Part I

A conjecture on Weyl's dimension polynomial

Last update: 1 April 2014

**1.1.** Let $\Delta $ be an indecomposable reduced root-system [Bou1968, p. 151] in a real vector space
$E$ with positive-definite inner product $\u27e8,\u27e9$
which is unique up to a scalar multiple. Let $W$ be the Weyl group of $\Delta $ generated by the
reflections ${R}_{\alpha}={R}_{-\alpha}$ on $E$ for
$\alpha \in \Delta ,$ where ${R}_{\alpha}$
sends $x$ to $x-\u27e8x,{\alpha}^{\vee}\u27e9\alpha $
with ${\alpha}^{\vee}$ defined to mean $2\alpha /\u27e8\alpha ,\alpha \u27e9\text{.}$
Let $X\prime $ denote the root-lattice $\mathbb{Z}\xb7\Delta $
(the $\mathbb{Z}\text{-span}$ of $\Delta \text{),}$ and
$X$ be the weight-lattice consisting of all $\lambda \in E$ such that
$\u27e8\lambda ,{\alpha}^{\vee}\u27e9\in \mathbb{Z}$
for every $\alpha \in \Delta \text{.}$ Let
$\{{\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{l}\}$
be a choice of fundamental $\text{(}=$ simple) roots in $\Delta ,$ and
${\Delta}_{+}$ (resp. ${\Delta}_{-}=-{\Delta}_{+}\text{)}$
the corresponding set of positive (resp. negative) roots with $\Delta ={\Delta}_{+}\cup {\Delta}_{-}$
(disjoint union). Define the fundamental weights ${\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{l}$
by $\u27e8{\lambda}_{l},{\alpha}_{j}^{\vee}\u27e9={\delta}_{ij}$
$\text{(}=$ the Kronecker delta), so that
$$\sum _{i}\mathbb{Z}{\lambda}_{i}=X\supseteq X\prime =\sum _{i}\mathbb{Z}{\alpha}_{i}\text{.}$$
The action of $W$ on the group $X$ maps each coset of $X\prime $ therein onto itself,
i.e. the natural action of $W$ on $X/X\prime $ is trivial. The elements of
${X}^{+}=\sum _{i}{\mathbb{Z}}^{+}{\lambda}_{i}$
(the set of non-negative integer linear combinations of ${\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{l}\text{)}$
are called dominant weights; each $W\text{-orbit}$ in $X$ contains a unique dominant weight. The set
${X}^{+}\cap \Delta $ of dominant roots has cardinality equal to the
number of $W\text{-orbits}$ in $\Delta ,$ which is 1 or 2
according as all elements of $\Delta $ have same length or not. The element(s) of the set
${X}^{+}\cap \Delta $ are therefore unambigously named "long dominant root"
(also called highest root) and "short dominant root", which do coincide if all roots have the same length. *We shall denote the short dominant root by*
$-{\alpha}_{0}\text{.}$ The set
${\Delta}^{\vee}=\left\{{\alpha}^{\vee}\hspace{0.17em}\right|\hspace{0.17em}\alpha \in \Delta \}$
of "coroots" is again a root-system (called the root-system inverse to $\Delta \text{)}$ such that
the set $\{{\alpha}_{1}^{\vee},{\alpha}_{2}^{\vee},\dots ,{\alpha}_{l}^{\vee}\}$
affords a choice of fundamental (co)roots in it; relative to this choice $-{\alpha}_{0}^{\vee}$
is the long dominant (co)root of ${\Delta}^{\vee}$ (and if $\beta $
were to denote the long dominant root of $\Delta ,{\beta}^{\vee}$
would be the short dominant (co)root in ${\Delta}^{\vee}\text{).}$ The set
$\{{\alpha}_{0},{\alpha}_{1},\dots ,{\alpha}_{l}\}$
has the property that $\u27e8{\alpha}_{i},{\alpha}_{j}^{\vee}\u27e9$
is a non-positive integer for distinct $i,j$ in $\{0,1,\dots ,l\}\text{.}$
The subset $\{{\alpha}_{0},{\alpha}_{1},\dots ,{\alpha}_{j-1},{\alpha}_{j+1},\dots ,{\alpha}_{l}\}$
obtained by deleting ${\alpha}_{j}$ is a fundamental system for a subroot-system of $\Delta $
which is all of $\Delta $ if and only if ${n}_{j}=1$ in the expansion
$$-{\alpha}_{0}^{\vee}=\sum _{i=1}^{l}{n}_{i}{\alpha}_{i}^{\vee}$$
of the long dominant coroot in the fundamental coroots. By the *extended Dynkin diagram* we shall mean the graph with $l+1$
nodes labelled $0,1,\dots ,l,$
in which the nodes labelled $i$ and $j$ (for $i\ne j\text{)}$
are joined by $\u27e8{\alpha}_{i},{\alpha}_{j}^{\vee}\u27e9\xb7\u27e8{\alpha}_{j},{\alpha}_{i}^{\vee}\u27e9$
$\text{(}=0,1,2,$ or
$3\text{)-fold}$ lines, omitting the arrows that are sometimes placed to distinguish the role of $i$ and
$j$ in case $\u27e8{\alpha}_{i},{\alpha}_{j}^{\vee}\u27e9\ne \u27e8{\alpha}_{j},{\alpha}_{i}^{\vee}\u27e9\text{.}$
It is known that this diagram determines $\Delta $ up to isomorphism. We remark that the common usage is to call it the
"extended Dynkin diagram of ${\Delta}^{\vee}\text{".}$

Let $D$ (resp. $D\prime \text{)}$ denote the group consisting of all translations ${T}_{\lambda}$ operating on $E$ or $X,$ for $\lambda \in X$ (resp. $\lambda \in X\prime \text{),}$ where ${T}_{\lambda}$ sends $x$ to $x+\lambda \text{.}$ We denote by $\stackrel{\sim}{W}$ (resp. $\stackrel{\sim}{W}\prime \text{)}$ the group $D\xb7W$ (resp. $D\prime \xb7W\text{)}$ of affine transformations of $E$ or $X,$ generated by $D$ (resp. $D\prime \text{)}$ and $W\text{.}$ It is clear that $\stackrel{\sim}{W}$ (resp. $\stackrel{\sim}{W}\prime \text{)}$ is the semidirect extension of $W$ by the normal subgroup $D=X$ (resp. $D\prime =X\prime \text{)}$ on which the action of $W$ is known. From the fact that $\lambda -{\lambda}^{\sigma}\in X\prime $ for $\lambda \in X$ and $\sigma \in W,$ it is easily seen that $\stackrel{\sim}{W}\prime $ is a normal subgroup of $\stackrel{\sim}{W}\text{.}$

For linear and affine transformations *we shall denote the operation on the right, and shall compose them accordingly;* thus for
$\lambda \in E$ its image under $w\in \stackrel{\sim}{W}$ will be written
${\lambda}^{w},$ or occasionally $\lambda w$ for typographical
reasons, and ${\left({\lambda}^{w}\right)}^{w\prime}={\lambda}^{ww\prime}\text{.}$
With this convention we introduce the abbreviations
$${R}_{0}={R}_{{\alpha}_{0}}\xb7{T}_{-{\alpha}_{0}},\phantom{\rule{1em}{0ex}}{R}_{i}={R}_{{\alpha}_{i}}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}1\leqq i\leqq l\text{.}$$
It is known that $\stackrel{\sim}{W}\prime $ (resp. $W\text{)}$
is a Coxeter group on generators ${R}_{0},{R}_{1},\dots ,{R}_{l}$
(resp. ${R}_{1},{R}_{2},\dots ,{R}_{l}\text{).}$
The order of the product ${R}_{i}{R}_{j},$ for distinct
$i,j$ in $\{0,1,\dots ,l\},$
is 2, 3, 4, or 6 according as the nodes labelled $i$ and $j$ in our extended Dynkin diagram are joined by 0, 1, 2, or 3-fold lines.
An element of $W$ whose fixpointspace has codimension 1 in $E$ is called a reflection, and two reflections are called parallel
if their fixed hyperplanes are parallel. Every reflection is of the form ${T}_{n\alpha}{R}_{\alpha}={R}_{\alpha}{T}_{-n\alpha}$
for unique $\alpha \in {\Delta}_{+}$ and $n\in \mathbb{Z}\text{;}$
one also knows that every reflection lies in $\stackrel{\sim}{W}\prime $ and is in fact conjugate to one of the
generators $\left\{{R}_{i}\hspace{0.17em}\right|\hspace{0.17em}0\leqq i\leqq l\}$
under an element of $\stackrel{\sim}{W}\prime \text{.}$ (More generally, it can be shown that
every finite subgroup of $\stackrel{\sim}{W}$ is isomorphic to a subgroup of $W,$
and every finite subgroup of $\stackrel{\sim}{W}\prime $ is conjugate under an element of
$\stackrel{\sim}{W}\prime $ to a subgroup of one of the $l+1$
finite subgroups generated by $\left\{{R}_{i}\hspace{0.17em}\right|\hspace{0.17em}0\leqq i\leqq l,\hspace{0.17em}i\ne j\}$
for $0\leqq j\leqq l\text{.)}$ We shall call a point of $E,$
$W\text{-regular}$ (resp. $\stackrel{\sim}{W}\prime \text{-regular,}$
$\stackrel{\sim}{W}\text{-regular)}$ if its stabilizer in $W$ (resp.
$\stackrel{\sim}{W}\prime ,$ $\stackrel{\sim}{W}\text{)}$
is trivial. The group $\stackrel{\sim}{W}\prime $ will be called the *affine Weyl group*. We remark that the
common usage is to call it the "affine Weyl group of the root-system ${\Delta}^{\vee}\text{".}$
The reader may refer to [Bou1968, (Chapter VI, § 2)] or [IMa1965] for details of some of the standard facts on affine Weyl groups which are stated here without proof.

**1.2.** Consider the *fundamental cone* (also called the dominant Weyl chamber)
$${E}^{+}=\sum _{i}{\mathbb{R}}^{+}{\lambda}_{i}=\{\lambda \in E\hspace{0.17em}|\hspace{0.17em}\u27e8\lambda ,{\alpha}_{i}^{\vee}\u27e9\geqq 0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}1\leqq i\leqq l\},$$
the *fundamental parallelepiped*
$${E}^{\u266f}=\{\lambda \in E\hspace{0.17em}|\hspace{0.17em}0\lneqq \ufe00\u27e8\lambda ,{\alpha}_{i}^{\vee}\u27e9\leqq 1\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}1\leqq i\leqq l\},$$
and the *fundamental simplex*
$${E}^{*}=\{\lambda \in {E}^{+}\hspace{0.17em}|\hspace{0.17em}\u27e8\lambda ,-{\alpha}_{0}^{\vee}\u27e9\leqq 1\}\text{.}$$
These are the fundamental regions for the actions of $W,$ $D,$ and
$\stackrel{\sim}{W}\prime $ respectively on $E,$
in the strict sense that every point of $E$ has a unique image in each of these under the action of the respective group. Note that
${E}^{*}$ lies in the closure of ${E}^{\u266f}$ but not in
${E}^{\u266f}\text{;}$ the reason for our *ad hoc* choice of omitting all walls through
$0$ (from the closure of ${E}^{\u266f}\text{)}$ in the definition of
${E}^{\u266f},$ would be clear only in Part II. The interior of
${E}^{*}$ is one of the connected components of the complement in $E$ of the union of the fixed hyperplanes
of all the reflections in $\stackrel{\sim}{W}\prime ,$ and the affine Weyl group
operates simply-transitively on the set of all connected components in this complement. Thus the collection
$\left\{{\left({E}^{*}\right)}^{W}\hspace{0.17em}\right|\hspace{0.17em}w\in \stackrel{\sim}{W}\prime \}$
of the corresponding closed simplices forms a tessellation $\text{(}=$ covering) of $E,$
which is parametrized by $\stackrel{\sim}{W}\prime $ itself; we shall call
${\left({E}^{*}\right)}^{w}$ the $w\text{-simplex.}$
The walls of ${E}^{*}$ are precisely the reflecting hyperplanes for the Coxeter generators
$\{{R}_{0},{R}_{1},\dots ,{R}_{l}\}$
of $\stackrel{\sim}{W}\prime \text{.}$ (In the more general situation when
$\Delta $ is the union of $k$ mutually orthogonal indecomposable reduced root-systems,
${E}^{*}$ would be the Cartesian product of $k$ simplices and thus would have
$l+k$ faces and $\geqq l+k$ vertices. It is merely to avoid the cumbersomeness
resulting from such a situation that we have assumed $\Delta $ indecomposable.) It is easy to see that
${n}_{i}^{-1}{\lambda}_{i}$ is the vertex of
${E}^{*}$ opposite the face fixed by ${R}_{i},$ where
${n}_{0}$ is the integer $1$ and ${\lambda}_{0}$ is the zero element of
$X$ (so that the ${n}_{i}\text{'s}$ satisfy
$\sum _{i=0}^{l}{n}_{i}{\alpha}_{i}^{\vee}=0\text{).}$ Thus
$${E}^{*}\cap X=\left\{{\lambda}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in {J}_{0}\}\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}{J}_{0}=\{j\in \{0,1,\dots ,l\}\hspace{0.17em}|\hspace{0.17em}{n}_{j}=1\}\text{.}$$
In view of the fact that the $\stackrel{\sim}{W}\prime \text{-orbit}$ of
$\lambda \in X$ is precisely the set $\lambda \in X\prime $
one finds that each coset of $X\prime $ in $X$ meets ${E}^{*}$
in a unique element of ${E}^{*}\cap X\text{.}$ Thus
$$X/X\prime =\{{\lambda}_{j}+X\prime \hspace{0.17em}|\hspace{0.17em}j\in {J}_{0}\}$$
is an enumeration of the group $X/X\prime ,$ with
$[X:X\prime ]=\left|{J}_{0}\right|\text{.}$
(We are using the standard notations for index and cardinality.) It is known that $X\prime =X$ (i.e. all the
${n}_{i}\text{'s'}$ in the expansion of $-{\alpha}_{0}^{\vee}$
in the fundamental coroots are greater than 1) if and only if $\Delta $ is of type
${G}_{2},$ ${F}_{4},$ or
${E}_{8}\text{.}$

The action of $\stackrel{\sim}{W}$ on $E$ permutes the simplices ${\left({E}^{*}\right)}^{w}$ in our tessellation parametrized by $w\in \stackrel{\sim}{W}\prime ,$ which shows that $\stackrel{\sim}{W}$ is the semidirect extension of the stabilizer $\Omega $ of ${E}^{*}$ by the normal subgroup $\stackrel{\sim}{W}\prime \text{.}$ Let ${\gamma}_{j}$ denote (as a temporary notation) the unique element of $\stackrel{\sim}{W}\prime $ which takes ${E}^{*}$ to its image ${E}^{*}-{\lambda}_{j}$ under the translation ${T}_{-{\lambda}_{j}}\in \stackrel{\sim}{W}\text{.}$ Since $0\in {E}^{*}-{\lambda}_{j}$ for $j\in {J}_{0},$ and since each simplex contains a unique point of $X\prime ,$ we find that ${\gamma}_{j}$ must lie in the stabilizer $W$ of the zero element of $E$ in $\stackrel{\sim}{W}\text{.}$ On the other hand, since the map $X/X\prime \to \stackrel{\sim}{W}/\stackrel{\sim}{W}\prime $ sending the coset $\lambda +X\prime $ to the coset $\stackrel{\sim}{W}\prime {T}_{\lambda}$ is clearly (well-defined and is) an isomorphism of groups, considering its composite with the canonical isomorphism $\stackrel{\sim}{W}/\stackrel{\sim}{W}\prime \to \Omega ,$ we see that for every $j\in {J}_{0}$ there is a unique element of $\Omega $ lying in the coset $\stackrel{\sim}{W}\prime {T}_{{\lambda}_{j}}$ (viz. the image of ${\lambda}_{j}+X\prime $ under this composite); it is clear that this element is simply ${\gamma}_{j}{T}_{{\lambda}_{j}}$ (which sends ${E}^{*}$ to $({E}^{*}-{\lambda}_{j})+{\lambda}_{j},$ and hence lies in $\Omega \text{).}$ Setting ${\omega}_{j}={\gamma}_{j}{T}_{{\lambda}_{j}},$ we have the enumeration $$\Omega =\left\{{\omega}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in {J}_{0}\},\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}\left|\Omega \right|=[X:X\prime ]=\left|{J}_{0}\right|\text{.}$$

For $j\in {J}_{0},$ the action of ${\omega}_{j}\in \Omega $ on $\stackrel{\sim}{W}\prime $ (through conjugation) permutes the Coxeter generators $\left\{{R}_{i}\hspace{0.17em}\right|\hspace{0.17em}0\leqq i\leqq l\}$ in the same manner in which it permutes the vertices $\left\{{n}_{i}^{-1}{\lambda}_{i}\hspace{0.17em}\right|\hspace{0.17em}0\leqq i\leqq l\}$ of ${E}^{*}\text{.}$ Thus, if ${\theta}_{j}$ denotes the permutation of $\{0,1,\dots ,l\}$ such that ${\omega}_{j}^{-1}{R}_{i}{\omega}_{j}={R}_{{\theta}_{j}\left(i\right)},$ then $${\left(\frac{{\lambda}_{i}}{{n}_{i}}\right)}^{{\omega}_{j}}=\frac{{\lambda}_{{\theta}_{j}\left(i\right)}}{{n}_{{\theta}_{j}\left(i\right)}},\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}{\lambda}_{i}^{{\gamma}_{j}}={\lambda}_{{\theta}_{j}\left(i\right)}-{n}_{i}{\lambda}_{j}={\lambda}_{{\theta}_{j}\left(i\right)}-{n}_{{\theta}_{j}\left(i\right)}{\lambda}_{j}\text{.}$$ Verifying directly the fact that the ordered basis $${\alpha}_{0}^{\vee},{\alpha}_{1}^{\vee},\dots ,{\alpha}_{j-1}^{\vee},{\alpha}_{j+1}^{\vee},\dots ,{\alpha}_{l}^{\vee}$$ is the dual, with respect to the $W\text{-invariant}$ pairing $\u27e8,\u27e9,$ of the ordered basis $${\lambda}_{0}-{n}_{0}{\lambda}_{j},\phantom{\rule{1em}{0ex}}{\lambda}_{1}-{n}_{1}{\lambda}_{j},\dots ,{\lambda}_{j-1}-{n}_{j-1}{\lambda}_{j},\phantom{\rule{1em}{0ex}}{\lambda}_{j+1}-{n}_{j+1}{\lambda}_{j},\dots ,{\lambda}_{l}-{n}_{l}{\lambda}_{j},$$ one concludes immediately that ${\alpha}_{i}^{{\gamma}_{j}}={\alpha}_{{\theta}_{j}\left(i\right)}\text{.}$ Since ${\lambda}_{0}^{{\gamma}_{j}}={\lambda}_{{\theta}_{j}\left(0\right)}-{\lambda}_{j},$ we have ${\theta}_{j}\left(0\right)=j\text{.}$ This shows that ${\gamma}_{j}^{-1}$ sends ${\alpha}_{j}$ to ${\alpha}_{0}\in {\Delta}_{-},$ and sends every other fundamental root into ${\Delta}_{+}\text{.}$ Thus ${\gamma}_{j}^{-1}$ sends all positive roots in the subroot-system ${\Delta}_{j}$ of rank $l-1$ spanned by $\left\{{\alpha}_{i}\hspace{0.17em}\right|\hspace{0.17em}1\leqq i\leqq l,\hspace{0.17em}i\ne j\}$ into ${\Delta}_{+}\text{;}$ conversely, if $\alpha $ is a positive root not in ${\Delta}_{j},$ say $\alpha =\sum _{i=1}^{l}{a}_{i}{\alpha}_{i}$ with ${a}_{j}\ne 0,$ then looking at $$\beta ={\left(\alpha \right)}^{{\gamma}_{j}^{-1}}=\sum _{i=1}^{l}{a}_{i}{\alpha}_{{\theta}_{j}^{-1}\left(i\right)}={a}_{j}{\alpha}_{0}+\sum _{\underset{i\ne j}{i=1}}^{l}{a}_{i}{\alpha}_{{\theta}_{j}^{-1}\left(i\right)}$$ we find at least one coefficient (in the expansion of $\beta $ in fundamental roots) negative, which gives ${\alpha}^{{\gamma}_{j}^{-1}}\in {\Delta}_{-}\text{.}$ If ${\sigma}_{0}$ (resp. ${\sigma}_{j}\text{)}$ denotes the unique element of maximal length in $W$ (resp. in the Weyl group $W\left({\Delta}_{j}\right)$ of ${\Delta}_{j}\text{),}$ verifying that ${\sigma}_{j}{\sigma}_{0}$ is the unique element of $W$ which makes precisely those positive roots negative which do not lie in ${\Delta}_{j}$ (here uniqueness follows from the standard theory of Weyl groups), we are forced to the conclusion ${\gamma}_{j}^{-1}={\sigma}_{j}{\sigma}_{0}\text{.}$ Thus we have the formula $${\omega}_{j}={\sigma}_{0}{\sigma}_{j}{T}_{{\lambda}_{j}}\phantom{\rule{1em}{0ex}}\text{(for}\hspace{0.17em}j\in {J}_{0}\text{)}$$ for the elements of $\Omega \text{;}$ note that every element of $W\left({\Delta}_{j}\right),$ and hence ${\sigma}_{j},$ fixes ${\lambda}_{j},$ which gives also ${\omega}_{j}={\sigma}_{0}{T}_{{\lambda}_{j}}{\sigma}_{j}\text{.}$ One calls ${\sigma}_{0}$ the "apposition element" of $W\text{.}$

Note that the permutation ${\theta}_{j}$ of $\{0,1,\dots ,l\}$ is an automorphism of the extended Dynkin diagram, and that its action is completely determined by its action on the subset ${J}_{0}\text{.}$ (The latter claim follows from the fact that ${J}_{0}$ is a proper subset of $\{0,1,\dots ,l\}$ in all cases except those of type ${A}_{l},$ and in such cases the extended Dynkin diagram is a tree and the nodes corresponding to ${J}_{0}$ are precisely the "end-nodes" of the diagram. Incidentally, this also shows that ${\gamma}_{j}$ is the unique element of $W$ sending ${\alpha}_{0}$ to ${\alpha}_{j}$ and preserving the set $\{{\alpha}_{0},{\alpha}_{1},\dots ,{\alpha}_{l}\}\text{.)}$ The action of ${\theta}_{j}$ on ${J}_{0}$ itself can be read-off from the group-structure of $\{{\lambda}_{j}+X\prime \hspace{0.17em}|\hspace{0.17em}j\in {J}_{0}\}$ (which is quite easy to write down in each given case), in view of the following straightforward equivalences on $i,j,k\in {J}_{0}\text{:}$ $$\begin{array}{c}{\theta}_{j}\left(i\right)=k\iff {\lambda}_{i}+{\lambda}_{j}\equiv {\lambda}_{k}\hspace{0.17em}\text{(mod}\hspace{0.17em}X\prime \text{)}\iff {\lambda}_{i}^{{\omega}_{j}}={\lambda}_{k}\iff {\alpha}_{i}^{{\gamma}_{j}}={\alpha}_{k}\\ \iff {\theta}_{i}{\theta}_{j}={\theta}_{k}\iff {\omega}_{i}{\omega}_{j}={\omega}_{k}\iff {\gamma}_{i}{\gamma}_{j}={\gamma}_{k}\text{.}\end{array}$$

The isomorphism ${\omega}_{j}\to {\gamma}_{j}$ (for $j\in {J}_{0}\text{)}$ of the group $\Omega $ onto the subgroup ${\Omega}_{0}=\left\{{\gamma}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in {J}_{0}\}$ of $W,$ given by the last equivalence above, can also realized as follows: Elements of $\Omega $ leave fixed the barycentre ${\beta}_{0}={\left|{J}_{0}\right|}^{-1}\sum _{j\in {J}_{0}}{\lambda}_{j}$ of the subsimplex of ${E}^{*}$ having vertices ${E}^{*}\cap X\text{;}$ note that ${\beta}_{0}$ does not lie in $X$ except in the three extreme cases (types ${G}_{2},$ ${F}_{4},$ ${E}_{8}\text{)}$ when $X\prime =X,$ ${J}_{0}=\left\{{\lambda}_{0}\right\}$ and ${\beta}_{0}={\lambda}_{0}=0\text{.}$ Thus we have ${\beta}_{0}^{{\gamma}_{j}}+{\lambda}_{j}={\beta}_{0},$ which gives $${\gamma}_{j}^{-1}{T}_{{\beta}_{0}}{\gamma}_{j}={T}_{{\beta}_{0}-{\lambda}_{j}}={T}_{{\beta}_{0}}{T}_{{\lambda}_{j}}^{-1},\phantom{\rule{1em}{0ex}}\text{i.e.}\phantom{\rule{1em}{0ex}}{\gamma}_{j}={T}_{{\beta}_{0}}{\gamma}_{j}{T}_{{\lambda}_{j}}{T}_{-{\beta}_{0}}={T}_{{\beta}_{0}}{\omega}_{j}{T}_{-{\beta}_{0}}\text{.}$$ Hence conjugation by ${T}_{-{\beta}_{0}}$ gives our isomorphism $\Omega \to {\Omega}_{0}\text{.}$

**1.3.** Now let $l\left(\right)$ denote the usual length-function on $\stackrel{\sim}{W}\prime $
in terms of its Coxeter generators $\{{R}_{0},{R}_{1},\dots ,{R}_{l}\}$
as in the standard theory of Coxeter groups. Thus for $w\in \stackrel{\sim}{W}\prime ,$
$l\left(w\right)$ is the smallest integer $k$ such that
$w={R}_{{i}_{1}}{R}_{{i}_{2}}\dots {R}_{{i}_{k}}$
for $0\leqq {i}_{q}\leqq l,$ $1\leqq q\leqq k\text{.}$
such an expression for $w$ (with $k=l\left(w\right)\text{)}$
is called a reduced expression. One knows that $l\left(w\right)$ equals the number of reflecting hyperplanes
(i.e. fixed hyperplanes of reflections in $\stackrel{\sim}{W}\prime \text{)}$ that separate
${E}^{*}$ from the $w\text{-simplex}$
${\left({E}^{*}\right)}^{w}\text{.}$ The restriction of
$l\left(\right)$ to $W$ coincides with the length-function on $W$ defined
in terms of its Coxeter generators $\{{R}_{1},{R}_{2},\dots ,{R}_{l}\}\text{;}$
for $\sigma \in W,$ $l\left(\sigma \right)$
also equals the number of elements of ${\Delta}_{+}$ that are carried into
${\Delta}_{-}$ under $\sigma \text{.}$

For $\lambda \in E$ and $\sigma \in W,$ let
${\lambda}^{\left(\sigma \right)}$ denote the unique translate under $X$ (i.e.
image under $D\text{)}$ of ${\lambda}^{\sigma}$ such that
${\lambda}^{\left(\sigma \right)}\in {E}^{\u266f}\text{.}$
In general ${\lambda}^{\left(\sigma \right)}$ depends very much on both $\lambda $
and $\sigma \text{.}$ But when $\lambda $ lies in the interior of ${E}^{*}$
the difference ${\lambda}^{\left(\sigma \right)}-{\lambda}^{\sigma}$
is independent of $\lambda ,$ and can be seen to be equal to
$${\delta}_{\sigma}=\sum _{i\in {I}_{\sigma}}{\lambda}_{i},\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}{I}_{\sigma}=\left\{i\hspace{0.17em}\right|\hspace{0.17em}1\leqq i\leqq l,\hspace{0.17em}l\left(\sigma {R}_{i}\right)<l\left(\sigma \right)\}\text{.}$$
Thus ${\lambda}^{\left(\sigma \right)}=\lambda {w}_{\sigma}$
where ${w}_{\sigma}=\sigma {T}_{{\delta}_{\sigma}}\in \stackrel{\sim}{W}\text{.}$
*It is this set* $\left\{{w}_{\sigma}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}$
*of affine transformations of* $E$ *that is basic to the formulation of our conjecture in Part I* (§ 2).

Let $\lambda $ be an interior of ${E}^{*}\text{;}$ thus $\lambda $ is $\stackrel{\sim}{W}\prime \text{-regular.}$ Setting $${W}^{\u266f}=\{\sigma \in W\hspace{0.17em}|\hspace{0.17em}{w}_{\sigma}\in \stackrel{\sim}{W}\prime \}=\{\sigma \in W\hspace{0.17em}|\hspace{0.17em}{\delta}_{\sigma}\in X\prime \},$$ it is clear that the $\stackrel{\sim}{W}\prime \text{-orbit}$ of $\lambda $ meets ${E}^{\u266f}$ precisely in the set $\left\{{\lambda}^{\left(\sigma \right)}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in {W}^{\u266f}\}$ of cardinality $\left|{W}^{\u266f}\right|=[W:{\Omega}_{0}]\text{.}$ The last equality follows from ${\left({\lambda}^{\left(\gamma \right)}\right)}^{\left(\sigma \right)}={\lambda}^{\left({\gamma}_{\sigma}\right)}$ for $\gamma \in {\Omega}_{0}$ and $\sigma \in W,$ the map $\gamma \to {w}_{\gamma}$ being simply the isomorphism ${\Omega}_{0}\to \Omega $ already encountered (with ${\gamma}_{j}\to {\omega}_{j}={w}_{{\gamma}_{j}}\text{).}$ This shows that ${W}^{\u266f}$ is a set of distinguished right-coset representatives for the right-coset space $${\Omega}_{0}\backslash W=\{{\Omega}_{0}\xb7\sigma \hspace{0.17em}|\hspace{0.17em}\sigma \in W\}\text{.}$$ Incidentally, this also gives the famous formula for the order of the Weyl group: $$\left|W\right|=l!{n}_{1}{n}_{2}\cdots {n}_{l}[X:X\prime ],$$ as one sees on making an elementary calculation verifying that the volume of ${E}^{\u266f}$ is $l!{n}_{1}{n}_{2}\cdots {n}_{l}$ times the volume of ${E}^{*}\text{.}$ Let us digress slightly at this point to mention another important formula involving the ${n}_{i}\text{'s}$ (this time their sum), which we shall require later: $$\sum _{i=1}^{l}{n}_{i}={h}_{\Delta}-1,\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}{h}_{\Delta}(=\text{the Coxeter no. of}\hspace{0.17em}\Delta )=\frac{2r}{l},\hspace{0.17em}r=\left|{\Delta}_{+}\right|\text{.}$$ An alternative interpretation of this formula is that the "height" $\sum _{i=1}^{l}{n}_{i}$ of the long dominant coroot $-{\alpha}_{0}^{\vee}$ equals the largest "exponent" of $W\text{;}$ this is a special case of a deep observation of Kostant and Steinberg on the "duality" between the two partitions of the number $r$ given by the exponents (which are known to add up to $r\text{)}$ on one hand, and by the number of elements of ${\Delta}_{+}$ of various heights on the other hand.

**1.4.** It is well known, and easily seen, that the set ${\stackrel{\sim}{W}}^{+}$ defined by
$${\stackrel{\sim}{W}}^{+}=\{w\in \stackrel{\sim}{W}\prime \hspace{0.17em}|\hspace{0.17em}l\left(w{R}_{i}\right)>l\left(w\right)\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}1\leqq i\leqq l\}$$
is a set of distinguished left-coset representatives for the left-coset space
$\stackrel{\sim}{W}\prime /W=\{w\xb7W\hspace{0.17em}|\hspace{0.17em}w\in W\prime \}\text{.}$
The cone ${E}^{+}$ is tessellated by the images of ${E}^{*}$ under
${\stackrel{\sim}{W}}^{+},$ just as the closure of ${E}^{\u266f}$
is tessellated by the images under
$${\stackrel{\sim}{W}}^{\u266f}=\left\{{w}_{\sigma}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in {W}^{\u266f}\}=\left\{{w}_{\sigma}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}\cap \stackrel{\sim}{W}\prime \text{.}$$
In particular, ${\stackrel{\sim}{W}}^{\u266f}$ is contained in
${\stackrel{\sim}{W}}^{+}\text{.}$

Let us note that in the tessellation $\left\{{\left({E}^{*}\right)}^{w}\hspace{0.17em}\right|\hspace{0.17em}w\in {\stackrel{\sim}{W}}^{\u266f}\}$ of the closure of ${E}^{\u266f}$ there is a unique simplex which contains $\delta =\sum _{i=1}^{l}{\lambda}_{i}\text{.}$ We call this the "top simplex" in ${E}^{\u266f},$ and the corresponding element ${w}_{0}$ of ${\stackrel{\sim}{W}}^{\u266f}$ the "top element" of ${\stackrel{\sim}{W}}^{\u266f}\text{.}$ Writing ${E}_{\text{top}}^{*}$ for the top simplex we have $${\left({E}^{*}\right)}^{{w}_{0}}={E}_{\text{top}}^{*}=-{E}^{*}+\delta ={\left({E}^{*}\right)}^{{\sigma}_{0}{T}_{\delta}},$$ which shows that ${w}_{0}{T}_{-\delta}{\sigma}_{0}\in \Omega \text{.}$ Denoting by ${j}_{0}$ the unique index in ${J}_{0}$ such that $\delta \equiv {\lambda}_{{j}_{0}}$ $\left(\text{mod}\hspace{0.17em}X\prime \right),$ we see that the zero element in $E$ goes under ${w}_{0}$ to $\delta -{\lambda}_{{j}_{0}}$ (this being the only point of ${E}_{\text{top}}^{*}$ in the root-lattice) and hence under ${w}_{0}{T}_{-\delta}{\sigma}_{0}$ to ${(-{\lambda}_{{j}_{0}})}^{{\sigma}_{0}}={\lambda}_{{j}_{0}}$ where this last equality is immediate from ${\delta}^{{\sigma}_{0}}=-\delta \text{.}$ Since ${\omega}_{{j}_{0}}$ is the only element of $\Omega $ which sends ${\lambda}_{0}=0$ to ${\lambda}_{{j}_{0}},$ we have ${w}_{0}={\omega}_{{j}_{0}}{\sigma}_{0}{T}_{\delta}\text{.}$ From this one easily gets ${w}_{0}={\sigma}_{{j}_{0}}{T}_{\delta -{\lambda}_{{j}_{0}}},$ on substituting ${\omega}_{{j}_{0}}^{-1}={\sigma}_{0}{T}_{{\lambda}_{{j}_{0}}}{\sigma}_{{j}_{0}}$ proved earlier. We call ${\omega}_{{j}_{0}}$ the "apposition element" of $\Omega \text{.}$

**1.5.** We shall now establish certain relativizations of the concepts and notations ${\stackrel{\sim}{W}}^{+}$
and ${\stackrel{\sim}{W}}^{\u266f},$ relative to a proper subset $J$ of
$\{0,1,\dots ,l\}$ (empty subset permitted). Let us write
$${E}_{J}^{*}=\{\lambda \in {E}^{*}\hspace{0.17em}|\hspace{0.17em}{\lambda}^{{R}_{j}}=\lambda \iff j\in J\}\text{.}$$
This is an open subset in its own affine-span (which passes through the origin if and only if $0\in J\text{).}$
When $J$ is empty this is precisely the interior of ${E}^{*},$ whose boundary is the
disjoint union of ${E}_{J}^{*}$ as $J$ ranges over the
${2}^{l+1}-2$ other subsets. The stabilizer of
$\lambda \in {E}_{J}^{*}$ in $\stackrel{\sim}{W}\prime $
is precisely the subgroup ${\stackrel{\sim}{W}}^{J}$ generated by
$\left\{{R}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in J\}\text{.}$
In the right-coset space ${\stackrel{\sim}{W}}^{J}\backslash \stackrel{\sim}{W}\prime $
each coset ${\stackrel{\sim}{W}}^{J}\xb7w$ has a unique element of minimal length,
all such elements forming the set ${\stackrel{\sim}{W}}_{J}$ of distinguished right-coset representatives, where
$${\stackrel{\sim}{W}}_{J}=\{w\in \stackrel{\sim}{W}\prime \hspace{0.17em}|\hspace{0.17em}l\left({R}_{j}w\right)>l\left(w\right)\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}j\in J\}\text{;}$$
and the double-coset space
$${\stackrel{\sim}{W}}^{J}\backslash \stackrel{\sim}{W}\prime /W=\{{\stackrel{\sim}{W}}^{J}\xb7w\xb7W\hspace{0.17em}|\hspace{0.17em}w\in \stackrel{\sim}{W}\prime \}$$
has ${\stackrel{\sim}{W}}_{J}^{+}={\stackrel{\sim}{W}}_{J}\cap {\stackrel{\sim}{W}}^{+}$
as a set of distinguished double-coset representatives. The $\stackrel{\sim}{W}\prime \text{-orbit}$
of $\lambda \in {E}_{J}^{*}$ is parametrized by ${\stackrel{\sim}{W}}_{J}$
(i.e. $w\to {\lambda}^{w}$ is a bijection from ${\stackrel{\sim}{W}}_{J}$
to this orbit), and the intersection of this orbit with ${E}^{+}$ is precisely
$\left\{{\lambda}^{w}\hspace{0.17em}\right|\hspace{0.17em}w\in {\stackrel{\sim}{W}}_{J}^{+}\}$
parametrized by ${\stackrel{\sim}{W}}_{J}^{+},$ Note that this intersection lies in the
interior of ${E}^{+}$ if and only if ${E}_{J}^{*}$ itself does, which happens
precisely when $J$ is either empty or the singleton $\left\{0\right\}\text{.}$ Setting
$${\stackrel{\sim}{W}}_{J}^{++}=\{w\in {\stackrel{\sim}{W}}^{J}\hspace{0.17em}|\hspace{0.17em}{\left({E}_{J}^{*}\right)}^{w}\hspace{0.17em}\text{lies in the interior of}\hspace{0.17em}{E}^{+}\},$$
and
$${\stackrel{\sim}{W}}_{J}^{\u266f}=\{w\in {\stackrel{\sim}{W}}^{J}\hspace{0.17em}|\hspace{0.17em}{\left({E}_{J}^{*}\right)}^{w}\hspace{0.17em}\text{lies in}\hspace{0.17em}{E}^{\u266f}\},$$
it is clear that ${\stackrel{\sim}{W}}_{J}^{++}$ (resp.
${\stackrel{\sim}{W}}_{J}^{\u266f}\text{)}$ is the subset of
${\stackrel{\sim}{W}}_{J}$ which parametrizes the intersection of the
$\stackrel{\sim}{W}\prime \text{-orbit}$ of
$\lambda \in {E}_{J}^{*}$ with the interior of ${E}^{+}$
(resp. with ${E}^{\u266f}\text{).}$ Thus
${\stackrel{\sim}{W}}_{J}^{\u266f}\subset {\stackrel{\sim}{W}}_{J}^{++}\u2ac5{\stackrel{\sim}{W}}_{J}^{+}\subset {\stackrel{\sim}{W}}_{J},$
where the middle inclusion is also strict whenever $J$ is nonempty and contains an index other than $0\text{.}$
It is also clear that ${\stackrel{\sim}{W}}_{J}^{\u266f}={\stackrel{\sim}{W}}^{\u266f}$
if and only if $J$ is empty. We claim that ${\stackrel{\sim}{W}}_{J}^{\u266f}\u2ac5{\stackrel{\sim}{W}}^{\u266f}$
in general. For, suppose $w$ lies in ${\stackrel{\sim}{W}}_{J}^{\u266f}$ but not in
${\stackrel{\sim}{W}}^{\u266f}\text{.}$ Then $w$ sends the interior of
${E}^{*}$ outside ${E}^{\u266f}\text{;}$ but since
${\lambda}^{w}\in {E}^{\u266f}$ for $\lambda \in {E}_{J}^{*}$
there is a wall of ${E}^{\u266f}$ on which ${\lambda}^{w}$ lies and which separates the
fundamental simplex from the $w\text{-simplex}$ ${\left({E}^{*}\right)}^{w}\text{.}$
Let $w\prime =wR$ where $R$ is the reflection in this wall. Then
${\lambda}^{w\prime}={\lambda}^{w}$ implies
$w\prime \in {\stackrel{\sim}{W}}^{J}\xb7w\text{;}$
and since the said wall of ${E}^{\u266f}$ does not separate the fundamental simplex and the
$w\prime \text{-simplex}$ we have $l\left(w\prime \right)<l\left(w\right),$
which contradicts the fact that $w\in {\stackrel{\sim}{W}}_{J}$ is the unique element of minimal
length in the coset ${\stackrel{\sim}{W}}^{J}\xb7w\text{.}$ Thus we have
$${\stackrel{\sim}{W}}_{J}^{\u266f}\u2ac5{\stackrel{\sim}{W}}^{\u266f}\cap {\stackrel{\sim}{W}}_{J}^{+}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\stackrel{\sim}{W}}_{J}^{\u266f}={\stackrel{\sim}{W}}^{\u266f}\cap {\stackrel{\sim}{W}}_{J}^{++}\text{.}$$

We remark that ${\stackrel{\sim}{W}}_{J}^{\u266f}$ could be empty, and that this situation does occur
for some $J$ if and only if $\Delta $ is not of type
${G}_{2},$ ${F}_{4},$ or
${E}_{8}\text{.}$ This follows from the more precise claim that
${\stackrel{\sim}{W}}_{J}^{\u266f}$ *is empty if and only if the complement of* $J$
*in* $\{0,1,\dots ,l\}$
*consists of a single index* $j$ *with* $j\in {J}_{0},$
$j\ne {j}_{0},$ and that *whenever*
${\stackrel{\sim}{W}}_{J}^{\u266f}$ *is nonempty the top element*
${w}_{0}$ *of* ${\stackrel{\sim}{W}}^{\u266f}$ *lies in*
$\text{(}{\stackrel{\sim}{W}}_{J}^{++}$ and therefore in)
${\stackrel{\sim}{W}}_{J}^{\u266f}\text{.}$ For this first consider the situation
when the complement of $J$ is the singleton $\left\{i\right\}\text{;}$ then
${E}_{J}^{*}$ consists of just one point, viz. the vertex
${n}_{i}^{-1}{\lambda}_{i}\text{.}$ If
$i\in {J}_{0}$ then ${n}_{i}=1$ and the
$\stackrel{\sim}{W}\prime \text{-orbit}$ of ${E}_{J}^{*}$
is contained in the set of ${2}^{l}$ vertices of the parallelepiped ${E}^{\u266f}$ (the
intersection of the closure of ${E}^{\u266f}$ with the weight-lattice); but none of these lie in
$\stackrel{\sim}{W}\prime \text{-orbit}$ of ${\lambda}_{i}$ unless
$\delta -{\lambda}_{i}\in X\prime ,$ i.e.
$i={j}_{0}\text{.}$ This proves the "if" part of the claim. Now note that the image of
${n}_{i}^{-1}{\lambda}_{i}$ under
${w}_{0}={\omega}_{{j}_{0}}{\sigma}_{0}{T}_{\delta}$
is of the form $\delta -{n}_{k}^{-1}{\lambda}_{k}$
with ${n}_{k}={n}_{i}$ (for ${n}_{i}^{-1}{\lambda}_{i}$
goes to ${n}_{j}^{-1}{\lambda}_{j}$ under
${\omega}_{{j}_{0}}$ if $j={\theta}_{{j}_{0}}\left(i\right),$
and so to ${n}_{k}^{-1}{\lambda}_{k}$ under
${\omega}_{{j}_{0}}{\tau}_{0}={\omega}_{{j}_{0}}\xb7(-{\sigma}_{0})$
if ${R}_{k}$ is the image of ${R}_{j}$ under the Dynkin outer automorphism of $W$
which is known to be given by conjugation under the endomorphism ${\tau}_{0}=-{\sigma}_{0}$
of $E$ sending $\alpha $ to $-{\alpha}^{{\sigma}_{0}}\text{).}$
It is clear that $\delta -{n}_{k}^{-1}{\lambda}_{k}\in {E}^{\u266f}$
if ${n}_{k}>1,$ which takes care of our assertion when $J$ has
cardinality $l\text{.}$ In other cases, the closure of ${E}_{J}^{*}$ has at
least two vertices ${n}_{i}^{-1}{\lambda}_{i}$ and
${n}_{i\prime}^{-1}{\lambda}_{i\prime}$
and its image under ${\omega}_{0}$ contains a point of the form
$\delta -a\left({n}_{k}^{-1}{\lambda}_{k}\right)-(1-a)\left({n}_{k\prime}^{-1}{\lambda}_{k\prime}\right)$
with $0<a<1,$ which clearly lies in ${E}^{\u266f}$
(all the coefficients in the expansion of this point in the basis ${\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{l}$
of $E$ being $>0$ and $\leqq 1\text{)}\text{;}$
this shows that ${\left({E}_{J}^{*}\right)}^{{w}_{0}}$
is contained in ${E}^{\u266f},$ i.e. ${w}_{0}\in {W}_{J}^{\u266f}$
as claimed.

**1.6.** We shall now introduce a canonical partial-order $\leqq $ on the set ${\stackrel{\sim}{W}}^{+},$
which would be needed in § 5 for streamlining Conjecture II and its applications. It is defined as the restriction to ${\stackrel{\sim}{W}}^{+}$
of the "Bruhat partial-order" $\stackrel{B}{\leqq}$ on $\stackrel{\sim}{W}\prime $
(see the next paragraph) which was introduced in [Ver1971] for arbitrary Coxeter groups in terms of given Coxeter generators (and earlier in [Ver1966] and [Ver1968] for
finite Weyl groups; the name "Bruhat partial-order" in that case was motivated by its relation with the "Bruhat decomposition" parametrized by a Weyl group). In these
references $\stackrel{B}{\leqq}$ was denoted simply $\leqq ,$ but here we have made this variation
because there is another partial-order $\stackrel{A}{\leqq}$ on $\stackrel{\sim}{W}\prime $
which we shall call the "affine partial-order" ("A" for "affine" and "B" for "Bruhat") for the definition of which one needs the afRne Weyl group structure of
$\stackrel{\sim}{W}\prime \text{.}$ This $\stackrel{A}{\leqq}$
is a more natural partial-order for the representation-theoretic purposes we have in mind in Part II, but on the set
${\stackrel{\sim}{W}}^{+}$ the two orderings coincide.

$\stackrel{B}{\leqq}$ is defined as the weakest partial-order on $\stackrel{\sim}{W}\prime $ such that $$l\left(wR\right)<l\left(w\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}wR\stackrel{B}{\leqq}w,$$ for arbitrary reflection $R$ and element $w$ in $\stackrel{\sim}{W}\prime \text{.}$ Thus, for $w,$ $w\prime $ in $\stackrel{\sim}{W}\prime ,$ if ${w}^{-1}w\prime $ is a reflection one has the dichotomy that one and exactly one of $w\stackrel{B}{\leqq}w\prime $ and $w\prime \stackrel{B}{\leqq}w$ holds. It has been shown in [Ver1972, Proposition 2] that $\stackrel{B}{\leqq}$ coincides with the (weakest) partial-order $\stackrel{B\prime}{\leqq}$ obtained by demanding $wR\stackrel{B\prime}{\leqq}w$ for only those pairs $(w,R)$ for which $l\left(wR\right)=l\left(w\right)-1\text{.}$ It is also known [Ver1972, Corollary 1] that if $w={R}_{{i}_{1}}{R}_{{i}_{2}}\cdots {R}_{{i}_{m}}$ is a reduced expression then $w\prime \stackrel{B}{\leqq}w$ if and only if $w\prime $ is of the form ${R}_{{j}_{1}}{R}_{{j}_{2}}\cdots {R}_{{j}_{n}}$ for some subsequence $({j}_{1},{j}_{2},\dots ,{j}_{n})$ of the sequence of indices $({i}_{1},{i}_{2},\dots ,{i}_{m})$ (all indices ranging in $\{0,1,\dots ,l\}\text{),}$ where this expression for $w\prime $ can also be assumed reduced by passing to a further subsequence.

The partial-order $\stackrel{A}{\leqq}$ on $\stackrel{\sim}{W}\prime $ is defined to be the weakest partial-order with the following property: Let $w\in \stackrel{\sim}{W}\prime $ and $R$ be a reflection parallel to ${R}_{\alpha}$ for $\alpha \in {\Delta}_{+}\text{;}$ for an interior point $\lambda $ of ${\left({E}^{*}\right)}^{w}$ the sign of the (nonzero) real number $c={c}_{\lambda}$ defined by $\lambda -{\lambda}^{R}=c\alpha $ is clearly independent of $\lambda $ (depends only on $w$ and $R\text{);}$ then we require that $$wR\stackrel{A}{\leqq}w\hspace{0.17em}\text{whenever}\hspace{0.17em}c>0\hspace{0.17em}\text{(for some, and hence all,}\hspace{0.17em}\lambda \text{).}$$

Once again we have the dichotomy $\text{"}w\stackrel{A}{\leqq}w\prime $ versus $w\prime \stackrel{A}{\leqq}w\text{"}$ in case ${w}^{-1}w\prime $ is a reflection. Let us observe that for arbitrary $(w,R)$ there is an odd number of reflections parallel to $R$ which separate the $w\text{-simplex}$ from the $wR\text{-simplex,}$ so that if $R\prime ,R\u2033,R\u2034,\dots $ is the enumeration of these reflections in the order of their closeness to the $w\text{-simplex}$ then $wR=wR\prime R\u2033R\u2034\dots \text{;}$ and for an interior point $\lambda $ in the $w\text{-simplex}$ ${\left({E}^{*}\right)}^{w},$ denoting its successive mirror-images by $\lambda \prime =\lambda R\prime ,$ $\lambda \u2033=\lambda \prime R\u2033,$ $\lambda \u2034=\lambda \u2033R\u2034,\dots ,$ one finds the consecutive differences $\lambda -\lambda \prime ,$ $\lambda \prime -\lambda \u2033,$ $\lambda \u2033-\lambda \u2034,\dots $ to be of the form $c\alpha ,$ $(1-c)\alpha ,$ $c\alpha ,\dots $ alternately, where $0<\left|c\right|<1\text{.}$ It follows that $\stackrel{A}{\leqq}$ coincides with the (weakest) partial-order $\stackrel{A\prime}{\leqq}$ obtained by demanding $wR\stackrel{A\prime}{\leqq}w$ whenever $0<c<1$ (where $c$ is defined as before by $\lambda -{\lambda}^{R}=c\alpha ,$ and $0<c<1$ holds either for all or for none $\lambda \in \text{interior}{\left({E}^{*}\right)}^{w}\text{).}$

It is easy to see that the condition $c>0$ the pair $(w,R)$ is equivalent to the following: $$\{\begin{array}{cc}l\left(wR\right)>l\left(w\right)& \text{if}\phantom{\rule{1em}{0ex}}\text{the}\hspace{0.17em}R\text{-plane}\hspace{0.17em}\text{does not meet}\hspace{0.17em}\text{interior}\hspace{0.17em}{E}^{+},\\ l\left(wR\right)<l\left(w\right)& \text{if}\phantom{\rule{1em}{0ex}}\text{the}\hspace{0.17em}R\text{-plane}\hspace{0.17em}\text{does meet}\hspace{0.17em}\text{interior}\hspace{0.17em}{E}^{+},\end{array}$$ where $R\text{-plane}$ means the fixed hyperplane of $R\text{.}$ It follows that for $w,w\prime $ in the Weyl group $W$ (resp. for $w,w\prime $ in the set ${\stackrel{\sim}{W}}^{+}$ of distinguished coset representatives for $\stackrel{\sim}{W}\prime /W\text{)},$ if ${w}^{-1}w\prime $ is a reflection then $$w\stackrel{A}{\leqq}w\prime \hspace{0.17em}\text{is equivalent to}\hspace{0.17em}w\prime \stackrel{B}{\leqq}w\hspace{0.17em}\text{(resp.}\hspace{0.17em}w\stackrel{B}{\leqq}w\prime \text{).}$$ This statement is true even without the restriction that ${w}^{-1}w\prime $ be a reflection; to deduce this generalization it suffices to show that for $w,w\prime \in W$ (resp. ${\stackrel{\sim}{W}}^{+}\text{)},$ if $w\stackrel{A}{\leqq}w\prime $ then one can find reflections ${S}_{1},{S}_{2},\dots ,{S}_{k}$ with $$\begin{array}{cc}w\stackrel{A}{\leqq}w{S}_{1}\stackrel{A}{\leqq}w{S}_{1}{S}_{2}\stackrel{A}{\leqq}\dots \stackrel{A}{\leqq}w{S}_{1}{S}_{2}\dots {S}_{k}=w\prime & {\text{(1.1)}}_{A}\end{array}$$ such that all the $w{S}_{1}{S}_{2}\dots {S}_{i}$ lie in $W$ (resp. ${\stackrel{\sim}{W}}^{+}\text{)},$ and also that the similar statement holds with $\stackrel{B}{\leqq}$ in place of $\stackrel{A}{\leqq}$ throughout. Of course, the existence of ${S}_{1},{S}_{2},\dots ,{S}_{k}$ such that ${\text{(1.1)}}_{A}$ (resp. ${\text{(1.1)}}_{B},$ obtained by replacing $A$ by $B$ in ${\text{(1.1)}}_{A}\text{)}$ is guaranteed from the definition of $\stackrel{A}{\leqq}$ (resp. $\stackrel{B}{\leqq}\text{).}$ When $w,w\prime $ are in $W$ one can prove that in both cases all the ${S}_{i},$ and hence the $w{S}_{1}{S}_{2}\dots {S}_{i},$ lie in $W\text{;}$ for the case of $\stackrel{A}{\leqq}$ this follows from the fact that the union of all the $\sigma \text{-simplices}$ for $\sigma $ running in $W$ lies inside the "strip" $\{\lambda \in E\hspace{0.17em}|\hspace{0.17em}-1\leqq \u27e8\lambda ,{\alpha}^{\vee}\u27e9\leqq 1\}$ for every $\alpha \in {\Delta}_{+}$ (in fact, it is precisely the intersection of all such strips for $\alpha $ running over the short roots only); and for the case of $\stackrel{B}{\leqq}$ this follows from the fact that if $\sigma \in W,$ $l\left(\sigma S\right)<l\left(\sigma \right)$ for a reflection $S$ implies $S\in W\text{.}$ When $w,w\prime $ are in ${\stackrel{\sim}{W}}^{+},$ the existence of ${S}_{i}\text{'s}$ satisfying ${\text{(1.1)}}_{B}$ with $w{S}_{1}{S}_{2}\dots {S}_{i}\in {\stackrel{\sim}{W}}^{+}$ is proved in [Ver1972, Proposition 3], which gives $w\stackrel{B}{\leqq}w\prime \Rightarrow w\stackrel{A}{\leqq}w\prime \text{.}$ We omit the proof of the converse, i.e. of the existence of ${S}_{i}\text{'s}$ satisfying ${\text{(1.1)}}_{A}$ with $w{S}_{1}{S}_{2}\dots {S}_{i}\in {\stackrel{\sim}{W}}^{+},$ as our proof is unduly involved and this result is not required here. This is why we consider the partial-order $\leqq $ on ${\stackrel{\sim}{W}}^{+}$ as the restriction of $\stackrel{B}{\leqq}\text{.}$

**1.7.** The identity is the unique smallest element of $\stackrel{\sim}{W}\prime $ under
$\stackrel{B}{\leqq},$ whereas there is no smallest or largest element under
$\stackrel{A}{\leqq}\text{.}$ The "atoms" of
$(\stackrel{\sim}{W}\prime ,\stackrel{B}{\leqq})$
are precisely the Coxeter generators, and it can be shown that when $l>1$ the only automorphisms of this partial-order are
the Dynkin outer automorphisms of the group $\stackrel{\sim}{W}\prime $ (coming from the automorphisms of the
extended Dynkin diagram) permuting the Coxeter generators. (See Footnote ${}^{1}\text{.)}$ On the other
hand, $(\stackrel{\sim}{W}\prime ,\stackrel{A}{\leqq})$
has lots of automorphisms: for $\lambda \in X\prime $ the right-multiplication by
${T}_{\lambda}\in \stackrel{\sim}{W}\prime $ preserves
$\stackrel{A}{\leqq}\text{;}$ and more generally, if
$\lambda \in {\lambda}_{j}+X\prime $ (for
$j\in {J}_{0}\text{)}$ then the map sending
$w\in \stackrel{\sim}{W}\prime $ to the unique element
${\omega}_{j}w{T}_{\lambda}$ lying in
$\stackrel{\sim}{W}\prime $ as well as the right-coset
$\Omega w{T}_{\lambda},$ is an automorphism of this
partial-order. Also, the Dynkin outer automorphism of the group $W$ permuting the generators
$\left\{{R}_{i}\hspace{0.17em}\right|\hspace{0.17em}1\leqq i\leqq l\}$
which lifts canonically to an automorphism of $\stackrel{\sim}{W}\prime $ (given by the unique automorphism of the
extended Dynkin diagram fixing the node labelled $0\text{),}$ gives a distinguished involution of the affine partial-order; it
sends $w\in \stackrel{\sim}{W}\prime $ to its conjugate
${\tau}_{0}w{\tau}_{0}$ where
${\tau}_{0}=-{\sigma}_{0}$ is the endomorphism of $E$ sending
$\lambda $ to $-{\lambda}^{{\sigma}_{0}}$ and hence permuting the
fundamental roots; thus this involution is nontrivial if and only if $-\text{id}\text{.}$
does not lie in $W\text{.}$ All the automorphisms of $(\stackrel{\sim}{W}\prime ,\stackrel{A}{\leqq})$
are composed of the ones just described; this is seen on observing that the preceding involution is the only one fixing the identity element.

One gets a better understanding of the affine partial-order on looking at it as a partial-order on the tessellation of $E,$ i.e. on the set of $w\text{-simplices}$ as $w$ runs in $\stackrel{\sim}{W}\prime \text{.}$ Then the automorphism $w\to {\omega}_{j}w{T}_{\lambda}$ (for $\lambda \in {\lambda}_{j}+X\prime \text{)}$ corresponds merely to the translation sending the $w\text{-simplex}$ ${\left({E}^{*}\right)}^{w}$ to ${\left({E}^{*}\right)}^{w}+\lambda \text{.}$ There is also an interesting anti-involution (order-reversing bijection of period 2) of $(\stackrel{\sim}{W}\prime ,\stackrel{A}{\leqq}),$ sending ${\left({E}^{*}\right)}^{w}$ to $-{\left({E}^{*}\right)}^{w},$ which commutes with the involution described above (which takes ${\left({E}^{*}\right)}^{w}$ to ${\left({E}^{*}\right)}^{{\tau}_{0}w{\tau}_{0}}={\left({E}^{*}\right)}^{w{\tau}_{0}}\text{)},$ the composite being the anti-involution $w\to w{\sigma}_{0}\text{.}$ When we compose this latter anti-involution with the "translation by $\delta \text{",}$ viz. $w\to {\omega}_{{j}_{0}}w{T}_{\delta}$ $\text{(}{\omega}_{{j}_{0}}$ being the apposition element of $\Omega \text{)},$ we get a distinguished anti-involution of the affine partial-order on $\stackrel{\sim}{W}\prime $ which maps ${\stackrel{\sim}{W}}^{\u266f}$ onto itself; it is given by $w\to {\omega}_{{j}_{0}}w{\sigma}_{0}{T}_{\delta}={\omega}_{{j}_{0}}w{\omega}_{{j}_{0}}{w}_{0},$ so that its effect on the tessellation is ${\left({E}^{*}\right)}^{w}\to {\left({E}^{*}\right)}^{w{\omega}_{{j}_{0}}{w}_{0}}\text{.}$ It seems likely that all the automorphisms of the partial-ordering $({\stackrel{\sim}{W}}^{\u266f},\stackrel{A}{\leqq})=({\stackrel{\sim}{W}}^{\u266f},\stackrel{B}{\leqq})$ are composed of this anti-involution and the involution $w\to {\tau}_{0}w{\tau}_{0}$ of the last paragraph.

It is clear from the preceding that $${\stackrel{\sim}{W}}^{\u266f}\u2ac5\{w\in {\stackrel{\sim}{W}}^{+}\hspace{0.17em}|\hspace{0.17em}w\leqq {w}_{0}\}=\{w\in \stackrel{\sim}{W}\prime \hspace{0.17em}|\hspace{0.17em}w\stackrel{B}{\leqq}{w}_{0}\}\text{.}$$ It can be seen that equality holds here if and only if $\Delta $ is of type ${A}_{1},$ ${A}_{2}$ or ${B}_{2}\text{.}$ The distinguishing feature of these types responsible for this phenomenon is that every positive root is either fundamental or dominant; for other types one can find even a short root $\alpha $ which is neither. Since $\alpha $ is short, $\u27e8\alpha ,{\beta}^{\vee}\u27e9$ lies between $-1$ and $1$ for all roots $\beta ,$ in particular for $\beta ={\alpha}_{i}$ which gives $\delta -\alpha \in {X}^{+}$ (with $\alpha \notin {X}^{+}\text{)}\text{;}$ one sees that if $R$ is the reflection parallel to ${R}_{\alpha}$ sending $\delta $ to $\delta -\alpha ,$ then the image of the top simplex ${\left({E}^{*}\right)}^{{w}_{0}}$ under $R$ lies in ${E}^{+}$ but not in ${E}^{\u266f}\text{.}$ Thus one has ${w}_{0}R\leqq {w}_{0}$ with ${w}_{0}R\notin {W}^{\u266f}\text{.}$ For type ${G}_{2},$ see the Tessellation Diagram in § 3, which indicates the four members of $\{w\in {\stackrel{\sim}{W}}^{+}\hspace{0.17em}|\hspace{0.17em}w\leqq {w}_{0}\}$ not lying in ${\stackrel{\sim}{W}}^{\u266f}\text{.}$

${}^{1}$ *(Added in Proof.)* It has been shown in a recent preprint of Mr. A. van den Hombergh (Katholieke
Universiteit, Nijmegen) through a somewhat complicated analysis, that if $\Gamma $ is an arbitrary indecomposable
Coxeter group of rank $>2$ and $\varrho $ an automorphism of the Bruhat partial order structure on
$\Gamma $ (which obviously permutes the atoms), then $\varrho $ is determined by its restriction
to the set of Coxeter generators of $\Gamma \text{;}$ in contrast, when
$\Gamma $ has rank 2 (which means it is finite or infinite dihedral), say with
$\left|\Gamma \right|=2m$
$\text{(}3\leqq m\leqq \infty \text{)},$
the structure of this partial-order is easily described upon noting that for $0<i<m$ there are exactly
2 elements of "height" $i$ (viz. the elements of length $i\text{)},$ which shows that the
automorphism group is the direct product of $m-1$ copies of the cyclic group of order 2.

This is an excerpt of the paper *The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras* by Daya-Nand Verma.
It appeared in Lie Groups and their Representations, *ed. I.M.Gelfand*, Halsted, New York, pp. 653–705, (1975).