FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS

R.M. Green and J. Losonczy

Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF England E-mail: r.m.green@lancaster.ac.uk

Department of Mathematics Long Island University Brookville, NY 11548 USA E-mail: losonczy@e-math.ams.org

Abstract. We investigate the compatibility of the set of fully commutative elements of a Coxeter group with the various types of Kazhdan–Lusztig cells using a canonical basis for a generalized version of the Temperley–Lieb algebra.

Cellules pleinement commutatives de Kazhdan–Lusztig

Nous étudions la compatibilité entre l’ensemble des éléments pleinement commutatifs d’un groupe de Coxeter et les divers types de cellules de Kazhdan–Lusztig, en utilisant une base canonique pour une version généralisée de l’algèbre de Temperley– Lieb.

Key Words: canonical basis, cell theory, Coxeter group, Hecke algebra, Kazhdan– Lusztig basis, Temperley–Lieb algebra

To appear in Annales de l’Institut Fourier

The first author was supported in part by a NUF–NAL award from the Nuffield Foundation.

Typeset by AMS-TEX 2 R.M. GREEN AND J. LOSONCZY

Introduction

The fully commutative elements, W_{c}, of a Coxeter group W may be defined, following [17], as the set of elements w with the property that any reduced expression for w may be obtained from any other by a sequence of interchanges of adjacent, commuting Coxeter generators. These elements arise naturally in connection with the generalized Temperley–Lieb algebras defined in the simply laced case by Fan [2] and in general by Graham [8].

Kazhdan and Lusztig [14] have defined partitions of a Coxeter group W arising from each of three equivalence relations, ∼_{L}, ∼_{R} and ∼_{LR}. The corresponding equivalence classes of W are known as left cells, right cells and two-sided cells, and it follows easily from the definitions that two-sided cells are unions of left (respectively, right) cells.

In this paper, we are concerned with the compatibility of the set W_{c} with the various Kazhdan–Lusztig cells. More precisely, we wish to know when W_{c} is a union of (left, right, or two-sided) cells. Our most general result is Theorem 2.2.3, where compatibility of W_{c} with the (left, right, or two-sided) Kazhdan–Lusztig cells is shown to be related to several other conditions having to do with the Temperley– Lieb quotient of the associated Hecke algebra.

It has been known at least since the theses of Fan and Graham ([2, Proposition 6], [8]) that W_{c} is a union of two-sided cells in type A. In Graham’s thesis a counterexample is given which shows that this is not the case in type E. (See also Example 2.2.5.) Our main task in §3 is to prove that there is full compatibility between W_{c} and the Kazhdan–Lusztig cells in type B. We have also verified this property for Coxeter groups of types F_{4}, H_{3}, and H_{4}.

These results are reminiscent of some work of Fan and Stembridge [4, §3], who showed that in types A, D, E and affine A, the set W_{c} is a union of Spaltenstein– Springer–Steinberg cells.

We point out that our methods of proof are combinatorial and based on our previous work on IC-type (“canonical”) bases for generalized Temperley–Lieb algebras [11, 12]. In types A, D and E, the canonical basis defined in [11] is a cellular basis, as defined by Graham in [8, §4].

As a consequence of the results in this paper, it becomes possible to describe

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 3

the fully commutative cells in type B very explicitly by using the diagram calculus for canonical basis elements in type B, which was given by the first author in [10, Theorem 2.2.5].

1. Kazhdan–Lusztig bases and cells

1.1 Kazhdan–Lusztig bases.

We begin by recalling the well-known basic properties of Hecke algebras arising from Coxeter systems. These properties all follow easily from the results of [14].

Let X be a Coxeter graph, of arbitrary type, and let W = W (X) be the associated Coxeter group with distinguished set of generating involutions S = S(X). Denote by < the Bruhat–Chevalley ordering on W. Let A = ℤ[v, v^{−1}], let A^{−} = ℤ[v^{−1}] and let q = v^{2}.

We denote by H = H(X) the Hecke algebra associated with W. As an A-module, the Hecke algebra has a basis consisting of elements T_{w}, with w ranging over W, that satisfy

T_{s}T_{w} = \{ \begin{matrix}T_{sw} & if ℓ(sw) > ℓ(w), \\ qT_{sw} + (q − 1)T_{w} & if ℓ(sw) < ℓ(w), \end{matrix}

where ℓ is the length function on the Coxeter group W, w ∈ W, and s ∈ S. It will be convenient to work with a second A-basis {T̃_{w} : w ∈ W } for H, which we obtain by defining T̃_{w} = v^{−ℓ(w)}T_{w}.

Definition 1.1.1. We denote by a ↦ ā the ℤ-linear involution on the ring A that exchanges v and v^{−1}. This can be extended to a ℤ-linear involution h ↦ h̄ on H, by defining \overline{∑_{w∈W} a_{w}T_{w}} = ∑_{w∈W} \overline{a_{w}} T_{w^{−1}}^{−1}.

The Kazhdan–Lusztig basis may be characterized as follows.

Theorem 1.1.2 (Kazhdan–Lusztig). There is a unique A-basis {C′_{w} : w ∈ W } for H such that

(i) \overline{C′_{w}} = C′_{w} for all w ∈ W; (ii) C′_{w} = ∑_{x∈W} P̃_{x,w}T̃_{x}, where each P̃_{x,w} lies in A^{−} and satisfies

P̃_{x,w} = \{ \begin{matrix}1 & if x = w, \\ 0 & if x ≰ w, \\ 0 mod v^{−1}A^{−} & if x < w. \end{matrix}

4 R.M. GREEN AND J. LOSONCZY

Proof. This is a restatement of [14, (1.1c)]. The relationship between P̃_{x,w} and the Kazhdan–Lusztig polynomials P_{x,w} of [14] is P̃_{x,w} = v^{ℓ(x)−ℓ(w)}P_{x,w}. □

Theorem 1.1.2 leads to the following reformulation of [14, Definition 1.2].

Definition 1.1.3. Let x, w ∈ W satisfy x < w. Define μ(x, w) ∈ ℤ to be coefficient of v^{−1} in P̃_{x,w} (i.e., the coefficient of v^{ℓ(w)−ℓ(x)−1} in the Kazhdan–Lusztig polynomial P_{x,w}). If x < w and μ(x, w) ≠ 0, then we write x ≺ w.

1.2 Kazhdan–Lusztig cells.

The structure constants of H with respect to the Kazhdan–Lusztig basis of §1.1 give rise to various natural partitions of the group W into “cells”.

Although these structure constants are subtle, the product of two Kazhdan– Lusztig basis elements may be computed in important special cases by appealing to the following well-known formula, which is implicit in [14, §2.2].

Proposition 1.2.1 (Kazhdan–Lusztig). Let s, w ∈ W with s ∈ S. Then

C′_{s}C′_{w} = \{ \begin{matrix}C′_{sw} + ∑_{{x≺w \atop sx<x}} μ(x, w)C′_{x} & if sw > w, \\ (v + v^{−1})C′_{w} & otherwise, \end{matrix}

where μ(x, w) is as in Definition 1.1.3. □

This formula motivates the following definitions.

Definition 1.2.2. Let x, w ∈ W. We write x ≤_{L} w if there is a chain

x = x_{0}, x_{1}, …, x_{r} = w,

possibly with r = 0, such that for each i < r, C′_{x_{i}} occurs with nonzero coefficient in the linear expansion of C′_{s}C′_{x_{i+1}} for some s ∈ S such that sx_{i+1} > x_{i+1}. (By Proposition 1.2.1, this implies sx_{i} < x_{i}.)

This transitive preorder yields an equivalence relation ∼_{L} on W (where x ∼_{L} w if and only if x ≤_{L} w and w ≤_{L} x) whose equivalence classes are called the left cells of W. The preorder ≤_{R} on W is defined by the condition x ≤_{R} w ⇔ x^{−1} ≤_{L} w^{−1}, and the preorder ≤_{LR} is that generated by ≤_{L} and ≤_{R}. These preorders yield equivalence relations ∼_{R} and ∼_{LR} on W whose equivalence classes are called right cells and two-sided cells, respectively.

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 5

Remark 1.2.3. It is well known that the definition of ≤_{L} given above agrees with the original definition in [14]. This follows from Proposition 1.2.1 and part (a) of the proof of [13, Proposition 7.15].

Remark 1.2.4. It is immediate from the construction of the left (respectively right, two-sided) cells that they are partially ordered via ≤_{L} (respectively ≤_{R}, ≤_{LR}).

2. Generalized Temperley–Lieb algebras

2.1 Canonical bases for generalized Temperley–Lieb algebras.

Let X be a Coxeter graph, of arbitrary type. Let J = J (X) be the two-sided ideal of H generated by the elements

∑_{w∈⟨s,s′⟩} T_{w},

where (s, s′) runs over all pairs of elements of S that correspond to adjacent nodes in the Coxeter graph. (If the nodes corresponding to (s, s′) are connected by a bond of infinite strength, then we omit the corresponding relation.)

Definition 2.1.1. The generalized Temperley–Lieb algebra, TL = TL(X), is the quotient A-algebra H/J. We denote the corresponding epimorphism of algebras by θ : H ⟶ TL.

The algebra TL may be of finite or infinite rank, and may be of finite rank even when it is the quotient of a Hecke algebra of infinite rank. Graham [8, Theorem 7.1] classified the algebras of finite rank into seven infinite families: A, B, D, E, F, H and I.

Definition 2.1.2. A product w_{1}w_{2} ⋯ w_{n} of elements w_{i} ∈ W is called reduced if ℓ(w_{1}w_{2} ⋯ w_{n}) = ∑_{i} ℓ(w_{i}). We reserve the terminology reduced expression for reduced products w_{1}w_{2} ⋯ w_{n} in which every w_{i} ∈ S.

We call an element w ∈ W fully commutative if it cannot be written as a reduced product x_{1}w_{ss′}x_{2}, where x_{1}, x_{2} ∈ W and w_{ss′} is the longest element of some parabolic subgroup ⟨s, s′⟩ such that s, s′ ∈ S do not commute. This definition is equivalent to the one given in the introduction (see [17, Proposition 1.1]).

We define the content of w ∈ W to be the set c(w) of Coxeter generators s ∈ S that appear in some (any) reduced expression for w.

6 R.M. GREEN AND J. LOSONCZY

Denote by W_{c} = W_{c}(X) the set of all elements of W that are fully commutative. Let t_{w} denote the image of the basis element T_{w} ∈ H in the quotient TL.

Proposition 2.1.3. [8, Theorem 6.2] The set {t_{w} : w ∈ W_{c}} is an A-basis for the algebra TL. □

We now recall a principal result of [11], which establishes a canonical basis for TL. This basis is a direct analogue of the Kazhdan–Lusztig basis in §1, although the precise relationship between the two is not immediate.

Definition 2.1.4. The involution h ↦ h̄ on H induces a ℤ-linear ring involution on TL [11, Lemma 1.4]. We use the bar notation to represent this map, as well: \overline{∑_{w∈W_{c}} a_{w}t_{w}} = ∑_{w∈W_{c}} \overline{a_{w}} t_{w^{−1}}^{−1}.

Let L be the free A^{−}-submodule of TL with basis {t̃_{w} : w ∈ W_{c}}, where t̃_{w} = v^{−ℓ(w)}t_{w}, and let π : L ⟶ L/v^{−1}L be the canonical projection. For each w ∈ W, we denote by L_{w} the free A^{−}-submodule of TL with basis {t̃_{x} : x ∈ W_{c}, x ≤ w}.

Proposition 2.1.5. [11, Theorem 2.3] There exists a unique basis {c_{w} : w ∈ W_{c}} for L such that \overline{c_{w}} = c_{w} and π(c_{w}) = π(t̃_{w}) for all w ∈ W_{c}. □

We call {c_{w} : w ∈ W_{c}} the canonical basis (or the IC basis) of TL. It depends on the t-basis, the involution on TL from above, and the A^{−}-lattice L.

Definition 2.1.6. If s ∈ S, we write b_{s} ∈ TL for the element v^{−1}t_{s} + v^{−1}. The elements b_{s} = θ(C′_{s}) generate TL as an A-algebra.

For each w ∈ W_{c}, it makes sense to define b_{w} = b_{s_{1}}b_{s_{2}} ⋯ b_{s_{n}}, where s_{1}s_{2} ⋯ s_{n} is any reduced expression for w. It is known that the set {b_{w} : w ∈ W_{c}} is an A-basis for TL; we call it the monomial basis. (In types A, D and E, this agrees with Graham’s cellular basis for TL as defined in [8].)

For each w ∈ W, we denote by L′_{w} the free A^{−}-submodule of TL with basis {b_{x} : x ∈ W_{c}, x ≤ w}. We shall study the A^{−}-lattices L′_{w} in §3.3.

2.2. Some general results.

One of the main obstructions to understanding the relationship between the Kazhdan–Lusztig basis of H(X) and the canonical basis of TL(X) is that the set W_{c}(X) may not be compatible with the two-sided cells. When a particular type of compatibility is present, the relationship between the two bases becomes transpar-

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 7

ent [12, Proposition 1.2.3]. It will be shown in §3 that this is the case when X is of type B_{n}.

Before restricting ourselves to type B_{n}, we shall say something more about the general problem. The following result is helpful in this context.

Lemma 2.2.1. The set {θ(C′_{w}) : w ∈ W_{c}} is an A-basis for TL.

Proof. This follows easily from [11, Lemma 1.5]. □

Lemma 2.2.2. Let c ∈ L be such that π(c) = 0 and \overline{c} = c. Then c = 0.

Proof. Express c = ∑_{w∈W_{c}} a_{w}c_{w} as an A^{−}-linear combination of the canonical basis. Since π(c) = 0, we must have a_{w} ∈ v^{−1}A^{−} for all w. But

c = \overline{c} = ∑_{w∈W_{c}} \overline{a_{w}} c_{w},

which implies that all a_{w} = 0. □

There is compatibility between the set W_{c} and the Kazhdan–Lusztig cells when the equivalent conditions of the following theorem are satisfied.

Theorem 2.2.3. Let X be an arbitrary Coxeter graph, and maintain the usual notation, e.g., J = J (X), W = W (X), etc. Then the following are equivalent:

(i) The ideal J is spanned by those elements C′_{w} that it contains. (ii) The ideal J is spanned by the set {C′_{w} : w ∈ W \W_{c}}. (iii) For each w ∈ W \W_{c}, one has θ(C′_{w}) = 0. (iv) If w ∈ W, then θ(C′_{w}) ∈ L and

π(θ(C′_{w})) = \{ \begin{matrix}π(c_{w}) & if w ∈ W_{c}; \\ 0 & otherwise. \end{matrix}

(v) For each w ∈ W \W_{c}, one has θ(T̃_{w}) ∈ v^{−1}L. (vi) The set W \W_{c} is closed under ≤_{L} and so is a union of left cells. (vii) The set W \W_{c} is closed under ≤_{LR} and so is a union of two-sided cells. (viii) The set W_{c} is closed under ≥_{LR} and so is a union of two-sided cells.

Note. If ≤_{Λ} is a preorder on a set Λ and Λ′ ⊆ Λ, then the statement “Λ′ is closed under ≤_{Λ}” means that whenever λ_{1} ∈ Λ′ and λ_{2} ∈ Λ are such that λ_{2} ≤_{Λ} λ_{1}, we have λ_{2} ∈ Λ′.

8 R.M. GREEN AND J. LOSONCZY

Proof. We begin with the equivalence of (i), (ii) and (iii). The statement (i) implies that the set {θ(C′_{w}) : θ(C′_{w}) ≠ 0} forms an A-basis for TL. It follows that this set must equal the set in Lemma 2.2.1, which implies (ii). It is clear that (ii) implies (iii). If (iii) holds then J must contain all elements C′_{w} with w ∈ W \W_{c}. However, J cannot be any bigger than the span of these elements by Lemma 2.2.1, so (i) follows.

(iii) ⇔ (iv). Assume (iii) holds. It is clear that if w ∉ W_{c}, then θ(C′_{w}) ∈ L and π(θ(C′_{w})) = 0. The remaining case is dealt with by the proof of [12, Proposition 1.2.3]. Now assume (iv) and let w ∉ W_{c}. Since the map θ is compatible (by [11, Lemma 1.4]) with the involutions on H and TL from above, we see that \overline{θ(C′_{w})} = θ(C′_{w}). Since π(θ(C′_{w})) = 0, Lemma 2.2.2 gives (iii).

(iv) ⇔ (v). Note that if w ∈ W_{c}, then π(θ(T̃_{w})) = π(t̃_{w}) = π(c_{w}) by the definition of the canonical basis. The equivalence of (iv) and (v) now follows from the fact that, relative to some total refinement of the Bruhat–Chevalley order, the (possibly infinite) change of basis matrices between the basis {C′_{w} : w ∈ W } and the basis {T̃_{w} : w ∈ W } are upper triangular with ones on the diagonal, and all the entries above the diagonal lie in v^{−1}A^{−}.

(ii) ⇒ (vi). By Definition 1.2.2, it is enough to check that if w ∉ W_{c} and s ∈ S with sw > w, then all the terms occurring in the expansion of C′_{s}C′_{w} in Proposition 1.2.1 are parametrized by elements x ∉ W_{c}. By (ii), w ∉ W_{c} implies C′_{w} ∈ J. Since J is an ideal, C′_{s}C′_{w} ∈ J. Another application of (ii) completes the proof.

(vi) ⇒ (vii). Suppose w ∉ W_{c} and x ≤_{R} w, meaning that x^{−1} ≤_{L} w^{−1}. By symmetry of the definition of W_{c}, we have w^{−1} ∉ W_{c}, and (vi) shows that x^{−1} ∉ W_{c}, meaning that x ∉ W_{c}. It follows that W \W_{c} is closed under ≤_{R}. Since W \W_{c} is closed under ≤_{L} and ≤_{R}, it is closed under ≤_{LR} and is therefore a union of two-sided cells.

(vii) ⇒ (ii). Since W \W_{c} is closed under ≤_{LR} and {C′_{s} : s ∈ S} is a set of algebra generators for H, it follows that {C′_{w} : w ∉ W_{c}} spans a two-sided ideal of H. The generators of J (see §2.1) are of the form v^{ℓ(w)}C′_{w} for certain w ∉ W_{c}, so J is contained in this ideal. On the other hand, J contains {C′_{w} : w ∉ W_{c}} by Lemma 2.2.1. Thus, condition (ii) holds.

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 9

The equivalence of (vii) and (viii) is obvious. □

Example 2.2.4. Consider the finite dihedral case, or in other words, take X = I_{2}(m) for m < ∞. Let w_{0} denote the longest element of W (I_{2}(m)). The ideal J (I_{2}(m)) is spanned by the single Kazhdan–Lusztig basis element C′_{w_{0}}. Thus, the equivalent conditions of Theorem 2.2.3 hold for finite dihedral groups.

In the next section, it will be shown that the conditions of Theorem 2.2.3 hold when the underlying graph is of type B_{n}. Our proof will also handle type A_{n} as a special case.

Example 2.2.5. Take the underlying graph X to be of type D_{n} (n ≥ 4). Let σ_{1}, σ_{2}, …, σ_{n} denote the Coxeter generators, labelled so that σ_{3} corresponds to the branch node and σ_{1}, σ_{2} commute with all generators except σ_{3}. Consider the elements w = σ_{2}σ_{3}σ_{4}σ_{3}σ_{1}σ_{2}σ_{3} ∉ W_{c}(D_{n}) and x = σ_{1}σ_{2}σ_{4}σ_{3} ∈ W_{c}(D_{n}). Observe that σ_{1}w > w. When C′_{σ_{1}}C′_{w} is written as a linear combination of Kazhdan–Lusztig basis elements, the element C′_{x} appears with coefficient 1. Thus, x ≤_{L} w, so that condition (vi) fails when the underlying graph is of type D_{n}. Further computation reveals that x ∼_{L} w, which shows that W_{c}(D_{n}) is not a union of left cells. This example also shows that condition (vi) is violated (and that W_{c} is not a union of left cells) in types E_{6}, E_{7} and E_{8}. The incompatibility of W_{c} with Kazhdan–Lusztig cells in type E was described explicitly in [8, §9.9].

We remark that it is nevertheless true that the image under θ of the set of all C′_{u} indexed by u ∈ W_{c}(D_{n}) equals the canonical basis of TL(D_{n}) (see [15, Theorem 3.4]). It is not known whether the corresponding statement holds in type E.

3. Type B

In this section, we study the compatibility of cells and fully commutative elements when the underlying Coxeter graph X is of type B_{n}.

3.1 Statement of results.

Our main objective in §3 is to prove the following

Theorem 3.1.1. When the underlying Coxeter graph is of type B_{n}, the equivalent conditions of Theorem 2.2.3 are satisfied. In particular, the set W_{c}(B_{n}) is closed under ≥_{LR} and so is a union of two-sided Kazhdan–Lusztig cells.

10 R.M. GREEN AND J. LOSONCZY

This result is a strengthening of [12, Theorem 2.2.1], where it was shown that θ(C′_{w}) ∈ L for all w ∈ W (B_{n}), and thus that the basis in Lemma 2.2.1 for TL(B_{n}) agrees with the canonical basis of TL(B_{n}).

Theorem 3.1.1 also implies the corresponding statement for type A_{n}, which was previously known (see [3, Proposition 3.1.1]). Corollary 3.1.2. When the underlying Coxeter graph is of type A_{n}, the equivalent conditions of Theorem 2.2.3 are satisfied. Proof. Identify the Coxeter group W (A_{n}) with the parabolic subgroup of the Coxeter group W (B_{n+1}) that corresponds to omission of the appropriate end generator.

Let s ∈ S(A_{n}) and w ∈ W (A_{n}) be such that w ∉ W_{c}(A_{n}) and sw > w. By considering C′_{s}C′_{w} and using condition (vi) of Theorem 2.2.3 applied to type B_{n+1}, we see that condition (vi) holds for type A_{n}. □

The remaining cases arising from finite irreducible Coxeter groups are F_{4}, H_{3}, and H_{4}. A series of computer calculations using du Cloux’s program “Coxeter” [1] shows that condition (vi) of Theorem 2.2.3 holds in each of these cases.

We can summarize our results as follows. Corollary 3.1.3. Let X be a Coxeter graph such that W (X) is finite and irreducible. Then W_{c}(X) is a union of two-sided Kazhdan–Lusztig cells if and only if X does not contain D_{4} as a subgraph. □ 3.2 Some combinatorial preparation.

The following proposition describes a useful way to parse certain reduced expressions. A proof can be found in [12, Lemma 2.1.2]. Proposition 3.2.1. Let w ∈ W_{c}(B_{n}) and s ∈ S(B_{n}) satisfy ws ∉ W_{c}(B_{n}). There exists a unique s′ ∈ S(B_{n}) such that any reduced expression for w can be parsed in one of the following two ways.

(i) w = w_{1}sw_{2}s′w_{3}, where ss′ has order 3, and s commutes with every member of c(w_{2}) ∪ c(w_{3});

(ii) w = w_{1}s′w_{2}sw_{3}s′w_{4}, where ss′ has order 4, s commutes with every member

of c(w_{3}) ∪ c(w_{4}), and s′ commutes with every member of c(w_{2}) ∪ c(w_{3}). □ The algebra TL(B_{n}) is known to be generated by the monomial elements b_{s}, with

s ranging over all Coxeter generators, subject to the following relations: b_{s}^{2} = q_{c}b_{s},

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 11

where q_{c} = [2] = v + v^{−1}; b_{s}b_{s′} = b_{s′}b_{s} if s, s′ commute; b_{s}b_{s′}b_{s} = b_{s} if ss′ has order 3; b_{s}b_{s′}b_{s}b_{s′} = 2b_{s}b_{s′} if ss′ has order 4 (see [9, §1]).

Lemma 3.2.2. Let w ∈ W (B_{n}) and let s_{1}s_{2} ⋯ s_{m} be a reduced expression for w. Given integers 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ m, we have b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k}}} = aq_{c}^{μ}b_{w′}, where a and μ are nonnegative integers and w′ ∈ W_{c}(B_{n}). Moreover, we have w′ ≤ w and ℓ(w′) ≤ k.

Proof. This follows by a simple induction on k, using the subexpression characterization of Bruhat–Chevalley order together with Proposition 3.2.1 and the relations for the monomial generators given in the previous paragraph. □

Remark 3.2.3. We usually apply Lemma 3.2.2 in the following way. Let w = xyz be a reduced product, and consider b_{x}t̃_{y}b_{z}. We would like to know that this is a linear combination of monomial basis elements b_{u} with u ≤ w. To see that this is the case, let s_{1}s_{2} ⋯ s_{m} be a reduced expression for y. Then b_{x}t̃_{y}b_{z} equals

b_{x}t̃_{s_{1}} t̃_{s_{2}} ⋯ t̃_{s_{m}}b_{z} = b_{x}(b_{s_{1}} − v^{−1})(b_{s_{2}} − v^{−1}) ⋯ (b_{s_{m}} − v^{−1})b_{z},

and we see that this expands into a combination of b_{u} with u ≤ w by Lemma 3.2.2.

The next result gives useful information concerning the structure constants for the monomial basis; it will be used repeatedly in §3.3.

Proposition 3.2.4. [12, Lemma 2.1.3] Let w ∈ W_{c}(B_{n}) and let s ∈ S(B_{n}). We have b_{w}b_{s} = aq_{c}^{μ}b_{w′} for some fully commutative w′ and some nonnegative integers a and μ. Furthermore, one has (i) μ ≤ 1; (ii) ℓ(w′s) < ℓ(w′); (iii) μ = 0 if ℓ(ws′) < ℓ(w) for some s′ ∈ S(B_{n}) that does not commute with s. □

Let σ_{1}, σ_{2}, …, σ_{n} be the elements of S(B_{n}), labelled so that σ_{1}σ_{2} has order 4 and σ_{i}σ_{i+1} has order 3 for all i > 1. Define, for each 1 ≤ r ≤ n, the set W^{(r)} = {w ∈ W (B_{r}) : i < r ⇒ ℓ(σ_{i}w) > ℓ(w)}. It is known that W^{(r)} is a system of right coset representatives for the parabolic subgroup W (B_{r−1}) of W (B_{r}). Moreover, one has ℓ(xy) = ℓ(x) + ℓ(y) for all x ∈ W (B_{r−1}) and y ∈ W^{(r)} (see [13, §5.12]). Thus, each y ∈ W^{(r)} is the unique element of minimum length in the coset W (B_{r−1})y. The elements of W^{(r)} are given as follows:

{e, σ_{r}, σ_{r}σ_{r−1}, …, σ_{r}σ_{r−1} ⋯ σ_{2}σ_{1}, σ_{r}σ_{r−1} ⋯ σ_{2}σ_{1}σ_{2}, …,

12 R.M. GREEN AND J. LOSONCZY

σ_{r}σ_{r−1} ⋯ σ_{2}σ_{1}σ_{2} ⋯ σ_{r−1}σ_{r}}.

Note that each element of W^{(r)} has a unique reduced expression and hence is fully commutative.

Any w ∈ W (B_{n}) can be written uniquely as a product w_{1}w_{2} ⋯ w_{n}, where each w_{i} ∈ W^{(i)}. By the previous paragraph, this product is reduced. Thus, if we delete each w_{i} that equals the identity, and then replace each of the remaining w_{i} with its unique reduced expression, we obtain a “normal” reduced expression for w.

We frequently use without comment the following consequence of the Exchange Condition, which is valid for any Coxeter system: if w ∈ W and s ∈ S, then w has a reduced expression ending in s if and only if ℓ(ws) < ℓ(w) (see [13, §5.8]). 3.3 The A^{−}-lattices L′_{w}.

In the following series of lemmas, we study the A^{−}-lattices in TL(B_{n}) of the form L′_{w} (recall Definition 2.1.6). Our goal, which is accomplished in Proposition 3.3.10, is to prove that t̃_{w} ∈ v^{−1}L′_{w} whenever w ∉ W_{c}(B_{n}). This will enable us to establish condition (v) of Theorem 2.2.3 for the case where X = B_{n}. Lemma 3.3.1. Let x ∈ W_{c}(B_{n−1}), let w ∈ W^{(n)} and let k ≥ 0. Suppose that there exist s, s′ ∈ S(B_{n}), with ℓ(wss′) < ℓ(ws) < ℓ(w), such that b_{x}t̃_{wss′} ∈ v^{−k}L′_{xwss′} and b_{x}t̃_{ws} ∈ v^{−k}L′_{xws}. Then b_{x}t̃_{w} ∈ v^{−k}L′_{xw}. Proof. First note that when b_{x}t̃_{w} is written as a linear combination of monomial basis elements b_{y}, the coefficient of b_{y} is nonzero only if y ≤ xw by Remark 3.2.3. Thus, we may turn our attention to the degrees of the various coefficients.

By hypothesis, we can write b_{x}t̃_{wss′} as a sum of terms a_{y}b_{y} with each a_{y} ∈ v^{−k}A^{−}. Since

b_{x}t̃_{ws} = b_{x}t̃_{wss′} t̃_{s′} = b_{x}t̃_{wss′}(b_{s′} − v^{−1}),

we see that b_{x}t̃_{ws} equals a sum of terms of the form a_{y}b_{y}(b_{s′} − v^{−1}), where a_{y} ∈ v^{−k}A^{−}. We can use this fact together with Proposition 3.2.4 (ii) and the hypothesis on b_{x}t̃_{ws} to deduce that when b_{x}t̃_{ws} is written as a linear combination of monomial basis elements b_{z}, either the coefficient of b_{z} lies in v^{−k−1}A^{−}, or the coefficient lies in v^{−k}A^{−} and ℓ(zs′) < ℓ(z).

Now consider the equalities

b_{x}t̃_{w} = b_{x}t̃_{ws}t̃_{s} = b_{x}t̃_{ws}(b_{s} − v^{−1}).

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 13

In view of the previous paragraph, together with the fact that s, s′ do not commute (since w has a unique reduced expression), parts (i) and (iii) of Proposition 3.2.4 enable us to conclude that b_{x}t̃_{w} is a linear combination of monomial basis elements b_{z′} (z′ ≤ xw) with coefficients in v^{−k}A^{−}, as desired. □

Lemma 3.3.2. Let x ∈ W_{c}(B_{n−1}) and let w ∈ W^{(n)}. Then b_{x}t̃_{w} ∈ L′_{xw}.

Proof. We argue by induction on ℓ(w). The lemma is obviously true for ℓ(w) = 0, and if ℓ(w) = 1, then w = σ_{n} ∉ c(x). Hence, b_{x}t̃_{w} = b_{x}(b_{σ_{n}} − v^{−1}) = b_{xσ_{n}} − v^{−1}b_{x}, and one sees that this last expression belongs to L′_{xw}.

Suppose that ℓ(w) > 1. There exist (uniquely determined) Coxeter generators s, s′ such that ℓ(wss′) < ℓ(ws) < ℓ(w). By the inductive hypothesis, b_{x}t̃_{wss′} ∈ L′_{xwss′} and b_{x}t̃_{ws} ∈ L′_{xws}. But then b_{x}t̃_{w} ∈ L′_{xw} by Lemma 3.3.1 (taking k = 0). The inductive step is complete. □

Proposition 3.3.3. We have t̃_{w} ∈ L′_{w} for all w ∈ W (B_{n}).

Proof. We proceed by induction on ℓ(w). If ℓ(w) = 0, then w = e and we have t̃_{e} = b_{e}. Suppose that ℓ(w) > 0. Let r > 0 be the smallest integer such that w ∈ W (B_{r}). Write w as a reduced product w = yz, where y ∈ W (B_{r−1}) and z ∈ W^{(r)}. We have t̃_{w} = t̃_{y}t̃_{z}. By the inductive hypothesis, we may write t̃_{y} as a linear combination of monomial basis elements b_{x} (x ≤ y) with coefficients in A^{−}. Thus, t̃_{w} equals a linear combination of products of the form b_{x}t̃_{z} (x ≤ y), with coefficients in A^{−}.

If we can show that any such product b_{x}t̃_{z} lies in L′_{w}, then the inductive step will be established. But Lemma 3.3.2 gives us b_{x}t̃_{z} ∈ L′_{xz}, and the subexpression characterization of Bruhat–Chevalley order gives xz ≤ w. The proof is complete. □

The following two lemmas are needed to handle certain cases that arise in the proofs of lemmas 3.3.6 and 3.3.8.

Lemma 3.3.4. Let x ∈ W_{c}(B_{n−1}) and let w, w′ ∈ W^{(n)}. Suppose that w = w′u (reduced) for some u ∈ W (B_{n}). Let u′ ∈ W (B_{n}) satisfy u′ ≤ u, and let k ≥ 1. If ℓ(u′) < k, then v^{−k}b_{x}t̃_{w′}b_{u′} ∈ v^{−1}L′_{xw}.

Proof. Fix a reduced expression s_{1}s_{2} ⋯ s_{m} for u′. We assume that m < k. By Lemma 3.3.2, the product b_{x}t̃_{w′} can be written as a sum of terms of the form a_{y}b_{y}, where a_{y} ∈ A^{−} and y ≤ xw′. Thus, v^{−k}b_{x}t̃_{w′}b_{u′} equals a sum of terms of the form

v^{−k}a_{y}b_{y}b_{u′} = v^{−k}a_{y}b_{y}b_{s_{1}}b_{s_{2}} ⋯ b_{s_{m}},

where again a_{y} ∈ A^{−} and y ≤ xw′.

By applying Proposition 3.2.4 (i) repeatedly (m times on the same term) and then applying Lemma 3.2.2, we find that v^{−k}a_{y}b_{y}b_{s_{1}}b_{s_{2}} ⋯ b_{s_{m}} lies in v^{−1}L′_{xw}. □

Lemma 3.3.5. Let x ∈ W_{c}(B_{n−1}) and let w, w′ ∈ W^{(n)}. Suppose that w = w′u (reduced) for some u ∈ W (B_{n}). Let u′ ∈ W (B_{n}) satisfy u′ ≤ u, and assume that u′ has a unique reduced expression s_{1}s_{2} ⋯ s_{m}. Then b_{x}t̃_{w′}b_{u′} ∈ L′_{xw} if there exists an s ∈ S(B_{n}) that does not commute with s_{1} and that satisfies either (i) ℓ(w′s) < ℓ(w′); or (ii) ℓ(xs) < ℓ(x) and sw′ = w′s.

Proof. We first point out that by Remark 3.2.3, when b_{x}t̃_{w′}b_{u′} is expressed as a linear combination of monomial basis elements, all nonzero terms correspond to y ∈ W_{c}(B_{n}) that satisfy y ≤ xw.

Suppose that (i) holds. Write w′ = w′′s (reduced). We have b_{x}t̃_{w′} ∈ L′_{xw′} and b_{x}t̃_{w′′} ∈ L′_{xw′′} by Lemma 3.3.2. The equalities b_{x}t̃_{w′} = b_{x}t̃_{w′′} t̃_{s} = b_{x}t̃_{w′′}(b_{s} − v^{−1}) together with Proposition 3.2.4 (ii) imply that when b_{x}t̃_{w′} is written as a sum of terms of the form a_{y}b_{y} (a_{y} ∈ A^{−}), for each y, either ℓ(ys) < ℓ(y) or else a_{y} ∈ v^{−1}A^{−}.

Thus, b_{x}t̃_{w′}b_{u′} is a sum of terms of the form a_{y}b_{y}b_{u′} = a_{y}b_{y}b_{s_{1}}b_{s_{2}} ⋯ b_{s_{m}}, with y and a_{y} as described above. If ℓ(ys) < ℓ(y) then, since ss_{1} ≠ s_{1}s and s_{i}s_{i+1} ≠ s_{i+1}s_{i} for all i < m, repeated applications of parts (ii) and (iii) of Proposition 3.2.4 give a_{y}b_{y}b_{s_{1}}b_{s_{2}} ⋯ b_{s_{m}} ∈ L′_{xw}. On the other hand, if a_{y} ∈ v^{−1}A^{−}, then a_{y}b_{y}b_{s_{1}} = a′b_{y′} with a′ ∈ A^{−} by Proposition 3.2.4 (i). Also, ℓ(y′s_{1}) < ℓ(y′) by part (ii) of the same proposition. Since s_{i}s_{i+1} ≠ s_{i+1}s_{i} for all i < m, we have a_{y}b_{y}b_{s_{1}}b_{s_{2}} ⋯ b_{s_{m}} = a′b_{y′}b_{s_{2}} ⋯ b_{s_{m}} ∈ L′_{xw} (again by repeated applications of parts (ii) and (iii) of Proposition 3.2.4).

Suppose now that (ii) holds. Write x = x′s (reduced). Since sw′ = w′s, we have b_{x}t̃_{w′} = b_{x′} t̃_{w′}b_{s}; hence, by Proposition 3.2.4 (ii), when b_{x}t̃_{w′} is written as a sum of terms of the form a_{y}b_{y}, we have ℓ(ys) < ℓ(y) whenever a_{y} ≠ 0. But then the reasoning from the previous paragraph gives a_{y}b_{y}b_{s_{1}}b_{s_{2}} ⋯ b_{s_{m}} ∈ L′_{xw}. □

The following lemma and Lemma 3.3.8 are needed to handle the inductive step of Lemma 3.3.9.

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 15

Lemma 3.3.6. Let x ∈ W_{c}(B_{n−1}) and let w ∈ W^{(n)}. Suppose that there exists s ∈ S(B_{n}), with ℓ(ws) < ℓ(w), such that xws is fully commutative but xw is not. Then b_{x}t̃_{w} ∈ v^{−1}L′_{xw}.

Proof. By the hypothesis of the lemma, together with Proposition 3.2.1, there are two possibilities concerning the nature of w and x: Either (a) w = σ_{n}σ_{n−1} ⋯ σ_{r+1}σ_{r} (2 ≤ r < n) and ℓ(xσ_{r}) < ℓ(x); or (b) w = σ_{n}σ_{n−1} ⋯ σ_{2}σ_{1}σ_{2} and ℓ(xσ_{1}) < ℓ(x).

Suppose that case (a) holds. Let w′ = wσ_{r}σ_{r+1}. We have

b_{x}t̃_{w} = b_{xσ_{r}}b_{σ_{r}} t̃_{w′} t̃_{σ_{r+1}} t̃_{σ_{r}} = b_{xσ_{r}} t̃_{w′}b_{σ_{r}}(b_{σ_{r+1}} − v^{−1})(b_{σ_{r}} − v^{−1}).

This last expression expands into the product

b_{xσ_{r}} t̃_{w′}(b_{σ_{r}}b_{σ_{r+1}}b_{σ_{r}} − v^{−1}b_{σ_{r}}b_{σ_{r+1}} − v^{−1}b_{σ_{r}}^{2} + v^{−2}b_{σ_{r}}).

Using the relations for the monomial generators given in §3.2, we may simplify this last expression to

b_{xσ_{r}} t̃_{w′}(−v^{−1}b_{σ_{r}}b_{σ_{r+1}}) = b_{x}t̃_{w′}(−v^{−1}b_{σ_{r+1}}).

Thus, b_{x}t̃_{w} = −v^{−1}b_{x}t̃_{w′}b_{σ_{r+1}}. This last expression is easily seen to lie in v^{−1}L′_{xw} by Lemma 3.3.5 (ii) (taking s = σ_{r}).

We turn to case (b), wherein w = σ_{n}σ_{n−1} ⋯ σ_{2}σ_{1}σ_{2} and x has a reduced expression ending in σ_{1}. Let w′′ = wσ_{2}σ_{1}σ_{2}. Observe that b_{x}t̃_{w} equals

b_{xσ_{1}}b_{σ_{1}} t̃_{w′′} t̃_{σ_{2}} t̃_{σ_{1}} t̃_{σ_{2}} = b_{xσ_{1}} t̃_{w′′}b_{σ_{1}}(b_{σ_{2}} − v^{−1})(b_{σ_{1}} − v^{−1})(b_{σ_{2}} − v^{−1}).

By expanding and then simplifying this last expression, we obtain

b_{xσ_{1}} t̃_{w′′}(−v^{−1}b_{σ_{1}}b_{σ_{2}}b_{σ_{1}} + v^{−1}b_{σ_{1}}) = b_{x}t̃_{w′′}(−v^{−1}b_{σ_{2}σ_{1}} + v^{−1}).

Thus, b_{x}t̃_{w} = −v^{−1}b_{x}t̃_{w′′}b_{σ_{2}σ_{1}} + v^{−1}b_{x}t̃_{w′′}. This last pair of terms lies in v^{−1}L′_{xw} by Lemma 3.3.5 (ii). □

The next lemma is required to address a certain term that arises in the proof of Lemma 3.3.8.

Lemma 3.3.7. Let x ∈ W_{c}(B_{n−1}) and let w = σ_{n}σ_{n−1} ⋯ σ_{r}σ_{r−1}, with 2 ≤ r < n. Let w′ = wσ_{r−1}σ_{r}σ_{r+1}. Suppose that ℓ(xσ_{r}) < ℓ(x). Then b_{x}t̃_{w′}b_{σ_{r−1}σ_{r+1}} ∈ L′_{xw}.

Proof. Note first that when b_{x}t̃_{w′}b_{σ_{r−1}σ_{r+1}} is written as a linear combination of monomial basis elements b_{y}, every nonzero term is indexed by a y satisfying y ≤ xw by Remark 3.2.3. If w′ = e, then b_{x}t̃_{w′}b_{σ_{r−1}σ_{r+1}} = b_{x}b_{σ_{r−1}}b_{σ_{r+1}}, which equals ab_{y}b_{σ_{r+1}} for some integer a and some y ∈ W_{c}(B_{n−1}) by Proposition 3.2.4 (iii). Since w′ = e, we have σ_{r+1} = σ_{n} ∉ c(y), hence the product yσ_{r+1} is reduced and fully commutative. Therefore, ab_{y}b_{σ_{r+1}} = ab_{yσ_{r+1}} ∈ L′_{xw}.

Now assume w′ ≠ e. Let w′′ = w′σ_{r+2} and let x′ = xσ_{r}. By Lemma 3.3.2, we can write b_{x}t̃_{w′} as a sum of terms of the form a_{y}b_{y}, where y ≤ xw′ and a_{y} ∈ A^{−}. Moreover, we have ℓ(yσ_{r}) < ℓ(y) whenever a_{y} ≠ 0, owing to the equality b_{x}t̃_{w′} = b_{x′} t̃_{w′}b_{σ_{r}} and Proposition 3.2.4 (ii). Thus, b_{x}t̃_{w′}b_{σ_{r−1}σ_{r+1}} is a sum of terms of the form a_{y}b_{y}b_{σ_{r−1}σ_{r+1}}, where y ≤ xw′ and ℓ(yσ_{r}) < ℓ(y) whenever a_{y} ≠ 0.

We can say more about the terms a_{y}b_{y}b_{σ_{r−1}σ_{r+1}}. Since b_{x}t̃_{w′} = b_{x}t̃_{w′′} t̃_{σ_{r+2}} = b_{x}t̃_{w′′}(b_{σ_{r+2}} − v^{−1}), we see (as in the proof of Lemma 3.3.1) that for each y ≤ xw′, either a_{y} ∈ v^{−1}A^{−} or else a_{y} ∈ A^{−} and ℓ(yσ_{r+2}) < ℓ(y). In the case where a_{y} ∈ v^{−1}A^{−} and a_{y} ≠ 0, we have a_{y}b_{y}b_{σ_{r−1}} = a′b_{y′} with a′ ∈ v^{−1}A^{−} by Proposition 3.2.4 (iii) (here, we are using the fact that ℓ(yσ_{r}) < ℓ(y)). But then a_{y}b_{y}b_{σ_{r−1}}b_{σ_{r+1}} = a′b_{y′}b_{σ_{r+1}} = a′′b_{y′′}, with a′′ ∈ A^{−} by Proposition 3.2.4 (i).

Consider now the other case, where a_{y} ∈ A^{−}, a_{y} ≠ 0 and ℓ(yσ_{r+2}) < ℓ(y). Here, we may write y as a fully commutative reduced product y = y_{1}σ_{r}σ_{r+2}. Notice that ℓ(yσ_{r−1}) > ℓ(y). If yσ_{r−1} is fully commutative, then, since yσ_{r−1} = y_{1}σ_{r}σ_{r−1}σ_{r+2} and σ_{r+1}σ_{r+2} ≠ σ_{r+2}σ_{r+1}, we have a_{y}b_{y}b_{σ_{r−1}}b_{σ_{r+1}} = a_{y}b_{yσ_{r−1}}b_{σ_{r+1}} = a′b_{y′} for some a′ ∈ A^{−} by Proposition 3.2.4 (iii).

If yσ_{r−1} is not fully commutative, then there are two possibilities to consider, depending on whether r − 1 is greater than or equal to 1. By Proposition 3.2.1 we have, after applying commutations to obtain a suitable reduced expression for y if necessary, either (a) y = y_{2}σ_{r−1}σ_{r}σ_{r+2} (reduced) if r − 1 > 1; or (b) y = y_{3}σ_{r}σ_{r−1}σ_{r}σ_{r+2} (reduced) if r − 1 = 1. The argument for (b) is essentially the same as that for (a), so we treat only (a).

The product a_{y}b_{y}b_{σ_{r−1}}b_{σ_{r+1}} equals a_{y}b_{y_{2}}b_{σ_{r−1}}b_{σ_{r}}b_{σ_{r+2}}b_{σ_{r−1}}b_{σ_{r+1}}. Since σ_{r+2} and σ_{r−1} commute, this last expression equals a_{y}b_{y_{2}}b_{σ_{r−1}}b_{σ_{r}}b_{σ_{r−1}}b_{σ_{r+2}}b_{σ_{r+1}} =

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 17

a_{y}b_{y_{2}}b_{σ_{r−1}}b_{σ_{r+2}}b_{σ_{r+1}}. Now, recall that y = y_{2}σ_{r−1}σ_{r}σ_{r+2} is a fully commutative reduced product. Since σ_{r+2} and σ_{r} commute, the product y_{2}σ_{r−1}σ_{r+2} is reduced and fully commutative. It follows that a_{y}b_{y}b_{σ_{r−1}}b_{σ_{r+1}} = a_{y}b_{y_{2}}b_{σ_{r−1}}b_{σ_{r+2}}b_{σ_{r+1}} equals a_{y}b_{y_{2}σ_{r−1}σ_{r+2}}b_{σ_{r+1}}, and this last expression equals a′′b_{y′′} for some a′′ ∈ A^{−} by Proposition 3.2.4 (iii). □

Lemma 3.3.8. Let x ∈ W_{c}(B_{n−1}) and let w ∈ W^{(n)}. Suppose that there exist s, s′ ∈ S(B_{n}), with ℓ(wss′) < ℓ(ws) < ℓ(w), such that xwss′ is fully commutative but xws is not. Then b_{x}t̃_{w} ∈ v^{−1}L′_{xw}.

Proof. As in the proof of Lemma 3.3.6, we can use Proposition 3.2.1 to divide the argument into two cases: Either (a) w = σ_{n}σ_{n−1} ⋯ σ_{r+1}σ_{r}σ_{r−1}, with 2 ≤ r < n and ℓ(xσ_{r}) < ℓ(x); or (b) n ≥ 3 and w = σ_{n}σ_{n−1} ⋯ σ_{2}σ_{1}σ_{2}σ_{3} with ℓ(xσ_{1}) < ℓ(x).

Suppose first that (a) holds. Let w′ = wσ_{r−1}σ_{r}σ_{r+1} and let x′ = xσ_{r}. The product b_{x}t̃_{w} equals b_{x′}b_{σ_{r}} t̃_{w′} t̃_{σ_{r+1}} t̃_{σ_{r}} t̃_{σ_{r−1}}, which in turn can be written as

b_{x′} t̃_{w′}b_{σ_{r}}(b_{σ_{r+1}} − v^{−1})(b_{σ_{r}} − v^{−1})(b_{σ_{r−1}} − v^{−1}).

After expanding and then simplifying this last expression, we obtain

b_{x′} t̃_{w′}(−v^{−1}b_{σ_{r}}b_{σ_{r+1}}b_{σ_{r−1}} + v^{−2}b_{σ_{r}}b_{σ_{r+1}}) = b_{x}t̃_{w′}(−v^{−1}b_{σ_{r+1}σ_{r−1}} + v^{−2}b_{σ_{r+1}}).

Thus, b_{x}t̃_{w} = −v^{−1}b_{x}t̃_{w′}b_{σ_{r+1}σ_{r−1}} + v^{−2}b_{x}t̃_{w′}b_{σ_{r+1}}. The first of these two terms lies in v^{−1}L′_{xw} by Lemma 3.3.7 and the second lies in v^{−1}L′_{xw} by Lemma 3.3.4.

Suppose now that (b) holds. Let w′ = wσ_{3}σ_{2}σ_{1}σ_{2}σ_{3} and let x′ = xσ_{1}. The product b_{x}t̃_{w} equals b_{x′}b_{σ_{1}} t̃_{w′} t̃_{σ_{3}} t̃_{σ_{2}} t̃_{σ_{1}} t̃_{σ_{2}} t̃_{σ_{3}}, which in turn equals

b_{x′} t̃_{w′}b_{σ_{1}}(b_{σ_{3}} − v^{−1})(b_{σ_{2}} − v^{−1})(b_{σ_{1}} − v^{−1})(b_{σ_{2}} − v^{−1})(b_{σ_{3}} − v^{−1}).

This last expression expands and then simplifies to

b_{x}t̃_{w′}(v^{−2}b_{σ_{3}σ_{2}σ_{1}} + v^{−2}b_{σ_{2}σ_{1}σ_{3}} − 2v^{−2}b_{σ_{3}} − v^{−3}b_{σ_{2}σ_{1}} + v^{−3}).

The products −2v^{−2}b_{x}t̃_{w′}b_{σ_{3}}, −v^{−3}b_{x}t̃_{w′}b_{σ_{2}σ_{1}}, and v^{−3}b_{x}t̃_{w′} belong to v^{−1}L′_{xw} by Lemma 3.3.4. Consider the product v^{−2}b_{x}t̃_{w′}b_{σ_{3}σ_{2}σ_{1}}. If w′ = e, then n = 3 and xσ_{3}σ_{2}σ_{1} is fully commutative; the latter holds because (1) x is fully commutative and has no reduced expression ending in σ_{2} (as ℓ(xσ_{1}) < ℓ(x) and σ_{1}, σ_{2} do

18 R.M. GREEN AND J. LOSONCZY

not commute) and (2) the presence of σ_{3} precludes the possibility of obtaining a substring of the form σ_{1}σ_{2}σ_{1}σ_{2} or σ_{2}σ_{1}σ_{2}σ_{1} through commutation moves. Hence, v^{−2}b_{x}t̃_{w′}b_{σ_{3}σ_{2}σ_{1}} = v^{−2}b_{xσ_{3}σ_{2}σ_{1}} ∈ v^{−1}L′_{xw}. If w′ ≠ e, then v^{−2}b_{x}t̃_{w′}b_{σ_{3}σ_{2}σ_{1}} ∈ L′_{xw} by Lemma 3.3.5 (i) (taking s = σ_{4}). Finally, we have b_{x}t̃_{w′}b_{σ_{2}σ_{1}} ∈ L′_{xw′σ_{3}σ_{2}σ_{1}} by Lemma 3.3.5 (ii) (taking s = σ_{1}), which, by Proposition 3.2.4 (i), shows that v^{−2}b_{x}t̃_{w′}b_{σ_{2}σ_{1}σ_{3}} ∈ v^{−1}L′_{xw}. □

Lemma 3.3.9. Let x ∈ W_{c}(B_{n−1}) and let w ∈ W^{(n)}. Suppose that xw is not fully commutative. Then b_{x}t̃_{w} ∈ v^{−1}L′_{xw}.

Proof. We proceed by induction on ℓ(w). The lemma is (vacuously) true for ℓ(w) < 2, since xw is fully commutative for such w ∈ W^{(n)}. Suppose that ℓ(w) ≥ 2. Let s, s′ be (the) Coxeter generators that satisfy ℓ(wss′) < ℓ(ws) < ℓ(w). If xwss′ is not fully commutative, then neither is xws; by the inductive hypothesis, b_{x}t̃_{wss′} ∈ v^{−1}L′_{xwss′} and b_{x}t̃_{ws} ∈ v^{−1}L′_{xws}. But then b_{x}t̃_{w} ∈ v^{−1}L′_{xw} by Lemma 3.3.1 (taking k = 1).

On the other hand, suppose that xwss′ is fully commutative. If xws is not fully commutative, then Lemma 3.3.8 gives us b_{x}t̃_{w} ∈ v^{−1}L′_{xw}. Finally, if xws is fully commutative, then Lemma 3.3.6 applies.

The inductive step is complete. □

Proposition 3.3.10. We have t̃_{w} ∈ v^{−1}L′_{w} for all w ∈ W (B_{n})\W_{c}(B_{n}).

Proof. Let w ∈ W (B_{n})\W_{c}(B_{n}). Write w = w_{1}w_{2} ⋯ w_{n}, where each w_{i} ∈ W^{(i)}. There exists a unique integer r > 1 such that w_{1}w_{2} ⋯ w_{r−1} is fully commutative and w_{1}w_{2} ⋯ w_{r} is not. Let y = w_{1}w_{2} ⋯ w_{r−1}. Given the nature of the representative w_{r}, it follows from Proposition 3.2.1 that y = y′s (reduced), where s ∈ S(B_{r−1}) is such that sw_{r} is not fully commutative. By Proposition 3.3.3, we have t̃_{y′} ∈ L′_{y′} and t̃_{y} ∈ L′_{y}. Combining this with the equality t̃_{y} = t̃_{y′}(b_{s} − v^{−1}), we find that if t̃_{y} is written as a linear combination of monomial basis elements b_{x} (x ≤ y), then for each x, either the coefficient of b_{x} lies in v^{−1}A^{−}, or the coefficient lies in A^{−} and ℓ(xs) < ℓ(x).

Thus, t̃_{yw_{r}} = t̃_{y}t̃_{w_{r}} equals a sum of terms of the form a_{x}b_{x}t̃_{w_{r}} (x ≤ y), where for each x, either a_{x} ∈ v^{−1}A^{−}, or else a_{x} ∈ A^{−} and xw_{r} is not fully commutative. Now, by Lemma 3.3.2, we have b_{x}t̃_{w_{r}} ∈ L′_{xw_{r}}. Therefore, in the case where a_{x} ∈ v^{−1}A^{−},

FULLY COMMUTATIVE KAZHDAN–LUSZTIG CELLS 19

we have a_{x}b_{x}t̃_{w_{r}} ∈ v^{−1}L′_{xw_{r}}. In the case where a_{x} ∈ A^{−} and xw_{r} ∉ W_{c}, Lemma 3.3.9 gives us a_{x}b_{x}t̃_{w_{r}} ∈ v^{−1}L′_{xw_{r}}. We have so far shown that t̃_{yw_{r}} ∈ v^{−1}L′_{yw_{r}}.

The product t̃_{yw_{r}} t̃_{w_{r+1}} must then lie in v^{−1}L′_{yw_{r}w_{r+1}}. To see this, expand t̃_{yw_{r}} into a sum of terms a′_{z}b_{z} (z ≤ yw_{r}), where each a′_{z} ∈ v^{−1}A^{−}. Then t̃_{yw_{r}} t̃_{w_{r+1}} equals a sum of terms a′_{z}b_{z}t̃_{w_{r+1}}, each of which must lie in v^{−1}L′_{yw_{r}w_{r+1}} by Lemma 3.3.2 (here, r +1 is playing the role of n in the lemma). By iterating this argument, we are able to conclude that t̃_{w} = t̃_{yw_{r}} t̃_{w_{r+1}} ⋯ t̃_{w_{n}} ∈ v^{−1}L′_{w}, as desired. □ 3.4 Proof of Theorem 3.1.1.

We are now in a position to give a proof that the conditions of Theorem 2.2.3 are satisfied when the underlying Coxeter graph is of type B_{n}. Proof of Theorem 3.1.1. We shall verify condition (v) of Theorem 2.2.3. It is required to prove that for each w ∈ W (B_{n})\W_{c}(B_{n}), the element θ(T̃_{w}) = t̃_{w} lies in v^{−1}L. Proposition 3.3.10 gives t̃_{w} ∈ v^{−1}L′_{w} for such w. Thus, the theorem will follow if we can show that L′_{w} = L_{w}.

Fix an arbitrary w. Proposition 3.3.3 gives L′_{w} ⊇ L_{w}. Let x ∈ W_{c} satisfy x ≤ w. By Proposition 3.3.3, we may write t̃_{x} = ∑ a_{y}b_{y}, where for all y we have y ≤ x, y ∈ W_{c} and a_{y} ∈ A^{−}. Moreover, given any reduced expression s_{1}s_{2} ⋯ s_{m} for x, we have

t̃_{x} = t̃_{s_{1}} t̃_{s_{2}} ⋯ t̃_{s_{m}} = (b_{s_{1}} − v^{−1})(b_{s_{2}} − v^{−1}) ⋯ (b_{s_{m}} − v^{−1}),

from which we can see that a_{x} = 1. It now follows by a straightforward induction on the Bruhat–Chevalley order that any b_{x} with x ≤ w can be written as an A^{−}- linear combination of basis elements t̃_{y} with y ≤ x. That is, we have L′_{w} ⊆ L_{w}, as well. □ Remark 3.4.1. Another possible approach to proving Theorem 3.1.1 would be to use the combinatorial classification of cells achieved by Garfinkle in [5, 6, 7].

We conclude with a discussion of an application: it is possible as a consequence of Theorem 3.1.1 to describe very explicitly the structure of the fully commutative left, right and two-sided cells in type B. By [12, Theorem 2.2.1], the canonical basis for TL(B_{n}) is the image under θ of the set {C′_{w} : w ∈ W_{c}(B_{n})}, and by condition (iii) of Theorem 2.2.3, all other C′_{w} are mapped to zero. The diagram calculus for the canonical basis of TL(B_{n}) given in [10, Theorem 2.2.5] can now be used to describe the various kinds of fully commutative Kazhdan–Lusztig cells.

By [16, Theorem 1.10], each left (respectively, right) cell in a crystallographic

Coxeter group W, such as W (B_{n}), contains a unique distinguished involution d.

In type A, any left cell and right cell from the same two-sided cell intersect in a

single element. Our approach can be used to describe the situation in type B in an

elementary way.

Let n ≥ 2, and let W′ ≅ W (A_{n−1}) be the parabolic subgroup of W (B_{n}) obtained

by omitting the first generator. Let d, d′ ∈ W (B_{n}) be distinguished involutions in

the same fully commutative two-sided cell, I_{T}. Let I_{R} be the right cell containing

d, and let I_{L} be the left cell containing d′. Let k = |I_{R} ∩ I_{L}|. Then we have k = 1

if exactly one of the elements d, d′ lies in W′, or if I_{T} ⊆ W′, or if I_{T} ∩ W′ = ∅.

Otherwise, we have k = 2.

References

[1] F. du Cloux, Coxeter Version 1.01, Université de Lyon, France, 1991.

[2] C.K. Fan, A Hecke algebra quotient and properties of commutative elements of a Weyl group, Ph.D. thesis, M.I.T., 1995.

[3] C.K. Fan and R.M. Green, Monomials and Temperley–Lieb algebras, J. Algebra 190 (1997), 498–517.

[4] C.K. Fan and J.R. Stembridge, Nilpotent orbits and commutative elements, J. Algebra 196 (1997), 490–498.

[5] D. Garfinkle, On the classification of primitive ideals for complex classical Lie algebras, I, Compositio Math. 75 (1990), 135–169.

[6] , On the classification of primitive ideals for complex classical Lie algebras, II, Compositio Math. 81 (1992), 307–336.

[7] , On the classification of primitive ideals for complex classical Lie algebras, III, Compositio Math. 88 (1993), 187–234.

[8] J.J. Graham, Modular representations of Hecke algebras and related algebras, Ph.D. thesis, University of Sydney, 1995.

[9] R.M. Green, Generalized Temperley–Lieb algebras and decorated tangles, J. Knot Theory Ramifications 7 (1998), 155–171.

[10] , Decorated tangles and canonical bases (preprint).

[11] R.M. Green and J. Losonczy, Canonical bases for Hecke algebra quotients, Math. Res. Lett. 6 (1999), 213–222.

[12] , A projection property for Kazhdan–Lusztig bases, Internat. Math. Res. Notices 1 (2000), 23–34.

[13] J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.

[14] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.

[15] J. Losonczy, The Kazhdan–Lusztig basis and the Temperley–Lieb quotient in type D, J. Algebra 233 (2000), 1–15.

[16] G. Lusztig, Cells in affine Weyl groups, II, J. Algebra 109 (1987), 536–548.

[17] J.R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), 353–385.