A PROJECTION PROPERTY FOR KAZHDAN–LUSZTIG BASES

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 compare the canonical basis for a generalized Temperley–Lieb algebra of type A or B with the Kazhdan–Lusztig basis for the corresponding Hecke algebra.

To appear in International Mathematics Research Notices

Introduction

Generalized Temperley–Lieb algebras arise as certain quotients of Hecke algebras

associated to Coxeter systems, in the same way that the ordinary Temperley–Lieb

algebra can be realised as a quotient of the Hecke algebra of type A (see [9]).

The finite dimensional generalized Temperley–Lieb algebras were classified by J.

Graham [5] into seven infinite families: types A, B, D, E, F, H and I.

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

The first author was supported in part by an award from the Nuffield Foundation.

In [7], we showed that a generalized Temperley–Lieb algebra arising from a Coxeter system of arbitrary type admits a canonical (more precisely, an IC) basis. Such a basis is by definition unique; furthermore, it is analogous, in a manner which can be made precise, to G. Lusztig’s canonical basis for the ±-part of a quantized enveloping algebra and also to the Kazhdan–Lusztig basis for a Hecke algebra. We determined the IC basis explicitly for the algebras of types A, D and E; in each case the basis coincides with a previously familiar basis with convenient properties [7, Theorem 3.6].

In this paper, we examine the relationship between the IC basis of a generalized Temperley–Lieb algebra and the corresponding Kazhdan–Lusztig basis, showing in particular that in type B, the projection of a certain natural subset of the Kazhdan– Lusztig basis agrees with the IC basis. Such a relationship is not obvious for a general Coxeter system. For example, the kernel of the relevant homomorphism need not be spanned by the Kazhdan–Lusztig basis elements which it contains. We point out that some of the results in this paper lead to a new, concise and more general proof of the main result of [4].

1. Preliminaries

1.1 Generalized Temperley–Lieb algebras and IC bases.

Let X be a Coxeter graph, of arbitrary type, and let W (X) be the associated Coxeter group with distinguished set of generating involutions S(X). Denote by H(X) the Hecke algebra associated to W (X). (The reader is referred to [8, §7] for the basic theory of Hecke algebras arising from Coxeter systems.) Let A be the ring of Laurent polynomials, ℤ[v, v^{−1}]. The A-algebra H(X) has a basis consisting of elements T_{w}, with w ranging over W (X), 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 (X), w ∈ W (X), and s ∈ S(X). The parameter q is equal to v^{2}.

Let J(X) be the two-sided ideal of H(X) generated by the elements

∑_{w∈⟨s,s′⟩} T_{w},

where (s, s′) runs over all pairs of elements of S(X) 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 1.1.1. The generalized Temperley–Lieb algebra, T L(X), is the quotient A-algebra H(X)/J(X). We denote the canonical epimorphism of algebras by θ : H(X) ⟶ T L(X).

Definition 1.1.2. A product w_{1}w_{2} ⋯ w_{n} of elements w_{i} ∈ W (X) 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(X).

Call an element w ∈ W (X) complex if it can be written as a reduced product x_{1}w_{ss′}x_{2}, where x_{1}, x_{2} ∈ W (X) and w_{ss′} is the longest element of some rank 2 parabolic subgroup ⟨s, s′⟩ such that s and s′ do not commute.

Denote by W_{c}(X) the set of all elements of W (X) that are not complex.

Let t_{w} denote the image of the basis element T_{w} ∈ H(X) in the quotient T L(X).

Theorem 1.1.3 (Graham). The set {t_{w} : w ∈ W_{c}} is an A-basis for the algebra T L(X).

Proof. See [5, Theorem 6.2]. □

The basis in Theorem 1.1.3 will be called the “t-basis”. The t-basis plays a rôle in the definition of other important bases for the generalized Temperley–Lieb algebra.

Definition 1.1.4. For each s ∈ S(X), we define b_{s} = v^{−1}t_{s} +v^{−1}t_{e}. If w ∈ W_{c}(X) and s_{1}s_{2} ⋯ s_{n} is a reduced expression for w, then we define b_{w} = b_{s_{1}}b_{s_{2}} ⋯ b_{s_{n}}.

Note that b_{w} does not depend on the choice of reduced expression for w.

4 R.M. GREEN AND J. LOSONCZY

It is known that {b_{w} : w ∈ W_{c}} is a basis for the A-module T L(X). We shall call it the monomial basis.

We now recall a principal result of [7], which establishes the canonical basis for T L(X). This basis is a direct analogue of the important Kazhdan–Lusztig basis of the Hecke algebra H(X).

Fix a Coxeter graph, X. Let (I, ≤) be the poset with I = W_{c}(X) and with ≤ defined as the restriction to I of the Bruhat–Chevalley order on W (X).

Let A^{−} = ℤ[v^{−1}]. Let ¯ be the involution on the ring A = ℤ[v, v^{−1}] which satisfies v̄ = v^{−1}.

By [7, Lemma 1.4], the algebra T L(X) has a ℤ-linear automorphism of order 2 which sends v to v^{−1} and t_{w} to t_{w^{−1}}^{−1}. We denote this map also by ¯ .

Let L be the free A^{−}-submodule of T L(X) with basis {v^{−ℓ(w)}t_{w} : w ∈ W_{c}}, and let π : L ⟶ L/v^{−1}L be the canonical projection. Theorem 1.1.5. There exists a unique basis {c_{w} : w ∈ W_{c}} for L such that \overline{c_{w}} = c_{w} and π(c_{w}) = π(v^{−ℓ(w)}t_{w}) for all w ∈ W_{c}. Proof. This is [7, Theorem 2.3]. □

The basis {c_{w} : w ∈ W_{c}} is called the IC basis (or the canonical basis) of T L(X). It depends on the t-basis, the involution ¯ , and the lattice L. 1.2 General questions.

It was shown in [7, Theorem 3.6] that the canonical basis of the generalized Temperley–Lieb algebra equals the monomial basis in types A, D and E. In this paper, we determine the IC basis for type B. Note that in any non-simply-laced case, the monomial basis does not equal the canonical basis [7, Remark 3.7 (1)]. It would be interesting to have a description of the canonical basis in each of the finite dimensional types.

A natural related problem concerns the relationship between the IC basis of a generalized Temperley–Lieb algebra and the Kazhdan–Lusztig basis for the corresponding Hecke algebra (see [7, Remark 2.4 (2)]). Recall from [10] that for each

A PROJECTION PROPERTY FOR KAZHDAN–LUSZTIG BASES 5

w ∈ W there exists a unique C′_{w} ∈ H such that \overline{C′_{w}} = C′_{w}, where ¯ is a certain ℤ-linear automorphism of H, and

C′_{w} = ∑_{{x∈W \atop x≤w}} P̃_{x,w}(v^{−ℓ(x)}T_{x}),

where P̃_{x,w} lies in v^{−1}A^{−} if x < w, and P̃_{w,w} = 1.

Let C denote the set of C′_{w} ∈ H indexed by w ∈ W_{c}. It is clear from [7, Lemma 1.5] that the set θ(C) is a basis for T L(X). Definition 1.2.1. We say that a Coxeter graph X satisfies the projection property if θ(C) equals the canonical basis of T L(X).

We do not know of an example of a Coxeter graph which fails to have the projection property. The following two propositions provide useful sufficient conditions for θ(C) to equal the canonical basis. Proposition 1.2.2. The following are equivalent for a Coxeter graph X:

(i) θ(C′_{w}) ∈ L for all w ∈ W (X), where L is the lattice defined in §1.1. (ii) θ(v^{−ℓ(w)}T_{w}) ∈ L for all w ∈ W (X).

If (i) or (ii) holds, then π(θ(C′_{w})) = π(θ(v^{−ℓ(w)}T_{w})) for all w ∈ W, so that X satisfies the projection property. Proof. For any w ∈ W (X), we have

C′_{w} = ∑_{{x∈W (X) \atop x≤w}} P̃_{x,w}(v^{−ℓ(x)}T_{x}),

where P̃_{x,w} lies in v^{−1}A^{−} if x < w, and P̃_{w,w} = 1. If we assume (ii), we see that θ(P̃_{x,w}(v^{−ℓ(x)}T_{x})) ∈ v^{−1}L if x < w, and θ(P̃_{w,w}(v^{−ℓ(w)}T_{w})) = θ(v^{−ℓ(w)}T_{w}) ∈ L. Thus θ(C′_{w}) ∈ L and π(θ(C′_{w})) = π(θ(v^{−ℓ(w)}T_{w})), which proves that (ii) ⇒ (i) and also that (ii) implies π(θ(C′_{w})) = π(θ(v^{−ℓ(w)}T_{w})) for all w ∈ W.

Next, we show that (i) implies (ii). One sees from [10] that for any w ∈ W (X) we have

v^{−ℓ(w)}T_{w} = ε_{w} ∑_{{x∈W \atop x≤w}} ε_{x}Q̃_{x,w}C′_{x},

6 R.M. GREEN AND J. LOSONCZY

where ε_{x} := (−1)^{ℓ(x)}, Q̃_{x,w} lies in v^{−1}A^{−} if x < w, and Q̃_{w,w} = 1. An argument similar to that of the previous paragraph gives (i) ⇒ (ii).

Finally, we verify that X satisfies the projection property if either (i) or (ii) holds. It was shown in [7, Lemma 1.4] that the automorphism ¯ of H given in [10] induces the automorphism ¯ of T L(X) defined in §1.1. It follows that \overline{θ(C′_{w})} = θ(C′_{w}). Since π(θ(C′_{w})) = π(θ(v^{−ℓ(w)}T_{w})), Theorem 1.1.5 shows that θ(C′_{w}) = c_{w} if w ∈ W_{c}, which completes the proof. □

Proposition 1.2.3. If X is a Coxeter graph such that the kernel of the canonical map θ : H(X) ⟶ T L(X) is spanned by the basis elements C′_{w} which it contains, then X has the projection property.

Proof. Suppose that the hypothesis is satisfied, and let w ∈ W_{c}. Recall from the proof of Proposition 1.2.2 that

v^{−ℓ(w)}T_{w} = ε_{w} ∑_{{x∈W \atop x≤w}} ε_{x}Q̃_{x,w}C′_{x}.

Applying θ to both sides yields

θ(v^{−ℓ(w)}T_{w}) = ε_{w} ∑_{{x∈Wc \atop x≤w}} ε_{x}Q̃_{x,w}θ(C′_{x}).

This means that the change of basis matrix from the basis {θ(C′_{w}) : w ∈ W_{c}} to the basis {v^{−ℓ(w)}t_{w} : w ∈ W_{c}} (with respect to some total refinement of the partial order ≤) is upper triangular with ones on the diagonal, and all the entries above the diagonal lie in v^{−1}A^{−}. The inverse of this matrix has the same properties, meaning that

θ(C′_{w}) = ∑_{{x∈Wc \atop x≤w}} P̃_{x,w}(v^{−ℓ(x)}t_{x}),

where P̃_{x,w} ∈ v^{−1}A if x < w and P̃_{w,w} = 1. Thus π(θ(C′_{w})) = π(v^{−ℓ(w)}t_{w}) = π(c_{w}), and since θ(C′_{w}) is fixed by ¯ , the proposition follows. □

We conclude this section with a conjecture (cf. [7, Remark 2.4 (1)]).

A PROJECTION PROPERTY FOR KAZHDAN–LUSZTIG BASES 7

Conjecture 1.2.4. Let X be an arbitrary Coxeter graph. Then the structure constants of the canonical basis for T L(X) lie in ℕ[v, v^{−1}].

The conjecture follows easily in types A, D and E from the results in [7, §3]. It is not difficult to see that when X = I_{2}(m) (the dihedral case), the hy-

pothesis of Proposition 1.2.3 is satisfied: the ideal J(I_{2}(m)) is spanned by the single Kazhdan–Lusztig basis element C′_{w_{0}}, where w_{0} denotes the longest element of W (I_{2}(m)). Moreover, it is known that the structure constants for the C′-basis of H(I_{2}(m)) have positive coefficients. Thus, one is able to deduce the positivity property for the IC basis of the generalized Temperley–Lieb algebra T L(I_{2}(m)).

2. Types A and B

Throughout this section, we assume that X is a Coxeter graph of type A or B. The algebra T L(X) is then generated by the monomial basis elements b_{s}, with s ranging over S(X), subject to the relations b_{s}^{2} = q_{c}b_{s}, b_{s}b_{s′} = b_{s′}b_{s} if ss′ has order 2, b_{s}b_{s′}b_{s} = b_{s} if ss′ has order 3, and b_{s}b_{s′}b_{s}b_{s′} = 2b_{s}b_{s′} if ss′ has order 4 (see [6, §1]). Here, q_{c} is the Laurent polynomial [2] = v + v^{−1}.

Our goal is to prove that X has the projection property. This will be accomplished in Theorem 2.2.1.

2.1 Reduced expressions.

Our first lemma describes a normal form reduced expression for elements in a Coxeter group of type A or B.

Let σ_{1}, σ_{2}, …, σ_{r} denote the Coxeter generators of the Coxeter group W (B_{r}), where σ_{1}σ_{2} has order 4 and σ_{i}σ_{i+1} has order 3 for i > 1. Define W^{(r)} = {w ∈ W (B_{r}) : 1 ≤ i < r ⇒ ℓ(σ_{i}w) > ℓ(w)}. Then W^{(r)} is a set of minimum length right coset representatives for the subgroup W (B_{r−1}) of W (B_{r}), and ℓ(xy) = ℓ(x) + ℓ(y) for all x ∈ W (B_{r−1}) and y ∈ W^{(r)} (see [8, §5.12]). Furthermore, it is known that the elements of W^{(r)} are given by

{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}, …,

8 R.M. GREEN AND J. LOSONCZY

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

(This can be established by working with the signed permutation representation of W (B_{r}).) Observe that each element of W^{(r)} has a unique reduced expression.

Lemma 2.1.1. Let w ∈ W (X). There exists a reduced expression s_{1}s_{2} ⋯ s_{n} for w such that for each k, either s_{k} does not appear to the left of the k-th factor in s_{1}s_{2} ⋯ s_{n}, or s_{k} does not commute with s_{k−1}.

Proof. Every Coxeter system of type B_{r} contains a Coxeter system of type A_{r−1}, the Coxeter generators of the latter being a subset of those of the former. Hence, it is enough to treat the case X = B_{r}.

Considering the chain {e} = W (B_{0}) ⊆ W (B_{1}) ⊆ ⋯ ⊆ W (B_{r}) = W of Coxeter groups, we see that any w ∈ W has a unique reduced decomposition w = w_{1}w_{2} ⋯ w_{r}, where each w_{i} ∈ W^{(i)}. By substituting for each w_{i} its unique reduced expression, we obtain a normal form reduced expression for w which satisfies the condition in the statement of the lemma. □

Let w ∈ W = W (X). It is known [1, §IV.1.5] that every reduced expression for w can be transformed into any other reduced expression for w by performing a sequence of braid moves. Given this fact, one may characterize W_{c} as the set of w ∈ W such that every reduced expression for w can be transformed into any other reduced expression for w by a sequence of commutation moves (see [11, 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.

The next result was stated without proof in [3, §7.1]. The proof that we give is similar to that of [2, Lemma 2].

Lemma 2.1.2. Let w ∈ W_{c} and s ∈ S satisfy ws ∉ W_{c}. There exists a unique s′ ∈ S such that any reduced expression for w can be parsed in one of the following two ways.

A PROJECTION PROPERTY FOR KAZHDAN–LUSZTIG BASES 9

(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}).

Proof. A sequence of commutation moves applied to any reduced expression x = s_{1}s_{2} ⋯ s_{n} results in a reduced expression x = s_{τ(1)}s_{τ(2)} ⋯ s_{τ(n)}, where the τ(i)-th factor of s_{1}s_{2} ⋯ s_{n} has been moved to the i-th position.

We remark that if the generators s_{τ(i)} and s_{τ(j)} do not commute and τ(i) < τ(j), then i < j.

Let s_{1}s_{2} ⋯ s_{n} be a reduced expression for w. We shall parse this reduced expression as in the statement of the lemma.

Define s_{n+1} = s. Since s_{1}s_{2} ⋯ s_{n+1} ∉ W_{c}, one can transform s_{1}s_{2} ⋯ s_{n+1} by a sequence of commutation moves into a reduced expression s_{τ(1)}s_{τ(2)} ⋯ s_{τ(n+1)} which possesses a substring of type: (1) s_{τ(k)}s_{τ(k+1)}s_{τ(k+2)}, where s_{τ(k)} = s_{τ(k+2)} and (s_{τ(k)}s_{τ(k+1)})^{3} = 1; or type (2) s_{τ(k−1)}s_{τ(k)}s_{τ(k+1)}s_{τ(k+2)}, where s_{τ(k−1)} = s_{τ(k+1)}, s_{τ(k)} = s_{τ(k+2)} and (s_{τ(k)}s_{τ(k+1)})^{4} = 1.

We claim that in either case, τ(k + 2) = n + 1. By the previous remark, we have n + 1 ≠ τ(k), τ(k + 1), and also n + 1 ≠ τ(k − 1) if we are considering a substring of type (2). Hence, if τ(k + 2) ≠ n + 1, then we may commute s_{n+1} to the end of s_{τ(1)}s_{τ(2)} ⋯ s_{τ(n+1)} and thereby produce a reduced expression for ws which contains a substring of type (1) or (2) among the first n factors, contradicting w ∈ W_{c}. Thus, we have s = s_{τ(k)} = s_{τ(k+2)}.

Again by the remark, we have τ(k) < τ(k+1) < τ(k+2), and also τ(k−1) < τ(k) if the substring is of type (2). Suppose that the substring is of type (1). Define w_{1} = s_{1}s_{2} ⋯ s_{τ(k)−1}, w_{2} = s_{τ(k)+1}s_{τ(k)+2} ⋯ s_{τ(k+1)−1}, s′ = s_{τ(k+1)} and w_{3} = s_{τ(k+1)+1}s_{τ(k+1)+2} ⋯ s_{n}. Observe that in order for the substring of type (1) to have occurred, it must be true that s commutes with every member of c(w_{2}) ∪ c(w_{3}). (To see why, consider the sequence of commutation moves required to transform the subexpression sw_{2}s′w_{3}s of ws into an expression where the occurrences of s,

10 R.M. GREEN AND J. LOSONCZY

s′, s are consecutive.)

If instead the substring is of type (2), then we define w_{1} = s_{1}s_{2} ⋯ s_{τ(k−1)−1}, s′ = s_{τ(k−1)}, w_{2} = s_{τ(k−1)+1}s_{τ(k−1)+2} ⋯ s_{τ(k)−1}, w_{3} = s_{τ(k)+1}s_{τ(k)+2} ⋯ s_{τ(k+1)−1} and w_{4} = s_{τ(k+1)+1}s_{τ(k+1)+2} ⋯ s_{n}. As above, in order for the substring of type (2) to have occurred, it must be the case that 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}).

Regardless of the type of substring that arises, inspection of the parsed form of the original reduced expression reveals that s′ is the rightmost factor which does not commute with s in any reduced expression for w. Thus, s′ is uniquely determined by w and s. □

Before presenting the next lemma, which furnishes some relevant information concerning the structure constants of the monomial basis, we remark that if w = s_{1}s_{2} ⋯ s_{n} is a reduced expression and 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n, then the product b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k}}} equals aq_{c}^{m}b_{x}, where a and m are nonnegative integers, and x ∈ W_{c} satisfies ℓ(x) ≤ k and x ≤ w. This follows easily from the subexpression characterization of Bruhat–Chevalley order and the presentation of T L(X) given at the beginning of this section. We shall invoke this fact freely in the remainder of §2.

We remind the reader that the elements b_{w} ∈ T L(X) are always understood to be indexed by group elements w ∈ W_{c} (see Definition 1.1.4).

Lemma 2.1.3. Let w ∈ W_{c} and s ∈ S. Then b_{w}b_{s} = aq_{c}^{m}b_{w′} for some w′ ∈ W_{c} and some nonnegative integers a and m, where m ≤ 1. We have ℓ(w′s) < ℓ(w′). If ℓ(ws′) < ℓ(w) for some s′ ∈ S which does not commute with s, then m = 0.

Proof. We first address the last assertion in the statement. Suppose that w has a reduced expression ending in a generator s′ which does not commute with s. Since w ∈ W_{c}, this implies ℓ(ws) > ℓ(w). If ws ∉ W_{c} then by Lemma 2.1.2, it is possible to write either w = uss′ (reduced), if (ss′)^{3} = 1, or w = us′ss′ (reduced), if (ss′)^{4} = 1. In either case, b_{w}b_{s} equals a positive integer multiple of some monomial

A PROJECTION PROPERTY FOR KAZHDAN–LUSZTIG BASES 11

basis element.

We now prove the rest of the lemma by induction on ℓ(w). The statement is true for ℓ(w) ≤ 1, so we may assume ℓ(w) > 1.

Suppose that ℓ(ws) < ℓ(w). We may write w = us (reduced) for some u ∈ W_{c}. We then have b_{w}b_{s} = b_{u}b_{s}b_{s} = q_{c}b_{u}b_{s} = q_{c}b_{w}, and the induction step follows.

Suppose instead that ℓ(ws) > ℓ(w). If ws ∈ W_{c}, the result is obvious: set m = 0, a = 1 and w′ = ws. If ws ∉ W_{c} then we choose a reduced expression for w as in the statement of Lemma 2.1.1, and parse it according to Lemma 2.1.2. We treat only the case where the reduced expression may be parsed as w = w_{1}s′w_{2}sw_{3}s′w_{4}, where (ss′)^{4} = 1, 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 reasoning for the other case is similar.

If ℓ(w_{4}) = 0, then b_{w}b_{s} = b_{w_{1}w_{2}w_{3}}b_{s′}b_{s}b_{s′}b_{s} = 2 b_{w_{1}w_{2}w_{3}s′s}, and the induction step follows.

If ℓ(w_{4}) = 1, then we have b_{w}b_{s} = b_{w_{1}w_{2}w_{3}}b_{s′}b_{s}b_{s′}b_{w_{4}}b_{s} = b_{w_{1}w_{2}w_{3}}b_{s′}b_{s}b_{s′}b_{s}b_{w_{4}} = 2 b_{w_{1}w_{2}w_{3}s′s}b_{w_{4}}. By induction, the last expression equals aq_{c}^{m}b_{w′}, where m ≤ 1. Also, since b_{w}b_{s} = 2b_{w_{1}w_{2}w_{3}s′}b_{w_{4}}b_{s}, induction gives ℓ(w′s) < ℓ(w′).

If ℓ(w_{4}) > 1, then we parse w_{4} = w_{5}s_{1}s_{2} (s_{1}, s_{2} ∈ S) according to the chosen reduced expression for w, and then we apply the inductive hypothesis to b_{ws_{2}}b_{s} (note that ℓ(ws_{2}) < ℓ(w) and so b_{ws_{2}}b_{s_{2}} = b_{w}), finding that it equals aq_{c}^{m}b_{w′}, where m ≤ 1 and ℓ(w′s) < ℓ(w′). Since our chosen reduced expression for w conforms to the condition in Lemma 2.1.1, either s_{2} ∉ c(ws_{2}) or s_{2} does not commute with s_{1}. In the first case, we have

b_{w}b_{s} = b_{ws_{2}}b_{s_{2}}b_{s} = b_{ws_{2}}b_{s}b_{s_{2}} = aq_{c}^{m}b_{w′}b_{s_{2}} = aq_{c}^{m}b_{w′s_{2}},

where the last equality holds because s_{2} ∉ c(w′). Since s commutes with s_{2}, we have ℓ(w′s_{2}s) < ℓ(w′s_{2}).

For the case where s_{1} and s_{2} do not commute, we observe that

aq_{c}^{m}b_{w′} = b_{ws_{2}}b_{s} = b_{w_{1}w_{2}w_{3}}b_{s′}b_{s}b_{s′}b_{w_{5}}b_{s_{1}}b_{s} = 2 b_{w_{1}w_{2}w_{3}s′s}b_{w_{5}}b_{s_{1}}.

12 R.M. GREEN AND J. LOSONCZY

By induction, w′ has a reduced expression ending in s_{1}. Hence, by the argument given in the first paragraph, if w′s_{2} ∉ W_{c}, then b_{w′}b_{s_{2}} equals a′b_{w′′} for some positive integer a′ and some w′′ ∈ W_{c}, so that b_{w}b_{s} = b_{ws_{2}}b_{s_{2}}b_{s} = b_{ws_{2}}b_{s}b_{s_{2}} = aq_{c}^{m}b_{w′}b_{s_{2}} = aa′q_{c}^{m}b_{w′′}. On the other hand, since

b_{w}b_{s} = b_{w_{1}w_{2}w_{3}}b_{s′}b_{s}b_{s′}b_{w_{4}}b_{s} = b_{w_{1}w_{2}w_{3}}b_{s′}b_{s}b_{s′}b_{s}b_{w_{4}} = 2 b_{w_{1}w_{2}w_{3}s′}b_{w_{4}}b_{s},

induction gives ℓ(w′′s) < ℓ(w′′).

The induction step is complete. □

If a reduced expression satisfies the condition in the statement of the following lemma, then we say that it has the deletion property.

Lemma 2.1.4. Let w ∈ W. There exists a reduced expression s_{1}s_{2} ⋯ s_{n} for w such that for any 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n, we have b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k}}} = aq_{c}^{m}b_{x} for some x ∈ W_{c} and some nonnegative integers a and m, where m ≤ n − k.

Proof. We shall prove by induction on n = ℓ(w) that if a reduced expression s_{1}s_{2} ⋯ s_{n} for w satisfies the condition in the statement of Lemma 2.1.1, then it has the deletion property. When n ≤ 2, this is evidently true, so we may assume n > 2. Given 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n, we consider two cases.

Case 1: i_{k−1} < n − 1. We apply the inductive hypothesis to s_{1}s_{2} ⋯ s_{n−2} and the integers 1 ≤ i_{1} < i_{2} < ⋯ < i_{k−1} ≤ n − 2, thereby obtaining b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k−1}}} = aq_{c}^{m}b_{x}, where m ≤ (n − 2) − (k − 1). By Lemma 2.1.3, we have b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k}}} = aq_{c}^{m}b_{x}b_{s_{i_{k}}} = a′q_{c}^{m′}b_{x′}, where m′ ≤ m + 1 ≤ n − k.

Case 2: i_{k−1} = n − 1. Here, we apply induction to s_{1}s_{2} ⋯ s_{n−1} and the integers 1 ≤ i_{1} < i_{2} < ⋯ < i_{k−1} = n − 1. We obtain b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k−1}}} = aq_{c}^{m}b_{x}, where m ≤ (n − 1) − (k − 1). If s_{n} ∉ c(x), then the conclusion follows after multiplying both sides of the last equation on the right by b_{s_{n}}. If instead s_{n} ∈ c(x), then because our reduced expression for w satisfies the condition in Lemma 2.1.1, s_{n−1} and s_{n} do not commute. Furthermore, by Lemma 2.1.3, the element x ∈ W_{c} has a reduced expression ending in s_{i_{k−1}} = s_{n−1}, hence by the last assertion of the

A PROJECTION PROPERTY FOR KAZHDAN–LUSZTIG BASES 13

same lemma, b_{x}b_{s_{n}} equals a nonnegative integer multiple of some monomial basis element. □

We note that Lemma 2.1.4 may also be proved using the diagram calculus for T L(B_{n}) described in [6, Theorem 4.1] in conjunction with Lemma 2.1.1.

2.2 Canonical basis in types A and B.

We are now prepared to state and prove the main result of this paper, which can be viewed as a generalization of [4, Theorem 3.8.2].

Theorem 2.2.1. Let X be a Coxeter graph of type A or B. Then the canonical basis of T L(X) equals the image of the set of Kazhdan–Lusztig basis elements C′_{w} ∈ H(X) indexed by w ∈ W_{c}(X).

Proof. By Proposition 1.2.2 (ii), it suffices to show that for all w ∈ W, the element v^{−ℓ(w)}t_{w} lies in the lattice L.

Every w ∈ W has a reduced expression s_{1}s_{2} ⋯ s_{n} as in the statement of Lemma 2.1.4. We have

\begin{aligned}v^{−ℓ(w)}t_{w} & = (v^{−1}t_{s_{1}})(v^{−1}t_{s_{2}}) ⋯ (v^{−1}t_{s_{n}}) \\ & = (b_{s_{1}} − v^{−1})(b_{s_{2}} − v^{−1}) ⋯ (b_{s_{n}} − v^{−1}). \end{aligned}

The last product expands to a sum of terms

±v^{k−ℓ(w)}b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k}}} ,

and each such term equals

av^{k−ℓ(w)}q_{c}^{m}b_{x},

where a ∈ ℤ, x ∈ W_{c} satisfies x ≤ w, and m ≤ ℓ(w) − k by Lemma 2.1.4. The coefficient of b_{x} in the expansion therefore lies in A^{−}. Also, one sees that if w ∈ W_{c}, then the coefficient of b_{w} equals 1.

We have shown that for any w ∈ W, the element v^{−ℓ(w)}t_{w} equals a linear combination of b_{x} (x ∈ W_{c} and x ≤ w) with coefficients in A^{−}; and if w ∈ W_{c}, then the coefficient of b_{w} equals 1. In particular, the change of basis matrix from the monomial basis {b_{x} : x ∈ W_{c}} to the basis {v^{−ℓ(x)}t_{x} : x ∈ W_{c}} (with respect to

some total refinement of ≤) is upper triangular with entries in A^{−} and with ones

on the diagonal. The inverse of this matrix must also have this property. That is,

for any x ∈ W_{c}, the element b_{x} equals a linear combination of v^{−ℓ(y)}t_{y} (y ∈ W_{c}

and y ≤ x) with coefficients in A^{−}. Combining this with the first statement of the

present paragraph, we conclude that every v^{−ℓ(w)}t_{w} lies in L. □

References

[1] N. Bourbaki, Groupes et Algèbres de Lie, Chapitres IV–VI, Masson, Paris, 1981.

[2] C.K. Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), 175–189.

[3] C.K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), 139–167.

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

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

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

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

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

[9] V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335–388.

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

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