $F(\Phi;\bq)$.
In this context it should be noted that Macdonald has given a coarse
refinement of $P(W;q)$ for any Weyl group $W$~\cite{M}.
\end{enumerate}
\end{remas}
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