10 The Affine Weyl Group

n−r− 1 2

−

r

X

i=1

λ_i,

and so the total number ofβ for this choice of S is q^r(^n−r−12)^−Pri=1λi. For such aβ, we have

f^◦(β) =δ^1/2(β)χ(β) = q^{−(n+1)r2+Pλi}t_λ1· · ·t_λr.

Recalling that θ_r is q^{−r(n−r)/2} times the characteristic function of the double coset K^◦τ_rK^◦, we see that (θ_rf^◦)(1) equals

q^−r(n−r)2 X

λ1<···<λ_r

q^r(^n−r−1₂ )^−Pri=1λi q^{−(n+1)r2 +Pλi} t_λ1· · ·t_λr

=e_r(t₁,· · · , t_r),

as required.

10 The Affine Weyl Group

We may extend the Weyl group by a group of translations, and obtain the so-called affine Weyl group.

Let Φ be a root system in the vector space V, and let W = W(Φ) be the Weyl group. We giveV a W-invariant inner product h, i. If α∈Φ andk ∈Z letP_α,k be

the hyperplaneP_α,k ={v ∈V| hα, vi=k}. A connected component of V − [

α∈Φ k∈Z

P_α,k

is called anopen alcove. They are relatively compact open subsets ofV. The closure of an open alcove is called an alcove.

LetC⁺ be the positive Weyl chamber, so that

C⁺ ={v ∈V| hαi, vi>0 (1 6i6r)}

where {α₁,· · · , α_r} are the simple roots. There is a unique alcove in C⁺ which contains the origin. This is the fundamental alcove F. Our immediate goal is to describe it more explicitly.

There is a partial order on V in which v >0 if v =P

c_iα_i with c_i >0. We call a root α highest if α⁰ > α for α⁰ ∈ Φ implies that α⁰ = α. We will see that if Φ is irreducible, this implies more: that actually α>α⁰ for every root α⁰ ∈Φ.

Exercise 9 Letα, β∈Φ be linearly independent. and letU be the two-dimensional space spanned byα and β. Show that U∩Φ is a root system inU.

Exercise 10 Assume that Φ is reduced and that α, β are distinct elements of Φ. Show that ifhα, βi>0 thenα−β∈Φ. (One way: you may use the previous exercise to reduce to the rank two case, and check this for the four rank two root systems A₁×A₁, A₂, B₂ and G2.)

Proposition 41 Suppose thatΦis irreducible. Then there is a unique root−α₀ that is highest with respect to the partial order. If α is any positive root, then α 6 −α0

andhα,−α₀i>0. If α is any root thenhα,−α₀i6hα₀, α₀i with equality if and only if α=−α₀.

It is most useful to use the notationα₀ for thenegative of the highest root, since then it will play a role exactly analogous to the simple roots{α₁,· · · , α_r}in certain situations.

Proof Suppose that β is a highest root. Since β >−β, β is positive.

We claim that hβ, α_ii > 0. If not, then s_i(β) = β − ^2hα_hα ^i,βi

i,αiiα_i > β, contradicting the assumption that β is a highest root.

Write β = P

k_iα_i. Clearly any highest root is positive, so k_i > 0. We will show that k_i > 0. Otherwise, let Σ = Σ₁ ∪ Σ₂, where Σ₁ = {α_i|k_i > 0} and Σ₂ ={α_i|k_i = 0}. Because Φ is irreducible, the simple roots may not be partitioned

into two disjoint mutually orthogonal sets. Therefore there is α_l ∈ Σ₁ and α_j ∈ Σ₂ such thathα_l, α_ji<0. Now

hβ, α_ji=

* X

αi∈Σ₁

k_iα_i, α_j +

= X

αi∈Σ₁

k_ihα_i, α_ji.

All terms on the right are non-positive sincekl >0, and one term (i=l) is negative.

This contradicts the fact that hβ, α_ii>0, proving that allk_i >0.

Now let γ be another highest weight. Consider hβ, γi = P

k_ihα_i, γi. We have hαi, γi>0 with strict inequality for somei, andki >0, so hβ, γi>0. It follows from Exercise 10 thatβ−γ is a root. Eitherβ−γ ∈Φ⁺, in which caseβ =γ+ (β−γ), contradicting the maximality of γ, or β −γ ∈ Φ⁻, in which case γ = β + (γ−β), contradicting the maximality ofβ. This contradiction shows that the highest root is unique. (We are therefore justified in naming it, and we call it −α₀.)

Now suppose that α is any positive root. We can write α=P

n_iα_i with n_i >0, and hβ, αi = P

nihβ, αii > 0. By Exercise 10, β − α is a root. It cannot be a negative root, since then α = β+ (α−β) would contradict the maximality of β.

Since β−α is a positive root, we have β>α.

Next we show that if γ is any root then hγ, βi6hβ, βiwith equality only in the case γ =β. We embed γ and β in a rank two root system Φ₀ = Φ∩V₀ where V₀ is the vector space they span. Then Φ₀ is one of A₁×A₁ orA₂, B₂ orG₂. Except in the first case β is the unique highest weight vector, and in every case, the assertion may be checked by inspection. We leave the verification to the reader.

Letα₀ be the negative of the highest root. We see that

hαi, αji>0 if αi, αj ∈ {α0, α1,· · · , αr}, i6=j.

Indeed, this is part of Proposition 12 if α_i, α_j ∈Σ and Proposition 41 if one of α_i is α₀.

Proposition 42 The fundamental alcove F is defined by the inequalities

hα_i, vi>0 (16i6r), hα₀, vi>−1. (53) Proof It is clear that the fundamental alcove is determined by the inequalities hα_i, vi > 0 and hα, vi 6 1 as α runs through the positive roots. We have to show that the inequalities hα, vi 6 1 all follow from the inequality h−α₀, vi 6 1, which is equivalent to the assumed inequality hα₀, vi > −1. Indeed, if α is any positive root, then α 6 −α₀, so we may write α = −α₀ − P

k_iα_i with k_i > 0. Then

hα, vi =h−α₀, vi −P

k_ihα_i, vi. However if α is already assumed to satisfy he first inequalities in (53), then hα_i, vi > 0, so hα, vi 6 h−α₀, vi. Hence the inequality h−α₀, vi 6 1 is sufficient to imply hα, vi 6 1 for all positive roots α, and thus the fundamental alcove is indeed determined by the given inequalities.

If α ∈ Φ and k ∈ Z we consider the reflection r_α,k in the hyperplane P_α,k. This is the map

rα,−k =v− hα^∨, viα+kα^∨ =v− hα, viα^∨ +kα^∨.

We haveP_α,k =P_−α,−k and r_−α,−k =r_α,k. Let s_i =r_αi,0 (16i6r) and s₀ =r_α0,−1. These are the reflections in the hyperplanes bounding the fundamental alcove.

Proposition 43 The group W_aff is generated by the s_i.

Proof Let w∈W_aff. ThenwF is an alcove. We consider a path pfrom an interior vertex ofF to an interior vertex ofwF. LetF₀ =F,F₁,· · · ,F_k =wFbe the series of alcoves through whichp passes. Each pair Fi,Fi+1 is separated by aPα,k.

Since F₁ is adjacent to F₀ = F, by Proposition 42 we have F₁ = s_i1F for some 0 6 i₁ 6 r. Now F₂ is adjacent to F₁, so s⁻¹_i1 F₂ is adjacent to s⁻¹_i1 F₁ = F. Thus s⁻¹_i1 F2 = si2F for some 0 6 i2 6 r, and so F2 = si1si2F. Continuing this way we obtain a sequencei₁, i₂,· · · such that F_l=s_i1· · ·s_ilF. Now wF=s_i1· · ·s_ikF implies that w=s_i1· · ·s_ik, proving that W_aff is generated by the s_i. Theorem 13 Waff is a Coxeter group with generators {s0, s1,· · · , sr}.

Proof Let us suppose that

s_i1· · ·s_ik =s_j1· · ·s_jl

are two words representing the same element w ∈ W_aff. Let G be the group with generators σ₀,· · ·σ_r and relations σ²_i = 1 and (σ_iσ_j)^m(i,j) = 1, where m(i, j) is the order of s_is_j. We want to show that σ_i1· · ·σ_ik =σ_j1· · ·σ_jl.

If t = 1,2,3,· · · let S_t be the set of all affine subspaces M of V that are the interesection of t or more hyperplanes P_α,k (α ∈ Φ, k ∈ Z) such that dim(M) = dim(V)−t. ThusS₁ consists of the set of hyperplanesP_α,k themselves. Observe that Ω = V −S

S₃ is simply-connected since the removed sets consists of closed affine spaces of codimenion 3 with no accumulation point.

The alcoves_i1Fis adjacent toF, separated by a hyperplane. Also,s_i2Fis adjacent toF so s_i1s_i2Fis adjacent to s_i1F. We therefore get a sequence of alcoves:

F, s_i1F, s_i1s_i2F,· · · , s_i1s_i2· · ·s_ikF=F₀,F₁,· · · ,F_k.

We take a path p from a point u in the interior of F to a point v in the interior of F_k=wF, passing through these alcoves in order. Similarly we have another path p⁰ from u to v passing through the alcoves F⁰₀,F⁰₁,· · · ,F⁰_l, where F⁰_t = s_j1· · ·s_jtF, with F⁰₀ = F₀ and F⁰_l = F_k = wF. The paths p and p⁰ both lie in the simply connected space Ω.

We deform p into p⁰ and observe how the set of alcoves changes as we make this deformation. We stay within Ω at every stage. We are allowed to cross elements of S₂ but notS₃. We arrange so that we cross elements ofS₂ one at a time and observe how the word s_i1· · ·s_ik representing w changes when we do. We will see that each such change corresponds to a use of the braid relation.

Since we may never pass through a space M in S, the only types of transitions that can occur have the path move across a subset M in S₂. We may visualize this by taking a cross section in a 2-dimensional affine space perpendicular to M, and projecting the path onto that path. The homotopy thus moves p₀ =p, one segment of which might look like this:

Ft

Ft+1

Ft+2

to an equivalent path p₁, whose corresponding segment looks like this:

F_t

F⁰_t+1

F⁰_t+2 F⁰_t+3

Ft+2

Now if we have F_t = s_i1· · ·s_itF, F_t+1 = s_i1· · ·s_it+1F, F_t+2 = s_i1· · ·s_it+2F. On the other hand (in this example)

F⁰_t+1 = s_i1· · ·s_its_it+2F, F⁰_t+2 = s_i1· · ·s_its_it+2s_it+1F, F⁰_t+3 = si1· · ·sitsit+2sit+1sit+2F,

F⁰_t+3 = s_i1· · ·s_its_it+2s_it+1s_it+2s_it+1F=F_t+2. So this homotopy replaces the words_i1· · ·s_ik by

s_i1s_i1· · ·s_its_it+2s_it+1s_it+2s_it+1s_it+3· · ·s_ik. In this example, the order of s_its_it+1 is 3, so (σ_itσ_it+1)³ = 1 and

σ_it+1σ_it =σ_it+1σ_itσ_it+1σ_it,

Thus this homotopy crossing the affine subspaceMof codimension 2 replacesσ_i1· · ·σ_it by

σ_i1σ_i1· · ·σ_it−1σ_it+1σ_itσ_it+1σ_itσ_it+1· · ·σ_ik,

but these are equal in the group G. Continuing in this way, we eventually get σ_i1· · ·σ_ik =σ_j1· · ·σ_jl.

We have done just one example of crossing an element M of S₂ but clearly any such crossing amounts to an application of a braid relation. We see thus that W_aff

is a Coxeter group.

We wish to have analogs of the roots. These will be theaffine roots, which in our interpretation are affine-linear functions onV that vanish on the hyperplanes P_α,k. Remark 1 Our definition of the affine roots is that given by Macdonald. However in view of the work of Kac and of Moody on infinite-dimensional Lie algebras, the affine roots of Macdonald should be supplemented by other “imaginary” roots. We will ignore the imaginary roots since they play no role for us.

If α∈Φ let

α^∨ = 2α hα, αi.

The α^∨ are called coroots, and the set ˆΦ of coroots is called thedual root system. If α∈Φ and k∈Z letL_α,k :V −→Rbe the linear functional

L_α,k(v) =hα, vi −k.

Then L_α,k vanishes onP_α,k, as does L−α,−k =−L_α,k.

The Weyl group acts on the roots by wL(v) =L(w⁻¹v).

We note that a rootL never vanishes on an alcove. Let us say that an affine root Lis positive (resp. negative) if its values are positive (negative) on the fundamental alcove F. Let Φ⁺_aff be the positive affine roots and Φ⁻_aff be the negative ones. If 16i6r letL_i =L_αi,0 and P_i =P_αi,0. Let L₀ =L_α0,−1 and P₀ =P_α0,−1.

Proposition 44 Let 06i6r and let L be a positive root. Then si(L)∈Φ⁻ if and only if L=L_i.

Proof If L = L_α,k, then s_i(L) is a negative root if and only if L(s_iv) < 0, where v ∈ F. Since L(v) > 0, v and si(v) must lie on opposite sides of the hyperplane L_α,k. But the only hyperplane among the P_α,k that separatesv and s_i(v) is P_i, and soP_α,k =P_i. SinceL is a positive root, L=L_i. Let Q^∨ be the lattice generated by the coroots. Ifλ ∈Q^∨ letτ(λ) :V −→V be the map τ(λ)v =v +λ. We may identify Q^∨ with its image under τ as a group of translations.

Proposition 45 The subgroupQ^∨ is normal in Waff andWaff is the semidirect prod-uct Q^∨oW.

Proof Translation by α^∨_i moves P_α,k to P_α,l where l = k +hα^∨_i, αi so l ∈ Z if k ∈ Z. Thus translation by an element of the coroot lattice permutes the alcoves, and corresponds to an element of W_aff.

The coroot lattice Q^∨ is invariant under W_aff, since the reflection in P_α,k moves any vectorv tov+ (k− hα, vi)α^∨; it sends a corootβ^∨ toβ^∨+ (k− hα, β^∨i)α^∨ which is also in the coroot lattice. If G is any alcove, then G = wF for some w ∈ W_aff, so G contains λ =w(0) which we see is an element of Q^∨. Therefore τ(−λ)w is an element ofW_aff that fixes 0, hence is an element ofW. This shows thatW_aff =Q^∨W. We will prove results about the length function that are analogous to those for the ordinary Weyl group in Section 4. As in that section, there are two definitions that are eventually shown to be equivalent.

As with the ordinary Weyl group the first definition makes l : W_aff −→ Z the minimal length of a decompositionw=s_i1· · ·s_ik. The second definition, temporarily denoted l⁰ until we prove that they are the same, is the number ofL∈Φ⁺ such that w(L)∈Φ⁻.

Lemma 8 l⁰(w) is the number of hyperplanes Pα,k that lie between F and w⁻¹F.

Proof If L = L_α,k ∈ Φ⁺, then w(L) ∈ Φ⁻ if and only if L(w⁻¹v) < 0 for v ∈ F.

This means that P_α,k is a hyperplane between F and w⁻¹F.

From the Lemma, l⁰(w)<∞.

Proposition 46 Let w∈W_aff and let s=s_i, L=L_i for 06i6r. Then l⁰(ws) =

l⁰(w) + 1 if w(L)∈Φ⁺, l⁰(w)−1 if w(L)∈Φ⁻.

Proof We have l⁰(w) = |Φ⁺ ∩ w⁻¹Φ⁻|. Therefore l⁰(ws) = |Φ⁺ ∩ s⁻¹w⁻¹Φ⁻|.

Applyingsreplaces this set with one of equal cardinality, sol⁰(ws) = |sΦ⁺∩w⁻¹Φ⁻|.

Now by Proposition 44,sΦ⁺ is obtained from Φ⁺ by removing L=L_i and replacing it by its negative. From this it is clear thatl⁰(ws) =l⁰(w) + 1 ifL∈w⁻¹Φ⁺, that is,

if wL∈Φ⁺, and l⁰(w)−1 otherwise.

Theorem 14 The Exchange property (Propositions 14 and 15) is true for the affine Weyl group. The two definitions of the length function are the same: l = l⁰. Tits’

Theorem 23 remains true.

Tits’ Theorem, in this context, says the following. Let B be the braid group with generators u₀, u₁,· · · , u_r subject to the same braid relations satisfied by the si. Then if w ∈ Waff has two reduced representations w =si1· · ·si_k = sj1· · ·sj_k as products of simple reflections withk =l(w), then u_i1· · ·u_ik =u_j1· · ·u_jk.

Proof The proofs of Section 4 go through without much change. We leave the details to the reader. (One also gets another proof thatW_aff is a Coxeter group.) Proposition 47 (Iwahori and Matsumoto)Let d∈Q^∨ andw∈W. Let L=L_i where 16i6r. Then

l(τ(d)ws_i) =











l(τ(d)w) + 1 if w(αi)∈Φ⁺ and hw(αi), di60, or w(α_i)∈Φ⁻ and hw(α_i), di<0.

l(τ(d)w)−1 if w(α_i)∈Φ⁺ and hw(α_i), di>0, or w(αi)∈Φ⁻ and hw(αi), di>0.

Moreover

l(τ(d)ws₀) =











l(τ(d)w) + 1 if w(α₀)∈Φ⁺ and hw(α₀), di61, or w(α₀)∈Φ⁻ and hw(α₀), di<1, l(τ(d)w)−1 if w(α₀)∈Φ⁺ and hw(α₀), di>1,

or w(α₀)∈Φ⁻ and hw(α₀), di>1.

Proof Let 1 6 i 6 r. By Proposition 46, a necessary and sufficient condition for l(τ(d)ws_i) = l(τ(d)w) + 1 is that τ(d)w(L_i) ∈ Φ⁺. This means that for v ∈ F we need h(τ(d)w)⁻¹(v), α_ii > 0, that is, hv−d, w(α_i)i > 0. We may take v near the origin. Thenhv, w(α_i)iwill be small, whileh−d, w(α_i)i ∈Z. Ifhw(α_i), diis nonzero, then hv−d, w(α_i)i >0 depending on whether hw(α_i), di <0 or > 0. On the other hand, if hw(α_i), di= 0, then hv−d, w(α_i)i>0 or <0 depending on whether w(α_i) is a positive or negative root, becausev is in the positive Weyl chamber. This proves the first case. We leave the second one (with i = 0) to the reader, but similar

considerations suffice.

Let us say that w ∈ Waff is dominant if w(F) is contained in the positive Weyl chamber. If d ∈ Q^∨ then clearly d is dominant in this sense if and only if it is dominant in the usual sense: hα, di>0 for all α∈Φ.

Proposition 48 Supposed ∈Q^∨ and that d is dominant. Then l(d) = X

α∈Φ⁺

hα, di=h2ρ, di. Here ρ= ¹₂P

α∈Φα is the Weyl vector.

Proof The length l(d) is equal to the number of hyperplanes H_α,k between F and τ(d)F. The hyperplane H_α,k lies between F and τ(d)F if and only if 0 < k 6hα, di.

There are hα, di of this, and summing over α, the statement follows.

Im Dokument Hecke Algebras (Seite 56-65)