ON TWO PRESENTATIONS OF THE AFFINE WEYL GROUPS OF THE CLASSICAL TYPES

Jian-yi Shi Department of Mathematics, East China Normal University, Shanghai, 200062, P.R.C.

and

School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia

Abstract. The main result of the paper is to get the transition formulae between the alcove form and the permutation form of w ∈ W_{a}, where W_{a} is an affine Weyl group of type B̃_{m} (m ⩾ 3), C̃_{n} (n ⩾ 2) or D̃_{k} (k ⩾ 4). On the other hand, we get a new characterization for the alcove form of an affine Weyl group element which has a much simpler form comparing with that in [10]. Our results give an affirmative answer to a conjecture of H. Eriksson and K. Eriksson in [4] concerning the characterization of the inverse table of w ∈ W_{a}.

Introduction.

Affine Weyl groups, as a family of infinite crystallographic Coxeter groups, play a more and more important role in various fields of mathematics, such as Kac-Moody algebras, algebraic groups and their representation theory, combinatorial and geometric group theory, etc. [2; 3; 5; 7; 9; 14].

Besides the presentations as Coxeter groups (i.e., the ones by generators and relations over the pairs of generators), there are many other presentations for the affine Weyl groups W_{a}, in particular for those of the classical types, i.e., types Ã_{l} (l ⩾ 1), B̃_{m} (m ⩾ 3), C̃_{n} (n ⩾ 2) and D̃_{k} (k ⩾ 4). Two of their presentations are particularly useful, one is to regard W_{a} as a certain permutation group over the integer set ℤ (only applied for the classical types); the other is to identify W_{a} with the set of alcoves in a euclidean space E after removing a certain set of hyperphanes (applied for all types) [1; 8; 9; 10; 11; 12; 13].

The relations between these two presentations has been explicitly described for the case of type Ã_{l} (see [9]). but not yet for the other classical types so far. The aim of the

Typeset by AMS-TEX

Supported by the Australia Research Council.

2 Jian-yi Shi

present paper is to study these relations for the cases of types B̃_{m}, C̃_{n} and D̃_{k}. Our main results is to obtain the transition formulae between the permutation form and the alcove form of w ∈ W_{a} when the type of W_{a} belongs to these three families (see Theorems 4.1, 5.2, 5.4 and 5.6).

The alcove form (k(w; α))_{α} of w ∈ W_{a} adopted here is essentially the same as that in [10], but with two changes as below. The one is that the relation k(w; −α) = −k(w; α) in [10] is replaced by k(w; −α) = −k(w; α)−1 for all the positive root (and hence for all the root) α. The other is that in dealing with the case of the classical types, we replace the root system Φ by the corresponding coroot system Φ^{∨} as the index set of the alcove form of w. The reason for these changes is to make our transition formulae simpler. The another reason for the second change is that we have got a new characterization for the alcove form of w ∈ W_{a}, which is equivalent to the original one in [10, Theorem 5.2] but has a much simpler form when stated in terms of coroot system instead of root system (see Theorem 1.3). We shall also show that when an affine Weyl group W_{a} has the classical type, the entries of the alcove form of w ∈ W_{a} exactly comprise the inverse table of w defined by H. Eriksson and K. Eriksson (see [4, §8.] and Theorem 4.1). Thus our new characterization for the alcove form of w ∈ W_{a} gives a characterization for the inverse table of w (see Theorem 4.5), the latter was conjectured by H. Eriksson and K. Eriksson in the case of type C̃_{l} in [4, 8.3.].

In general, the permutation form of w ∈ W_{a} is not unique. This depends on the way of embedding the group W_{a} into the permutation group on ℤ (see [1; 13]). The permutation form of w ∈ W_{a} we take in the present paper has the advantage that when restricting to the corresponding Weyl group, the permutation form of an element is just the usual one.

Thoughout this paper, an affine Weyl group is always assumed irreducible, i.e., its Coxeter graph is connected.

The contents of the paper are organized as follows. In Section 1, we recall some results of [10] concerning the alcove form of an affine Weyl group element and give a new characterization for the alcoves of a euclidean space. Then from Section 2 on, we pay our attention only to the affine Weyl groups W_{a} of the classical types. We describe the alcove form and the permutation form of w ∈ W_{a} in Sections 2 and 3 respectively. Then we show our main results of the paper in Sections 4 and 5, where we obtain the transition formulae between the alcove form and the permutation form of w ∈ W_{a} in the cases of the classical types.

§1. The alcove forms.

On two presentations of the affine Weyl groups 3

In this section, we shall recall some results concerning the alcove form of an element w in an affine Weyl group W_{a}. As mentioned in the Introduction, we make some formal changes in the definition of the alcove form of w. This forces us to reformulate some results in [10]. There is one new result in this section (i.e., Theorem 1.3), which gives a new characterization for an alcove in a euclidean space. 1.1. Let Φ be an irreducible root system of rank l > 1. Let E be the euclidean space spanned by Φ with inner product ⟨ , ⟩ such that |α|^{2} = ⟨α, α⟩ = 1 for any short root α of Φ. Choose a simple root system Π = {α_{1}, ⋯, α_{l}} of Φ and let Φ^{+}, Φ^{−} be the corresponding positive and negative root systems. Let −α_{0} be the highest short root of Φ. Denote by α^{∨} = 2α/⟨α, α⟩ the coroot of α ∈ Φ and by Φ^{∨} the coroot system {α^{∨} | α ∈ Φ}. Note that a coroot system Φ^{∨} itself is also a root system, but with its type dual to that of Φ. This fact will be important later when we make a change for the index set of the alcove form of an affine Weyl group element.

For any α ∈ Φ and k ∈ ℤ, define a hyperplane

(1.1.1)  H_{α;k} = {v ∈ E | ⟨v, α^{∨}⟩ = k}

and a stripe

(1.1.2)  H_{α;k}^{1} = {v ∈ E | k < ⟨v, α^{∨}⟩ < k + 1}.

Then we have H_{−α,k} = H_{α;−k} and H_{−α;k}^{1} = H_{α;−k−1}^{1} for any α ∈ Φ and k ∈ ℤ. Note that the definition of a stripe H_{α;k}^{1} for α ∈ Φ^{−} given here slightly differs from that in [10]. According to [10], we would have H_{−α;k}^{1} = H_{α;−k}^{1}. Thus by this new definition, we have to re-examine some results of [10]. Note that it makes no change for the results only involving positive roots. We call any non-empty connected simplex of

(1.1.3)  E − ⋃_{\begin{matrix}α∈Φ^{+} \\ k∈ℤ \end{matrix}} H_{α;k}

an alcove of E. Each alcove of E has the form ⋂_{α∈Φ^{+}} H_{α;k_{α}}^{1} for some Φ^{+}-tuple (k_{α})_{α∈Φ^{+}}

over ℤ. Since H_{−α;k_{α}}^{1} = H_{α;−k_{α}−1}^{1}, sometimes it is more convenient to denote the alcove ⋂_{α∈Φ^{+}} H_{α;k_{α}}^{1} by ⋂_{α∈Φ} H_{α;k_{α}}^{1} with the convention that k_{−α} = −k_{α} − 1 for any α ∈ Φ^{+}.

1.2. One should note that not every Φ^{+}-tuple (k_{α})_{α∈Φ^{+}} over ℤ gives rise to an alcove of E as above. In fact, it is well known that ⋂_{α∈Φ^{+}} H_{α;k_{α}}^{1} is an alcove of E if and only if

for any α, β ∈ Φ^{+} with α + β ∈ Φ^{+}, the inequality

(1.2.1) |α|^{2}k_{α}+|β|^{2}k_{β} +1 ⩽ |α+β|^{2}(k_{α+β} +1) ⩽ |α|^{2}(k_{α}+1)+|β|^{2}(k_{β} +1)+|α+β|^{2}−1

holds (see [10, Theorem 5.2]). Now we give a new and simpler form for this result.

Theorem 1.3. ⋂_{α∈Φ^{+}} H_{α;k_{α}}^{1} is an alcove of E if and only if for any α, β ∈ Φ^{+} with

γ = (α^{∨} + β^{∨})^{∨} ∈ Φ^{+}, the inequality

(1.3.1)  k_{α} + k_{β} ⩽ k_{γ} ⩽ k_{α} + k_{β} + 1.

holds.

Proof. By [10, Theorem 5.2], it is enough to show that the inequality systems (1.2.1) and (1.3.1) are equivalent for any irreducible positive root system Φ^{+} of rank 2. Note that when α, β, α + β ∈ Φ^{+} satisfy |α| = |β| = |α + β|, the inequality (1.2.1) becomes

(1.3.2)  k_{α} + k_{β} ⩽ k_{α+β} ⩽ k_{α} + k_{β} + 1,

and γ = (α^{∨} + β^{∨})^{∨} = α + β. Thus the inequalities (1.2.1) and (1.3.1) are the same in this case. So we need only consider the case that Φ^{+} has type B_{2} or G_{2} and that the lengths of the three roots involved in the inequalities are not all the same. We shall only deal with the case of type B_{2} here. The case of type G_{2} can be done similarly and hence is left to the reader. Now let Φ^{+} = {α, β, α + β, 2α + β}. Restricting on the triples of roots with the lengths not all equal, the inequality system (1.2.1) becomes

(1.3.3)  \{ \begin{matrix}k_{α} + 2k_{β} ⩽ k_{α+β} ⩽ k_{α} + 2k_{β} + 2, \\ k_{α} + k_{α+β} ⩽ 2k_{2α+β} + 1 ⩽ k_{α} + k_{α+β} + 2. \end{matrix}

and (1.3.1) becomes

(1.3.4)  \{ \begin{matrix}k_{α} + k_{β} ⩽ k_{2α+β} ⩽ k_{α} + k_{β} + 1, \\ k_{β} + k_{2α+β} ⩽ k_{α+β} ⩽ k_{β} + k_{2α+β} + 1. \end{matrix}

We have to show the equivalence of the inequality systems (1.3.3) and (1.3.4). Now assume (1.3.3). Adding two inequalities of (1.3.3) together on the corresponding sides and then subtracting k_{α+β} from all sides of the resulting inequality, we get

(1.3.5)  2(k_{α} + k_{β}) ⩽ 2k_{2α+β} + 1 ⩽ 2(k_{α} + k_{β}) + 4.

Then the first inequality of (1.3.4) follows easily from (1.3.5). Rewrite the second inequality of (1.3.3) into the form

(1.3.6)  2k_{2α+β} ⩽ k_{α} + k_{α+β} + 1 ⩽ 2k_{2α+β} + 2.

Then adding (1.3.6) to the first inequality of (1.3.3) on the corresponding sides and subtracting k_{α} from all sides of the resulting inequality, we get

(1.3.7)  2(k_{2α+β} + k_{β}) ⩽ 2k_{α+β} + 1 ⩽ 2(k_{2α+β} + k_{β}) + 4.

On two presentations of the affine Weyl groups 5

The second inequality of (1.3.4) follows easily from (1.3.7). Next assume (1.3.4). Then the first inequality of (1.3.3) is obtained by adding two inequalities of (1.3.4) together on the corresponding sides, followed by subtracting k_{2α+β} from all sides of the resulting inequality. Rewrite the second inequality of (1.3.4) into the form

(1.3.8)  k_{α+β} ⩽ k_{β} + k_{2α+β} + 1 ⩽ k_{α+β} + 1.

Then the second inequality of (1.3.3) is obtained by adding (1.3.8) to the first inequality of (1.3.4) on the corresponding sides, followed by subtracting k_{β} from all sides of the resulting inequality. □

Note that the result still holds if we replace all the notations Φ^{+} by Φ in Theorem 1.3. 1.4. Let W be the Weyl group of Φ generated by all the reflections s_{α}, α ∈ Φ, on E, where s_{α} sends x to x − ⟨x, α^{∨}⟩α. Let Q denote the root lattice ℤΦ, and N the group consisting of all the translations T_{λ} operating on E for λ ∈ Q, where T_{λ} sends x to x + λ. We denote by W_{a} the group of affine transformations of E generated by N and W. It is well known that W_{a} is the semidirect extension of W by the normal subgroup N on which the action of W is known.

For linear and affine transformations, we shall denote the operation on the right and compose them accordingly. With this convention, we define s_{0} = s_{α_{0}}T_{−α_{0}} and s_{i} = s_{α_{i}}, 1 ⩽ i ⩽ l. It is known that W_{a} (resp. W) is a Coxeter group on the generators s_{0}, s_{1}, ⋯, s_{l} (resp. s_{1}, ⋯, s_{l}). We write S̃ = {s_{0}, s_{1}, ⋯, s_{l}} and S = {s_{1}, ⋯, s_{l}}. The group W_{a} is called an affine Weyl group of type X̃, where X is the type of the coroot system Φ^{∨}. 1.5. It is well known that the right action of W_{a} on E gives rise to the permutations of the set {H_{α;k} | α ∈ Φ, k ∈ ℤ}. So it induces the permutations of the set 𝔘 of alcoves of E. It is also well known that 𝔘 is simply transitive under W_{a} (see [2; 14]).

Let w = w̄ · T_{λ} ∈ W_{a} with w̄ ∈ W and λ ∈ Q. We consider the alcove

⋂_{α∈Φ} H_{α;h_{α}}^{1} = ( ⋂_{α∈Φ} H_{α;k_{α}}^{1}) w.

For v ∈ ⋂_{α∈Φ} H_{α;k_{α}}^{1}, we have (v)w = (v)w̄ + λ. This implies that

⟨(v)w, α^{∨}⟩ = ⟨v, ((α)w̄^{−1})^{∨}⟩ + ⟨λ, α^{∨}⟩.

So we get

(1.5.1)  h_{α} = k_{(α)w̄^{−1}} + ⟨λ, α^{∨}⟩

6 Jian-yi Shi

for α ∈ Φ.

We see that A_{1} = ⋂_{α∈Φ} H_{α;ϵ_{α}}^{1} is an alcove of E, where ϵ_{α} = 0 if α ∈ Φ^{+}, and ϵ_{α} = −1

if α ∈ Φ^{−} (comparing with [10, Lemma 1.1]). Denote A_{w} = (A_{1})w for any w ∈ W_{a}. Thus any alcove of 𝔘 has the form A_{w} = ⋂_{α∈Φ^{+}} H_{α;k(w;α)}^{1} or A_{w} = ⋂_{α∈Φ} H_{α;k(w;α)}^{1} for some

w ∈ W_{a}, with the convention that

(1.5.2)  k(w; −α) = −k(w; α) − 1 for any α ∈ Φ^{+}

(comparing with [10, §1.]). Note that (1.5.2) actually holds for any α ∈ Φ. We shall identify W_{a} with 𝔘 as sets under the correspondence w ↦ A_{w}, and call (k(w; α))_{α∈Φ^{+}} or (k(w; α))_{α∈Φ} the alcove form of w. From (1.5.1), we get

(1.5.3)  k(w; α) = ϵ_{(α)w̄^{−1}} + ⟨λ, α^{∨}⟩

for α ∈ Φ, where the decomposition w = w̄·T_{λ} is as above (comparing with [10, Theorem 3.3]). This is also equivalent to a formula given by Iwahori and Matsumoto [6]. 1.6. The action of an element s_{j} ∈ S̃ on 𝔘 can be described as follows. For w ∈ W_{a}, we have by (1.5.1) that

(1.6.1)  k(ws_{j}, α) = \{ \begin{matrix}k(w; (α)s_{j}), & if j ≠ 0, \\ k(w; (α)s_{α_{0}}) + ⟨−α_{0}, α^{∨}⟩, & if j = 0. \end{matrix}

for any α ∈ Φ (comparing with [10, Proposition 4.2]). 1.7. We can define the left action of W_{a} on 𝔘 as below. For y, w ∈ W_{a}, we set y(A_{w}) = A_{yw}. Then for w ∈ W_{a} and s_{j} ∈ S̃, we have

(1.7.1)  k(s_{j}w, α) = \{ \begin{matrix}k(w; α), & if α ≠ ±(α_{j})w̄, \\ k(w; α) ∓ 1, & if α = ±(α_{j})w̄, \end{matrix}

where w̄ is the image of w under the natural map W_{a} ⟶ W_{a}/N ≅ W (comparing with [10, Proposition 4.1]). 1.8. To any w ∈ W_{a}, we associate two subsets of S̃: L(w) = {s ∈ S̃ | sw ⩽ w} and R(w) = {s ∈ S̃ | ws ⩽ w}, where the notation ⩽ stands for the Chevalley-Bruhat order on W_{a} regarded as a Coxeter group with respect to S̃. These two subsets of S̃, together with the length function ℓ(w), can be described in terms of the k(w; α)’s as below.

Proposition 1.9 (comparing with [10, Propositions 3.4 and 4.3]). Let (k(w; α))_{α∈Φ} be the alcove form of w ∈ W_{a}. Then

(1) ℓ(w) = ∑_{α∈Φ^{+}} |k(w; α)|.

(2) R(w) = {s_{j} ∈ S̃ | either j ≠ 0, k(w; α_{j}) < 0, or j = 0, k(w; −α_{0}) > 0}.

(3) L(w) = {s_{j} ∈ S̃ | k(w; (α_{j})w̄) > 0}.

(4) k(w^{−1}, α) = \{ \begin{matrix}k(w; −(α)w̄), & if either α, −(α)w̄ ∈ Φ^{+} or α, −(α)w̄ ∈ Φ^{−}, \\ k(w; −(α)w̄) + 1, & if α, (α)w̄ ∈ Φ^{+}, \\ k(w; −(α)w̄) − 1, & if α, (α)w̄ ∈ Φ^{−}. \end{matrix}

§2. The cases of the classical types.

In this section, we shall first give an explicit description for all the irreducible root systems of the classical types. Then we reformulate some results on the alcove forms of w ∈ W_{a} in these cases. A significant change is made for the index set of the alcove form of w ∈ W_{a}, where the root system Φ is replaced by the corresponding coroot system Φ^{∨}. 2.1. We set the notations Ω_{l} = {A_{l−1}, B_{l}, C_{l}, D_{l}} and Ω′_{l} = {B_{l}, C_{l}, D_{l}} for l ⩾ 2, which will be used quite often later on. 2.2. We know that there is a natural bijective map α ↦ α^{∨} from a root system Φ to the corresponding coroot system Φ^{∨}. By Theorem 1.3, it is more convenient to express the alcove form (k(w; α))_{α∈Φ} of w ∈ W_{a} by (k(w; α))_{α∈Φ^{∨}}, where we set k(w; α) = k(w; α^{∨}) for any α ∈ Φ^{∨} (note (α^{∨})^{∨} = α). We make such a change from now on. Then for an affine Weyl group W_{a} of type X̃, X ∈ Ω_{l}, the index set of the alcove form of w ∈ W_{a} will be the root system of type X. Note that −α_{0}^{∨} is the highest coroot in Φ^{∨}. 2.3. Let e_{1}, e_{2}, ⋯, e_{l} be an orthonormal basis of a euclidean space E. Set e_{−i} = −e_{i}, 1 ⩽ i ⩽ l, and e_{0} = 0. Define (i, j) = e_{j} − e_{i} for i, j ∈ [−l, l], where the notation [a, b] stands for the interval {a, a + 1, ⋯, b} for any a ⩽ b in ℤ. Then the relations

(2.3.1)  (i, j) = (i, t) + (t, j) and (i, j) = (−j, −i)

hold for any i, t, j ∈ [−l, l]. 2.4. The root system of type A_{l−1}, l ⩾ 2, can be described by

Φ(A_{l−1}) = {(i, j) | 1 ⩽ i, j ⩽ l; i ≠ j}.

The roots (t − 1, t), 1 < t ⩽ l, form a simple root system, and Φ^{+}(A_{l−1}) = {(i, j) | 1 ⩽ i < j ⩽ l} is the corresponding positive root system. The highest root is (1, l). 2.5. The root system of type B_{l}, l ⩾ 3, can be described by

Φ(B_{l}) = {(i, j) | i, j ∈ [−l, l]; i ≠ ±j}.

The roots (t−1, t), 1 ⩽ t ⩽ l, form a simple root system, and Φ^{+}(B_{l}) = {(i, j) ∈ Φ(B_{l}) | i < j} is the corresponding positive root system. The highest root is (−l + 1, l).

8 Jian-yi Shi

2.6. The root system of type C_{l}, l ⩾ 2, can be described by

Φ(C_{l}) = {(i, j) | i, j ∈ [−l, l] \ {0}; i ≠ j}.

The roots (t − 1, t), 1 < t ⩽ l, and (−1, 1) form a simple root system, and Φ^{+}(C_{l}) = {(i, j) ∈ Φ(C_{l}) | i < j} is the corresponding positive root system. The highest root is (−l, l). 2.7. The root system of type D_{l}, l ⩾ 4, can be described by

Φ(D_{l}) = {(i, j) | i, j ∈ [−l, l] \ {0}; i ≠ ±j}.

The roots (t − 1, t), 1 < t ⩽ l, and (−1, 2) form a simple root system, and Φ^{+}(D_{l}) = {(i, j) ∈ Φ(D_{l}) | i < j} is the corresponding positive root system. The highest root is (−l + 1, l). 2.8. Fix w ∈ W_{a}(X̃), X ∈ Ω_{l}. Let Φ be the root system of type X. In the alcove form (k(w; α))_{α∈Φ} of w, we shall denote the entry k(w; (i, j)) simply by k(w; i, j) for (i, j) ∈ Φ. Then the entry k(w; (i, j)x̄) becomes k(w; (i)x̄, (j)x̄) for any element x̄ of the corresponding Weyl group, the latter is regarded as a certain permutation group on the set [−l, l]. By (1.5.2), (2.3.1), Theorem 1.3 and the notice immediately after the Theorem, we get

Lemma 2.9. Let X ∈ Ω_{l} for an appropriate integer l and let Φ be the root system of type X. Then for any w ∈ W_{a}(X̃) and (i, j), (i, t), (t, j) ∈ Φ, we have

(1) k(w; j, i) = −k(w; i, j) − 1 = k(w; −i, −j),

(2) k(w; i, t) + k(w; t, j) ⩽ k(w; i, j) ⩽ k(w; i, t) + k(w; t, j) + 1.

§3. The permutation forms. 3.1. Let A_{l} be the affine Weyl group of type Ã_{l−1}, l ⩾ 2. It is well known that A_{l} could be regarded as a group of certain permutations on ℤ:

A_{l} = {w : ℤ ⟶ ℤ |(i + l)w = (i)w + l for i ∈ ℤ; ∑_{i=1}^{l}(i)w = ∑_{i=1}^{l} i} .

Its simple reflection set S̃ = {r_{0}, r_{1}, ⋯, r_{l−1}} is given by

(i)r_{t} = \{ \begin{matrix}i, & if i ≢ t, t + 1 ( mod l), \\ i + 1, & if i ≡ t ( mod l), \\ i − 1, & if i ≡ t + 1 ( mod l), \end{matrix}

for i ∈ ℤ and 0 ⩽ t < l. Each element w ∈ A_{l} is determined entirely by the images (a_{1})w, (a_{2})w, ⋯, (a_{l})w of any given l integers a_{1}, ⋯, a_{l} under w, which are pairwisely incongruent modulo l.

3.2. As a Coxeter group, the affine Weyl group W_{a}(X̃) of type X ∈ Ω′_{l} has the following presentation:

W_{a}(X̃) = ⟨s_{i} | 0 ⩽ i ⩽ l, (s_{i}s_{j})^{m_{ij}} = 1, for 0 ⩽ i ⩽ j ⩽ l⟩,

where m_{ij} = 1 for i = j; m_{ij} = 3 in one of the following cases:

(i) 2 < i + 1 = j ⩽ l,

(ii) (i, j) = (0, l − 1) and X = B_{l},

(iii) (i, j) = (1, 3), (0, l − 1) and X = D_{l};

m_{ij} = 4 in one of the following cases:

(i) (i, j) = (1, 2) and X = B_{l},

(ii) (i, j) = (1, 2), (0, l) and X = C_{l};

m_{ij} = 2 in all the other cases. Note that the labelings of the nodes in the corresponding extended Dynkin diagrams slightly differ from the usual ones (see [2; 5] for examples). 3.3. The affine Weyl group W_{a}(C̃_{l}) (l ⩾ 2) can be embedded into A_{2l+2} by an injective homomorphism which sends s_{t}, 1 < t ⩽ l, to r_{t−1}r_{2l+2−t}, s_{1} to r_{0}r_{2l+1}r_{0}, and s_{0} to r_{l}r_{l+1}r_{l}. Thus by identifying with its image, we can regard W_{a}(C̃_{l}) as a subgroup of A_{2l+2}. The group W_{a}(C̃_{l}) can also be regarded as the set of all the fixed points of A_{2l+2} under the involutive automorphism ϕ which sends r_{i} to r_{2l+1−i} for all i, 0 ⩽ i ⩽ 2l + 1:

(3.3.1)  W_{a}(C̃_{l}) ={w ∈ A_{2l+2} | ϕ(w) = w}
={w ∈ A_{2l+2} | (−i)w = −(i)w, ∀i ∈ ℤ}.

The simple reflections s_{t}, 0 ⩽ t ⩽ l, of W_{a}(C̃_{l}) are the permutations on ℤ satisfying that for i ∈ ℤ and 1 < t ⩽ l, we have

(i)s_{t} = \{ \begin{matrix}i, & if i ≢ ±t, ±(t − 1) ( mod 2l + 2), \\ i + 1, & if i ≡ −t, t − 1 ( mod 2l + 2), \\ i − 1, & if i ≡ t, −t + 1 ( mod 2l + 2). \end{matrix}

(i)s_{1} = \{ \begin{matrix}i, & if i ≢ ±1 ( mod 2l + 2), \\ i + 2, & if i ≡ −1 ( mod 2l + 2), \\ i − 2, & if i ≡ 1 ( mod 2l + 2). \end{matrix}

(i)s_{0} = \{ \begin{matrix}i, & if i ≢ ±l ( mod 2l + 2), \\ i + 2, & if i ≡ l ( mod 2l + 2), \\ i − 2, & if i ≡ −l ( mod 2l + 2). \end{matrix}

10 Jian-yi Shi

3.4. The group W_{a}(B̃_{l}) (l ⩾ 3) can be embedded into A_{2l+2} by an injective homomorphism which sends s_{t}, 1 < t ⩽ l, to r_{t−1}r_{2l+2−t}, s_{1} to r_{0}r_{2l+1}r_{0} and s_{0} to r_{l}r_{l+1}r_{l}r_{l−1}r_{l+2}r_{l}r_{l+1}r_{l}. Thus we can regard W_{a}(B̃_{l}) as a subgroup of A_{2l+2} by identifying with its image. We can also regard W_{a}(B̃_{l}) as a certain fixed point set of A_{2l+2} under the automorphism ϕ (see 3.3):

(3.4.1) W_{a}(B̃_{l}) ={w ∈ A_{2l+2} | ϕ(w) = w; N_{l}(w) ≡ 0 ( mod 2)},
={w ∈ A_{2l+2} | (−i)w = −(i)w, ∀i ∈ ℤ; N_{l}(w) ≡ 0 ( mod 2)},

where N_{l}(w) is the number of the integers j with j < l + 1 and (j)w > l + 1. The simple reflections s_{t}, 0 ⩽ t ⩽ l, of W_{a}(B̃_{l}) are the permutations on ℤ as below. For i ∈ ℤ and 1 < t ⩽ l, we have

(i)s_{t} = \{ \begin{matrix}i, & if i ≢ ±t, ±(t − 1) ( mod 2l + 2), \\ i + 1, & if i ≡ −t, t − 1 ( mod 2l + 2), \\ i − 1, & if i ≡ t, −t + 1 ( mod 2l + 2). \end{matrix}

(i)s_{1} = \{ \begin{matrix}i, & if i ≢ ±1 ( mod 2l + 2), \\ i + 2, & if i ≡ −1 ( mod 2l + 2), \\ i − 2, & if i ≡ 1 ( mod 2l + 2). \end{matrix}

(i)s_{0} = \{ \begin{matrix}i, & if i ≢ ±(l − 1), ±l ( mod 2l + 2), \\ i + 3, & if i ≡ l − 1, l ( mod 2l + 2), \\ i − 3, & if i ≡ −l, −l + 1 ( mod 2l + 2). \end{matrix}

3.5. The affine Weyl group W_{a}(D̃_{l}) (l ⩾ 4) can be regarded as a subgroup of A_{2l+2} by an injective homomorphism which sends s_{t}, 1 < t ⩽ l, to r_{t−1}r_{2l+2−t}, s_{1} to r_{0}r_{2l+1}r_{0}r_{1}r_{2l}r_{0}r_{2l+1}r_{0} and s_{0} to r_{l}r_{l+1}r_{l}r_{l−1}r_{l+2}r_{l}r_{l+1}r_{l}, or equivalently, regarded as a certain fixed point set of A_{2l+2} under the automorphism ϕ (see 3.3):

W_{a}(D̃_{l}) = {w ∈ A_{2l+2} | ϕ(w) = w; N_{0}(w) ≡ N_{l}(w) ≡ 0 ( mod 2)}, (3.5.1)
= {w ∈ A_{2l+2} | (−i)w = −(i)w, ∀i ∈ ℤ; N_{0}(w) ≡ N_{l}(w) ≡ 0 ( mod 2)},

where N_{0}(w) is the number of the integers j with j < 0 and (j)w > 0. The simple reflections s_{t}, 0 ⩽ t ⩽ l, of W_{a}(D̃_{l}) are the permutations on ℤ as below. For 1 < t ⩽ l and i ∈ ℤ, we have

(i)s_{t} = \{ \begin{matrix}i, & if i ≢ ±t, ±(t − 1) ( mod 2l + 2), \\ i + 1, & if i ≡ −t, t − 1 ( mod 2l + 2), \\ i − 1, & if i ≡ t, −t + 1 ( mod 2l + 2). \end{matrix}

(i)s_{1} = \{ \begin{matrix}i, & if i ≢ ±1, ±2 ( mod 2l + 2), \\ i + 3, & if i ≡ −1, −2 ( mod 2l + 2), \\ i − 3, & if i ≡ 1, 2 ( mod 2l + 2). \end{matrix}

(i)s_{0} = \{ \begin{matrix}i, & if i ≢ ±l, ±(l − 1) ( mod 2l + 2), \\ i + 3, & if i ≡ l − 1, l ( mod 2l + 2), \\ i − 3, & if i ≡ −l, −l + 1 ( mod 2l + 2). \end{matrix}

From the above discussion, we get the following result easily. Lemma 3.6. Let X ∈ Ω_{l} and w ∈ W_{a}(X̃) for an appropriate integer l. Define an integer m_{X} to be l if X = A_{l−1}, and 2l + 2 if X ∈ Ω′_{l}. Then regarding w as an element of A_{m_{X}}, we have

(1) (i)w ≢ (j)w ( mod m_{X}) for any i, j ∈ ℤ with i ≢ j ( mod m_{X}).

In the case of X ∈ Ω′_{l}, we have

(2) (−i)w = −(i)w for any i ∈ ℤ.

(3) (h(l + 1))w = h(l + 1) for any h ∈ ℤ.

(4) (i)w ≢ 0, l + 1 ( mod 2l + 2) for any i ∈ ℤ with i ≢ 0, l + 1 ( mod 2l + 2). Remark 3.7. (i) Our definition of W_{a}(C̃_{l}) as a group of permutations on ℤ is slightly different from that given by R. Bédard (see [1]). According to R. Bédard, the group W_{a}(C̃_{l}) was embedded into the group A_{2l+1} instead of A_{2l+2}. The advantage of our definition is that the symmetry between the generators s_{0} and s_{1} in W_{a}(C̃_{l}) could be exposed more explicitly in form.

(ii) The descriptions (3.3.1), (3.4.1), (3.5.1) of the groups W_{a}(X̃), X ∈ Ω′_{l}, can be shown easily by applying induction on ℓ(w) ⩾ 0 and by the fact that the number of inversions of w is increasing when ℓ(w) is getting larger.

(iii) Let X ∈ Ω′_{l}. Then regarded as a permutation on ℤ, an element w ∈ W_{a}(X̃) is determined uniquely by the images (a_{1})w, ⋯, (a_{l})w of any given l integers a_{1}, ⋯, a_{l} under w, provided that they are pairwisely incongruent modulo 2l + 2, none of which is divisible by l + 1, and none pair of which have the sum divisible by 2l+2. §4. Transition from the permutation forms to the alcove forms.

Let X ∈ Ω_{l} for an appropriate integer l (see 2.1.). In this section, we shall show the transition formulae from the permutation form to the alcove form of w ∈ W_{a}(X̃). Let Φ = Φ(X) be the root system of type X. Let m_{X} be the integer defined as in Lemma 3.6. Theorem 4.1. For any (i, j) ∈ Φ and w ∈ W_{a}(X̃), we have

(4.1.1)  k(w; i, j) = [\frac{(j)w^{−1} − (i)w^{−1}}{m_{X}} ] ,

where [x] denotes the largest integer not greater than x for a rational number x.

Proof. For any i, j ∈ ℤ with i ≢ j ( mod m_{X}), we have the relation

(4.1.2)  [\frac{(i)w^{−1} − (j)w^{−1}}{m_{X}} ] = − [\frac{(j)w^{−1} − (i)w^{−1}}{m_{X}} ] − 1

by Lemma 3.6, (1). Also, we have (i, j) ∈ Φ^{+} ⟺ (j, i) ∈ Φ^{−}. Thus by Lemma 2.9, (1), we need only show our result in the case when (i, j) is a positive root. The result is known for the case of X = A_{l−1} (see [9, Proposition 6.2.1]). Now assume X ≠ A_{l−1} and hence m_{X} = 2l + 2. We see that the set

Φ^{+} = {(i, j) ∈ Φ | i < j; |i| ⩽ |j|}.

forms a positive root system of Φ. Apply induction on ℓ(w) ⩾ 0. When ℓ(w) = 0, i.e., w = 1, we have ((1)1^{−1}, ⋯, (l)1^{−1}) = (1, 2, ⋯, l) and k(w; i, j) = 0 for all the positive root (i, j). The result is obviously true. Now assume ℓ(w) > 0. Take s_{t} ∈ R(w) (see 1.8). By inductive hypothesis, the result is true for ws_{t}. First assume 1 < t ⩽ l. When (i, j) ≠ (t − 1, t), (−t + 1, t) with (i, j) ∈ Φ^{+}, we have (i)s_{t} < (j)s_{t} and |(i)s_{t}| ⩽ |(j)s_{t}|. Thus in this case, we get from (1.6.1) that

k(w; i, j) = k(ws_{t}; (i)s_{t}, (j)s_{t}) = [\frac{(j)s_{t}(ws_{t})^{−1} − (i)s_{t}(ws_{t})^{−1}}{2l + 2} ]

= [\frac{(j)w^{−1} − (i)w^{−1}}{2l + 2} ] .

On the other hand, by (1.6.1), (2.3.1), (4.1.2), Lemmas 2.9 and 3.6, we have

k(w; t − 1, t) = −k(ws_{t}; t − 1, t) − 1 = − [\frac{(t)(ws_{t})^{−1} − (t − 1)(ws_{t})^{−1}}{2l + 2} ] − 1

= − [\frac{(t − 1)w^{−1} − (t)w^{−1}}{2l + 2} ] − 1 = [\frac{(t)w^{−1} − (t − 1)w^{−1}}{2l + 2} ]

and

k(w; −t + 1, t) = k(ws_{t}; −t, t − 1) = k(ws_{t}; −t + 1, t)

= [\frac{(t)(ws_{t})^{−1} − (−t + 1)(ws_{t})^{−1}}{2l + 2} ]

= [\frac{(t − 1)w^{−1} − (−t)w^{−1}}{2l + 2} ] = [\frac{(t)w^{−1} − (−t + 1)w^{−1}}{2l + 2} ] .

Next consider the cases t = 0, 1. Here we only deal with the case of X = B_{l}. The proofs for the cases of X = C_{l}, D_{l} are similar and hence are left to the reader. In the following calculation, we shall repeatedly apply (1.6.1), (2.3.1), (4.1.2), Lemmas 2.9 and 3.6.

On two presentations of the affine Weyl groups 13

Now assume t = 1. The result can be checked easily when i, j ≠ ±1 . On the other hand, for 1 < j ⩽ l, we have

k(w; ±1, j) = k(ws_{1}; ∓1, j) = [\frac{(j)(ws_{1})^{−1} − (∓1)(ws_{1})^{−1}}{2l + 2} ]

= [\frac{(j)w^{−1} − (±1)w^{−1}}{2l + 2} ] .

We also have

k(w; 0, 1) = k(ws_{1}; 0, −1) = −k(ws_{1}; 0, 1) − 1

= − [\frac{(1)(ws_{0})^{−1} − (0)(ws_{0})^{−1}}{2l + 2} ] − 1 = − [\frac{(−1)w^{−1}}{2l + 2} ] − 1

= [\frac{(1)w^{−1} − (0)w^{−1}}{2l + 2} ] .

Finally assume t = 0. We need only check the cases when {i, j} ∩ {l − 1, l} ≠ ∅ for otherwise the result is obvious. For i ∈ [−l + 2, l − 2], we have

k(w; i, l − 1) = k(ws_{0}; i, −l) + 1 = −k(ws_{0}; −i, l)

= − [\frac{(l)(ws_{0})^{−1} − (−i)(ws_{0})^{−1}}{2l + 2} ]

= − [\frac{(l + 3)w^{−1} − (−i)w^{−1}}{2l + 2} ]

= − [1 − \frac{(l − 1)w^{−1} − (i)w^{−1}}{2l + 2} ]

= [\frac{(l − 1)w^{−1} − (i)w^{−1}}{2l + 2} ] .

The equation k(w; i, l) = [\frac{(l)w^{−1}−(i)w^{−1}}{2l+2} ] can be shown similarly. Also, we have

k(w; l − 1, l) = k(ws_{0}; −l, −l + 1) = k(ws_{0}; l − 1, l)

= [\frac{(l)(ws_{0})^{−1} − (l − 1)(ws_{0})^{−1}}{2l + 2} ] = [\frac{(l + 3)w^{−1} − (l + 2)w^{−1}}{2l + 2} ]

= [\frac{(−l + 1)w^{−1} − (−l)w^{−1}}{2l + 2} ] = [\frac{(l)w^{−1} − (l − 1)w^{−1}}{2l + 2} ]

and

k(w; −l + 1, l) = k(ws_{0}; l, −l + 1) + 2 = −k(ws_{0}; −l + 1, l) + 1

= − [\frac{(l)(ws_{0})^{−1} − (−l + 1)(ws_{0})^{−1}}{2l + 2} ] + 1

= − [\frac{(l + 3)w^{−1} − (−l − 2)w^{−1}}{2l + 2} ] + 1

= − [\frac{(−l + 1)w^{−1} − (l)w^{−1}}{2l + 2} ] − 1 = [\frac{(l)w^{−1} − (−l + 1)w^{−1}}{2l + 2} ] .

14 Jian-yi Shi

Therefore the equation (4.1.1) holds in all the cases. □

4.2. By Theorem 4.1, it is natural to extend the alcove form of w ∈ W_{a}(X̃) to (k(w; i, j))_{i,j∈ℤ} by setting

(4.2.1)  k(w; i, j) = [\frac{(j)w^{−1} − (i)w^{−1}}{m_{X}} ]

for any X ∈ Ω_{l} and i, j ∈ ℤ. Call (k(w; i, j))_{i,j∈ℤ} the extended alcove form of w. The relations in Lemma 2.9 remain valid for any i, t, j ∈ ℤ with i ≢ j ( mod m_{X}). Clearly, we have

(4.2.2)  k(w; i − m_{X}, j) = k(w; i, j + m_{X}) = k(w; i, j) + 1

for any i, j ∈ ℤ. We call k(w; i, j) a basic entry of the alcove form of w if (i, j) ∈ Φ(X). It is straightforward to deduce the following

Corollary 4.3. Let X ∈ Ω_{l} for an appropriate integer l. Then we have

(4.3.1)  k(wy; i, j) = k(w; (i)y^{−1}, (j)y^{−1})

for any i, j ∈ ℤ and w, y ∈ W_{a}(X̃).

Note that in the above corollary, the element y is regarded as a permutation on ℤ. 4.4. The right-hand side of (4.1.1) was named the inverse table of w by H. Eriksson and K. Eriksson in [4, §8.] as (i, j) ranges over a certain subset of ℤ × ℤ. They proposed a conjecture in the case of X = C_{l} in order to characterize the inverse table of w ∈ W_{a}(C̃_{l}) as a family of integers. Then they claimed that they would be able to formulate a similar conjecture in the cases of X = B_{l}, D_{l}. Now Theorems 1.3, 4.1 and Lemma 2.9 together give the following characterization for the inverse table of w ∈ W_{a}(X̃), X ∈ Ω_{l}.

Theorem 4.5. Let X ∈ Ω_{l} for an appropriate integer l and let Φ be the root system of type X. Then a family of integers (k_{ij})_{(i,j)∈Φ} is the inverse table of some element in W_{a}(X̃) if and only if the following conditions are satisfied:

(1) k_{ji} = −k_{ij} − 1 for any (i, j) ∈ Φ;

(2) k_{it} + k_{tj} ⩽ k_{ij} ⩽ k_{it} + k_{tj} + 1 for any i, t, j, with (i, j), (i, t), (t, j) ∈ Φ.

This theorem gives an affirmative answer to the conjecture in [4, 8.3].

§5. Transition from the alcove forms to the permutation forms.

Keep the notations in 4.2. Denote the integer m = m_{X} (see Lemma 3.6) and the root system Φ = Φ(X) (see 2.4-2.7) for X ∈ Ω_{l}. Fix w ∈ W_{a}(X̃). The aim of the present

On two presentations of the affine Weyl groups 15

section is to express the numbers (t)w^{−1}, 1 ⩽ t ⩽ l, in terms of the basic entries of the alcove form of w. We actually go further by expressing these numbers in terms of k(w; i, j)’s with (i, j) the positive roots. 5.1. By (4.1.1), we can write

(5.1.1)  (j)w^{−1} − (i)w^{−1} = m · k(w; i, j) + r(w; i, j)

for any (i, j) ∈ ℤ × ℤ, where the integer r(w; i, j) is taken from the interval [0, m − 1]. Fix t, 1 ⩽ t ⩽ l. By taking the sums over j = 1, 2, ⋯, m on both sides of (5.1.1), we get

(5.1.2)  ∑_{j=1}^{m} ((j)w^{−1} − (t)w^{−1}) = m · ∑_{j=1}^{m} k(w; t, j) + ∑_{j=1}^{m} r(w; t, j).

By the fact

(5.1.3)  m + ∑_{j=1}^{m} r(w; t, j) = ∑_{j=1}^{m} j = ∑_{j=1}^{m}(j)w^{−1},

we get from (5.1.2) that

(t)w^{−1} =1 − ∑_{j=1}^{m} k(w; t, j) (5.1.4)
=1 − ∑_{j=t+1}^{l} k(w; t, j) + (t − 1 + ∑_{j=1}^{t−1} k(w; j, t)) − ∑_{h=0}^{m−l−1} k(w; t, m − h)
=t + ∑_{j=1}^{t−1} k(w; j, t) − ∑_{j=t+1}^{l} k(w; t, j) + ∑_{h=0}^{m−l−1} k(w; −h, t).

The last equality follows by the fact k(w; t, m − h) = k(w; t, −h) + 1 = −k(w; −h, t) for any t ≢ h ( mod m) in ℤ. When X = A_{l−1}, we have m = l and then the last sum of the right-hand side of (5.1.4) vanishes. We get the formula in [9, Corollary 6.2.2, (i)]. Now assume X ∈ Ω′_{l}. Then m = 2l + 2 and (5.1.4) becomes

(5.1.5) (t)w^{−1} = t + ∑_{j=1}^{t−1} k(w; j, t) − ∑_{j=t+1}^{l} k(w; t, j) + ∑_{\begin{matrix}1⩽h⩽l \\ h≠t \end{matrix}} k(w; −h, t) + 2k(w; −t, t),

where we use the relation

(5.1.6)  k(w; 0, t) + k(w; −l − 1, t) = k(w; −t, t).

which can be deduced easily from (4.2.1). On the other hand, we have

(5.1.7)  (t)w^{−1} = k(w; 0, t)(2l + 2) + r(w; 0, t)

by (5.1.1) and Lemma 3.6, (3). Note that k(w; 0, t) is a basic entry of the alcove form of w only when X = B_{l}.

We see that all the k(w; a, b)’s occurring on the right-hand side of (5.1.5) are the basic entries of the alcove form of w except for k(w; −t, t). On the other hand, k(w; −t, t) is a basic entry of the alcove form of w only when X = C_{l}. So we get

Theorem 5.2. Let w ∈ W_{a}(X̃), l ⩾ 2 and t ∈ [1, l]. Then we have the following transition formulae on w.

(1) When X = A_{l−1}, we have

(5.2.1)  (t)w^{−1} = t + ∑_{j=1}^{t−1} k(w; j, t) − ∑_{j=t+1}^{l} k(w; t, j),

(2) When X = C_{l}, we have

(5.2.2) (t)w^{−1} = t + ∑_{j=1}^{t−1} k(w; j, t) − ∑_{j=t+1}^{l} k(w; t, j) + ∑_{h=1}^{l} k(w; −h, t) + k(w; −t, t).

Since k(w; −t, t) is not a basic entry of the alcove form of w when X = B_{l}, D_{l}. So in these cases, we have to calculate k(w; −t, t) via the basic entries of the alcove form of w in order to get the required transition formulae. 5.3. Assume X = B_{l}, l ⩾ 3. By (5.1.7), we need only determine r(w; 0, t) in order to calculate (t)w^{−1}. By (4.2.1), we have

(5.3.1)  k(w; −t, t) = 2k(w; 0, t) + ϵ(w, t),

where ϵ(w, t) = 0 or 1 according to r(w; 0, t) < l + 1 or > l + 1 respectively (note that r(w; 0, t) ≠ l + 1 by Lemma 3.6, (3), (4)). Let

(5.3.2) Δ(w, t) = t + ∑_{j=1}^{t−1} k(w; j, t) − ∑_{j=t+1}^{l} k(w; t, j) + ∑_{\begin{matrix}1⩽h⩽l \\ h≠t \end{matrix}} k(w; −h, t) + 4k(w; 0, t).

Then by (5.1.5) and (5.3.1), we have

(5.3.3)  (t)w^{−1} = Δ(w, t) + 2ϵ(w, t),

where Δ(w, t) is determined entirely by the basic entries of the alcove form of w. Let δ(w, t) be the remainder of Δ(w, t) divided by 2l + 2. Then

(5.3.4)  r(w; 0, t) = δ(w, t) + 2ϵ(w, t)

by the definition of ϵ(w, t) and by the fact 1 ⩽ r(w; 0, t) ⩽ 2l + 1 (see Lemma 3.6, (3), (4)). So ϵ(w, t) can also be determined uniquely by δ(w, t) except for the case of δ(w, t) = l. When δ(w, t) = l, r(w; 0, t) could be either l or l + 2 and hence ϵ(w, t) could be either 0 or 1. But we see from Lemma 3.6, (1), (2) that there is exactly one integer t with 1 ⩽ t ⩽ l satisfying r(w; 0, t) ∈ {l, l + 2} for a given w ∈ W_{a}(B̃_{l}). This tells us that all the (t)w^{−1}, 1 ⩽ t ⩽ l, but one are determined uniquely by Δ(w, t). For this exceptional t, we have δ(w, t) = l, r(w; 0, t) ∈ {l, l + 2} and (t)w^{−1} = Δ(w, t) or Δ(w, t) + 2. But the evenness of the value N_{l}(w^{−1}) (see 3.4) can determine which value we should take for this exceptional (t)w^{−1}.

Note that we have 1 ⩽ δ(w, t) ⩽ 2l − 1 for any t ∈ [1, l] by (5.3.4). So by (5.1.7) and (5.3.4), the above results can be summarized as follows.

Theorem 5.4. In the above setup, we have

(5.4.1)  (t)w^{−1} = \{ \begin{matrix}k(w; 0, t)(2l + 2) + δ(w, t), & if 1 ⩽ δ(w, t) < l, \\ k(w; 0, t)(2l + 2) + δ(w, t) + 2, & if l < δ(w, t) ⩽ 2l − 1, \\ k(w; 0, t)(2l + 2) + l + 1 ± 1, & if δ(w, t) = l. \end{matrix}

for any w ∈ W_{a}(B̃_{l}), l ⩾ 3. In the last case, the value (t)w^{−1} is determined eventually by the evenness of the number N_{l}(w^{−1}).

5.5. Next assume X = D_{l}, l ⩾ 4. Let

(5.5.1)  Δ′(w, t) = t + ∑_{j=1}^{t−1} k(w; j, t) − ∑_{j=t+1}^{l} k(w; t, j) + ∑_{\begin{matrix}1⩽i⩽l \\ i≠t \end{matrix}} k(w; −i, t).

Then Δ′(w, t) is determined entirely by the basic entries of the alcove form of w. By (5.1.5) and (4.2.1), we have

(5.5.2)  (t)w^{−1} − 2 [\frac{2 · (t)w^{−1}}{2l + 2} ] = Δ′(w, t).

This implies by (5.1.7) that

(5.5.3)  k(w; 0, t)(2l − 2) + r(w; 0, t) − 2ϵ′(w, t) = Δ′(w, t),

where ϵ′(w, t) = 0 or 1 according to r(w; 0, t) < l + 1 or > l + 1 respectively. Note that k(w; 0, t) is not a basic entry of the alcove form of w. We see from Lemma 3.6, (1) that r(w; 0, t) − 2ϵ′(w, t) is in [1, 2l − 1]. Let δ′(w, t) be the remainder of Δ′(w, t) divided by 2l − 2. Then we see from (5.5.3) that when the remainder δ′(w, t) ≠ 1, the numbers k(w; 0, t) and r(w; 0, t) − 2ϵ′(w, t) are determined uniquely by Δ′(w, t). On the other hand, when δ′(w, t) ≠ 1, l, the number r(w; 0, t) is also determined uniquely by Δ′(w, t). If δ′(w, t) = 1, then r(w; 0, t) = 1 or 2l + 1; also, if δ′(w, t) = l, then r(w; 0, t) = l or l + 2. By Lemma 3.6, (1), (2), we see that there are exactly two integers t′, t′′ ∈ [1, l] satisfying r(w; 0, t′) ∈ {1, 2l + 1} and r(w; 0, t′′) ∈ {l, l + 2} respectively. Thus all the (t)w^{−1} (1 ⩽ t ⩽ l) but two are determined entirely by Δ′(w, t) and hence by the basic entries of the alcove form of w. Then by (5.5.3) and (5.1.7), we see that the evenness of the values N_{0}(w^{−1}) and N_{l}(w^{−1}) (see 3.5 and 3.4) can determine the values of these two exceptional (t′)w^{−1} and (t′′)w^{−1}.

Note that we have δ′(w, t) ∈ [0, 2l − 3] for any t ∈ [1, l]. Let λ(w, t) = [\frac{Δ′(w,t)}{2l−2}]. Then by (5.1.7) and (5.5.3), we can summarize our results as follows.

Theorem 5.6. In the above setup, we have

(5.6.1)  (t)w^{−1} = \{ \begin{matrix}λ(w, t)(2l + 2) + δ′(w, t), & if 1 < δ′(w, t) < l, \\ λ(w, t)(2l + 2) + δ′(w, t) + 2, & if l < δ′(w, t) ⩽ 2l − 3, \\ λ(w, t)(2l + 2) − 2, & if δ′(w, t) = 0, \\ λ(w, t)(2l + 2) ± 1, & if δ′(w, t) = 1, \\ λ(w, t)(2l + 2) + l + 1 ± 1, & if δ′(w, t) = l. \end{matrix}

for any w ∈ W_{a}(D̃_{l}), l ⩾ 4. In the last two cases, the values (t)w^{−1} are determined by the evenness of the numbers N_{0}(w^{−1}) and N_{l}(w^{−1}) respectively.

5.7. Finally we make some discussion on the functions N_{0}(w) and N_{l}(w) of W_{a}(X̃) (X ∈ Ω′_{l}). Write, for 1 ⩽ t ⩽ l,

(5.7.1)  (t)w = k_{t}(2l + 2) + r_{t} = k′_{t}(2l + 2) − (l + 1) + r′_{t}

with k_{t}, k′_{t}, r_{t}, r′_{t} ∈ ℤ and 0 ⩽ r_{t}, r′_{t} < 2l + 2 (By Lemma 3.6 (4), we have 1 ⩽ r_{i}, r′_{i} < 2l + 2). Then

(5.7.2)  N_{0}(w) = ∑_{i=1}^{l} |k_{t}| and N_{l}(w) = ∑_{i=1}^{l} |k′_{t}|.

By (5.7.1), we get the inequality

(5.7.3)  N_{0}(w) − l ⩽ N_{l}(w) ⩽ N_{0}(w) + l.

By the relations (−i)w = −(i)w and (i)w + (2l + 2 − i)w = 2l + 2 (i ∈ ℤ), we also get N_{0}(w) = N_{0}(w^{−1}) and N_{l}(w) = N_{l}(w^{−1}) for any w ∈ W_{a}(X̃).

When X = C_{l}, N_{0}(w) (resp. N_{l}(w)) is equal to the multiplicity of s_{1} (resp. s_{0}) occurring as the factors in any reduced expression of w. The same holds for N_{0}(w) in the case of X = B_{l}. When X = D_{l}, let b_{1}, b_{2}, ⋯, b_{l} (resp. b′_{1}, b′_{2}, ⋯, b′_{l}) be the rearrangement of |k_{1}|, |k_{2}|, ⋯, |k_{l}| (resp. |k′_{1}|, |k′_{2}|, ⋯, |k′_{l}|) in (weakly) decreasing order. Denote c_{0}(w) = b_{1}, c′_{0}(w) = b′_{1}, c_{1}(w) = ∑_{i=2}^{l} b_{i} and c′_{1}(w) = ∑_{i=2}^{l} b′_{i}. Let c(w) = max{0, c_{0}(w) − c_{1}(w)} and c′(w) = max{0, c′_{0}(w) − c′_{1}(w)}. Then c(w), c′(w) ∈ 2ℤ. Let π_{1}(w) (resp. π_{0}(w)) be the minimal possible multiplicity of s_{1} (resp. s_{0}) occurring as the factors in a reduced expression of w. Then we can deduce the following formulae by applying induction on ℓ(w) ⩾ 0.

(5.7.4)  2π_{1}(w) = N_{0}(w) + c(w) and 2π_{0}(w) = N_{l}(w) + c′(w).

The same holds for N_{l}(w) in the case of X = B_{l}.

By (5.7.2) and (5.7.4), we get

(5.7.5)  π_{1}(w) = \{ \begin{matrix}\frac{1}{2}N_{0}(w), & if c(w) = 0, \\ c_{0}(w), & if c(w) > 0. \end{matrix}

or equivalently,

(5.7.6)  π_{1}(w) = max\{\frac{1}{2}N_{0}(w), c_{0}(w)\}.

Similarly, we have

(5.7.7)  π_{0}(w) = max\{\frac{1}{2}N_{l}(w), c′_{0}(w)\}.

This, together with (5.7.1) and (5.7.3), implies

(5.7.8)  π_{1}(w) − \frac{l}{2} ⩽ π_{0}(w) ⩽ π_{1}(w) + \frac{l}{2}.

Acknowledgement. I would like to express my deep gratitude to Bob Howlett for his arranging me to visit Australia as an ARC Senoir Research Associate, and to School of Mathematics and Statistics, Sydney University for the hospitality during my writing of this paper. I am also grateful to H. Eriksson and K. Eriksson for sending me their preprints which stimulates me to write out the present paper.

20 Jian-yi Shi

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