The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras
Part I
A conjecture on Weyl's dimension polynomial

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 1 April 2014

§ 1. Affine Weyl groups

1.1. Let $\Delta$ be an indecomposable reduced root-system [Bou1968, p. 151] in a real vector space $E$ with positive-definite inner product $⟨,⟩$ 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 $x-⟨x,{\alpha }^{\vee }⟩\alpha$ with ${\alpha }^{\vee }$ defined to mean $2\alpha /⟨\alpha ,\alpha ⟩\text{.}$ Let $X\prime$ denote the root-lattice $\mathbb{Z}·\Delta$ (the $\mathbb{Z}\text{-span}$ of $\Delta \text{),}$ and $X$ be the weight-lattice consisting of all $\lambda \in E$ such that $⟨\lambda ,{\alpha }^{\vee }⟩\in \mathbb{Z}$ 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 $⟨{\lambda }_l,{\alpha }_j^{\vee }⟩={\delta }_{ij}$ $\text{(}=$ 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 $⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$ 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 $⟨{\alpha }_i,{\alpha }_j^{\vee }⟩·⟨{\alpha }_j,{\alpha }_i^{\vee }⟩$ $\text{(}=0,1,2,$ or $3\text{)-fold}$ lines, omitting the arrows that are sometimes placed to distinguish the role of $i$ and $j$ in case $⟨{\alpha }_i,{\alpha }_j^{\vee }⟩\ne ⟨{\alpha }_j,{\alpha }_i^{\vee }⟩\text{.}$ 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 $D·W$ (resp. $D\prime ·W\text{)}$ 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 $$R_0=R_{{\alpha }_0}·T_{-{\alpha }_0},R_i=R_{{\alpha }_i}\text{for}1\leqq i\leqq l\text{.}$$ 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) $$E^{+}=\sum _i{ℝ}^{+}{\lambda }_i=\{\lambda \in E\hspace{0.17em}|\hspace{0.17em}⟨\lambda ,{\alpha }_i^{\vee }⟩\geqq 0\text{for}1\leqq i\leqq l\},$$ the fundamental parallelepiped $$E^{♯}=\{\lambda \in E\hspace{0.17em}|\hspace{0.17em}0\lneqq ︀⟨\lambda ,{\alpha }_i^{\vee }⟩\leqq 1\text{for}1\leqq i\leqq l\},$$ and the fundamental simplex $$E^{*}=\{\lambda \in E^{+}\hspace{0.17em}|\hspace{0.17em}⟨\lambda ,-{\alpha }_0^{\vee }⟩\leqq 1\}\text{.}$$ 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 $E^{♯}$ but not in $E^{♯}\text{;}$ the reason for our ad hoc choice of omitting all walls through $0$ (from the closure of $E^{♯}\text{)}$ in the definition of $E^{♯},$ 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 $$E^{*}\cap X=\left\{{\lambda }_j\hspace{0.17em}\right|\hspace{0.17em}j\in J_0\}\text{where}{J}_0=\{j\in \{0,1,\dots ,l\}\hspace{0.17em}|\hspace{0.17em}{n}_j=1\}\text{.}$$ 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 $$\Omega =\left\{{\omega }_j\hspace{0.17em}\right|\hspace{0.17em}j\in J_0\},\text{with}\left|\Omega \right|=[X:X\prime ]=\left|J_0\right|\text{.}$$

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 $${\left(\frac{{\lambda }_i}{n_i}\right)}^{{\omega }_j}=\frac{{\lambda }_{{\theta }_j\left(i\right)}}{n_{{\theta }_j\left(i\right)}},\text{i.e.}{\lambda }_i^{{\gamma }_j}={\lambda }_{{\theta }_j\left(i\right)}-n_i{\lambda }_j={\lambda }_{{\theta }_j\left(i\right)}-n_{{\theta }_j\left(i\right)}{\lambda }_j\text{.}$$ 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 $⟨,⟩,$ of the ordered basis $${\lambda }_0-n_0{\lambda }_j,{\lambda }_1-n_1{\lambda }_j,\dots ,{\lambda }_{j-1}-n_{j-1}{\lambda }_j,{\lambda }_{j+1}-n_{j+1}{\lambda }_j,\dots ,{\lambda }_l-n_l{\lambda }_j,$$ 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 $${\omega }_j={\sigma }_0{\sigma }_j{T}_{{\lambda }_j}\text{(for}\hspace{0.17em}j\in J_0\text{)}$$ 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 $${\gamma }_j^{-1}{T}_{{\beta }_0}{\gamma }_j=T_{{\beta }_0-{\lambda }_j}=T_{{\beta }_0}{T}_{{\lambda }_j}^{-1},\text{i.e.}{\gamma }_j=T_{{\beta }_0}{\gamma }_j{T}_{{\lambda }_j}{T}_{-{\beta }_0}=T_{{\beta }_0}{\omega }_j{T}_{-{\beta }_0}\text{.}$$ 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 ${\lambda }^{\left(\sigma \right)}\in E^{♯}\text{.}$ 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 $${\delta }_{\sigma }=\sum _{i\in I_{\sigma }}{\lambda }_i,\text{where}{I}_{\sigma }=\left\{i\hspace{0.17em}\right|\hspace{0.17em}1\leqq i\leqq l,\hspace{0.17em}l\left(\sigma R_i\right)<l\left(\sigma \right)\}\text{.}$$ 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 $$W^{♯}=\{\sigma \in W\hspace{0.17em}|\hspace{0.17em}{w}_{\sigma }\in \stackrel{\sim }{W}\prime \}=\{\sigma \in W\hspace{0.17em}|\hspace{0.17em}{\delta }_{\sigma }\in X\prime \},$$ it is clear that the $\stackrel{\sim }{W}\prime \text{-orbit}$ of $\lambda$ meets $E^{♯}$ precisely in the set $\left\{{\lambda }^{\left(\sigma \right)}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W^{♯}\}$ of cardinality $\left|W^{♯}\right|=[W:{\Omega }_0]\text{.}$ 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 $W^{♯}$ is a set of distinguished right-coset representatives for the right-coset space $${\Omega }_0\backslash W=\{{\Omega }_0·\sigma \hspace{0.17em}|\hspace{0.17em}\sigma \in W\}\text{.}$$ 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 $E^{♯}$ 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: $$\sum _{i=1}^l{n}_i=h_{\Delta }-1,\text{where}{h}_{\Delta }(=\text{the Coxeter no. of}\hspace{0.17em}\Delta )=\frac{2r}{l},\hspace{0.17em}r=\left|{\Delta }_{+}\right|\text{.}$$ 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 $${\stackrel{\sim }{W}}^{+}=\{w\in \stackrel{\sim }{W}\prime \hspace{0.17em}|\hspace{0.17em}l\left(w{R}_i\right)>l\left(w\right)\text{for}1\leqq i\leqq l\}$$ is a set of distinguished left-coset representatives for the left-coset space $\stackrel{\sim }{W}\prime /W=\{w·W\hspace{0.17em}|\hspace{0.17em}w\in W\prime \}\text{.}$ The cone $E^{+}$ is tessellated by the images of $E^{*}$ under ${\stackrel{\sim }{W}}^{+},$ just as the closure of $E^{♯}$ is tessellated by the images under $${\stackrel{\sim }{W}}^{♯}=\left\{w_{\sigma }\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W^{♯}\}=\left\{w_{\sigma }\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}\cap \stackrel{\sim }{W}\prime \text{.}$$ In particular, ${\stackrel{\sim }{W}}^{♯}$ is contained in ${\stackrel{\sim }{W}}^{+}\text{.}$

Let us note that in the tessellation $\left\{{\left(E^{*}\right)}^w\hspace{0.17em}\right|\hspace{0.17em}w\in {\stackrel{\sim }{W}}^{♯}\}$ of the closure of $E^{♯}$ there is a unique simplex which contains $\delta =\sum _{i=1}^l{\lambda }_i\text{.}$ We call this the "top simplex" in $E^{♯},$ and the corresponding element $w_0$ of ${\stackrel{\sim }{W}}^{♯}$ the "top element" of ${\stackrel{\sim }{W}}^{♯}\text{.}$ 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 ${\stackrel{\sim }{W}}^{♯},$ 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 ${\stackrel{\sim }{W}}^J·w$ has a unique element of minimal length, all such elements forming the set ${\stackrel{\sim }{W}}_J$ of distinguished right-coset representatives, where $${\stackrel{\sim }{W}}_J=\{w\in \stackrel{\sim }{W}\prime \hspace{0.17em}|\hspace{0.17em}l\left(R_{j}w\right)>l\left(w\right)\text{for}j\in J\}\text{;}$$ and the double-coset space $${\stackrel{\sim }{W}}^J\backslash \stackrel{\sim }{W}\prime /W=\{{\stackrel{\sim }{W}}^J·w·W\hspace{0.17em}|\hspace{0.17em}w\in \stackrel{\sim }{W}\prime \}$$ 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 $${\stackrel{\sim }{W}}_J^{♯}=\{w\in {\stackrel{\sim }{W}}^J\hspace{0.17em}|\hspace{0.17em}{\left(E_J^{*}\right)}^w\hspace{0.17em}\text{lies in}\hspace{0.17em}{E}^{♯}\},$$ it is clear that ${\stackrel{\sim }{W}}_J^{++}$ (resp. ${\stackrel{\sim }{W}}_J^{♯}\text{)}$ 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 $E^{♯}\text{).}$ Thus ${\stackrel{\sim }{W}}_J^{♯}\subset {\stackrel{\sim }{W}}_J^{++}⫅{\stackrel{\sim }{W}}_J^{+}\subset {\stackrel{\sim }{W}}_J,$ where the middle inclusion is also strict whenever $J$ is nonempty and contains an index other than $0\text{.}$ It is also clear that ${\stackrel{\sim }{W}}_J^{♯}={\stackrel{\sim }{W}}^{♯}$ if and only if $J$ is empty. We claim that ${\stackrel{\sim }{W}}_J^{♯}⫅{\stackrel{\sim }{W}}^{♯}$ in general. For, suppose $w$ lies in ${\stackrel{\sim }{W}}_J^{♯}$ but not in ${\stackrel{\sim }{W}}^{♯}\text{.}$ Then $w$ sends the interior of $E^{*}$ outside $E^{♯}\text{;}$ but since ${\lambda }^w\in E^{♯}$ for $\lambda \in E_J^{*}$ there is a wall of $E^{♯}$ 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 $w\prime \in {\stackrel{\sim }{W}}^J·w\text{;}$ and since the said wall of $E^{♯}$ 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 ${\stackrel{\sim }{W}}^J·w\text{.}$ Thus we have $${\stackrel{\sim }{W}}_J^{♯}⫅{\stackrel{\sim }{W}}^{♯}\cap {\stackrel{\sim }{W}}_J^{+}\text{and}{\stackrel{\sim }{W}}_J^{♯}={\stackrel{\sim }{W}}^{♯}\cap {\stackrel{\sim }{W}}_J^{++}\text{.}$$

We remark that ${\stackrel{\sim }{W}}_J^{♯}$ 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 ${\stackrel{\sim }{W}}_J^{♯}$ 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 ${\stackrel{\sim }{W}}_J^{♯}$ is nonempty the top element $w_0$ of ${\stackrel{\sim }{W}}^{♯}$ lies in $\text{(}{\stackrel{\sim }{W}}_J^{++}$ and therefore in) ${\stackrel{\sim }{W}}_J^{♯}\text{.}$ 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 $E^{♯}$ (the intersection of the closure of $E^{♯}$ 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 ${\omega }_{j_0}{\tau }_0={\omega }_{j_0}·(-{\sigma }_0)$ 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 $\delta -n_k^{-1}{\lambda }_k\in E^{♯}$ 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 $E^{♯}$ (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 $E^{♯},$ i.e. $w_0\in W_J^{♯}$ 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 $$l\left(wR\right)<l\left(w\right)\text{implies}wR\stackrel{B}{\leqq }w,$$ 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 $R\prime ,R″,R‴,\dots$ is the enumeration of these reflections in the order of their closeness to the $w\text{-simplex}$ then $wR=wR\prime R″R‴\dots \text{;}$ and for an interior point $\lambda$ in the $w\text{-simplex}$ ${\left(E^{*}\right)}^w,$ denoting its successive mirror-images by $\lambda \prime =\lambda R\prime ,$ $\lambda ″=\lambda \prime R″,$ $\lambda ‴=\lambda ″R‴,\dots ,$ one finds the consecutive differences $\lambda -\lambda \prime ,$ $\lambda \prime -\lambda ″,$ $\lambda ″-\lambda ‴,\dots$ 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: $$\{\begin{array}{cc}l\left(wR\right)>l\left(w\right)& \text{if}\text{the}\hspace{0.17em}R\text{-plane}\hspace{0.17em}\text{does not meet}\hspace{0.17em}\text{interior}\hspace{0.17em}{E}^{+},\\ l\left(wR\right)<l\left(w\right)& \text{if}\text{the}\hspace{0.17em}R\text{-plane}\hspace{0.17em}\text{does meet}\hspace{0.17em}\text{interior}\hspace{0.17em}{E}^{+},\end{array}$$ 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" $\{\lambda \in E\hspace{0.17em}|\hspace{0.17em}-1\leqq ⟨\lambda ,{\alpha }^{\vee }⟩\leqq 1\}$ 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 ${\stackrel{\sim }{W}}^{♯}$ 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 $({\stackrel{\sim }{W}}^{♯},\stackrel{A}{\leqq })=({\stackrel{\sim }{W}}^{♯},\stackrel{B}{\leqq })$ 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 $${\stackrel{\sim }{W}}^{♯}⫅\{w\in {\stackrel{\sim }{W}}^{+}\hspace{0.17em}|\hspace{0.17em}w\leqq w_0\}=\{w\in \stackrel{\sim }{W}\prime \hspace{0.17em}|\hspace{0.17em}w\stackrel{B}{\leqq }{w}_0\}\text{.}$$ 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, $⟨\alpha ,{\beta }^{\vee }⟩$ 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 $E^{♯}\text{.}$ Thus one has $w_{0}R\leqq w_0$ with $w_{0}R\notin W^{♯}\text{.}$ 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 ${\stackrel{\sim }{W}}^{♯}\text{.}$

${}^1$ (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.

Notes and References

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).

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