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

Last update: 1 April 2014

§ 1. Affine Weyl groups

1.1. Let Δ 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 Δ generated by the reflections Rα=R-α on E for αΔ, where Rα sends x to x-x,αα with α defined to mean 2α/α,α. Let X denote the root-lattice ·Δ (the -span of Δ), and X be the weight-lattice consisting of all λE such that λ,α for every αΔ. Let {α1,α2,,αl} be a choice of fundamental (= simple) roots in Δ, and Δ+ (resp. Δ-=-Δ+) the corresponding set of positive (resp. negative) roots with Δ=Δ+Δ- (disjoint union). Define the fundamental weights λ1,λ2,,λl by λl,αj=δij (= the Kronecker delta), so that iλi=X X=iαi. The action of W on the group X maps each coset of X therein onto itself, i.e. the natural action of W on X/X is trivial. The elements of X+=i+λi (the set of non-negative integer linear combinations of λ1,λ2,,λl) are called dominant weights; each W-orbit in X contains a unique dominant weight. The set X+Δ of dominant roots has cardinality equal to the number of W-orbits in Δ, which is 1 or 2 according as all elements of Δ have same length or not. The element(s) of the set X+Δ 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 -α0. The set Δ={α|αΔ} of "coroots" is again a root-system (called the root-system inverse to Δ) such that the set {α1,α2,,αl} affords a choice of fundamental (co)roots in it; relative to this choice -α0 is the long dominant (co)root of Δ (and if β were to denote the long dominant root of Δ,β would be the short dominant (co)root in Δ). The set {α0,α1,,αl} has the property that αi,αj is a non-positive integer for distinct i,j in {0,1,,l}. The subset {α0,α1,,αj-1,αj+1,,αl} obtained by deleting αj is a fundamental system for a subroot-system of Δ which is all of Δ if and only if nj=1 in the expansion -α0= i=1lni αi 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,,l, in which the nodes labelled i and j (for ij) are joined by αi,αj·αj,αi (=0,1,2, or 3)-fold lines, omitting the arrows that are sometimes placed to distinguish the role of i and j in case αi,αjαj,αi. It is known that this diagram determines Δ up to isomorphism. We remark that the common usage is to call it the "extended Dynkin diagram of Δ".

Let D (resp. D) denote the group consisting of all translations Tλ operating on E or X, for λX (resp. λX), where Tλ sends x to x+λ. We denote by W (resp. W) the group D·W (resp. D·W) of affine transformations of E or X, generated by D (resp. D) and W. It is clear that W (resp. W) is the semidirect extension of W by the normal subgroup D=X (resp. D=X) on which the action of W is known. From the fact that λ-λσX for λX and σW, it is easily seen that W is a normal subgroup of W.

For linear and affine transformations we shall denote the operation on the right, and shall compose them accordingly; thus for λE its image under wW will be written λw, or occasionally λw for typographical reasons, and (λw)w=λww. With this convention we introduce the abbreviations R0=Rα0· T-α0, Ri=Rαi for1il. It is known that W (resp. W) is a Coxeter group on generators R0,R1,,Rl (resp. R1,R2,,Rl). The order of the product RiRj, for distinct i,j in {0,1,,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 TnαRα=RαT-nα for unique αΔ+ and n; one also knows that every reflection lies in W and is in fact conjugate to one of the generators {Ri|0il} under an element of W. (More generally, it can be shown that every finite subgroup of W is isomorphic to a subgroup of W, and every finite subgroup of W is conjugate under an element of W to a subgroup of one of the l+1 finite subgroups generated by {Ri|0il,ij} for 0jl.) We shall call a point of E, W-regular (resp. W-regular, W-regular) if its stabilizer in W (resp. W, W) is trivial. The group W 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 Δ". 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+=i+λi= { λE| λ,αi 0for1il } , the fundamental parallelepiped E= { λE|0 ≨︀λ,αi 1for1il } , and the fundamental simplex E*= { λE+| λ,-α0 1 } . These are the fundamental regions for the actions of W, D, and W 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; the reason for our ad hoc choice of omitting all walls through 0 (from the closure of E) 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 W, and the affine Weyl group operates simply-transitively on the set of all connected components in this complement. Thus the collection {(E*)W|wW} of the corresponding closed simplices forms a tessellation (= covering) of E, which is parametrized by W itself; we shall call (E*)w the w-simplex. The walls of E* are precisely the reflecting hyperplanes for the Coxeter generators {R0,R1,,Rl} of W. (In the more general situation when Δ 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 l+k vertices. It is merely to avoid the cumbersomeness resulting from such a situation that we have assumed Δ indecomposable.) It is easy to see that ni-1λi is the vertex of E* opposite the face fixed by Ri, where n0 is the integer 1 and λ0 is the zero element of X (so that the ni's satisfy i=0lniαi=0). Thus E*X= {λj|jJ0} whereJ0= {j{0,1,,l}|nj=1}. In view of the fact that the W-orbit of λX is precisely the set λX one finds that each coset of X in X meets E* in a unique element of E*X. Thus X/X= {λj+X|jJ0} is an enumeration of the group X/X, with [X:X]=|J0|. (We are using the standard notations for index and cardinality.) It is known that X=X (i.e. all the ni's' in the expansion of -α0 in the fundamental coroots are greater than 1) if and only if Δ is of type G2, F4, or E8.

The action of W on E permutes the simplices (E*)w in our tessellation parametrized by wW, which shows that W is the semidirect extension of the stabilizer Ω of E* by the normal subgroup W. Let γj denote (as a temporary notation) the unique element of W which takes E* to its image E*-λj under the translation T-λjW. Since 0E*-λj for jJ0, and since each simplex contains a unique point of X, we find that γj must lie in the stabilizer W of the zero element of E in W. On the other hand, since the map X/XW/W sending the coset λ+X to the coset WTλ is clearly (well-defined and is) an isomorphism of groups, considering its composite with the canonical isomorphism W/WΩ, we see that for every jJ0 there is a unique element of Ω lying in the coset WTλj (viz. the image of λj+X under this composite); it is clear that this element is simply γjTλj (which sends E* to (E*-λj)+λj, and hence lies in Ω). Setting ωj=γjTλj, we have the enumeration Ω={ωj|jJ0}, with|Ω|= [X:X]=|J0|.

For jJ0, the action of ωjΩ on W (through conjugation) permutes the Coxeter generators {Ri|0il} in the same manner in which it permutes the vertices {ni-1λi|0il} of E*. Thus, if θj denotes the permutation of {0,1,,l} such that ωj-1Riωj=Rθj(i), then (λini)ωj= λθj(i) nθj(i) ,i.e.λiγj= λθj(i)-ni λj=λθj(i)- nθj(i)λj. Verifying directly the fact that the ordered basis α0,α1,, αj-1, αj+1,,αl is the dual, with respect to the W-invariant pairing ,, of the ordered basis λ0-n0λj, λ1-n1λj,, λj-1-nj-1 λj,λj+1- nj+1λj,, λl-nlλj, one concludes immediately that αiγj=αθj(i). Since λ0γj=λθj(0)-λj, we have θj(0)=j. This shows that γj-1 sends αj to α0Δ-, and sends every other fundamental root into Δ+. Thus γj-1 sends all positive roots in the subroot-system Δj of rank l-1 spanned by {αi|1il,ij} into Δ+; conversely, if α is a positive root not in Δj, say α=i=1laiαi with aj0, then looking at β=(α)γj-1= i=1lai αθj-1(i)= ajα0+ i=1ijl aiαθj-1(i) we find at least one coefficient (in the expansion of β in fundamental roots) negative, which gives αγj-1Δ-. If σ0 (resp. σj) denotes the unique element of maximal length in W (resp. in the Weyl group W(Δj) of Δj), verifying that σjσ0 is the unique element of W which makes precisely those positive roots negative which do not lie in Δj (here uniqueness follows from the standard theory of Weyl groups), we are forced to the conclusion γj-1=σjσ0. Thus we have the formula ωj=σ0σjTλj (forjJ0) for the elements of Ω; note that every element of W(Δj), and hence σj, fixes λj, which gives also ωj=σ0Tλjσj. One calls σ0 the "apposition element" of W.

Note that the permutation θj of {0,1,,l} is an automorphism of the extended Dynkin diagram, and that its action is completely determined by its action on the subset J0. (The latter claim follows from the fact that J0 is a proper subset of {0,1,,l} in all cases except those of type Al, and in such cases the extended Dynkin diagram is a tree and the nodes corresponding to J0 are precisely the "end-nodes" of the diagram. Incidentally, this also shows that γj is the unique element of W sending α0 to αj and preserving the set {α0,α1,,αl}.) The action of θj on J0 itself can be read-off from the group-structure of {λj+X|jJ0} (which is quite easy to write down in each given case), in view of the following straightforward equivalences on i,j,kJ0: θj(i)=kλi +λjλk(mod X)λiωj =λkαiγj= αk θiθj=θk ωiωj=ωkγiγj =γk.

The isomorphism ωjγj (for jJ0) of the group Ω onto the subgroup Ω0={γj|jJ0} of W, given by the last equivalence above, can also realized as follows: Elements of Ω leave fixed the barycentre β0=|J0|-1jJ0λj of the subsimplex of E* having vertices E*X; note that β0 does not lie in X except in the three extreme cases (types G2, F4, E8) when X=X, J0={λ0} and β0=λ0=0. Thus we have β0γj+λj=β0, which gives γj-1Tβ0 γj=Tβ0-λj =Tβ0Tλj-1 ,i.e.γj=Tβ0 γjTλjT-β0 =Tβ0ωjT-β0. Hence conjugation by T-β0 gives our isomorphism ΩΩ0.

1.3. Now let l() denote the usual length-function on W in terms of its Coxeter generators {R0,R1,,Rl} as in the standard theory of Coxeter groups. Thus for wW, l(w) is the smallest integer k such that w=Ri1Ri2Rik for 0iql, 1qk. such an expression for w (with k=l(w)) is called a reduced expression. One knows that l(w) equals the number of reflecting hyperplanes (i.e. fixed hyperplanes of reflections in W) that separate E* from the w-simplex (E*)w. The restriction of l() to W coincides with the length-function on W defined in terms of its Coxeter generators {R1,R2,,Rl}; for σW, l(σ) also equals the number of elements of Δ+ that are carried into Δ- under σ.

For λE and σW, let λ(σ) denote the unique translate under X (i.e. image under D) of λσ such that λ(σ)E. In general λ(σ) depends very much on both λ and σ. But when λ lies in the interior of E* the difference λ(σ)-λσ is independent of λ, and can be seen to be equal to δσ=iIσ λi,whereIσ= { i|1il, l(σRi)<l(σ) } . Thus λ(σ)=λwσ where wσ=σTδσW. It is this set {wσ|σW} of affine transformations of E that is basic to the formulation of our conjecture in Part I (§ 2).

Let λ be an interior of E*; thus λ is W-regular. Setting W= {σW|wσW}= {σW|δσX}, it is clear that the W-orbit of λ meets E precisely in the set {λ(σ)|σW} of cardinality |W|=[W:Ω0]. The last equality follows from (λ(γ))(σ)=λ(γσ) for γΩ0 and σW, the map γwγ being simply the isomorphism Ω0Ω already encountered (with γjωj=wγj). This shows that W is a set of distinguished right-coset representatives for the right-coset space Ω0\W= {Ω0·σ|σW}. Incidentally, this also gives the famous formula for the order of the Weyl group: |W|=l!n1n2 nl[X:X], as one sees on making an elementary calculation verifying that the volume of E is l!n1n2nl times the volume of E*. Let us digress slightly at this point to mention another important formula involving the ni's (this time their sum), which we shall require later: i=1lni= hΔ-1,where hΔ (=the Coxeter no. ofΔ)= 2rl,r= |Δ+|. An alternative interpretation of this formula is that the "height" i=1lni of the long dominant coroot -α0 equals the largest "exponent" of W; 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) on one hand, and by the number of elements of Δ+ of various heights on the other hand.

1.4. It is well known, and easily seen, that the set W+ defined by W+= { wW| l(wRi)>l(w) for1il } is a set of distinguished left-coset representatives for the left-coset space W/W={w·W|wW}. The cone E+ is tessellated by the images of E* under W+, just as the closure of E is tessellated by the images under W= {wσ|σW} ={wσ|σW} W. In particular, W is contained in W+.

Let us note that in the tessellation {(E*)w|wW} of the closure of E there is a unique simplex which contains δ=i=1lλi. We call this the "top simplex" in E, and the corresponding element w0 of W the "top element" of W. Writing Etop* for the top simplex we have (E*)w0= Etop*=-E*+δ =(E*)σ0Tδ, which shows that w0T-δσ0Ω. Denoting by j0 the unique index in J0 such that δλj0 (modX), we see that the zero element in E goes under w0 to δ-λj0 (this being the only point of Etop* in the root-lattice) and hence under w0T-δσ0 to (-λj0)σ0=λj0 where this last equality is immediate from δσ0=-δ. Since ωj0 is the only element of Ω which sends λ0=0 to λj0, we have w0=ωj0σ0Tδ. From this one easily gets w0=σj0Tδ-λj0, on substituting ωj0-1=σ0Tλj0σj0 proved earlier. We call ωj0 the "apposition element" of Ω.

1.5. We shall now establish certain relativizations of the concepts and notations W+ and W, relative to a proper subset J of {0,1,,l} (empty subset permitted). Let us write EJ*= {λE*|λRj=λjJ}. This is an open subset in its own affine-span (which passes through the origin if and only if 0J). When J is empty this is precisely the interior of E*, whose boundary is the disjoint union of EJ* as J ranges over the 2l+1-2 other subsets. The stabilizer of λEJ* in W is precisely the subgroup WJ generated by {Rj|jJ}. In the right-coset space WJ\W each coset WJ·w has a unique element of minimal length, all such elements forming the set WJ of distinguished right-coset representatives, where WJ= { wW| l(Rjw)>l(w) forjJ } ; and the double-coset space WJ\W /W={WJ·w·W|wW} has WJ+=WJW+ as a set of distinguished double-coset representatives. The W-orbit of λEJ* is parametrized by WJ (i.e. wλw is a bijection from WJ to this orbit), and the intersection of this orbit with E+ is precisely {λw|wWJ+} parametrized by WJ+, Note that this intersection lies in the interior of E+ if and only if EJ* itself does, which happens precisely when J is either empty or the singleton {0}. Setting WJ++= { wWJ| (EJ*)w lies in the interior ofE+ } , and WJ= { wWJ| (EJ*)wlies in E } , it is clear that WJ++ (resp. WJ) is the subset of WJ which parametrizes the intersection of the W-orbit of λEJ* with the interior of E+ (resp. with E). Thus WJWJ++WJ+WJ, where the middle inclusion is also strict whenever J is nonempty and contains an index other than 0. It is also clear that WJ=W if and only if J is empty. We claim that WJW in general. For, suppose w lies in WJ but not in W. Then w sends the interior of E* outside E; but since λwE for λEJ* there is a wall of E on which λw lies and which separates the fundamental simplex from the w-simplex (E*)w. Let w=wR where R is the reflection in this wall. Then λw=λw implies wWJ·w; and since the said wall of E does not separate the fundamental simplex and the w-simplex we have l(w)<l(w), which contradicts the fact that wWJ is the unique element of minimal length in the coset WJ·w. Thus we have WJW WJ+and WJ=W WJ++.

We remark that WJ could be empty, and that this situation does occur for some J if and only if Δ is not of type G2, F4, or E8. This follows from the more precise claim that WJ is empty if and only if the complement of J in {0,1,,l} consists of a single index j with jJ0, jj0, and that whenever WJ is nonempty the top element w0 of W lies in (WJ++ and therefore in) WJ. For this first consider the situation when the complement of J is the singleton {i}; then EJ* consists of just one point, viz. the vertex ni-1λi. If iJ0 then ni=1 and the W-orbit of EJ* is contained in the set of 2l vertices of the parallelepiped E (the intersection of the closure of E with the weight-lattice); but none of these lie in W-orbit of λi unless δ-λiX, i.e. i=j0. This proves the "if" part of the claim. Now note that the image of ni-1λi under w0=ωj0σ0Tδ is of the form δ-nk-1λk with nk=ni (for ni-1λi goes to nj-1λj under ωj0 if j=θj0(i), and so to nk-1λk under ωj0τ0=ωj0·(-σ0) if Rk is the image of Rj under the Dynkin outer automorphism of W which is known to be given by conjugation under the endomorphism τ0=-σ0 of E sending α to -ασ0). It is clear that δ-nk-1λkE if nk>1, which takes care of our assertion when J has cardinality l. In other cases, the closure of EJ* has at least two vertices ni-1λi and ni-1λi and its image under ω0 contains a point of the form δ-a(nk-1λk)-(1-a)(nk-1λk) with 0<a<1, which clearly lies in E (all the coefficients in the expansion of this point in the basis λ1,λ2,,λl of E being >0 and 1); this shows that (EJ*)w0 is contained in E, i.e. w0WJ as claimed.

1.6. We shall now introduce a canonical partial-order on the set W+, which would be needed in § 5 for streamlining Conjecture II and its applications. It is defined as the restriction to W+ of the "Bruhat partial-order" B on W (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 B was denoted simply , but here we have made this variation because there is another partial-order A on W 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 W. This A is a more natural partial-order for the representation-theoretic purposes we have in mind in Part II, but on the set W+ the two orderings coincide.

B is defined as the weakest partial-order on W such that l(wR)<l(w) implieswRBw, for arbitrary reflection R and element w in W. Thus, for w, w in W, if w-1w is a reflection one has the dichotomy that one and exactly one of wBw and wBw holds. It has been shown in [Ver1972, Proposition 2] that B coincides with the (weakest) partial-order B obtained by demanding wRBw for only those pairs (w,R) for which l(wR)=l(w)-1. It is also known [Ver1972, Corollary 1] that if w=Ri1Ri2Rim is a reduced expression then wBw if and only if w is of the form Rj1Rj2Rjn for some subsequence (j1,j2,,jn) of the sequence of indices (i1,i2,,im) (all indices ranging in {0,1,,l}), where this expression for w can also be assumed reduced by passing to a further subsequence.

The partial-order A on W is defined to be the weakest partial-order with the following property: Let wW and R be a reflection parallel to Rα for αΔ+; for an interior point λ of (E*)w the sign of the (nonzero) real number c=cλ defined by λ-λR=cα is clearly independent of λ (depends only on w and R); then we require that wRAwwheneverc>0 (for some, and hence all,λ).

Once again we have the dichotomy "wAw versus wAw" in case w-1w 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-simplex from the wR-simplex, so that if R,R,R, is the enumeration of these reflections in the order of their closeness to the w-simplex then wR=wRRR; and for an interior point λ in the w-simplex (E*)w, denoting its successive mirror-images by λ=λR, λ=λR, λ=λR,, one finds the consecutive differences λ-λ, λ-λ, λ-λ, to be of the form cα, (1-c)α, cα, alternately, where 0<|c|<1. It follows that A coincides with the (weakest) partial-order A obtained by demanding wRAw whenever 0<c<1 (where c is defined as before by λ-λR=cα, and 0<c<1 holds either for all or for none λinterior(E*)w).

It is easy to see that the condition c>0 the pair (w,R) is equivalent to the following: { l(wR)>l(w) iftheR-plane does not meetinterior E+, l(wR)<l(w) iftheR-plane does meetinteriorE+, where R-plane means the fixed hyperplane of R. It follows that for w,w in the Weyl group W (resp. for w,w in the set W+ of distinguished coset representatives for W/W), if w-1w is a reflection then wAwis equivalent to wBw(resp. wBw). This statement is true even without the restriction that w-1w be a reflection; to deduce this generalization it suffices to show that for w,wW (resp. W+), if wAw then one can find reflections S1,S2,,Sk with wAwS1AwS1 S2AAwS1S2 Sk=w (1.1)A such that all the wS1S2Si lie in W (resp. W+), and also that the similar statement holds with B in place of A throughout. Of course, the existence of S1,S2,,Sk such that (1.1)A (resp. (1.1)B, obtained by replacing A by B in (1.1)A) is guaranteed from the definition of A (resp. B). When w,w are in W one can prove that in both cases all the Si, and hence the wS1S2Si, lie in W; for the case of A this follows from the fact that the union of all the σ-simplices for σ running in W lies inside the "strip" {λE|-1λ,α1} for every αΔ+ (in fact, it is precisely the intersection of all such strips for α running over the short roots only); and for the case of B this follows from the fact that if σW, l(σS)<l(σ) for a reflection S implies SW. When w,w are in W+, the existence of Si's satisfying (1.1)B with wS1S2SiW+ is proved in [Ver1972, Proposition 3], which gives wBwwAw. We omit the proof of the converse, i.e. of the existence of Si's satisfying (1.1)A with wS1S2SiW+, as our proof is unduly involved and this result is not required here. This is why we consider the partial-order on W+ as the restriction of B.

1.7. The identity is the unique smallest element of W under B, whereas there is no smallest or largest element under A. The "atoms" of (W,B) 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 W (coming from the automorphisms of the extended Dynkin diagram) permuting the Coxeter generators. (See Footnote 1.) On the other hand, (W,A) has lots of automorphisms: for λX the right-multiplication by TλW preserves A; and more generally, if λλj+X (for jJ0) then the map sending wW to the unique element ωjwTλ lying in W as well as the right-coset ΩwTλ, is an automorphism of this partial-order. Also, the Dynkin outer automorphism of the group W permuting the generators {Ri|1il} which lifts canonically to an automorphism of W (given by the unique automorphism of the extended Dynkin diagram fixing the node labelled 0), gives a distinguished involution of the affine partial-order; it sends wW to its conjugate τ0wτ0 where τ0=-σ0 is the endomorphism of E sending λ to -λσ0 and hence permuting the fundamental roots; thus this involution is nontrivial if and only if -id. does not lie in W. All the automorphisms of (W,A) 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-simplices as w runs in W. Then the automorphism wωjwTλ (for λλj+X) corresponds merely to the translation sending the w-simplex (E*)w to (E*)w+λ. There is also an interesting anti-involution (order-reversing bijection of period 2) of (W,A), sending (E*)w to -(E*)w, which commutes with the involution described above (which takes (E*)w to (E*)τ0wτ0=(E*)wτ0), the composite being the anti-involution wwσ0. When we compose this latter anti-involution with the "translation by δ", viz. wωj0wTδ (ωj0 being the apposition element of Ω), we get a distinguished anti-involution of the affine partial-order on W which maps W onto itself; it is given by wωj0wσ0Tδ=ωj0wωj0w0, so that its effect on the tessellation is (E*)w(E*)wωj0w0. It seems likely that all the automorphisms of the partial-ordering (W,A)=(W,B) are composed of this anti-involution and the involution wτ0wτ0 of the last paragraph.

It is clear from the preceding that W {wW+|ww0} ={wW|wBw0}. It can be seen that equality holds here if and only if Δ is of type A1, A2 or B2. 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 α which is neither. Since α is short, α,β lies between -1 and 1 for all roots β, in particular for β=αi which gives δ-αX+ (with αX+); one sees that if R is the reflection parallel to Rα sending δ to δ-α, then the image of the top simplex (E*)w0 under R lies in E+ but not in E. Thus one has w0Rw0 with w0RW. For type G2, see the Tessellation Diagram in § 3, which indicates the four members of {wW+|ww0} not lying in W.

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 Γ is an arbitrary indecomposable Coxeter group of rank >2 and ϱ an automorphism of the Bruhat partial order structure on Γ (which obviously permutes the atoms), then ϱ is determined by its restriction to the set of Coxeter generators of Γ; in contrast, when Γ has rank 2 (which means it is finite or infinite dihedral), say with |Γ|=2m (3m), 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), 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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