REPRESENTATIONS OF THE HECKE ALGEBRA ON A FAMILY OF RATIONAL FUNCTIONS

KATSUHISA MIMACHI AND MASATOSHI NOUMI

Abstract. We give a representation of the Hecke algebra on the linear space spanned by a family of rational functions. This representation is isomorphic to the induced representation from a trivial representation. Our family of rational functions arose from the study of the quantum Knizhnik-Zamolodchikov equations.

The first author introduced a family of rational functions {φ_{F} } in [8] for the study of the quantum Knizhnik-Zamolodchikov (QKZ) equations due to Cherednik [2]. These functions φ_{F} are parametrized by the flags F of indexing sets defined below, and the action of the symmetric groups on them was used to derive the QKZ equation. On the other hand, the QKZ equation is also formulated by the extended affine or double affine Hecke algebras besides the Weyl groups. See [2] and also [3],[4]. Hence it is important and indispensable to clarify the aspect from the Hecke algebras of these rational functions. Such problem was also proposed in [8]. This paper is devoted to this point: We show this family gives a representation of the Hecke algebra H(𝕊_{n}) via Lusztig operator T_{i}. This representation is isomorphic to the induced representation from a one dimensional “ trivial ” representation t . We believe such aspects should be important for the study in these topics. Indeed, to prove the Macdonald conjecture about the integrality of the (q, t)-Kostka coefficients, Kirillov and the second author used our Corollary 10 effectively [5].

We treat in this paper the Hecke algebra associated with the symmetric group 𝕊_{n} . The connection with the extended affine Hecke algebras will be discussed elsewhere in the context of the q-de Rham cohomology in the sense of Aomoto [1].

1. Preliminary and main theorem

Let n be a positive integer and fix a sequence m = (m_{0}, m_{1}, …, m_{d}) of positive integers such that

n = m_{0} ≥ m_{1} ≥ ⋯ ≥ m_{d} > m_{d+1} = 0.

We say that a sequence F = (F^{(0)}, F^{(1)}, …, F^{(d)}) of subsets of the interval [1, n] = {1, …, n} is a flag of type m if

[1, n] = F^{(0)} ⊃ F^{(1)} ⊃ ⋯ ⊃ F^{(d)} ⊃ F^{(d+1)} = ∅ and |F^{(i)}| = m_{i}.

We denote by F(m) the set of all flags of type m in [1, n] and take the flag

B = (B^{(0)}, …, B^{(d)}), B^{(i)} = [n − m_{i} + 1, n]

as the base point of F(m). The symmetric group W = 𝕊_{n} acts on this set F(m) transitively and its stablizer at B is given by

W_{0} = 𝕊_{n_{0}} × 𝕊_{n_{1}} × ⋯ × 𝕊_{n_{d}}

where n_{i} = m_{i} −m_{i+1} (i = 0, 1, …, d), and so n_{0} +⋯+n_{d} = n. There is a bijective mapping

W/W_{0} = 𝕊_{n}/𝕊_{n_{0}} × 𝕊_{n_{1}} × ⋯ 𝕊_{n_{d}} ⟶^{∼} F(m).

Given a flag F = (F^{(0)}, …, F^{(d)}) ∈ F(m), for each i = 1, …, d, we define the strictly increasing mapping a^{(i)} : [1, m_{i}] → [1, n] by Im a^{(i)} = F^{(i)}. Associated with the function a^{(i)}, we also define the strictly increasing mappings p^{(i)} : [1, m_{i}] → [1, m_{i−1}] so that the following diagram should be commutative.

[figure: [1, n] = [1, m_{0}]; [1, n] = F^{(0)}; a^{(0)} = id; ⟵; p^{(1)}; ↩; [1, m_{1}]; F^{(1)}; a^{(1)}; ←; ⋯; ←; ↩; ⋯; ↩; [1, m_{i−1}]; F^{(i−1)}; a^{(i−1)}; ⟵; p^{(i)}; ↩; [1, m_{i}]; F^{(i)}; a^{(i)}; [1, m_{d}]; F^{(d)}; a^{(d)}; ←; ⋯; ←; ↩; ⋯; ↩; p^{(d)}]

In other words, the sequence p = (p^{(1)}, …, p^{(d)}) of strictly increasing mappings p^{(i)} : [1, m_{i}] → [1, m_{i−1}] is determined by the condition

F^{(i)} = Im(p^{(1)} ∘ ⋯ ∘ p^{(i)} : [1, m_{i}] → [1, n]).

We introduce a family of rational functions φ_{F} = φ_{F} (x^{(0)}; …; x^{(d)}) indexed by F ∈ F(m) in the variables

x^{(0)} = (x_{1}^{(0)}, …, x_{m_{0}}^{(0)}), x^{(1)} = (x_{1}^{(1)}, …, x_{m_{1}}^{(1)}), …, x^{(d)} = (x_{1}^{(d)}, …, x_{m_{d}}^{(d)})

as

φ_{F} = ∑_{σ} σ ∏_{i=1}^{d}\{∏_{j=1}^{m_{i}} \frac{∏_{k=p^{(i)}(j)+1}^{m_{i−1}}(1 − \frac{x_{k}^{(i−1)}}{x_{j}^{(i)}} )}{∏_{k=p^{(i)}(j)}^{m_{i−1}}(1 − t\frac{x_{k}^{(i−1)}}{x_{j}^{(i)}} )} ∏_{1≤j<k≤m_{i}} \frac{(1 − t\frac{x_{k}^{(i)}}{x_{j}^{(i)}} )}{(1 − \frac{x_{k}^{(i)}}{x_{j}^{(i)}} )}\}.

Here the summation is taken over all σ = (σ^{(1)}, …, σ^{(d)}) ∈ 𝕊_{m_{1}} × ⋯ × 𝕊_{m_{d}} and the action of σ = (σ^{(1)}, …, σ^{(d)}) is determined by σ(x_{j}^{(i)}) = x_{σ^{(i)}(j)}^{(i)} for i = 1, …, d and j = 1, …, m_{i}.

We denote by V (m) the vector subspace of ℂ(x^{(0)}, …, x^{(d)}) spanned by all φ_{F} = φ_{F} (x^{(0)}; …; x^{(d)}) :

V (m) = ∑_{F ∈F(m)} ℂ φ_{F} ⊂ ℂ(x^{(0)}, …, x^{(d)}).

Next we recall the definition of the Hecke algebra H(𝕊_{n}) associated with the symmetric group 𝕊_{n}. We denote by H(𝕊_{n}) the ℂ-algebra generated by the elements T_{1}, …, T_{n−1} with fundamental relations

(T_{k} − t)(T_{k} + 1) = 0 (k = 1, …, n − 1),
T_{k}T_{k+1}T_{k} = T_{k+1}T_{k}T_{k+1} (k = 1, …, n − 2),  (1.1)
T_{k}T_{l} = T_{l}T_{k} (|k − l| ≥ 2),

where t is a fixed nonzero complex parameter.

We call the representation defined by T_{k} ↦ t the “ trivial ” representaion, and denote it by t = t_{H(𝕊_{n})} . It is noted as a reference that the representation defined by T_{k} ↦ −1 should be called the “ parity ” representation. These names are after the case of the symmetric groups.

It is known [6] that this algebra H(𝕊_{n}) can be realized by the following Lusztig operators T_{k}^{y} (k = 1, …, n−1) acting on functions in the n variables y = (y_{1}, …, y_{n}) :

T_{k}^{y} = t + \frac{1 − ty_{k}/y_{k+1}}{1 − y_{k}/y_{k+1}} (s_{k}^{y} − 1) (k = 1, …, n − 1).  (1.2)

Here s_{k} denotes the transposition (k, k + 1) in 𝕊_{n}, so that s_{k}^{y} interchanges the variables y_{k} and y_{k+1}. The superscript y here indicates the variables on which the symmetric group 𝕊_{n} acts. Similar notations will be frequently used below without further remarks.

In what follows we identify the variaibles y = (y_{1}, …, y_{n}) with x^{(0)} = (x_{1}^{(0)}, …, x_{n}^{(0)}).

Theorem 1. (1) The vector space V (m) spanned by the rational functions φ_{F} (F ∈ F(m)) becomes a left H(𝕊_{n})-module through the action of the Lusztig operators T_{k}^{y} (k = 1, …, n − 1) in the variables y = x^{(0)} .

(2) The H(𝕊_{n})-module V (m) is isomorphic to the induced representation

Ind_{H(W_{0})}^{H(W)} t_{H(W_{0})} = H(W) ⊗_{H(W_{0})} t_{H(W_{0})}

for the pair (W, W_{0}) = (𝕊_{n}, 𝕊_{n_{0}} × ⋯ × 𝕊_{n_{d}}). In fact there is a H(𝕊_{n})-isomorphism

Ind_{H(W_{0})}^{H(W)} t_{H(W_{0})} = H(𝕊_{n})/J ⟶^{∼} V (m)

which sends the canonical generator to φ_{B}, where J is the left ideal of H(𝕊_{n}) generated by T_{k} − t (k ≠ n_{0}, n_{0} + n_{1}, …, n_{0} + ⋯ + n_{d−1}). The dimension of V (m) is n!/n_{0}!n_{1}! ⋯ n_{d}!.

Moreover we will describe the action of the Hecke algebra on the rational functions {φ_{F} }, precicely. For that purpose, we prepare some notation.

With each sequence p = (p^{(1)}, …, p^{(d)}) of strictly increasing mappings, we associate a mapping μ : [1, n] → [0, d] defined by

μ(k) = max{i ∈ [0, d]; k ∈ Im(p_{1} ∘ ⋯ ∘ p_{i})} for k = 1, …, n,

and μ(k) = 0 if k ∉ Im p_{1}. The set of all sequences p has a bijection to the set of all mappings μ such that |μ^{−1}(i)| = n_{i} = m_{i} − m_{i+1} for i ∈ [0, d]. As a result, we obtain a bijection between F(m) and the set of such μ ’s. Thus we can set

φ_{μ} = φ_{F} .

Then we can state the following:

Theorem 2. For each l = 1, …, n − 1, we have

(1) T_{l}^{y}φ_{μ} = φ_{μ} (μ(l) = μ(l + 1))
(2) T_{l}^{y}φ_{μ} = t φ_{μs_{l}} (μ(l) < μ(l + 1))
(3) T_{l}^{y}φ_{μ} = φ_{μs_{l}} + (t − 1)φ_{μ} (μ(l) > μ(l + 1))

We note that the canonical generator of the induced representation coresponds to the φ_{μ_{min}}(x^{(0)}, …, x^{(d)}), where μ_{min} is the unique non-decreasing mapping μ : [1, n] → [0, d] such that |μ^{−1}(i)| = n_{i} for i ∈ [0, d]. In terms of the flag F, this corresponds to the case F = B, where B is the base point of F(m) defined before.

Corollary 3. The representation V (n, n − 1, …, 1) is isormorphic to the regular representation of H(𝕊_{n}). In this case, we have

T_{w}φ_{μ_{min}}(x^{(0)}, …, x^{(n−1)}) = t^{l(w)}φ_{μ_{min}w^{−1}}(x^{(0)}, …, x^{(n−1)}).

Remark. In [8], this rational function φ_{μ_{min}w^{−1}} is denoted by φ_{w}, so that T_{w}φ_{e} = t^{l(w)}φ_{w}.

2. Proof of Theorems

First we consider the d = 1 case and set m_{1} = m and x^{(1)} = x = (x_{1}, …, x_{m}) for brevity. For each injective mapping p : [1, m] → [1, n], we define a rational function ψ_{p}(y; x) in ℂ(y, x) to be

ψ_{p}(y; x) = ∏_{1≤s≤m} \frac{1}{1 − ty_{p(s)}/x_{s}} ∏_{p(s)<r≤n} \frac{1 − y_{r}/x_{s}}{1 − ty_{r}/x_{s}} ∏_{1≤p(r)<p(s)≤m} \frac{1 − tx_{s}/x_{r}}{1 − x_{s}/x_{r}} . (2.1)

For each subset K ⊂ [1, n] with |K| = m, we also use the symbol ψ_{K}(y; x) to express ψ_{p_{K}} (y; x), where p_{K} : [1, m] → [1, n] is a unique strictly increasing mapping such that p_{K}([1, m]) = K, namely K = {p_{K}(1) < … < p_{K}(m)} ⊂ [1, n]. Then we have

Lemma 4.

(1) T_{l}^{y}(ψ_{K}(y; x)) = \{t + (s_{j}^{x} − 1)\frac{1 − tx_{j}/x_{j+1}}{1 − x_{j}/x_{j+1}} \} (ψ_{K}(y; x))
(l, l + 1 ∈ K; p_{K}(j) = l),
(2) T_{l}^{y}(ψ_{K}(y; x)) = t ψ_{s_{l}K}(y; x)  (l ∉ K, l + 1 ∈ K),
(3) T_{l}^{y}(ψ_{K}(y; x)) = ψ_{s_{l}K}(y; x) + (t − 1)ψ_{K}(y; x)  (l ∈ K, l + 1 ∉ K),
(4) T_{l}^{y}(ψ_{K}(y; x)) = t ψ_{K}(y; x)  (l, l + 1 ∉ K).

Proof. In the case (1), we have

\frac{1 − ty_{l}/y_{l+1}}{1 − y_{l}/y_{l+1}} (s_{l}^{y} − 1)(ψ_{K}) = \frac{(y_{l+1} − ty_{l})(x_{j+1} − tx_{j})}{(x_{j+1} − ty_{l})(x_{j} − y_{l+1})}ψ_{K}

and

(s_{j}^{x} − 1) (\frac{1 − tx_{j}/x_{j+1}}{1 − x_{j}/x_{j+1}} ψ_{K}) = \frac{(y_{l+1} − ty_{l})(x_{j+1} − tx_{j})}{(x_{j+1} − ty_{l})(x_{j} − y_{l+1})}ψ_{K} ,

which leads to the formula. The formulas for the other cases (2) − (4) are derived similarly.

Let K be an arbitrary field containing ℂ and regard K(x) as a left H(𝕊_{m})- module through the Lusztig operators T_{j}^{x} (j = 1, …, m − 1), and K(y, x) as a left H(𝕊_{n})-modules through T_{l}^{y} (l = 1, …, n − 1). Let S^{x} = ∑_{σ∈𝕊_{m}} σ^{x} represent the symmetrization in the variables x = (x_{1}, …, x_{m}). For each subset K ⊂ [1, n] with |K| = m, we define a linear mapping I_{K} : K(x) → K(y, x) by

I_{K}(u) = S^{x}(ψ_{K}(y, x)u(x)) = ∑_{σ∈𝕊_{m}} ψ_{p_{K}σ^{−1}}(y; x)σ^{x}(u(x)) (u ∈ K(x)),

noting the relation σ^{x}ψ_{p} = ψ_{pσ^{−1}} for σ ∈ S_{m} . Then Lemma 4 implies the following.

Proposition 5. Let K be a subset of [1, n] with |K| = m, and l = 1, …, n − 1. For each u ∈ K(x), we have

(1) T_{l}^{y}I_{K}(u) = I_{K}(T_{j}^{x}u)  (l, l + 1 ∈ K; p_{K}(j) = l),
(2) T_{l}^{y}I_{K}(u) = t I_{s_{l}K}(u)  (l ∉ K, l + 1 ∈ K),
(3) T_{l}^{y}I_{K}(u) = I_{s_{l}K}(u) + (t − 1) I_{K}(u) (l ∈ K, l + 1 ∉ K),
(4) T_{l}^{y}I_{K}(u) = t I_{K}(u)  (l, l + 1 ∉ K).

Proof. The equality

∑_{σ^{x}} σ^{x} \{(s_{j}^{x} · \frac{1 − tx_{j}/x_{j+1}}{1 − x_{j}/x_{j+1}} · ψ_{K}(y; x)) u(x)\}

= ∑_{σ^{x}} σ^{x} \{ψ_{K}(y; x) · (\frac{1 − tx_{j}/x_{j+1}}{1 − x_{j}/x_{j+1}} · (s_{j}^{x}u(x)))\}

shows the relation (1). The other relations directly follow from Lemma 4.

Remark. The formula in case (3) is equivalent to (T_{l}^{y})^{−1}I_{K}(u) = t^{−1}I_{s_{l}K}(u).

The mapping I_{K} can be shown to be injective. To see it we introduce an ordering on the set of all subsets K ⊂ [1, n] with |K| = m .

Given two distinct subsets K, L ⊂ [1, n] such that |K| = |L| = m, we take the minimal element r of the symmetric difference (K\L) ∪ (L\K). We write K ≺ L or L ≺ K according as r ∈ K or r ∈ L. In terms of the strictly increasing mappings p_{K}, p_{L} : [1, m] → [1, n] such that K = p_{K}([1, m]), L = p_{L}([1, m]), the condition “K ⪯ L” is equivalent to saying that (p_{K}(1), …, p_{K}(m)) ⪯ (p_{L}(1), …, p_{L}(m)) under the lexicographic order of [1, n]^{m}.

Fix a subset K of [1, n] with |K| = m. When a rational function f(y; x) ∈ ℂ(y, x) is regular along x_{i} = y_{p_{K}(i)} for all i = 1, …, m, we can consider the substitution x_{i} ↦ y_{p_{K}(i)}. With the abbreviation

y_{K} = (y_{p_{K}(1)}, …, y_{p_{K}(m)}) (K = {p_{K}(1) < … < p_{K}(m)}),

we simply denote by f(y; y_{K}) the funtion obtained from f(y; x) by this substitution.

Lemma 6. Let K be a subset of [1, n] with |K| = m and q : [1, m] → [1, n] any injective mapping with q([1, m]) = L.

(1) If L ≺ K, then ψ_{q}(y; y_{K}) = 0.

(2) If L = K and q ≠ p_{K}, then ψ_{q}(y; y_{K}) = 0.

(3) If L = K and q = p_{K}, then

ψ_{K}(y; y_{K}) = \frac{1}{(1 − t)^{m}} ∏_{\begin{matrix}k<l \\ k∈K,l∉K \end{matrix}} \frac{1 − y_{l}/y_{k}}{1 − ty_{l}/y_{k}} .

Proof. If q(i) ≥ p_{K}(i) for all i = 1, …, m, we have p_{K}(i) ≤ q(i) and p_{K} < q(i) (i < j ≤ n) for each i = 1, …, m. Thus, under the condition L ⪯ K, we have p_{K}(i) ≤ q(i) (1 ≤ i ≤ m). This implies that if L ≺ K or L = K with p_{K} ≠ q there exists a number i such that q(i) < p_{K}(i) and 1 ≤ i ≤ m. It implies (1) and (2). The assertion (3) follows from a direct computation.

Lemma 6 implies the injectivity as follows:

Lemma 7. I_{K} : K(x) → K(y, x) is injective mapping for any K ∈ [1, n].

Proof. By Lemma 6, we have

I_{K}(u)(y; y_{K}) = ψ_{K}(y; y_{K})u(y_{K}),

which implies the assertion.

When M is a left H(𝕊_{m})-submodule of K(x) , we have an isomorphism

M ≃ I_{K}(M)

as a left H(𝕊_{m})-submodule by Lemma 7.

On the other hand, by Proposition 5 we have

T_{l}^{y}I_{[n−m+1,n]}(u) = t I_{[n−m+1,n]}(u) (l = 1, …, n − m − 1),
T_{n−m+j}^{y}I_{[n−m+1,n]}(u) = I_{[n−m+1,n]}(T_{j}^{x}u) (j = 1, …, m − 1),

which imply the vector subspace I_{[n−m+1,n]}(K(x)) ⊂ K(y, x) is stable under the action of T_{l}^{y} (l ≠ n − m). Combining these, we obtain the isomorphism

t_{H(𝕊_{n−m})} ⊗_{ℂ}K(x) →^{∼} I_{[n−m+1,n]}(K(x))

as a H(𝕊_{n−m} × 𝕊_{m})-module, via the embedding H(𝕊_{m}) ↪ H(𝕊_{n−m} × 𝕊_{m}) which sends T_{j} to T_{n−m+j} for j = 1, …, m − 1.

Hereafter, we regard

H(𝕊_{n−m} × 𝕊_{m}) = ℂ⟨T_{1}, …, T_{n−m−1}, T_{n−m+1}, …, T_{n−1}⟩

as a subalgebra of H(𝕊_{n}) and will investigate the left H(𝕊_{n})-module structure of I_{K}(K(x)) ⊂ K(y, x) via the Lusztig operator T_{j}^{y}.

The element T_{w} ∈ H(𝕊_{n}) for a general w ∈ 𝕊_{n} is defined to be

T_{w} = T_{k_{1}} ⋯ T_{k_{l}},

using a reduced expression w = s_{k_{1}} ⋯ s_{k_{l}}. The length l and the definition of T_{w} do not depend on the choice of a reduced expression.

Let K be a subset of [1, n] with |K| = m, and p_{K} : [1, m] → [1, n] the strictly increasing mapping such that p_{K}([1, m]) = K. Then we denote by w_{K} the following permutation:

w_{K} = ( \begin{matrix}1 & ⋯ & n − m & n − m + 1 & ⋯ & n \\ q_{K}(1) & ⋯ & q_{K}(n − m) & p_{K}(1) & ⋯ & p_{K}(m) \end{matrix} ) ,

where q_{K} : [1, n − m] → [1, n] is the strictly increasing mapping corresponding to the complement [1, n]\K of K: q_{K}([1, n−m]) = [1, n]\K. Here a reduced expression of w_{K} is given by

w_{K} = (s_{p_{K}(m)}s_{p_{K}(m)+1} ⋯ s_{n−1}) (s_{p_{K}(m−1)} ⋯ s_{n−2}) ⋯
⋯ (s_{p_{K}(1)}s_{p_{K}(1)+1} ⋯ s_{n−m})

and its length is

l(w_{K}) = ∑_{i=1}^{m}(n − m + i − p_{K}(i)) = mn − \binom{m}{2} − ∑_{k∈K} k.

For a fixed number m, each permutation w ∈ S_{n} = ⟨s_{1}, …, s_{n−1}⟩ can be expressed uniquely in the form w = w_{K}w_{1}w_{2} with some w_{1} ∈ ⟨s_{1}, …, s_{n−m−1}⟩, w_{2} ∈ ⟨s_{n−m+1}, …, s_{n−1}⟩ and K ⊂ [1, n] (|K| = m), so that l(w) = l(w_{K})+l(w_{1})+l(w_{2}). It implies that T_{w} = T_{w_{K}} T_{w_{1}}T_{w_{2}}, and moreover that the Hecke algebra H(𝕊_{n}) is a free right H(𝕊_{n−m} × 𝕊_{m})-module of rank \binom{n}{m} with basis \{T_{w_{K}} ; K ⊂ [1, n], |K| = m}.

The expression

T_{w_{K}} = (T_{p_{K}(m)}T_{p_{K}(m)+1} ⋯ T_{n−1}) (T_{p_{K}(m−1)} ⋯ T_{n−2}) ⋯
⋯ (T_{p_{K}(1)}T_{p_{K}(1)+1} ⋯ T_{n−m})

and the formula (2) in Proposition 5 imply the relation

T_{w_{K}} I_{[n−m+1,n]}(u) = t^{l(w_{K})}I_{K}(u) (u ∈ K(x)).

If we define the vector subspace I_{m}^{n}(M) of K(y, x) by

I_{m}^{n}(M) = ∑_{\begin{matrix}K⊂[1,n] \\ |K|=m \end{matrix}} I_{K}(M),  (2.2)

it is easily seen by Proposition 5 that I_{m}^{n}(M) is a left H(𝕊_{n})-submodule of K(y, x). Moreover we have the following:

Proposition 8. There is a natural H(𝕊_{n})-isomorphism

H(𝕊_{n}) ⊗_{H(𝕊_{n−m}×𝕊_{m})} (t_{H(𝕊_{n−m})} ⊗_{ℂ}M) →^{∼} I_{m}^{n}(M),

which maps T_{w_{K}} ⊗v⊗u to t^{l(w_{K})}I_{K}(u) = T_{w_{K}} I_{[n−m+1,n]}(u), where v is the canonical generator of the trivial module t_{H(𝕊_{n−m})} and u is an element of M.

Proof. It is enough to show that the sum in (2.2) is actually a direct sum.

Suppose that there exists a nontrivial relation

∑_{\begin{matrix}L⊂[1,n] \\ |L|=m \end{matrix}} I_{L}(u_{L}) = 0 (u_{L} ∈ M).  (2.3)

If we take the maximum K under ⪯ among all L such that u_{L} ≠ 0, the substitution x ↦ y_{K} in (2.3) implies I_{K}(u_{K})(y; y_{K}) = ψ_{K}(y; y_{K})u(y_{K}) = 0, contrary to the choice of K. This completes the proof of Proposition 8.

In particular, our rational functions φ_{F} (y; x) with F = ([1, n], K), which will be denoted by φ_{K}(y; x) for simplicity, can be expressed as

φ_{K}(y; x) = I_{K}(1) = ∑_{p([1,m])=K} ψ_{p}(y; x).

Accordingly, the vector subspace V (n, m) is expressed as

V (n, m) = ∑_{\begin{matrix}K⊂[1,n] \\ |K|=m \end{matrix}} ℂφ_{K}(y; x) ⊂ ℂ(y, x).

Then Proposition 8 implies the following, which is a special case (d = 1) of Theorem 1.

Corollary 9. There is an isomorphism

V (n, m) ≃ H(𝕊_{n}) ⊗_{H(𝕊_{n−m}×𝕊_{m})} t_{H(𝕊_{n−m}×𝕊_{m})}

as a left H(𝕊_{n})-module, whose dimension is \binom{n}{m}.

The action of the Lusztig operator is given by Proposition 5 as follows, which is also a special case of Theorem 2.

Corollary 10.

(1) T_{l}^{y}φ_{K}(y; x) = φ_{s_{l}K}(y; x) + (t − 1)φ_{K}(y; x) (l ∈ K, l + 1 ∉ K),
(2) T_{l}^{y}φ_{K}(y; x) = t φ_{s_{l}K}(y; x)  (l ∉ K, l + 1 ∈ K),
(3) T_{l}^{y}φ_{K}(y; x) = t φ_{K}(y; x)  (l, l + 1 ∉ K; l, l + 1 ∈ K).

As is noted in Introduction, these relations played a crucial role in the proof of the integrality of the (q, t)-Kostka coefficients (See Theorem 4.1 in [5]).

The remainder will be devoted to the general d case.

Let M be a H(𝕊_{m_{d}})-submodule of ℂ(x^{(d)}). Application of Proposition 8 to the variables (x^{(d−1)}, x^{(d)}) gives a H(𝕊_{m_{d−1}})-submodule I_{m_{d}}^{m_{d−1}}(M) of ℂ(x^{(d−1)}, x^{(d)}). Similar application of Proposition 8 to the variables (x^{(d−2)}, x^{(d−1)}) with K = ℂ(x^{(d)}) gives a H(𝕊_{m_{d−2}})-submodule I_{m_{d−1}}^{m_{d−2}}I_{m_{d}}^{m_{d−1}}(M) of ℂ(x^{(d−2)}, x^{(d−1)}, x^{(d)}). Repeating this procedure, we finally obtain a H(𝕊_{n})-submodule (n = m_{0})

I_{F} (M) = I_{m_{1}}^{m_{0}} ⋯ I_{m_{d}}^{m_{d−1}}(M) ⊂ ℂ(x^{(0)}, x^{(1)}, …, x^{(d)}).

Proposition 8 guarantees there is a natural H(𝕊_{n})-isomorphism

H(𝕊_{n}) ⊗_{H(𝕊_{n_{0}} ×···×𝕊_{n_{d}})} (t_{H(𝕊_{n_{0}} ×···×𝕊_{n_{d−1}} )} ⊗M) →^{∼} I_{F} (M),

where n_{i} = m_{i} − m_{i+1} (i = 0, 1, …, d) with m_{d+1} = 0.

Proposition 5 implies the following:

Proposition 11. For each l = 1, …, n − 1 and u = u(x^{(d)}) ∈ ℂ(x^{(d)}), we have (1) T_{l}^{y}I_{μ}(u) = I_{μ}(T_{j}^{x^{(d)}}u) if μ(l) = μ(l + 1) = d, and j ∈ [1, m − 1] such that s_{l}p_{1} ⋯ p_{d} = p_{1} ⋯ p_{d}s_{j} ,

(2) T_{l}^{y}I_{μ}(u) = t I_{μ}(u) if μ(l) = μ(l + 1) < d ,

(3) T_{l}^{y}I_{μ}(u) = t I_{μs_{l}}(u) if μ(l) < μ(l + 1) ,

(4) T_{l}^{y}I_{μ}(u) = I_{μs_{l}}(u) + (t − 1)I_{μ}(u) if μ(l) > μ(l + 1) .

Noting φ_{F} = I_{F} (1), the proof of our Theorems completes.

References

[1] Aomoto, K.: Finiteness of a cohomology associated with certain Jackson integrals, Tôhoku Math. J. 43(1991), 75–101.

[2] Cherednik, I.: Quantum Knizhnik-Zamolodchikov equations and affine root systems, Commn. Math. Phys. 150 (1992), 109–136.

[3] Cherednik, I.: Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators. Internat. Math. Res. Notices 1992, No.9, 171-180

[4] Kato, S.: R-matrix arising from affine Hecke algebras and its application to Macdonald’s difference operators. Commun. Math. Phys. 165, 533-553 (1994)

[5] Kirillov, A.N. and Noumi, M.: Raising operators for Macdonald polynomials and the integrality of double Kostka coefficients, Duke Math. J. (to appear).

[6] Lusztig, G.: Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635.

[7] Macdonald, I.G.: Symmetric Functions and Hall Polynomials (Second Edition), Oxford Mathematical Monographs, Clarendon Press, Oxford, 1995.

[8] Mimachi, K.: A solution to quantum Knizhnik-Zamolodchikov equations and its application to eigenvalue problems of the Macdonald type, Duke Math. J. (Dec.,1996).

(K. Mimachi) Department of Mathematics, Kyushu University, Hakozaki, Fukuoka 812, Japan

  E-mail address: mimachi@math.kyushu-u.ac.jp

(M. Noumi) Department of Mathematics, Kobe University, Rokko, Kobe 657, Japan

  E-mail address: noumi@math.s.kobe-u.ac.jp