On certain bases for Ariki-Koike algebras arising from canonical bases for U_{v}(𝔰𝔩_{m})

Kiyotaka Ohkura and Toshiaki Shoji

Abstract. Frenkel, Khovanov and Kirillov showed that the parabolic Kazhdan-Lusztig basis of Iwahori-Hecke algebra associated to 𝔖_{n} can be obtained as the canonical basis of a weight subspace of V^{⊗n}, where V is the vector representation of the quantum group U_{v}(𝔰𝔩_{m}). In this paper, a similar problem for the case of Ariki-Koike algebra H_{n,r} is discussed. We construct a certain basis of H_{n,r}, which is fixed by the involution and is closely related to the canonical basis of V^{⊗n}, by making use of the representation of H_{n,r} on V^{⊗n}. In the case where r = 2, i.e., in the case of Iwahori-Hecke algebra of type B_{n}, this gives a basis different from the Kazhdan-Lusztig basis of H_{n,r}.

AMS 1991 Mathematics Subject Classification. 17B37, 16G99

Key words and phrases. Ariki-Koike algebra, quantum group, canonical basis, Kazhdan-Lusztig basis

Contents

0. Introduction

1. Review on Ariki-Koike algebras

2. Involutions associated to H_{n,r} and U_{v}(𝔰𝔩_{m})

3. Kazhdan-Lusztig basis and canonical basis

4. H_{n,r}-submodules of V^{⊗n}

5. The case of Iwahori-Hecke algebras of type B_{n}

§0. Introduction

Let U_{v} = U_{v}(𝔰𝔩_{m}) be the quantum group associated to the Lie algebra 𝔰𝔩_{m}, and V the vector representation of U_{v}. Let H_{n} be the Iwahori-Hecke algebra associated to the symmetric group 𝔖_{n}. Then the n-fold tensor space V^{⊗n} turns out to be a U_{v} ⊗ H_{n}-module. Each weight subspace V_{λ}^{⊗n} of V^{⊗n} is H_{n}- stable, and is naturally isomorphic to an induced module M_{J} from a linear representation of some parabolic subalgebra H_{J} of H_{n}. A parabolic Kazhdan-Lusztig basis on M_{J} was defined by Deodhar [D], by generalizing the notion of Kazhdan-Lusztig basis of H_{n} introduced by Kazhdan and Lusztig [KL].

The notion of canonical basis for highest weight modules of U_{v} was introduced by Lusztig [L], which is a union of canonical bases for each weight subspace. In the case of highest weight module V^{⊗n}, Frenkel, Khovanov and Kirillov [FKK] showed that the canonical basis of the weight subspace V_{λ}^{⊗n} coincides with the Kazhdan-Lusztig basis of M_{J} under the above isomorphism. Note that H_{n} has a standard basis {T_{σ} | σ ∈ 𝔖_{n}}, and the Kazhdan-Lusztig basis of H_{n} is characterized by the property that the transition matrix between this basis and the standard basis is of the unitriangular shape, and that it is fixed by a certain involution on H_{n}, called the bar involution. In turn, V^{⊗n} has also a standard basis consisting of the tensor product of the given basis of V, and the canonical basis on V^{⊗n} is characterized by a certain involution ψ on it, together with some additional property related to the standard basis. The important step for proving the result in [FKK] is to show that these two involutions coincide with under the isomorphism M_{J} ≃ V_{λ}^{⊗n}.

Let W_{n,r} be the complex reflection group 𝔖_{n} ⋉ (ℤ/rℤ)^{n}, and H_{n,r} the associated cyclotomic Hecke algebra, i.e., the Ariki-Koike algebra associated to W_{n,r}. In the case where r = 1, H_{n,r} ≃ H_{n}, and H_{n,r} is isomorphic to the Iwahori-Hecke algebra of type B_{n} if r = 2. H_{n,r} contains H_{n} as a subalgebra, and in [SS] the action of H_{n} on V^{⊗n} was extended to the action of H_{n,r}. Each weight space V^{⊗n} is again H_{n,r}-stable. The aim of this paper is to extend the result of [FKK] to the case of certain induced H_{n,r}-modules. One of our main results is Theorem 2.4, which asserts that the bar involution of H_{n,r} is compatible with the involution ψ on V^{⊗n}. By making use of this fact, one can show, in Theorem 4.3, that the weight subspace V_{λ}^{⊗n} is isomorphic to an H_{n,r}-module M_{J} induced from a “non-parabolic” subalgebra H_{J} of H_{n,r}, and that the canonical basis of V_{λ}^{⊗n} determines a basis of M_{J} fixed by the bar involution of H_{n,r}. This may be regarded as a non-parabolic analogue of the result of [FKK].

However, if one focuses on the H_{n,r}-module M_{J} induced from the parabolic subalgebra H_{J} of H_{n,r}, for example H_{n,r} itself, the situation is much more complicated. There is no natural notion of standard basis nor Kazhdan-Lusztig basis of H_{n,r} for r > 2. Moreover, M_{J} turns out to be a direct sum of various weight subspaces V_{λ}^{⊗n}. In order to treat these cases, we make use of the new generators of H_{n,r} introduced by [S]. By using the direct sum decomposition M_{J} = ⨁_{λ} V_{λ}^{⊗n}, one can define two bases of M_{J} inherited from the standard basis and the canonical basis of ⨁_{λ} V_{λ}^{⊗n}. As a special case, we can construct two bases of H_{n,r} in Theorem 4.7; the one has a property that the action of generators of H_{n,r} on this basis is explicitly described, and the other has a property that it is fixed by the bar involution on H_{n,r}, and the transition matrix between these two bases is described by various parabolic Kazhdan-Lusztig polynomials of type A.

We remark that even in the case where r = 2 (i.e., the case of Iwahori-Hecke algebras of type B_{n}), our basis does not coincide with the Kazhdan-Lusztig basis of H_{n,r}. In section 5, we discuss the relationship between these two bases, with the standard basis and the Kazhdan-Lusztig basis of H_{n,r}. In particular we show in Proposition 5.2 that the parabolic Kazhdan-Lusztig polynomials of type B_{n} can be determined uniquely by various parabolic Kazhdan-Lusztig polynomials of type A, together with the information on the transition matrix between the standard basis of H_{n,r} and the standard basis of ⨁_{λ} V_{λ}^{⊗n}.

§1. Review on Ariki-Koike algebras

1.1. Let K = ℚ(v, u_{1}, …, u_{r}) be a field of rational functions in variables v, u_{1}, …, u_{r}. Let W = W_{n,r} be the complex reflection group 𝔖_{n} ⋉ (ℤ/rℤ)^{n}, and H_{n,r} the Ariki-Koike algebra associated to W. H_{n,r} is the associative algebra over K with generators a_{1}, …, a_{n}, and relations

(1.1.1)  \begin{matrix}(a_{1} − u_{1})(a_{1} − u_{2}) ⋯ (a_{1} − u_{r}) = 0, \\ (a_{i} − v)(a_{i} + v^{−1}) = 0 & (2 ≤ i ≤ n), \\ a_{1}a_{2}a_{1}a_{2} = a_{2}a_{1}a_{2}a_{1}, \\ a_{i}a_{i+1}a_{i} = a_{i+1}a_{i}a_{i+1} & (2 ≤ i < n), \\ a_{i}a_{j} = a_{j}a_{i} & (|i − j| ≥ 2). \end{matrix}

It is known that the subalgebra H_{n} of H_{n,r} generated by a_{2}, …, a_{n} is isomorphic to the Hecke algebra associated to the symmetric group 𝔖_{n} with standard generators.

1.2. Let U_{v} = U_{v}(𝔰𝔩_{m}) be the quantum group associated to the Lie algebra 𝔰𝔩_{m} with generators E_{i}, F_{i}, K_{i} (1 ≤ i ≤ m − 1) and standard relations. Apriori, U_{v} is an associative algebra over ℚ(v), but for later discussion, we regard them as an algebra over K by an extension of scalars.

Let V be an m-dimensional vector space over K with basis e_{1}, …, e_{m}. The vector representation of U_{v} on V is defined by

\begin{aligned}E_{i}e_{i+1} = e_{i}, & E_{i}e_{j} & = 0 & j & ≠ i + 1, \\ F_{i}e_{i} = e_{i+1}, & F_{i}e_{j} & = 0 & j & ≠ i, \end{aligned}

K_{i}e_{j} = \begin{cases}v^{−1}e_{i} & j = i, \\ ve_{i+1} & j = i + 1, \\ e_{j} & j ≠ i, i + 1. \end{cases}

It is known that U_{v} has a Hopf algebra structure with comultiplication Δ : U_{v} → U_{v} ⊗ U_{v} given by

Δ(K_{i}) = K_{i} ⊗ K_{i},
Δ(E_{i}) = E_{i} ⊗ 1 + K_{i} ⊗ E_{i},
Δ(F_{i}) = F_{i} ⊗ K_{i}^{−1} + 1 ⊗ F_{i}.

For a positive integer n, we consider the tensor space V^{⊗n} on which U_{v}^{⊗n} acts naturally. We define inductively an algebra homomorphism Δ^{(k)} : U_{v} → U_{v}^{⊗k} by Δ^{(k)} = (Δ^{(k−1)} ⊗ id) ∘ Δ for each k ≥ 3, where Δ^{(2)} = Δ. By using Δ^{(n)}, one can define an action of U_{v} on V^{⊗n}.

1.3. In [Ji], Jimbo constructed an action of H_{n} on V^{⊗n}, commuting with the action of U_{v}(𝔰𝔩_{m}). Let us fix integers m_{1}, …, m_{r} such that ∑ m_{i} = m, and consider a Levi subalgebra 𝔤 = 𝔰𝔩_{m_{1}} ⊕ ⋯ ⊕ 𝔰𝔩_{m_{r}} of 𝔰𝔩_{m}. The action of H_{n} was extended by [SS] to the action of H_{n,r} on V^{⊗n} so that it commutes with the action of the subalgebra U_{v}(𝔤) of U_{v}(𝔰𝔩_{m}). We consider the decomposition V = ⨁_{i} V_{i} with dim V_{i} = m_{i}. We assume that a basis {e_{j}^{(k)}} (1 ≤ j ≤ m_{k}) of V_{k} is chosen for k = 1, …, r, and that

e_{1}^{(1)}, …, e_{m_{1}}^{(1)}, e_{1}^{(2)}, …, e_{m_{2}}^{(2)}, …, e_{1}^{(r)}, …, e_{m_{r}}^{(r)}

gives the basis e_{1}, …, e_{m} of V in this order. The construction of the action of H_{n,r} on V^{⊗n} is given as follows. Let T be the element in End(V ⊗ V) defined by

T (e_{i} ⊗ e_{j}) = \begin{cases}ve_{j} ⊗ e_{i} & if i = j, \\ e_{j} ⊗ e_{i} & if i > j, \\ e_{j} ⊗ e_{i} + (v − v^{−1})e_{i} ⊗ e_{j} & if i < j. \end{cases} (1.3.1)

Next we define a map b : {1, 2, …, m} → ℕ by b(j) = k whenever e_{j} ∈ V_{k}. Let wt : V → V be a linear operator defined by wt(e_{j}) = u_{b(j)}e_{j}. Let us define linear operators, σ, S on V^{⊗2} as follows.

σ(e_{i} ⊗ e_{j}) = e_{j} ⊗ e_{i},

S(e_{i} ⊗ e_{j}) = \begin{cases}T (e_{i} ⊗ e_{j}) & if b(i) = b(j), \\ σ(e_{i} ⊗ e_{j}) & if b(i) ≠ b(j). \end{cases}

Using these operators, we define operators T_{i}, σ_{i}, S_{i}, ω_{j} ∈ End V^{⊗n}, (2 ≤ i ≤ n), (1 ≤ j ≤ n), by the condition,

T_{i} = id_{V}^{⊗(i−2)} ⊗T ⊗ id_{V}^{⊗(n−i)},
(1.3.2)  σ_{i} = id_{V}^{⊗(i−2)} ⊗σ ⊗ id_{V}^{⊗(n−i)},
S_{i} = id_{V}^{⊗(i−2)} ⊗S ⊗ id_{V}^{⊗(n−i)},
ω_{j} = id_{V}^{⊗(j−1)} ⊗wt ⊗ id_{V}^{⊗(n−j)} .

We now define an operator T_{1} on V^{⊗n} by

(1.3.3)  T_{1} = T_{2}^{−1} ⋯ T_{n}^{−1}S_{n} ⋯ S_{2}ω_{1}.

Then it is shown in [SS, Th.3.2] that τ : a_{i} ↦ T_{i} (1 ≤ i ≤ n) gives rise to a representation of H_{n,r} on V^{⊗n}.

Let^{−} : K → K be the unique ℚ-algebra involution such that v̄ = v^{−1}, ū_{i} = u_{i}^{−1} for i = 1, …, r. We say that a map ϕ on a K vector space X is antilinear if ϕ(λx) = λ̄ϕ(x) for λ ∈ K, x ∈ X. One can check by (1.1.1) that there exists a unique antilinear ℚ-algebra automorphism a ↦ ā on H_{n,r} such that ā_{i} = a_{i}^{−1} (1 ≤ i ≤ n). We call this map the bar involution on H_{n,r}.

1.4. Recall that H_{n,r} has an alternative presentation as given in [S, Th.3.7]. H_{n,r} is generated by {a_{2}, …, a_{n}, ξ_{1}, …, ξ_{n}}, subject to the following relations.

(1.4.1)  \begin{aligned}(a_{i} − v)(a_{i} + v^{−1}) & = 0 & (2 ≤ i ≤ n) \\ (ξ_{i} − u_{1}) ⋯ (ξ_{i} − u_{r}) & = 0 & (1 ≤ i ≤ n) \\ a_{i}a_{i+1}a_{i} & = a_{i+1}a_{i}a_{i+1} & (2 ≤ i ≤ n) \\ a_{i}a_{j} & = a_{j}a_{i} & (|i − j| ≥ 2) \\ ξ_{i}ξ_{j} & = ξ_{j}ξ_{i} & (1 ≤ i, j ≤ n), \end{aligned}

(1.4.2)  a_{j}ξ_{j} = ξ_{j−1}a_{j} + Δ^{−2} ∑_{c_{1}<c_{2}}(u_{c_{2}} − u_{c_{1}})(v − v^{−1})F_{c_{1}}(ξ_{j−1})F_{c_{2}}(ξ_{j}),
(1.4.3) a_{j}ξ_{j−1} = ξ_{j}a_{j} − Δ^{−2} ∑_{c_{1}<c_{2}}(u_{c_{2}} − u_{c_{1}})(v − v^{−1})F_{c_{1}}(ξ_{j−1})F_{c_{2}}(ξ_{j}),
(1.4.4)  a_{j}ξ_{k} = ξ_{k}a_{j} (k ≠ j − 1, j),

where Δ = ∏_{i>j}(u_{i} −u_{j}) is the Vandermonde determinant with respect to the parameters u_{1}, …, u_{r}, and the sum in (1.4.2) or (1.4.3) is taken for all integers 1 ≤ c_{1}, c_{2} ≤ r. For each integer 1 ≤ c ≤ r, F_{c}(X) is a certain polynomial in a variable X with coefficients in ℤ[u_{1}, …, u_{r}], defined in [S, 3.3.2]. Note that the generators a_{2}, …, a_{n} above may be identified with the generators appeared in 1.1.

Under the representation τ : H_{n,r} → End V^{⊗n}, the generator ξ_{i} is mapped to ω_{i} for each i.

§2. Involutions associated to H_{n,r} and U_{v}(𝔰𝔩_{m})

2.1 The operator T in (1.3.1) has its origin in the study of the universal Rmatrix Θ attached to U_{v}. Let E be the orthogonal complement of ∑ ε_{i} in the Euclidean space ℝ^{m} with the standard basis ε_{1}, …, ε_{m}. The root system Φ of 𝔰𝔩_{m} is given by the set {ε_{i} − ε_{j} | 1 ≤ i ≠ j ≤ m} with Φ^{+} = {ε_{i} − ε_{j} | i < j}. Thus the root lattice ℤΦ is given by the ℤ-submodule of E consisting of integral vectors. Let ( , ) be the inner product of ℝ^{m}. The weight lattice Λ is given by the set of λ ∈ E such that (λ, μ) ∈ ℤ for any μ ∈ Φ. For 1 ≤ i ≤ m, put ε̄_{i} = ε_{i} − \frac{1}{m} ∑_{j=1}^{m} ε_{j} ∈ Λ. Then ∑ ε̄_{i} = 0, and ε̄_{i} is a weight with weight vector e_{i}. The weight lattice Λ is identified with the set ℤ^{m}/ℤ(1, …, 1) by the correspondence λ = ∑ c_{i}ε̄_{i} ↔ (c_{1}, …, c_{m}). For a U_{v}-module M, we denote by M_{λ} the weight subspace of M corresponding to λ ∈ Λ.

Let U_{v}^{+} (resp. U_{v}^{−}) be the subalgebra of U_{v} generated by E_{i}, K_{i} (resp. F_{i}, K_{i}), respectively. For each μ ∈ ℤΦ, μ ≥ 0, we denote by U_{μ}^{+}, U_{−μ}^{−} the weight subspace of U_{v}^{±} with respect to μ or −μ, respectively. Then there exists an element Θ_{μ} ∈ U_{−μ}^{−} ⊗ U_{μ}^{+} with Θ_{0} = 1 ⊗ 1, for each μ, and Θ = ∑_{μ≥0} Θ_{μ} (an element in a completion of U_{v} ⊗ U_{v}, see [L, 4.1]) can be defined.

Let M and M′ be finite dimensional U_{v}-modules. We fix an m-th root v^{1/m} of v, and consider the extension field K(v^{1/m}) of K. (Accordingly, we regard U_{v} as the algebra over K(v^{1/m}) if needed). Following [Ja, 7.3, 7.9], we introduce a linear map C′ ∈ End M ⊗ M′ (f̃ in the notation of [Ja]). We define a map f : Λ × Λ → K(v^{1/m})^{∗} by

f(λ, μ) = (v^{1/m})^{−m(λ,μ)}

for all λ, μ ∈ Λ. Note that (λ, μ) ∈ \frac{1}{m}ℤ. In particular, we have

(2.1.1)  f(ε̄_{i}, ε̄_{j}) = v^{1/m−δ_{ij}}.

Now C′ is defined, for λ, μ ∈ Λ, by

C′(x ⊗ y) = f(λ, μ)x ⊗ y

for all x ∈ M_{λ} and y ∈ M_{μ}.

The element Θ induces a well-defined map Θ_{M,M′} ∈ End M ⊗ M′. It is known ([Ja, Th. 7.3]) that the map Θ_{M,M′}C′σ : M′ ⊗ M → M ⊗ M′ gives rise to an isomorphism of U_{v}-modules, where σ : M′ ⊗ M → M ⊗ M′ is the permutation of factors.

2.2 The bar involution on K can be extended obviously to an involution on K(v^{1/n}). The bar involution ^{−} on U_{v} is an antilinear ℚ-algebra automorphism on U_{v} defined on the generators by

\overline{E}_{i} = E_{i}, \overline{F}_{i} = F_{i}, \overline{K}_{i} = K_{i}^{−1}.

The bar involution is extended to U_{v} ⊗ U_{v} by \overline{x ⊗ y} = x̄ ⊗ ȳ. Let \overline{Θ} = ^{−} ∘ Θ ∘^{−} be the bar conjugate of Θ. Then it is known by [L, 4.1] that Θ\overline{Θ} = 1 ⊗ 1.

We consider the special case where M = M′ = V, and write Θ_{V,V} ∈ End V ⊗ V simply as Θ. Then as is well-known (cf. [FKK, Prop. 2.1]), we have

(2.2.1)  (ΘC′σ)^{−1} = v^{−1/m}T.

More precisely, the action of C′ and Θ = ∑ Θ_{μ} on V ⊗ V are described as follows. Since e_{i} ∈ V is a weight vector with weight ε̄_{i} ∈ Λ, by the property of Θ_{μ} (cf. [Ja, Chap. 7]), we have

(2.2.2) Θ_{μ}(e_{i} ⊗ e_{j}) = \begin{cases}(v^{−1} − v)e_{j} ⊗ e_{i} & if μ = ε_{i} − ε_{j} with i < j, \\ e_{i} ⊗ e_{j} & if μ = 0, \\ 0 & otherwise. \end{cases}

It follows that

(2.2.3)  Θ(e_{i} ⊗ e_{j}) = \begin{cases}e_{i} ⊗ e_{j} & if i ≥ j, \\ e_{i} ⊗ e_{j} − (v − v^{−1})e_{j} ⊗ e_{i} & if i < j. \end{cases}

Put C = v^{−1/m}C′. Then by (2.1.1), we have

(2.2.4)  C(e_{i} ⊗ e_{j}) = \begin{cases}e_{i} ⊗ e_{j} & i ≠ j, \\ v^{−1}e_{i} ⊗ e_{j} & i = j. \end{cases}

We define an antilinear involution ψ on V^{⊗n} inductively as follows: First define ψ on V by

ψ(∑ c_{i}e_{i}) = ∑ c̄_{i}e_{i}.

Next let W_{1}, W_{2} be tensor powers of V, and assume that the involutions ψ on W_{1}, W_{2} are already defined. We define ψ on W_{1} ⊗ W_{2} by

ψ(w_{1} ⊗ w_{2}) = Θ(ψ(w_{1}) ⊗ ψ(w_{2})).

Then it is shown in [L, Chap. 27] that ψ on V^{⊗n} does not depend on the decomposition V^{⊗n} = W_{1} ⊗ W_{2}, and it is compatible with the U_{v}-module structure of V^{⊗n} in the following sense: ψ(ux) = ūψ(x) for u ∈ U_{v}, x ∈ V^{⊗n}.

In [FKK], Frenkel, Khovanov and Kirillov studied the relationship between Kazhdan-Lusztig basis of H_{n} and canonical basis of U_{v} by making use of the H_{n} ⊗ U_{v}-module V^{⊗n}. In particular, they showed

Proposition 2.3 ([FKK, Prop. 2.4]). The bar involution of H_{n} is compatible with the involution ψ on V^{⊗n}, i.e., for any a ∈ H_{n}, we have

ψ ∘ a = \overline{a} ∘ ψ.

The main objective in this section is to extend this result to the case of H_{n,r}. We shall show that

Theorem 2.4. The bar involution on H_{n,r} is compatible with the involution ψ on V^{⊗n}, i.e., for any a ∈ H_{n,r}, we have

(2.4.1)  ψ ∘ a = \overline{a} ∘ ψ.

2.5. The remainder of this section is devoted to the proof of Theorem 2.4. We denote by e_{I} with I = (i_{1}, …, i_{n}) the vector e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}} of V^{⊗n}. Hence {e_{I} | I ∈ [1, m]^{n}} gives a basis of V^{⊗n}. (Here [1, m] means the set {1, 2, …, m}). The symmetric group 𝔖_{n} acts on [1, m]^{n} by permuting the factors, compatible with the action on V^{⊗n}, i.e. σ(e_{I}) = e_{σI} for σ ∈ 𝔖_{n}. If we denote by m_{I}(i) the multiplicity of i occurring in I = (i_{1}, …, i_{n}), then e_{I} is a weight vector of U_{v}-module V^{⊗n} with weight ∑_{i} m_{I}(i)ε̄_{i}.

We define an antilinear involution ^{−} on V^{⊗n} by x̄ = ∑_{I} c̄_{I}e_{I} for x = ∑_{I} c_{i}e_{I}. Let Ψ_{i} be a linear map on V^{⊗n} defined by

Ψ_{i} = (Δ^{(i−1)} ⊗ 1)Θ) ⊗ 1^{⊗(n−i)}.

Then it follows from the definition that the involution ψ can be expressed as

(2.5.1)  ψ = Ψ_{n}Ψ_{n−1} ⋯ Ψ_{2} ∘^{−}.

In order to describe the involution ψ, first we shall concentrate on the map Ψ_{n} = (Δ^{(n−1)} ⊗ 1)Θ. We prepare some notation. By z ↦ z_{ij}, we denote the embedding U_{v}^{⊗2} → U_{v}^{⊗n} subject to the i-th and j-th factors, i.e., for z = a ⊗ b ∈ U_{v} ⊗ U_{v}, we put

z_{ij} = x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n}

with x_{i} = a, x_{j} = b and x_{k} = 1 for k ≠ i, j.

For α, β ∈ ℤΦ with α ≥ β ≥ 0, we define X_{α,β}^{i} ∈ U_{v}^{⊗n} by

(2.5.2) X_{α,β}^{i} = (1^{⊗(n−i)} ⊗ K_{α}^{−1} ⊗ 1^{⊗(i−1)})(Θ_{α−β})_{n−i,n}  (2 ≤ i ≤ n − 1),

where K_{α} = ∏ K_{i}^{m_{i}} if α = ∑_{i} m_{i}(ε_{i} − ε_{i+1}). The following lemma is a generalization of [J, Lemma 7.4]. The proof is reduced to the case n = 3, which case follows from Lemma 7.4 in [J].

Lemma 2.6. For all μ ∈ ℤΦ with μ ≥ 0, we have

(Δ^{(n−1)} ⊗ 1)Θ_{μ} = ∑(Θ_{μ−ν_{1}})_{n−1,n}X_{ν_{1},ν_{2}}^{2} ⋯ X_{ν_{n−2},ν_{n−1}}^{n−1},

where the sum is taken over all the sequences μ ≥ ν_{1} ≥ ⋯ ≥ ν_{n−1} = 0 such that ν_{i} ∈ ℤΦ.

2.7. We shall describe Ψ_{n} = (Δ^{(n−1)} ⊗ 1)Θ. For I = (i_{1}, …, i_{n}), let (η_{1}, …, η_{n}) be the sequence defined by η_{k} = ε_{i_{k}}. Let us define a linear map Θ_{k,n}^{♯} on V^{⊗n}, for k = 1, …, n − 1, by

Θ_{k,n}^{♯}(e_{I}) = \begin{cases}e_{I} & if i_{k} ≥ i_{n}, \\ e_{I} + v^{−(η_{k}−η_{n},η_{k+1}+···+η_{n−1})}(v^{−1} − v)e_{I′} & if i_{k} < i_{n}, \end{cases}

where I′ = (k, n)I. (In the case where k = n − 1, we understand that the inner product in the second formula is equal to 0).

We have the following lemma.

Lemma 2.8. As operators on V^{⊗n}, we have

(2.8.1)  (Δ^{(n−1)} ⊗ 1)Θ = Θ_{n−1,n}^{♯}Θ_{n−2,n}^{♯} ⋯ Θ_{1,n}^{♯}.

Proof. First we compute X_{α,β}^{n−k}(e_{I}) for I = (i_{1}, …, i_{n}). By (2.2.2) and (2.5.2), we see that

(2.8.2) X_{α,β}^{n−k}(e_{I}) = \begin{cases}v^{−(α,η_{k+1})}(v^{−1} − v)e_{I′} & if α − β = η_{k} − η_{n} > 0, \\ v^{−(α,η_{k+1})}e_{I} & if α = β, \\ 0 & otherwise, \end{cases}

with I′ = (k, n)I.

Next we compute (Δ^{(n−1)} ⊗ 1)Θ(e_{I}). For a fixed I = (i_{1}, …, i_{n}), let P_{I} be the set of subsets p = {p_{1} < ⋯ < p_{k}} of {1, …, n − 1} such that i_{p_{k}} < ⋯ < i_{p_{2}} < i_{p_{1}} < i_{n}. We put k = |p|. For p ∈ P_{I}, let

I(p) = (p_{k}, n) ⋯ (p_{2}, n)(p_{1}, n)I = (n, p_{1}, p_{2}, …, p_{k})I.

Then it follows from Lemma 2.6 and (2.8.2) that

(Δ^{(n−1)} ⊗ 1)Θ(e_{I}) = ∑_{p∈P_{I}} v^{−c_{p}}(v^{−1} − v)^{|p|}e_{I(p)},

where

c_{p} = (η_{p_{1}} − η_{n}, η_{p_{1}+1} + ⋯ + η_{p_{2}})
+ (η_{p_{2}} − η_{n}, η_{p_{2}+1} + ⋯ + η_{p_{3}})
+ · · · · · ·
+ (η_{p_{k}} − η_{n}, η_{p_{k}+1} + ⋯ + η_{n−1}).

On the other hand, the right hand side of (2.8.1) is easily computed. We have

Θ_{n−1,n}^{♯}Θ_{n−2,n}^{♯} ⋯ Θ_{1,n}^{♯}(e_{I}) = ∑_{p∈P_{I}} v^{−d_{p}}(v^{−1} − v)^{|p|}e_{I(p)}

with

d_{p} = (η_{p_{1}} − η_{n}, η_{p_{1}+1} + ⋯ + η_{n−1})
+ (η_{p_{2}} − η_{p_{1}}, η_{p_{2}+1} + ⋯ + η_{n−1})
+ · · · · · ·
+ (η_{p_{k}} − η_{p_{k−1}}, η_{p_{k}+1} + ⋯ + η_{n−1}).

But then we have

d_{p} = ∑_{j=1}^{k}(η_{p_{j}}, η_{p_{j}+1} + ⋯ + η_{p_{j+1}}) − (η_{n}, η_{p_{1}+1} + ⋯ + η_{n−1}),

where we use the convention that p_{k+1} = n − 1. This implies that c_{p} = d_{p} for any p ∈ P_{I}, and the lemma follows.

2.9. For a fixed 1 ≤ i, j ≤ n with i ≠ j, we define an embedding End V^{⊗2} → End V^{⊗n}, x ↦ x_{ij}, in a similar way as in 2.5; x_{ij} denotes the transformation on V^{⊗n} which acts on i-th and j-th factors of V^{⊗n} via the map x, and acts trivially on other factors. Then it is easy to see for any σ ∈ 𝔖_{n} that

(2.9.1)  σx_{ij}σ^{−1} = x_{σ(i)σ(j)}.

In later discussions, we consider the operators Θ_{ij}, C_{ij}, T_{ij}, S_{ij} for Θ, C, T, S ∈ End V^{⊗2}, respectively. In particular, we note that T_{i−1,i} = T_{i} (resp. S_{i−1,i} = S_{i}) for i = 2, …, n in the notation of (1.3.2). We also note that Θ_{i−1,i} = Θ_{i−1,i}^{♯} by (2.2.3) and 2.7. However, Θ_{ij}^{♯} does not mean the embedding in general.

Let \overline{Θ}_{ij}^{♯} be the linear transformation on V^{⊗n} defined by \overline{Θ}_{ij}^{♯} = ^{−} ∘ Θ_{ij}^{♯} ∘^{−}. Then \overline{Θ}_{ij}^{♯} coincides with the map defined in 2.7, but by replacing v by v^{−1}. The bar operation on V^{⊗2} is compatible with the bar operation on Θ. It follows that \overline{Θ}_{ij} = ^{−} ∘ Θ_{ij} ∘^{−}. The following relations are easily verified.

− ∘ ω_{i} ∘^{−} = ω_{i}^{−1},
(2.9.2)  ^{−} ∘ C_{ij} ∘^{−} = C_{ij}^{−1},
− ∘ T_{ij} ∘^{−} = T_{ji}^{−1},
− ∘ S_{ij} ∘^{−} = S_{ji}^{−1}.

For a pair k, n such that 1 ≤ k ≤ n − 1, we put

D_{k,n} = σ_{k,n}C_{k,n}C_{k+1,n} ⋯ C_{n−1,n},

where σ_{k,n} denotes the cyclic permutation (k, k + 1, …, n). We have the following lemma.

Lemma 2.10. For 1 ≤ k ≤ n − 1, we have

(2.10.1)  \overline{Θ}_{k,n}^{♯}\overline{Θ}_{k,n−1}^{♯} ⋯ \overline{Θ}_{k,k+1}^{♯} = D_{k,n}T_{n}T_{n−1} ⋯ T_{k+1}.

Proof. First consider the case where k = n − 1. It follows from (2.2.1) that we have

(2.10.2)  T_{n} = (n − 1, n)C_{n−1,n}^{−1}\overline{Θ}_{n−1,n}^{♯}

since Θ\overline{Θ} = 1 and Θ_{n−1,n} = Θ_{n−1,n}^{♯}. Since D_{n−1,n} = (n − 1, n)C_{n−1,n} = C_{n−1,n}(n − 1, n), we have \overline{Θ}_{n−1,n}^{♯} = D_{n−1,n}T_{n} as asserted.

Next we show that

(2.10.3)  \overline{Θ}_{k,n}^{♯}D_{k,n−1} = D_{k,n}T_{n}  for 1 ≤ k ≤ n − 2.

By (2.9.1) we have

(n − 1, n)C_{k,n}C_{k+1,n} ⋯ C_{n−1,n}T_{n}
= C_{k,n−1}C_{k+1,n−1} ⋯ C_{n−2,n−1}C_{n−1,n}(n − 1, n)T_{n}
= C_{k,n−1}C_{k+1,n−1} ⋯ C_{n−2,n−1}\overline{Θ}_{n−1,n}^{♯}.

The last formula follows from (2.10.2). In order to show (2.10.3), we have only to check that

\overline{Θ}_{k,n}^{♯}σ_{k,n−1}C_{k,n−1} ⋯ C_{n−2,n−1} = σ_{k,n−1}C_{k,n−1} ⋯ C_{n−2,n−1}\overline{Θ}_{n−1,n}^{♯}.

It is easy to evaluate the maps on both sides at e_{I}. For e_{I} with I = (i_{1}, …, i_{n}), they have the common values

v^{−(η_{k}+η_{k+1}+···+η_{n−2},η_{n−1})}e_{I′}

if i_{n−1} ≥ i_{n}, and

v^{−(η_{k}+η_{k+1}+···+η_{n−2},η_{n−1})}e_{I′} + v^{−(η_{k}+η_{k+1}+···+η_{n−2},η_{n})}(v − v^{−1})e_{I′′}

if i_{n−1} < i_{n}, with I′ = σ_{k,n−1}I and I′′ = σ_{k,n}I. This proves (2.10.3).

Now the lemma is immediate by substituting \overline{Θ}_{k,i}^{♯} = D_{k,i}T_{i}D_{k,i−1}^{−1} for i ≥ k + 2 by (2.10.3), and \overline{Θ}_{k,k+1}^{♯} = D_{k,k+1}T_{k+1} into the left hand side of (2.10.1).

By using Lemma 2.10, we can describe the involution ψ as follows.

Proposition 2.11. Let σ_{0} = (1, n)(2, n − 1) ⋯ be the longest length element in 𝔖_{n}, and put

Ĉ = ∏_{1≤i<j≤n} C_{ij}.

(Note that the operators C_{ij} commute with each other). Then we have

ψ = ^{−} ∘ σ_{0}ĈT_{2}(T_{3}T_{2}) ⋯ (T_{n}T_{n−1} ⋯ T_{2}).

Proof. By (2.5.1) and Lemma 2.8, we have (cf. 2.9)

− ∘ ψ = ^{−} ∘ (Θ_{n−1,n}^{♯}Θ_{n−2,n}^{♯} ⋯ Θ_{1,n}^{♯})(Θ_{n−2,n−1}^{♯}Θ_{n−3,n−1}^{♯} ⋯ Θ_{1,n−1}^{♯}) ⋯ (Θ_{12}^{♯}) ∘^{−}
= (\overline{Θ}_{n−1,n}^{♯}\overline{Θ}_{n−2,n}^{♯} ⋯ \overline{Θ}_{1,n}^{♯})(\overline{Θ}_{n−2,n−1}^{♯}\overline{Θ}_{n−3,n−1}^{♯} ⋯ \overline{Θ}_{1,n−1}^{♯}) ⋯ (\overline{Θ}_{12}^{♯}).

It is clear that \overline{Θ}_{ij}^{♯} and \overline{Θ}_{i′j′}^{♯} commute with each other when \{i, j\}∩\{i′, j′\} = ∅. Hence we have

− ∘ ψ = (\overline{Θ}_{n−1,n}^{♯})(\overline{Θ}_{n−2,n}^{♯}\overline{Θ}_{n−2,n−1}^{♯}) ⋯ (\overline{Θ}_{1,n}^{♯}\overline{Θ}_{1,n−1}^{♯} ⋯ \overline{Θ}_{12}^{♯})
= (D_{n−1,n}T_{n})(D_{n−2,n}T_{n}T_{n−1}) ⋯ (D_{1,n}T_{n}T_{n−1} ⋯ T_{2}),

where the second equality follows from Lemma 2.10. By definition of D_{ij}, and by using (2.9.1), the last formula is modified to

(2.11.1)

− ∘ ψ = σ_{0}(C_{12}T_{2})(C_{13}C_{23}T_{3}T_{2}) ⋯ (C_{1,n}C_{2,n} ⋯ C_{n−1,n}T_{n}T_{n−1} ⋯ T_{2}).

Here we note that

(2.11.2) The product C_{1,k}C_{2,k} ⋯ C_{k−1,k} commutes with T_{2}, T_{3}, …, T_{k−1}.

In fact, (2.11.2) is reduced to showing that C_{a−1,k}C_{a,k} commutes with T_{a}, and this follows from the fact that C_{a−1,k}C_{a,k} acts on the subspace of V^{⊗n} generated by e_{I} and e_{(a−1,a)I} by a scalar multiplication v^{−(η_{a−1}+η_{a},η_{k})}.

Now, by using (2.11.2), (2.11.1) is further modified to

− ∘ ψ = σ_{0} · ∏_{i<j} C_{ij} · T_{2}(T_{3}T_{2}) ⋯ (T_{n} ⋯ T_{2}).

This proves the proposition.

2.12. We now proceed to the proof of Theorem 2.4. For the proof, it is enough to show (2.4.1) for the generators a_{1}, …, a_{n}. By Proposition 2.3, we know already that (2.4.1) holds for a_{2}, …, a_{n}. So, we have only to show it for a_{1}, i.e., to show that

(2.12.1)  ψT_{1} = T_{1}^{−1}ψ.

We shall show (2.12.1). Let Ĉ = ∏ C_{ij} be as in Proposition 2.11. First we note that

(2.12.2) Ĉ commutes with T_{ij}, S_{ij}, σ for any i ≠ j and any σ ∈ 𝔖_{n}.

In fact, let V_{I}^{⊗n} be the subspace of V^{⊗n} generated by {e_{σI} | σ ∈ 𝔖_{n}} for a fixed I = (i_{1}, …, i_{n}). Then Ĉ acts on V_{I}^{⊗n} as a scalar multiplication by v^{−c} with c = ∑_{i<j}(η_{i}, η_{j}). (2.12.2) follows from this.

By definition (1.3.3) and Proposition 2.11, we can write (2.12.3)

ψT_{1} = Zω_{1} with Z = ^{−} ∘ σ_{0}ĈT_{2}(T_{3}T_{2}) ⋯ (T_{n−1} ⋯ T_{2})S_{n}S_{n−1} ⋯ S_{2}.

We show that

(2.12.4)  Z = Z^{−1}.

In fact, by (2.9.2), (2.9.1) and (2.12.2), we have

Z^{−1} = ^{−} ∘ (S_{21}S_{32} ⋯ S_{n,n−1})(T_{21}T_{32} ⋯ T_{n−1,n−2}) ⋯ (T_{21}T_{32})T_{21}Ĉσ_{0}
= ^{−} ∘ σ_{0}Ĉ(S_{n}S_{n−1} ⋯ S_{2})(T_{n}T_{n−1} ⋯ T_{3}) ⋯ (T_{n}T_{n−1})T_{n}.

It is known by [SS, Lemma 3.8] that

(2.12.5)  (S_{n}S_{n−1} ⋯ S_{2})T_{j} = T_{j−1}(S_{n}S_{n−1} ⋯ S_{2})

for j = 3, …, n. Therefore we have

Z^{−1} = ^{−} ∘ σ_{0}Ĉ(T_{n−1} ⋯ T_{2}) ⋯ (T_{n−1}T_{n−2})T_{n−1}(S_{n} ⋯ S_{2}).

Now by using the relations

(T_{n−1}T_{n−2} ⋯ T_{n−a})T_{i} = T_{i−1}(T_{n−1}T_{n−2} ⋯ T_{n−a})

for i ≥ n − a + 1, which follows from the braid relations of H_{n}, it is easy to see that

(T_{n−1} ⋯ T_{2}) ⋯ (T_{n−1}T_{n−2})T_{n−1} = T_{2}(T_{3}T_{2}) ⋯ (T_{n−1} ⋯ T_{2}).

Hence (2.12.4) holds.

Since ψ is an involution, by using (2.12.4), we have

T_{1}^{−1}ψ = (ψT_{1})^{−1} = ω_{1}^{−1}Z^{−1} = ω_{1}^{−1}Z.

Hence to prove (2.12.1), it is enough to show that ω_{1}^{−1}Z = Zω_{1}. Note that σ_{0}ω_{1}σ_{0}^{−1} = ω_{n} and that ω_{n} commutes with Ĉ and T_{2}, …, T_{n−1}. Thus, by (2.9.2), we have

(2.12.6)  ω_{1}^{−1}Z = ^{−} ∘ σ_{0}ĈT_{2}(T_{3}T_{2}) ⋯ (T_{n−1} ⋯ T_{2})ω_{n}(S_{n} ⋯ S_{2}).

Here we note the following formula.

(2.12.7)  ω_{i}S_{i} = S_{i}ω_{i−1}  for i = 2, …, n.

In fact, it is enough to see the formula for the case where n = i = 2, and S_{i} = S. Now ω_{2} and ω_{1} act as a (common) scalar multiplication on e_{j} ⊗ e_{k} and e_{k} ⊗ e_{j} if b(j) = b(k). S permutes e_{j} ⊗ e_{k} and e_{k} ⊗ e_{j} if b(j) ≠ b(k). (2.12.7) follows easily from these facts.

Now by applying (2.12.7), we have ω_{n}(S_{n} ⋯ S_{2}) = (S_{n} ⋯ S_{2})ω_{1}. Hence (2.12.6) implies that ω_{1}^{−1}Z = Zω_{1}, and (2.12.1) holds. The theorem is proved.

2.13. By making use of Theorem 2.4, combined with Proposition 2.11, one can describe the bar involution for generators {a_{2}, …, a_{n}, ξ_{1}, …, ξ_{n}} of H_{n,r} given in 1.4.

Proposition 2.14. Let {a_{2}, …, a_{n}, ξ_{1}, …, ξ_{n}} be the generators of H_{n,r} given in 1.4, and put x = a_{2}(a_{3}a_{2}) ⋯ (a_{n}a_{n−1} ⋯ a_{2}). Then we have

\begin{aligned}ā_{i} & = a_{i}^{−1} & (2 ≤ i ≤ n), \\ ξ̄_{j} & = x^{−1}ξ_{n−j+1}^{−1}x & (1 ≤ j ≤ n). \end{aligned}

Proof. It is enough to show the formula for ξ_{j}. By Theorem 2.4, we have

\overline{ω}_{j} = ψ^{−1} ∘ ω_{j} ∘ ψ.

Note that ^{−} ∘ ω_{j} ∘^{−} = ω_{j}^{−1}, σ_{0}ω_{j}σ_{0} = ω_{n−j+1}, and that ω_{j} commutes with Ĉ. Then by Proposition 2.11, we see that

ψ^{−1} ∘ ω_{j} ∘ ψ = X^{−1}ω_{n−j+1}^{−1}X

with X = T_{2}(T_{3}T_{2}) ⋯ (T_{n}T_{n−1} ⋯ T_{2}). Since the representation τ is faithful if we choose m_{k} ≥ n for k = 1, …, r, and since τ(ξ_{j}) = ω_{j}, τ(x) = X, this gives the required formula for ξ_{j}.

§3. Kazhdan-Lusztig basis and Canonical basis

3.1. Let W be a Weyl group with a set of generators S. We denote by H the Hecke algebra associated to W. It is an associative algebra over ℚ(v) defined by generators a_{s} (s ∈ S) and relations

(3.1.1)  (a_{s} − v)(a_{s} + v^{−1}) = 0

together with usual braid relations. H has a basis {a_{w} | w ∈ W }, where a_{w} = a_{s_{1}} ⋯ a_{s_{q}} for a reduced expression w = s_{1} ⋯ s_{q}.

For any subset J of S, we denote by W_{J} the parabolic subgroup in W generated by s ∈ J. Let W^{J} be the set of distinguished representatives in W/W_{J}. Hence W^{J} is the set of minimal elements w in wW_{J} with respect to the length l(w) of W. Let H_{J} be the the subalgebra of H generated by a_{s} with s ∈ J. Then H_{J} is isomorphic to the Hecke algebra of W_{J}. Let φ be a homomorphism from H to ℚ(v) defined by a_{s} ↦ v for any s ∈ S. We denote by φ_{J} the restriction of φ on H_{J}. Let M_{J} be the induced H-module Ind_{H_{J}}^{H} φ_{J}. Then by Deodhar [D], it is known that M_{J} has a basis {m_{w} | w ∈ W^{J}} with the following properties,

(3.1.2)  a_{s}m_{w} = \begin{cases}m_{sw} + (v − v^{−1})m_{w} & if l(sw) < l(w), \\ m_{sw} & if l(sw) > l(w), sw ∈ W^{J}, \\ vm_{w} & if l(sw) > l(w), sw ∉ W^{J}, \end{cases}

and a_{w}m_{1} = m_{w} for an identity element 1 ∈ W. Note that w ∈ W^{J} and l(sw) < l(w) imply that sw ∈ W^{J}.

Let us define a bar involution on H by \overline{v} = v^{−1} and \overline{a}_{s} = a_{s}^{−1} as in 1.3. We also define a bar involution on M_{J} by the condition that \overline{m}_{e} = m_{e} and that \overline{hm} = \overline{h}\overline{m} for h ∈ H, m ∈ M_{J}. Let ≤ be the partial order on W^{J} induced from the Bruhat order on W. The Kazhdan-Lusztig basis {C_{w}^{J} | w ∈ W^{J}} of M_{J} was introduced by Kazhdan-Lusztig [KL] for M_{∅} ≃ H, and then extended by Deodhar [D] for the case M_{J}. They are characterized by the following two properties.

(3.1.3)  C_{w}^{J} ∈ m_{w} + ∑_{\begin{matrix}x∈W^{J} \\ x<w \end{matrix}} v^{−1}ℤ[v^{−1}]m_{x}

(3.1.4)  \overline{C}_{w}^{J} = C_{w}^{J}.

The parabolic Kazhdan-Lusztig polynomial P_{x,w}^{J} ∈ ℤ[q] is defined, following Deodhar, in terms of the coefficient p_{x,w} of m_{x} in the expression of C_{w}^{J} as follows.

p_{x,w} = q^{l(w)−l(x))/2}\overline{P_{x,w}^{J}}(q) with v = q^{−1/2}.

Note that in [KL], [D], H is defined by the quadratic relation (T_{s}−q)(T_{s}+1) for an indeterminate q instead of (3.1.1). Then the relationship with our situation is given as follows; v = q^{−1/2}, and our a_{s} corresponds to −vT_{s} in their setup. In particular, our C_{σ}^{J} corresponds to (−1)^{l(σ)}C_{σ}^{J} under the notation of [KL], [D], and our m_{w} corresponds to (−v)^{l(w)}m_{w} of [D].

3.2. For x, w ∈ W^{J} such that x < w, we denote by μ(x, w) the coefficient of q^{(l(w)−l(x)−1)/2} in P_{x,w}^{J}(q). Note that deg P_{x,w}^{J} ≤ \frac{1}{2}(l(w)−l(x)−1). Let s ∈ S be such that l(sw) < l(w) for w ∈ W^{J}. Then C_{w}^{J} is determined inductively, with respect to the Bruhat order, by

(3.2.1)  C_{w}^{J} = (a_{s} + v^{−1})C_{sw}^{J} − ∑_{\begin{matrix}y∈W^{J} ,y≤sw \\ sy<y or sy∉W^{J} \end{matrix}}(−1)^{l(w)−l(y)}μ(y, sw)C_{y}^{J}

Now the action of a_{s} on C_{w}^{J} is given as follows; for s ∈ S and w ∈ W^{J},

(3.2.2) a_{s}C_{w}^{J} = \begin{cases}−v^{−1}C_{w}^{J} + C_{sw}^{J} − ∑_{\begin{matrix}y<w \\ sy≤y or sy∉W^{J} \end{matrix}} (−1)^{l(w)−l(x)}μ(y, w)C_{y}^{J}, \\ vC_{w}^{J}, \end{cases}

where the first equality occurs when sw > w and sw ∈ W^{J}, and the second occurs when sw < w or sw ∉ W^{J}. In fact, (3.2.2) can be shown as in a similar way in [KL, 2.3] once we know that

(3.2.3)  a_{s}C_{w}^{J} = vC_{w}^{J} if sw > w and sw ∉ W^{J}.

We show (3.2.3). By comparing the coefficients of m_{w} on both sides, we see that (3.2.3) is equivalent to the identities,

(3.2.4)  P_{sx,w}^{J} = P_{x,w}^{J} if sx ∈ W^{J}.

(Note that sx ∈ W^{J} if sx < x and x ∈ W^{J}). Since we are in a setting u = −1 in the notation of [D], we have by [D, Prop. 3.4],

(3.2.5)  P_{x,w}^{J} = P_{xw_{J} ,ww_{J}}

for x, w ∈ W^{J}, where w_{J} is the longest element in W^{J}, and P_{x,y} is the original Kazhdan-Lusztig polynomial for W. By our assumption, sw > w and sw ∉ W^{J}. Then there exists s′ ∈ W_{J} such that sw = ws′ (e.g., [H, Lemma 7.2]), and we have sww_{J} < ww_{J}. It follows that a_{s}C_{ww_{J}} = vC_{ww_{J}} by [KL, 2.3] (C_{w} is a Kazhdan-Lusztig basis for H). This induces similar identities for P_{xw_{J} ,ww_{J}} and P_{sxw_{J} ,ww_{J}} as in (3.2.4). Now (3.2.4) follows from these identities in view of (3.2.5).

3.3. We assume that W ≃ 𝔖_{n}, and consider the U_{v} ⊗ H_{n}-module V^{⊗n}. The weights of U_{v} on V^{⊗n} are given by λ = (λ_{1}, ⋯, λ_{m}) ∈ ℤ_{≥0}^{m} with ∑ λ_{i} = n. The weight subspace V_{λ}^{⊗n} has a basis {e_{I}}, with I = (i_{1}, …, i_{n}) such that ♯{a | i_{a} = k} = λ_{k}. The involution ψ on V^{⊗n} stabilizes the subspace V_{λ}^{⊗n}. Moreover V_{λ}^{⊗n} is an H_{n}-submodule of V^{⊗n} generated by a single element e_{I_{λ}}, where

(3.3.1)  I_{λ} = (\underbrace{m, …, m}_{λ_{m}−times}, …, \underbrace{1, …, 1}_{λ_{1}−times}).

Let 𝔖_{λ} ≃ 𝔖_{λ_{m}} × ⋯ × 𝔖_{λ_{1}} be the stabilizer of I_{λ} in 𝔖_{n}. Then 𝔖_{λ} is a parabolic subgroup W_{J} of 𝔖_{n}, and we denote by H_{λ} the parabolic subalgebra H_{J} corresponding to 𝔖_{λ}. It is easy to see that H_{n}-module V_{λ}^{⊗n} is isomorphic to M_{J} = Ind_{H_{λ}}^{H_{n}} φ.

Recall that {e_{I} | I ∈ [1, m]^{n}} is the basis of V^{⊗n}, which we call the standard basis of V^{⊗n}. The canonical basis {b_{I} | I ∈ [1, m]^{n}} of U_{v}-module V^{⊗n} is characterized by the following two properties ([L, Chap. 27]).

(3.3.2)  b_{I} ∈ e_{I} + ∑_{I′} v^{−1}ℤ[v^{−1}]e_{I′},
ψ(b_{I}) = b_{I},

where the sum in the first formula is taken over all I′ having the same weight as I.

It is shown in [FKK] that the map f : m_{σ} ↦ e_{σ(I_{λ})} gives an isomorphism M_{J} ≃ V_{λ}^{⊗n}, which transfers the bar involution on M_{J} to the involution ψ on V_{λ}^{⊗n}. We identify M_{J} with V_{λ}^{⊗n}.

We define a partial order I < I′ on [1, m]^{n} as the transitive closure of the relation

(…, a, …, b, …) < (…, b, …, a, …) if a > b.

Then we have the following.

Lemma 3.4. Let σ, τ ∈ 𝔖_{n}^{J}, and assume that σ < τ. Then we have σ(I_{λ}) < τ(I_{λ}).

Proof. The proof is reduced to the case where τ = σs with a (not necessarily simple) reflection s ∈ 𝔖_{n}. So we assume that s is a transvection (p, q) with 1 ≤ p < q ≤ n. Then it is easy to check that σ^{−1}(p) < σ^{−1}(q) if l(σ^{−1}) < l(sσ^{−1}). If we write I_{λ} = (i_{1}, …, i_{n}), we have

σ(I_{λ}) = (…, i_{σ^{−1}(p)}, …, i_{σ^{−1}(q)}, …), σs(I_{λ}) = (…, i_{σ^{−1}(q)}, …, i_{σ^{−1}(p)}, …).

Now by (3.3.1) and by our assumption, we have i_{σ^{−1}(q)} ≤ i_{σ^{−1}(p)}. The lemma follows from this.

The following special case is worth mentioning.

Lemma 3.5 ([FKK, Lemma 2.1]). Let σ ∈ W^{J} and s ∈ S be a transvection (i, i + 1). Let a and b be i-th and (i + 1)-th entries of σ(I_{λ}), respectively. Then we have

if a > b, then sσ > σ and sσ ∈ W^{J},
if a = b, then sσ > σ and sσ ∉ W^{J},
if a < b, then sσ < σ.

The following result shows that the Kazhdan-Lusztig basis is obtained as a special case of the canonical basis of V^{⊗n}.

Theorem 3.6 ([FKK, Th. 2.5]). Assume that W ≃ 𝔖_{n}. Then, under the identification M_{J} ≃ V_{λ}^{⊗n}, we have for each σ ∈ W^{J},

C_{σ}^{J} = b_{σ(I_{λ})}.

Combining Theorem 3.6 with Lemma 3.4, we have the following refinement of (3.3.2).

Corollary 3.7. Under the above notation, we have

b_{I} ∈ e_{I} + ∑_{I′<I} v^{−1}ℤ[v^{−1}]e_{I′}.

The following corollary is also immediate from (3.2.2), Lemma 3.5 and Theorem 3.6.

Corollary 3.8. Let s = (a, a+1) be a transvection. Then for I = (i_{1}, …, i_{n}), we have

T_{s}b_{I} = vb_{I} if i_{a} ≤ i_{a+1}.

§4. H_{n,r}-submodules of V^{⊗n}

4.1. We now return to the setup in section 1, and assume that W = W_{n,r}. We consider the H_{n,r}-module V^{⊗n} with the graded vector space V = ⨁_{i=1}^{r} V_{i} as before. We prepare some notation in addition to 3.1. Let S = {s_{1}, …, s_{n}} be the set of generators of W_{n,r}, where t_{1} = s_{1} has order r, and s_{2}, …, s_{n} are generators of 𝔖_{n} corresponding to transvections (1, 2), …, (n − 1, n). We define t_{i} ∈ W_{n,r} by t_{i} = s_{i} ⋯ s_{2}t_{1}s_{2} ⋯ s_{i} for i = 2, …, n. Then t_{1}, …, t_{n} gives rise to a set of generators of the group (ℤ/rℤ)^{n}.

The basis vector of V^{⊗n} is given by e_{I} with I = (i_{1}, …, i_{n}) as before. By 1.3, e_{I} can also be written as e_{j_{1}}^{(ε_{1})} ⊗ ⋯ ⊗ e_{j_{n}}^{(ε_{n})}. In this case, we write I as I = (j_{1}^{(ε_{1})}, …, j_{n}^{(ε_{n})}). The weight λ of U_{v} on V^{⊗n} is expressed as λ = (λ_{1}, …, λ_{m}) as in 3.3. In our situation, I determines a multi-composition λ = (λ^{(1)}, …, λ^{(r)}), with λ^{(k)} = (λ_{1}^{(k)}, …, λ_{m_{k}}^{(k)}) ∈ ℤ_{≥0}^{m_{k}}, such that ∑_{j,k} λ_{j}^{(k)} = n; the correspondence is given by λ_{j}^{(k)} = ♯{a | j_{a} = j, ε_{a} = k}. If one ignores the superscripts of λ_{j}^{(k)}, λ reduces to λ. We call λ the weight of e_{I}. We denote by V_{λ}^{⊗n} the subspace of V^{⊗n} generated by e_{I} whose weight is λ. It is easy to check that the action of H_{n,r} on V^{⊗n} stabilizes the subspace V_{λ}^{⊗n}. For the weight λ, put

(4.1.1)  e_{λ^{(k)}} = \underbrace{e_{m_{k}}^{(k)} ⊗ ⋯ ⊗ e_{m_{k}}^{(k)}}_{λ_{m_{k}}^{(k)} −times} ⊗ ⋯ ⊗ \underbrace{e_{1}^{(k)} ⊗ ⋯ ⊗ e_{1}^{(k)}}_{λ_{1}^{(k)}−times}

and define a vector e_{λ} ∈ V_{λ}^{⊗n} by

(4.1.2)  e_{λ} = e_{λ^{(r)}} ⊗ e_{λ^{(r−1)}} ⊗ ⋯ ⊗ e_{λ^{(1)}}.

The stabilizer of e_{λ} in 𝔖_{n} is isomorphic to

𝔖_{λ} = 𝔖_{λ^{(r)}} × 𝔖_{λ^{(r−1)}} × ⋯ × 𝔖_{λ^{(1)}}

with 𝔖_{λ^{(k)}} = 𝔖_{λ_{m_{k}}^{(k)}} × ⋯ × 𝔖_{λ_{1}^{(k)}}. We define a subgroup W_{λ} of W_{n,r} by W_{λ} = 𝔖_{λ} ⋉ (ℤ/rℤ)^{n}, i.e.,

W_{λ} ≃ W_{λ^{(r)}} × W_{λ^{(r−1)}} × ⋯ × W_{λ^{(1)}}

with W_{λ^{(k)}} = W_{λ_{m_{k}}^{(k)} ,r} × ⋯ × W_{λ_{1}^{(k)},r}. Let H_{λ} be the Ariki-Koike algebra associated to W_{λ}, i.e.,

(4.1.3)  H_{λ} = H_{λ^{(r)}} ⊗ H_{λ^{(r−1)}} ⊗ ⋯ ⊗ H_{λ^{(1)}}

with H_{λ^{(k)}} = H_{λ_{m_{k}}^{(k)} ,r} ⊗ ⋯ ⊗ H_{λ_{1}^{(k)},r}. One can regard H_{λ} as a subalgebra of H_{n,r} in a natural way, by making use of generators {ξ_{1}, …, ξ_{n}} as discussed in [S, 4.2]. (For example, if a + b = n, then H_{a,r} ⊗ H_{b,r} ↪ H_{n,r}, where H_{a,r} (resp. H_{b,r}) is the subalgebra of H_{n,r} generated by s_{2}, …, s_{a}, ξ_{1}, …, ξ_{a}, (resp. s_{a+2}, …, s_{n}, ξ_{a+1}, …, ξ_{n}), respectively.)

We can define a linear character φ_{n}^{(k)} : H_{n,r} → K by
φ_{n}^{(k)}(a_{i}) = v  (2 ≤ i ≤ n),
φ_{n}^{(k)}(ξ_{j}) = u_{k}  (1 ≤ j ≤ n),

and define φ_{λ^{(k)}} : H_{λ^{(k)}} → K by φ_{λ^{(k)}} = φ_{λ_{m_{k}}^{(k)}}^{(k)} ⊗ ⋯ ⊗ φ_{λ_{1}^{(k)}}^{(k)}. Then we define a

linear character φ_{λ} : H_{λ} → K by

(4.1.4)  φ_{λ} = φ_{λ^{(r)}} ⊗ φ_{λ^{(r−1)}} ⊗ ⋯ ⊗ φ_{λ^{(1)}}

according to the embedding into H_{n,r} given in (4.1.3). Put V_{λ}^{⊗n} = M_{λ}. Then we have the following result.

Proposition 4.2. Let the notations be as above.

(i) M_{λ} is generated by e_{λ} as H_{n,r}-module, and we have

M_{λ} = H_{n,r}e_{λ} ≃ Ind_{H_{λ}}^{H_{n,r}} φ_{λ}

as H_{n,r}-modules.

(ii) M_{λ} has a basis {e_{σ}} indexed by the set 𝔖_{n}^{J} (here we regard 𝔖_{λ} as a parabolic subgroup (𝔖_{n})_{J} of 𝔖_{n}). The action of H_{n,r} on this basis is given as follows:

a_{s}e_{σ} = \begin{cases}e_{sσ} + (v − v^{−1})e_{σ} & if l(sσ) < l(σ), \\ e_{sσ} & if l(sσ) > l(σ), sσ ∈ 𝔖_{n}^{J}, \\ ve_{σ} & if l(sσ) > l(σ), sσ ∉ 𝔖_{n}^{J}, \end{cases}

ξ_{j}e_{σ} = u_{ε(j,σ)}e_{σ},

where ε(j, σ) ∈ {1, …, r} is given as follows; write e_{λ} = e_{I} as in 4.1, and put ε(j, σ) = ε_{j} for σ(I) = (j_{1}^{(ε_{1})}, …, j_{n}^{(ε_{n})}).

(iii) There exists an involution ^{−} : M_{λ} → M_{λ} satisfying the property that \overline{hm} = h̄\overline{m} for h ∈ H_{n,r}, m ∈ M_{λ}, and that ē_{σ} = e_{σ} for σ = 1.

Proof. Let λ be the weight of e_{λ} as U_{v}-module. Then M_{λ} coincides with V_{λ}^{⊗n} and e_{λ} is nothing but e_{I_{λ}} given in (3.3.1). Then by [FKK, Prop. 2.1], V_{λ}^{⊗n} is generated by e_{I_{λ}} as H_{n}-module, and is isomorphic to M_{J} as in 3.3, for a parabolic subgroup 𝔖_{λ} = 𝔖_{n}^{J}. In particular, we see that M_{λ} = H_{n,r}e_{λ}, and that

dim V_{λ}^{⊗n} = |𝔖_{n}^{J}| = dim Ind_{H_{λ}}^{H_{n,r}} φ_{λ}.

Since it is easy to see that Ke_{λ} is a one-dimensional H_{λ}-module affording φ_{λ}, the first assertion follows.

Now we define a basis {e_{σ} | σ ∈ 𝔖_{n}^{J}} in M_{λ} by using the basis {m_{σ}} in M_{J}. Then we have e_{σ} = m_{σ} = e_{σ(I_{λ})} by [FKK]. The first three formula in (ii) now follows from (3.1.2). The last formula in (ii) follows by considering the action of ω_{j} on e_{σ(I_{λ})} ∈ V^{⊗n}.

The involution ψ on V^{⊗n} stabilizes the subspace V_{λ}^{⊗n}. We define the bar involution^{−} on M_{λ} = V_{λ}^{⊗n} in terms of ψ. Then we have \overline{hm} = h̄\overline{m} by Theorem 2.4. Since ψ(e_{λ}) = e_{λ}, we have ē_{1} = e_{1}. The proposition is proved.

The following result is an analogue to the case of H_{n,r} of the result of Frenkel, Khovanov and Kirillov (cf. Theorem 3.6) concerning the Kazhdan-Lusztig basis of H_{n} and canonical basis of U_{q}, and also of the parabolic Kazhdan-Lusztig basis of Deodhar (cf. 3.1). But note that H_{λ} is no longer a parabolic subalgebra of H_{n,r}.

Theorem 4.3. Let M_{λ} ≃ Ind_{H_{λ}}^{H_{n,r}} φ_{λ} be the induced H_{n,r}-module. Then there exists a unique basis {b_{σ} | σ ∈ 𝔖_{n}^{J}} in M_{λ} satisfying the following properties.

b_{σ} ∈ e_{σ} + ∑_{\begin{matrix}τ∈𝔖_{n}^{J} \\ τ<σ \end{matrix}} v^{−1}ℤ[v^{−1}]e_{τ} ,

\overline{b}_{σ} = b_{σ}.

The coefficient p_{τ,σ} of e_{τ} in the expression of b_{σ} is given by the parabolic Kazhdan-Lusztig polynomial for the case of 𝔖_{n}^{J} ⊂ 𝔖_{n} just as in 3.1.

Proof. By Theorem 3.6, canonical basis {b_{σ(I_{λ})} | σ ∈ 𝔖_{n}^{J}} gives rise to a basis of V_{λ}^{⊗n}, which corresponds to the parabolic Kazhdan-Lusztig basis {C_{σ}^{J}} in M_{J}. Hence, if we define the basis {b_{σ}} in M_{λ} = V_{λ}^{⊗n} in terms of {b_{σ(I_{λ})}}, the assertions in the theorem follow from 3.1 and Proposition 4.2.

4.4. We now pass to a more general situation. Take an integer t ≥ 0 such that t ≤ m_{k} for k = 1, …, r. Let λ = (λ^{(1)}, …, λ^{(r)}) be an r-tuple of compositions as in 4.1, but here we assume that λ^{(k)} = ∅ for k ≠ r, and that λ^{(r)} = (λ_{t+1}^{(r)}, …, λ_{m_{r}}^{(r)}) ∈ ℤ_{≥0}^{m_{r}−t}. We consider a pair (λ; c), with c = (c_{1}, …, c_{t}) ∈ ℤ_{>0}^{t} a composition such that ∑_{j,k} λ_{j}^{(k)} + ∑_{i} c_{i} = n. We put c = ∑ c_{i}. We denote by M_{λ,c} the subspace of V^{⊗n} generated by e_{I} with I = (j_{1}^{(ε_{1})}, …, j_{n}^{(ε_{n})}) such that λ_{j}^{(r)} = ♯{a | j_{a} = j, ε_{a} = r} and that c_{i} = ♯{a | j_{a} = i}. Then M_{λ,c} is a direct sum of various weight spaces V_{ν}^{⊗n}, and so has a structure of H_{n,r}-module. The decomposition of M_{λ,c} into V_{ν}^{⊗n} is described more precisely as follows. Let ν be a pair (λ; μ), where λ is as above, and μ = (μ^{(1)}, …, μ^{(r)}) is an r-tuples of compositions μ^{(k)} = (μ_{1}^{(k)}, …, μ_{t}^{(k)}) ∈ ℤ_{≥0}^{t} such that ∑_{k=1}^{r} μ_{i}^{(k)} = c_{i} for 1 ≤ i ≤ t. We denote by P_{λ,c} the set of such ν = (λ; μ). Note that ν can be written, by rearranging the entries, as (ν^{(1)}, …, ν^{(r)}) with ν^{(r)} = (μ_{1}^{(r)}, …, μ_{t}^{(r)}, λ_{t+1}^{(r)}, …, λ_{m_{r}}^{(r)}), and ν^{(k)} = (μ_{1}^{(k)}, …, μ_{t}^{(k)}, 0, …, 0) for k ≠ r. Hence it determines an H_{n,r}- subspace V_{ν}^{⊗n}. It is easy to check that

M_{λ,c} = ⨁_{ν∈P_{λ,c}} V_{ν}^{⊗n}.

We shall investigate the H_{n,r}-module structure of M_{λ,c}. For each ν ∈ P_{λ,c}, we define e^{ν} ∈ V_{ν}^{⊗n} by e^{ν} = e_{λ} ⊗ e^{μ}, where e_{λ} = e_{λ^{(r)}} is defined just as in (4.1.1), by restricting the factors in between e_{m_{r}}^{(r)} and e_{t+1}^{(r)}. e^{μ} ∈ V^{⊗c} is defined by e^{μ} = E_{1} ⊗ E_{2} ⊗ ⋯ ⊗ E_{t}, with

(4.4.1)  E_{i} = (e_{t−i+1}^{(1)})^{μ_{i}^{(1)}} ⊗ ⋯ ⊗ (e_{t−i+1}^{(r)})^{μ_{i}^{(r)}} ∈ V^{⊗c_{i}}.

Now e^{ν} can be written as e^{ν} = e_{I} for some I, and we denote by b^{ν} the canonical basis b_{I} ∈ V_{ν}^{⊗n} corresponding to e_{I}. We define m_{λ,c} ∈ M_{λ,c} by

(4.4.2)  m_{λ,c} = ∑_{ν∈P_{λ,c}} b^{ν}.

We define a subalgebra H_{λ,c} of H_{n,r} by H_{λ,c} = H_{λ}⊗H_{c}, where H_{λ} = H_{λ^{(r)}} is defined as in (4.1.3), by modifying the definition of H_{λ^{(r)}} appropriately, and H_{c} is defined by

(4.4.3)  H_{c} = H_{c_{1}} ⊗ ⋯ ⊗ H_{c_{t}}.

(Remember that H_{i} is the Iwahori-Hecke algebra of type A_{i−1}). We define a linear character φ_{λ,c} of H_{λ,c} by φ_{λ,c} = φ_{λ} ⊗ φ_{c}, where φ_{λ} = φ_{λ^{(r)}} is given as in (4.1.4). φ_{c} is given by φ_{c} = φ_{c_{1}} ⊗ ⋯ ⊗ φ_{c_{t}}, where φ_{n} is the linear character of H_{n} defined by φ_{n}(a_{j}) = v for all generators a_{j}.

Under these notations, we have the following result.

Proposition 4.5. M_{λ,c} is generated by m_{λ,c} as H_{n,r}-module, and we have

M_{λ,c} = H_{n,r}m_{λ,c} ≃ Ind_{H_{λ,c}}^{H_{n,r}} φ_{λ,c}, ψ(m_{λ,c}) = m_{λ,c}.

Proof. It is clear that m_{λ,c} is fixed by ψ. We show the first two equalities. First we note that

(4.5.1)  hm_{λ,c} = φ_{λ,c}(h)m_{λ,c} for h ∈ H_{λ,c}.

In fact to show (4.5.1), it is enough to see, for each ν ∈ P_{λ,c}, that

(4.5.2)  hb^{ν} = \begin{cases}φ_{λ}(h)b^{ν} & if h ∈ H_{λ}, \\ φ_{c}(h)b^{ν} & if h ∈ H_{c}. \end{cases}

We show (4.5.2). In view of (4.4.1) and Corollary 3.8, we see that a_{j}b^{ν} = vb^{ν} for all generators a_{j} ∈ H_{c_{i}}. This implies the second equality in (4.5.2).

Next we consider the first equality. By (modified form of) (4.1.1), (4.1.2), together with (4.4.1), we see that e^{ν} = e_{I}, where I is of the form I = (i_{1}, …, i_{n}) with i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{n−c}, and with i_{n−c} > i_{k} for all c + 1 ≤ k ≤ n. Then by Corollary 3.7, b^{ν} is written as a linear combination of e_{I′}, where e_{I′} is of the form e_{λ}⊗ e^{μ′}, for some e^{μ′} ∈ V^{⊗c}. Then as in the case of Proposition 4.2, one can check that he_{I′} = φ_{λ}(h)e_{I′} for h ∈ H_{λ}. The first equality follows from this, and so (4.5.2) holds.

Next we show that

(4.5.3)  M_{λ,c} = H_{n,r}m_{λ,c}.

Let ζ be a primitive r-th root of unity. By the specialization v ↦ 1, u_{i} ↦ ζ^{i}, H_{n,r} turns out to be the group algebra ℂW_{n,r}. (Note that in order to apply the specialization argument, one has to replace H_{n,r} by its “integral form” defined over a subring R_{1} = ℤ[v, v^{−1}, u_{1}, …, u_{r}, Δ^{−1}] of K as in [S, 3.6]. Accordingly one needs to replace V by its R_{1}-lattice with basis e_{i}. All the ingredients up to now make sense for this setup, and we use them freely without referring R_{1} in the discussion below.)

Let \overline{V} = ⨁ \overline{V}_{i} be the ℂ-vector space with dim \overline{V}_{i} = m_{i}. We denote by \{\overline{e}_{j}^{(i)}\} the basis of V_{i}. Then the H_{n,r}-module V^{⊗n} is specialized to the ℂW_{n,r}-module \overline{V}^{⊗n}. Let t_{i} be as in 4.1. Then the action of t_{i} on \overline{V}^{⊗n} is given by t_{i}e_{I} = ζ^{ε_{i}}e_{I} for I = (j_{1}^{(ε_{1})}, …, j_{n}^{(ε_{n})}), which is the specialization of ξ_{i} on V^{⊗n}. The previous construction for M_{λ,c} = ⨁_{ν} V_{ν}^{⊗n} makes sense, and by the specialization we have a W_{n,r}-module \overline{M}_{λ,c} = ⨁_{ν} \overline{V} _{ν}^{⊗n}. Let \overline{e}^{ν}, \overline{b}^{ν}, \overline{m}_{λ,c} be the elements in \overline{M}_{λ,c} obtained from e^{ν}, b^{ν}, m_{λ,c} by the specialization.

To show (4.5.3), it is enough to see that

(4.5.4)  \overline{M}_{λ,c} = ℂW_{n,r}\overline{m}_{λ,c}.

We show (4.5.4). We prepare some notation. For I = (j_{1}^{(ε_{1})}, …, j_{n}^{(ε_{n})}), we call ε = (ε_{1}, …, ε_{n}) the signature of e_{I}, and j = (j_{1}, …, j_{n}) the foot of e_{I}. Put U = ℂW_{n,r}\overline{m}_{λ,c}. Now \overline{m}_{λ,c} can be written as \overline{m}_{λ,c} = ∑_{ε∈[1,r]^{n}} \overline{m}(ε), where \overline{m}(ε) is a linear combination of vectors ē_{I} whose signature is ε. Note that t_{1}, …, t_{n} are generators of the subgroup (ℤ/rℤ)^{n} of W_{n,r}, and e_{j_{1}}^{(ε_{1})}⊗⋯⊗e_{j_{n}}^{(ε_{n})} generates a one dimensional representation φ_{ε} of (ℤ/rℤ)^{n} given by t_{i} ↦ ζ^{ε_{i}}. It follows that each \overline{m}(ε) belongs to U.

Let us consider the partial order < on [1, m]^{n} defined in 3.3. Let F (ε) be the set of vectors in \overline{M}_{λ,c} consisting of ē_{I} with signature ε, together with the vectors e_{I′} obtained from those e_{I} by permuting the factors. Clearly ⋃_{ε}F (ε) gives rise to a basis of \overline{M}_{λ,c}. We show, by backward induction on the partial order of the set of signatures, that F (ε) ⊂ U. Take ε = (ε_{1}, …, ε_{n}) and assume that F (ε′) ⊂ U holds for any ε′ > ε. Now \overline{m}(ε) can be written as

(4.5.5)  \overline{m}(ε) ∈ \overline{e}_{I} + ∑_{I′} ℂ\overline{e}_{I′},

where the foot j of I is given by

(4.5.6)  j = (\underbrace{m_{r}, …, m_{r}}_{λ_{mr}^{(r)} −times}, ⋯, \underbrace{t + 1, …, t + 1}_{λ_{t+1}^{(r)}−times}, \underbrace{t, …, t}_{c_{1}−times}, …, \underbrace{1, …, 1}_{c_{t}−times}),

and e_{I′} is a summand of some b^{ν} = b_{I′′}, not equal to e^{ν}. Thus e_{I′} is obtained from e_{I′′} by permuting the factors. Note that e_{I′′} has the same foot as (4.5.6), and we have I′ < I′′ by Corollary 3.7. Let ε′′ be the signature of I′′. As in the proof of (4.5.2), one can write e_{I′′} = e_{λ} ⊗ e^{μ} and e_{I′} = e_{λ} ⊗ e^{μ′}. Since (t, …, t, …, 1, …, 1) is ordered decreasingly, the condition that I′ < I′′ implies that ε < ε′′. It follows, by induction, that e_{I′′} ∈ U. By operating 𝔖_{n}, we see that e_{I′} ∈ U also. This implies that e_{I} and all its permutations of factors belong to U. Hence we have F (ε) ⊂ U. Thus (4.5.4), and so (4.5.3) holds.

It is easy to see that dim M_{λ,c} = dim \overline{M}_{λ,c} = |W_{n,r}|/|W_{λ,c}|, where W_{λ,c} is the subgroup of W_{n,r} corresponding to the subalgebra H_{λ,c} of H_{n,r}. Then (4.5.1) and (4.5.3) implies that M_{λ,c} ≃ Ind_{H_{λ,c}}^{H_{n,r}} φ_{λ,c}. The proposition is proved.

4.6. The space M_{λ,c} can be decomposed into a direct sum of weight spaces V_{ν}^{⊗n}. Hence in view of Proposition 4.2 and Theorem 4.3, M_{λ,c} have bases inherited from the basis {e_{I}} and {b_{I}} of various V_{ν}^{⊗n}. In particular, a bar involution on M_{λ,c} can be defined, and one obtains a basis invariant under the bar involution.

Here we consider the special case where M_{λ,c} is isomorphic to the regular representation of H_{n,r}. Hence we assume that λ = ∅, and c = (1^{n}). So H_{λ,c} ≃ K and φ_{λ,c} = 1_{K}. P_{λ,c} is in bijection with the set [1, r]^{n}, and the vector e^{ν} ∈ V_{ν}^{⊗n} corresponding to ν ∈ P_{λ,c} in 4.4 is given by

(4.6.1)  e^{ν} = e_{n}^{(ε_{1})} ⊗ e_{n−1}^{(ε_{2})} ⊗ ⋯ ⊗ e_{1}^{(ε_{n})}

under the correspondence ν ↔ ε = (ε_{1}, …, ε_{n}) ∈ [1, r]^{n}. The basis of V_{ν}^{⊗n} is obtained by permuting the factors of e^{ν}. If we write e^{ν} = e_{I} and e_{σ(I)} = e_{σ,ε}, then {e_{σ,ε} | σ ∈ 𝔖_{n}} forms a basis of V_{ν}^{⊗n}, and

{e_{σ,ε} | σ ∈ 𝔖_{n}, ε ∈ [1, r]^{n}}

gives rise to a basis of M_{λ,c}. We define a partial order on the set {(σ, ε) | σ ∈ 𝔖_{n}} (for a fixed ε) as follows. Let τ_{0} ∈ 𝔖_{n} be an element such that e^{ν} = e_{τ_{0}(I)}, where I = (i_{1}, …, i_{n}) with i_{1} ≥ ⋯ ≥ i_{n}. Then we put (τ, ε) < (σ, ε) if ττ_{0} < στ_{0}.

We write m_{0} = m_{λ,c} = ∑ b^{ν} as in (4.4.2). Then the map h ↦ hm_{0} gives an isomorphism H_{n,r} \underrightarrow{∼} M_{λ,c}. We denote by the same symbol the basis of H_{n,r} obtained from the basis {e_{σ,ε}} of M_{λ,c}. Since m_{0} is ψ-invariant, it follows from Theorem 2.4 that the bar involution on H_{n,r} can be identified, under the above isomorphism, with the involution ψ on M_{λ,c}. Let {b_{σ,ε}} be the basis of H_{n,r} obtained by transferring the canonical basis of M_{λ,c} attached to {e_{σ,ε}} ⊂ M_{λ,c}. Then the following result is immediate from Theorem 4.3.

Theorem 4.7. There exists a unique basis {b_{σ,ε} | σ ∈ 𝔖_{n}, ε ∈ [1, r]^{n}} of H_{n,r} satisfying the following properties.

b_{σ,ε} ∈ e_{σ,ε} + ∑_{\begin{matrix}τ∈𝔖_{n} \\ (τ,ε)<(σ,ε) \end{matrix}} v^{−1}ℤ[v^{−1}]e_{τ,ε},

\overline{b}_{σ,ε} = b_{σ,ε}.

The coefficient p_{(τ,ε),(σ,ε)} of e_{τ,ε} in the expression of b_{σ,ε} is described by the parabolic Kazhdan-Lusztig polynomials of type A for the weight space V_{ν}^{⊗n} under the correspondence ν ↔ ε in (4.6.1).

§5. The case of Iwahori-Hecke algebras of type B_{n}

5.1. We consider the case where W = W_{n,2} is the Weyl group of type B_{n}. We specify the parameters of H_{n,2} by putting u_{1} = −v^{−1}, u_{2} = v, so that H_{n,2} is the Hecke algebra H of W as given in 3.1. We discuss the relationship between Kazhdan-Lusztig basis of H and the previous basis.

Let us consider the subalgebra H_{λ,c} of H as in 4,4, and assume that H_{λ,c} is the subalgebra H_{J} associated to a parabolic subgroup W_{J} of W. We also assume that the linear character φ_{λ,c} : H_{J} → K in 4.4 is of the form φ_{J} in 3.1 (hence λ = (λ^{(1)}; λ^{(2)}) = (−; k) for some k ≥ 0). Then the H-submodule M_{λ,c} = ⨁ V_{ν}^{⊗n} of V^{⊗n} can be identified with M_{J} in 3.1, where m_{λ,c} ∈ M_{λ,c} corresponds to m_{e} ∈ M_{J}. By Theorem 2.4 and Proposition 4.5, the bar involution on M_{J} given in 3.1 coincides with the involution ψ on M_{λ,c}. We shall compare various bases on M_{J}. Put m_{σ} = T_{σ}m_{e} for σ ∈ W^{J}. Then M = {m_{σ} | σ ∈ W^{J}} gives a basis of M_{J}. Let C = {C_{σ}^{J} | σ ∈ W^{J}} be the Kazhdan-Lusztig basis of M_{J} given in 3.1. We also put E = {e_{I} | I ∈ I_{J}} and B = {b_{I} | I ∈ I_{J}} the bases of M_{J} arising from the standard basis and the canonical basis of ⨁_{ν} V_{ν}^{⊗n}. For a two bases X = (x_{i}) and Y = (y_{j}) of M_{J} indexed by the set I_{J} ≃ W^{J}, we denote by M(X, Y) = (a_{ij}) the transition matrix from X to Y given by

y_{i} = ∑_{j∈I} a_{ji}x_{j}.

Put

P_{B} = M(M, C), P_{A} = M(E, B), X = M(B, C), Y = M(E, M).

We define a total order on W^{J} which is compatible with the converse of the Bruhat order on W^{J}, and consider the matrix P_{B} = (p_{τ,σ})_{τ,σ∈W^{J}} with respect to this order. Then P_{B} is a lower unitriangular matrix. Moreover, p_{τ,σ} ∈ v^{−1}ℤ[v^{−1}], and p_{τ,σ} represents the parabolic Kazhdan-Lusztig polynomials of type B_{n} associated to W_{J} up to a power of v. If we fix a total order on the set I_{J} compatible with the weight decomposition M_{J} = ⨁_{ν} V_{ν}^{⊗n}, the matrix P_{A} is a block-wise diagonal matrix, and diagonal blocks correspond to the weights on M_{J}. The diagonal block P_{A}^{ν} corresponding to the weight ν is the matrix of the parabolic Kazhdan-Lusztig polynomials of type A associated to the parabolic subgroup 𝔖_{ν} (up to powers of v), where 𝔖_{ν} is the stabilizer of e^{ν} in 𝔖_{n}.

We have the following.

Proposition 5.2. The matrices P_{A}, P_{B}, X, Y satisfy the following relation.

(5.2.1)  P_{B} = Y^{−1}P_{A}X.

Moreover, the matrices P_{B} and X are determined uniquely by P_{A} and Y. In other words, the parabolic Kazhdan-Lusztig polynomials of type B_{n} can be determined by various parabolic Kazhdan-Lusztig polynomials of type A and by the matrix Y.

Proof. It is clear that P_{A}, P_{B}, X, Y satisfy (5.2.1). We show that P_{A} and Y determine P_{B} and X uniquely. Write the equation (5.2.1) as

(5.2.2)  P_{B}X^{−1} = Y^{−1}P_{A}

and consider (5.2.2) as the matrix equation with unknown matrices P_{B} and X. We fix a bijection W^{J} ≃ I_{J}, and write the matrices as P_{B} = (p_{ij}), X^{−1} = (x_{ij}) with i, j ∈ I_{J} along the order inherited from the order on W^{J}. Here p_{ij} ∈ v^{−1}ℤ[v^{−1}] and x_{ij} ∈ ℚ(v) such that x̄_{ij} = x_{ij}. We determine the matrices P_{B} and X^{−1} row wisely. Suppose that the first (i − 1)-rows of P_{B} and X^{−1} are determined. Since P_{B} is lower unitriangular, one can write

(5.2.3)  ∑_{j=1}^{i−1} p_{ij}x_{j} + x_{i} = α_{i},

where x_{j} (resp. α_{j}) denotes the j-th row of X^{−1} (resp. Y^{−1}P_{A}), respectively. By applying the bar involution on (5.2.3), and by subtracting each other, one has

(5.2.4)  ∑_{j=1}^{i−1}(p_{ij} − p̄_{ij})x_{j} = α_{i} − \overline{α}_{i}.

Here x_{1}, …, x_{i−1}, α_{i} − \overline{α}_{i} are known vectors. Since x_{1}, …, x_{i−1} are linearly independent, (5.2.4) determines d_{ij} = p_{ij} − p̄_{ij} uniquely. But since p_{ij} ∈ v^{−1}ℤ[v^{−1}], d_{ij} determines p_{ij} uniquely. Thus the i-th row of P_{B} is determined. By substituting p_{ij} into (5.2.3), the i-th row x_{i} is also determined. Thus the matrices P_{B} and X^{−1} are determined.

5.3. The bases {e_{σ,ε}} and {b_{σ,ε}} appeared in Theorem 4.7 are nothing but the bases E and B, respectively. In order to relate these bases to the Kazhdan-Lusztig basis, it is essential to know about the matrix X since the matrix Y is more or less simpler than X. It would be an interesting problem to study the matrix X. One might expect that X has a relatively simple form compared to the matrix P_{B}. We give below a simple example of the matrix X, i.e., the relation between parabolic Kazhdan-Lusztig basis and the canonical basis.

Assume that W is the Weyl group of type B_{n}, and let W_{J} be the parabolic subgroup of type B_{n−1}. We put J = {t_{1}, s_{2}, …, s_{n−1}}. Then the distinguished representatives W^{J} are given as

W^{J} = {s_{i} ⋯ s_{n} | 2 ≤ i ≤ n + 1} ∪ {s_{i} ⋯ s_{2}t_{1}s_{2} ⋯ s_{n} | 1 ≤ i ≤ n}

under the convention that s_{1} = s_{n+1} = 1. Assume that V = V_{1} ⊕ V_{2} with dim V_{1} = 1, dim V_{2} = 2. We fix bases e_{1}^{(1)} of V_{1} and e_{1}^{(2)}, e_{2}^{(2)} of V_{2}, respectively. We also write e_{3} = e_{2}^{(2)}, e_{2} = e_{1}^{(2)}, e_{1} = e_{1}^{(1)}. Let us consider M_{λ,c} as in 4.4, where λ = (λ^{(1)}, λ^{(2)}) = (−; n − 1) and c = (1) (i.e., t = 1). Then M_{λ,c} is isomorphic to the induced representation Ind_{H_{J}}^{H} φ_{J}, where H_{J} is the parabolic subalgebra of H = H_{n,2} of type B_{n−1} and φ_{J} is as in 3.1. It can be decomposed into the direct sum of weight spaces M_{λ,c} = V_{ν}^{⊗n} ⨁ V_{ν′}^{⊗n}, where ν = (0, 1, n − 1), ν′ = (1, 0, n − 1) as weights for U_{v}. We define, for 1 ≤ i ≤ n, I_{i}, I′_{i} by

I_{i} = (\underbrace{2^{(2)}, …, 2^{(2)}}_{i−1-times}, 2^{(1)}, 2^{(2)}, …, 2^{(2)}),

I′_{i} = (\underbrace{2^{(2)}, …, 2^{(2)}}_{i−1-times}, 1^{(1)}, 2^{(2)}, …, 2^{(2)}).

Then b^{ν} = b_{I_{n}} = e_{I_{n}} and b^{ν′} = b_{I′_{n}} = e_{I′_{n}}, and we have m_{λ,c} = b_{I_{n}} + b_{I′_{n}}. The Kazhdan-Lusztig basis C_{σ}^{J} for σ ∈ W^{J} can be expressed in terms of canonical basis, as

C_{σ}^{J} = \begin{cases}b_{I_{i−1}} + b_{I′_{i−1}} & if σ = s_{i} ⋯ s_{n}, \\ (v^{i} + v^{−i})b_{I_{1}} − b_{I_{i+1}} + b_{I′_{i+1}} & if σ = s_{i} ⋯ s_{2}ts_{2} ⋯ s_{n}. \end{cases}

This determines the matrix X completely.

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Department of Mathematics Science University of Tokyo Noda, Chiba 278-8510, Japan