MACDONALD FUNCTIONS ASSOCIATED TO COMPLEX REFLECTION GROUPS

TOSHIAKI SHOJI

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

To Robert Steinberg

Abstract. Let W be the complex reflection group 𝔖_{n} ⋉ (ℤ/eℤ)^{n}. In the author’s previous paper [S1], Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_{n}, they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall-Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture.

0. Introduction

Macdonald functions P_{λ}(x; q, t), which were introduced by I.G. Macdonald [M2] in 1987, are two variables versions of Hall-Littlewood functions P_{λ}(x; t). Those Hall-Littlewood functions and Macdonald functions are parametrized by partitions λ of n. Since partitions of n parameterize irreducible characters of the symmetric group 𝔖_{n}, we may say that these functions are associated to symmetric groups. On the other hand, Hall-Littlewood functions are closely related to Green polynomials of a finite general linear group GL_{n}(𝔽_{q}). In this direction, partitions λ of n occur as unipotent classes of GL_{n}(𝔽_{q}). Unipotent classes of other finite classical groups such as Sp_{2n}(𝔽_{q}), SO_{2n+1}(𝔽_{q}) have more complicated patterns. Lusztig introduced in [L2] (unipotent) symbols, as a generalization of the notion of partitions, to describe such unipotent classes in connection with Springer representations of Weyl groups. He also introduced in [L1] a notion of symbols to parameterize unipotent characters of finite classical groups.

In [S1], the author constructed Hall-Littlewood functions associated to complex reflection groups W ≃ 𝔖_{n} ⋉ (ℤ/eℤ)^{n}. In [S2], [S3], some related topics are discussed. Our Hall-Littlewood functions are parametrized by e-tuples of partitions (which parameterize irreducible characters of W), or rather by various types of e-symbols. In the case where e = 2, W is the Weyl group of type B_{n}. In this case, Hall-Littlewood functions attached to unipotent symbols are closely related to Green functions of Sp_{2n}(𝔽_{q}) or SO_{2n+1}(𝔽_{q}). It is also expected that our Hall-Littlewood functions attached to symbols have some connection with unipotent characters.

This paper is an attempt to generalize Hall-Littlewood functions P_{Λ}^{±}(x; t) (where Λ is an e-symbol) to the two variables version P_{Λ}^{±}(x; q, t), just as in the case of P_{λ}(x; q, t). We call such functions P_{Λ}^{±}(x; q, t) Macdonald functions associated to W. In the case of original Macdonald functions, they are characterized as simultaneous eigenfunctions of various Macdonald operators. We also construct Macdonald operators having (conjecturally) good properties. However, we note that our construction of Macdonald operators works only for a special type of symbols (in the case where e = 2, this is exactly the symbols used to parameterize unipotent characters), though Macdonald functions can be constructed for any type of symbols. In the case of original Macdonald operators, the representation matrix with respect to the basis of Schur functions is a triangular matrix, with distinct eigenvalues. In our case, the matrix of Macdonald operators turns out to be a block triangular matrix, where the blocks correspond to families of symbols. One can conjecture that the matrices appearing in the diagonal blocks have no common eigenvalues. Assuming this conjecture, we show that the Macdonald operator characterizes Macdonald functions, not as eigenfunction, but as a unique solution of linear systems attached to the above diagonal blocks.

The properties of Macdonald functions discussed in this paper are just a part of those established for the original Macdonald functions. We hope to discuss more about them in a subsequent paper.

1. Symmetric functions with two parameters

1.1. An e-tuple of partitions α = (α^{(0)}, …, α^{(e−1)}) is called an e-partition. We define the size |α| of α by |α| = ∑_{k=0}^{e−1} |α^{(k)}|, where |α^{(k)}| is the size of the partition α^{(k)}. For a partition α : α_{1} ≥ α_{2} ≥ ⋯ ≥ α_{k} ≥ 0, let l(α) be the number of non-zero parts α_{k}. We denote by P_{n} the set of partitions of n, and P_{n,e} the set of e-partitions of size n. Let W be the complex reflection group 𝔖_{n} ⋉ (ℤ/eℤ)^{n}. Then the set of conjugacy classes in W is in one to one correspondence with the set P_{n,e}.

Let us fix a sequence of positive integers m = (m_{0}, …, m_{e−1}), and consider indeterminates x_{j}^{(k)} (0 ≤ k ≤ e−1, 1 ≤ j ≤ m_{k}). We denote by x = x_{m} the whole variables (x_{j}^{(k)}), and also denote by x^{(k)} the variables x_{1}^{(k)}, …, x_{m_{k}}^{(k)}. Let ζ be a primitive e-th root of unity in ℂ. For each integer r ≥ 1 and i such that 0 ≤ i ≤ e − 1, put

p_{r}^{(i)}(x) = ∑_{j=0}^{e−1} ζ^{ij}p_{r}(x^{(j)}),

where p_{r}(x^{(j)}) denotes the r-th power sum symmetric function with respect to the variables x^{(j)}. We put p_{r}^{(i)}(x) = 1 for r = 0. For an e-partition α = (α^{(0)}, …, α^{(e−1)}) with α^{(k)} : α_{1}^{(k)} ≥ ⋯ ≥ α_{m_{k}}^{(k)}, we define a function p_{α}(x) by

p_{α}(x) = ∏_{k=0}^{e−1} ∏_{j=1}^{m_{k}} p_{α_{j}^{(k)}}^{(k)}(x).

Next, we define the Schur function s_{α}(x) and monomial symmetric functions m_{α}(x) associated to α by

s_{α}(x) = ∏_{k=0}^{e−1} s_{α^{(k)}}(x^{(k)}), m_{α}(x) = ∏_{k=0}^{e−1} m_{α^{(k)}}(x^{(k)}),

where s_{α^{(k)}}(x^{(k)}) (resp. m_{α^{(k)}}(x^{(k)} ) denotes the usual Schur function (resp. monomial symmetric function) associated to the partition α^{(k)} with respect to the variables x^{(k)}. These are the symmetric functions associated to complex reflection groups W ≃ 𝔖_{n} ⋉ (ℤ/eℤ)^{n}, as given in [M1, Appendix B].

1.2. Put 𝔖_{m} = 𝔖_{m_{0}} ×⋯×𝔖_{m_{e−1}}. We denote by Ξ_{m} = ⨂_{k=0}^{e−1} ℤ[x_{1}^{(k)}, …, x_{m_{k}}^{(k)}]^{𝔖_{m_{k}}} the ring of symmetric polynomials (with respect to 𝔖_{m}) with variables x = (x_{j}^{(k)}). Ξ_{m} has a structure of a graded ring Ξ_{m} = ⨁_{i≥0} Ξ_{m}^{i}, where Ξ_{m}^{i} consists of homogeneous symmetric polynomials of degree i, together with the zero polynomial. We consider the inverse limit

Ξ^{i} = lim_{\overleftarrow{m}} Ξ_{m}^{i}

with respect to homomorphisms ρ_{m′,m} : Ξ_{m′}^{i} → Ξ_{m}^{i}, where m′ = (m′_{0}, …, m′_{e−1}) with m′_{k} = m_{k} + l for some integer l ≥ 0, and ρ_{m′,m} is induced from the homomorphism ⨂_{k} ℤ[x_{1}^{(k)}, …, x_{m′_{k}}^{(k)}] → ⨂_{k} ℤ[x_{1}^{(k)}, …, x_{m_{k}}^{(k)}] given by sending x_{i}^{(k)} to 0 for i > m_{k}, and

leaving the other x_{j}^{(k)} invariant. Ξ = ⨁_{i≥0} Ξ^{i} is called the space of symmetric functions. Schur functions s_{α}(x) with infinitely many variables x_{1}^{(k)}, x_{2}^{(k)} … are regarded as elements in Ξ^{n} with n = |α|, and the set {s_{α}(x) | α ∈ P_{n,e}} forms a ℤ-basis of Ξ^{n}. Similarly, {m_{α}(x) | α ∈ P_{n,e}} gives a ℤ-basis of Ξ^{n}. Put Ξ_{ℂ} = ℂ ⊗ Ξ. Then {p_{α}(x) | α ∈ P_{n,e}} gives rise to a basis of Ξ_{ℂ}.

1.3. Let q, t be independent indeterminates and let F = ℂ(q, t) be the field of rational functions in q, t. We consider the F -algebra of symmetric functions Ξ_{F} = F ⊗_{ℤ} Ξ with coefficients in F. Let α = (α^{(0)}, …, α^{(e−1)}) be an e-partition. For each partition α^{(k)} : α_{1}^{(k)} ≥ α_{2}^{(k)} ≥ ⋯, put

z_{α^{(k)}}(q, t) = ∏_{j=1}^{l(α^{(k)})} \frac{1 − ζ^{k}q^{α_{j}^{(k)}}}{1 − ζ^{k}t^{α_{j}^{(k)}}} .

We then define z_{α}(q, t) ∈ F by

(1.3.1)  z_{α}(q, t) = z_{α} ∏_{k=0}^{e−1} z_{α}_{(k)}(q, t),

where z_{α} is the order of the centralizer of w_{α} in W (a representative of the conjugacy class of W corresponding to α ∈ P_{n,e}). Explicitly, z_{α} is given as follows. For α ∈ P_{n,e}, put l(α) = ∑_{k=0}^{e−1} l(α^{(k)}). For a partition α = (1^{n_{1}}, 2^{n_{2}}, …), put z_{α} = ∏_{i≥1} i^{n_{i}}n_{i}!. Then z_{α} = e^{l(α)} ∏_{k=0}^{e−1} z_{α^{(k)}}.

We define a sesquilinear form on Ξ_{F} by

(1.3.2)  ⟨p_{α}, p_{β}⟩ = δ_{α,β}z_{α}(q, t)

for α, β ∈ P_{n,e}. Let x = (x_{j}^{(k)}), y = (y_{j}^{(k)}) (0 ≤ k ≤ e−1) be two sequences of infinitely many variables, and we define an infinite product of x and y by

(1.3.3)  Π(x, y; q, t) = ∏_{k=0}^{e−1} ∏_{i,j=1}^{∞} ∏_{r=0}^{∞} \frac{1 − tx_{i}^{(k−r−1)}y_{j}^{(k)}q^{r}}{1 − x_{i}^{(k−r)}y_{j}^{(k)}q^{r}}

= ∏_{k=0}^{e−1} ∏_{i,j=1}^{∞} ∏_{r=0}^{∞} \frac{1 − tx_{i}^{(k)}y_{j}^{(k+r+1)}q^{r}}{1 − x_{i}^{(k)}y_{j}^{(k+r)}q^{r}} .

(Here the upper indices of the variables in the formula should be read mod e). Note that in the case where q = 0, the product Π(x, y; q, t) reduces to the product Ω(x, y; t) introduced in [S1, 2.5], (or rather [S2, (5.7.1)], see the remark there).

Lemma 1.4. Π(x, y; q, t) has the following expansion.

Π(x, y; q, t) = ∑_{α}z_{α}(q, t)^{−1}p_{α}(x)\overline{p_{α}(y)}

where α runs over all the e-partitions of any size.

Proof. Taking the log on both sides of the first formula of (1.3.3),

\begin{matrix}log Π( & x, y; q, t) \\ = ∑_{k} ∑_{i,j} ∑_{r=0}^{∞} ∑_{m=1}^{∞}\{\frac{1}{m}(x_{i}^{(k−r)}y_{j}^{(k)}q^{r})^{m} − \frac{t^{m}}{m} (x_{i}^{(k−r−1)}y_{j}^{(k)}q^{r})^{m}\}. \end{matrix}

By making use of the equation

\frac{1}{e} ∑_{a=0}^{e−1}(ζ^{k}ζ^{−k′})^{a} = δ_{k,k′},

the above formula can be written as

\begin{aligned}log Π(x, y; q, t) = ∑_{a=0}^{e−1} & ∑_{m≥1} ∑_{r≥0} ∑_{k,k′,i,j}\{\frac{1}{em}ζ^{ar}ζ^{a(k−r)}ζ^{−ak′}(x_{i}^{(k−r)}y_{j}^{(k′)}q^{r})^{m} \\ & − \frac{t^{m}}{em}ζ^{a(r+1)}ζ^{a(k−r−1)}ζ^{−ak′}(x_{i}^{(k−r−1)}y_{j}^{(k′)}q^{r})^{m}\}. \end{aligned}

MACDONALD FUNCTIONS 5

It follows that

log Π(x, y; q, t) = ∑_{a=0}^{e−1} ∑_{m≥1} ∑_{r≥0} ∑_{k,k′,i,j} \frac{1 − ζ^{a}t^{m}}{em} ζ^{ar}q^{rm}(x_{i}^{(k)}y_{j}^{(k′)})^{m}ζ^{ak}ζ^{−ak′}

= ∑_{a=0}^{e−1} ∑_{m≥1} ∑_{k,k′,i,j} \frac{1}{em} · \frac{1 − ζ^{a}t^{m}}{1 − ζ^{a}q^{m}} ζ^{ak}ζ^{−ak′}(x_{i}^{(k)}y_{j}^{(k′)})^{m}

= ∑_{a=0}^{e−1} ∑_{m=1}^{∞} \frac{1}{em} · \frac{1 − ζ^{a}t^{m}}{1 − ζ^{a}q^{m}} p_{m}^{(a)}(x)\overline{p_{m}^{(a)}(y)}.

Hence we have

Π(x, y; q, t) = ∏_{a=0}^{e−1} ∏_{m=1}^{∞} exp\{\frac{1}{em} · \frac{1 − ζ^{a}t^{m}}{1 − ζ^{a}q^{m}} p_{m}^{(a)}(x)\overline{p_{m}^{(a)}(y)}\}

= ∑_{α}z_{α}(q, t)^{−1}p_{α}(x)\overline{p_{α}(y)}.

□

1.5. For a fixed y_{j}^{(k)}, we define a function g_{m,±}^{(k)}(x; q, t) as the coefficient of (y_{j}^{(k)})^{m} in

(1.5.1)  ∏_{i} ∏_{r≥0} \frac{1 − tx_{i}^{(k∓r∓1)}y_{j}^{(k)}q^{r}}{1 − x_{i}^{(k∓r)}y_{j}^{(k)}q^{r}} = ∑_{m≥0} g_{m,±}^{(k)}(x; q, t)(y_{j}^{(k)})^{m}

and put, for each α = (α_{j}^{(k)}) ∈ P_{n,e},

g_{α,±}(x; q, t) = ∏_{j,k} g_{α_{j}^{(k)},±}^{(k)}.

Then we have

(1.5.2)  Π(x, y; q, t) = ∑_{α}g_{α,+}(x; q, t)m_{α}(y) = ∑_{α}m_{α}(x)g_{α,−}(y; q, t).

In fact, by comparing the first formula of (1.3.3) and (1.5.1), we have

Π(x, y; q, t) = ∏_{k,j} ∑_{α_{j}^{(k)}≥0} g_{α_{j}^{(k)},+}^{(k)}(x; q, t)(y_{j}^{(k)})^{α_{j}^{(k)}}

= ∑_{α}g_{α,+}(x; q, t)m_{α}(y).

This shows the first equality. If we compare the second formula of (1.3.3) and (1.5.1) by replacing x and y, the second equality is obtained in a similar way.

Now by using a similar argument as in [M1, VI, 2.7], we see that

(1.5.3)  ⟨g_{α,+}(x; q, t), m_{β}(x)⟩ = ⟨m_{α}(x), g_{β,−}(x; q, t)⟩ = δ_{α,β}.

In particular, the functions g_{α,±}(x; q, t) form a basis of Ξ_{F} dual to m_{α}. Hence {g_{n}(x; q, t) | n ≥ 0} are algebraically independent over F, and Ξ_{F} = F [g_{1}, g_{2}, …].

The following lemma can be proved in a similar way as [M1, VI, 2.13] by using (1.5.2) and (1.5.3).

Lemma 1.6. Let E^{±} : Ξ_{F} → Ξ_{F} be two F -linear operators. Then the following conditions are equivalent.

(i) ⟨E^{+}f, g⟩ = ⟨f, E^{−}g⟩ for any f, g ∈ Ξ_{F} (ii) E_{x}^{+}Π(x, y; q, t) = E_{y}^{−}Π(x, y; q, t), where the suffix x indicates the action of E^{±}

on the x variables, and similarly for y.

1.7. We shall give an explicit form of the function g_{α,±}(x; q, t). For this, we prepare some notation. For μ = (μ_{0}, …, μ_{e−1}) ∈ ℤ_{≥0}^{e}, put

f_{μ,±}^{(k,i)}(x; q, t) = ∏_{a=0}^{e−1} ∏_{j=1}^{μ_{a}} \frac{x_{i}^{(k∓a)} − tx_{i}^{(k∓a∓1)}q^{e(j−1)}}{1 − q^{ej}} .

Let M_{m} be the set of sequences μ = (μ^{(1)}, μ^{(2)}, …) such that μ^{(i)} ∈ ℤ_{≥0}^{e} and that ∑_{i} |μ^{(i)}| = m. For μ = (μ_{1}, μ_{2}, …, μ_{e}) ∈ ℤ_{≥0}^{e}, put n(μ) = ∑_{j}(j − 1)μ_{j}. Then

Proposition 1.8. For each m ≥ 0, we have

(1.8.1)  g_{m,±}^{(k)}(x; q, t) = ∑_{μ∈M_{m}} ∏_{i} f_{μ^{(i)},±}^{(k,i)}(x; q, t)q^{n(μ^{(i)})}.

Proof. By [M1, I, §2, Ex. 5], the following identity of formal power series is known.

∏_{i=0}^{∞} \frac{1 − bq^{i}t}{1 − aq^{i}t} = ∑_{m≥0}(∏_{i=1}^{m} \frac{a − bq^{i−1}}{1 − q^{i}} )t^{m}.

Substituting this into

A = ∏_{r≥0} \frac{1 − tx_{i}^{(k∓r∓1)}yq^{r}}{1 − x_{i}^{(k∓r)}yq^{r}} = ∏_{a=0}^{e−1} ∏_{r≥0} \frac{1 − tx_{i}^{(k∓a∓1)}yq^{re+a}}{1 − x_{i}^{(k∓a)}yq^{re+a}} .

with a = x_{i}^{(k∓a)}, b = tx_{i}^{(k∓a∓1)}, t = q^{a}y, q = q^{e}, we see that

A = ∏_{a=0}^{e−1} ∑_{μ_{a}=0}^{∞} ∏_{j=1}^{μ_{a}} \frac{x_{i}^{(k∓a)} − tx_{i}^{(k∓a∓1)}q^{e(j−1)}}{1 − q^{ej}} (q^{a}y)^{μ_{a}}

= ∑_{μ∈ℤ_{≥0}^{e}} f_{μ,±}^{(k,i)}(x; q, t)q^{n(μ)}y^{|μ|}.

It follows that

∏_{i≥1} ∏_{r≥0} \frac{1 − tx_{i}^{(k∓r∓1)}y_{j}^{(k)}q^{r}}{1 − x_{i}^{(k∓r)}y_{j}^{(k)}q^{r}} = ∑_{m≥0} ∑_{μ∈M_{m}} ∏_{i≥1} f_{μ^{(i)},±}^{(k,i)}(x; q, t)q^{n(μ^{(i)})}(y_{j}^{(k)})^{m}.

By comparing this with (1.5.1), we obtain the required formula. □

Remark 1.9. g_{m,±}^{(k)}(x; 0, t) coincides with the function q_{m,±}^{(k)}(x; t) introduced in [S1, 2.2]. By using a similar, but much simpler arguments as above, one obtains an alternative expression of q_{m,±}^{(k)}(x; t) as follows.

q_{m,±}^{(k)}(x; t) = ∑_{μ∈P_{m}} |𝔖_{μ}|^{−1} ∑_{w∈𝔖_{μ}} w{∏_{i=1}^{l(μ)}(x_{i}^{(k)} − tx_{i}^{(k∓1)})(x_{i}^{(k)})^{μ_{i}−1}},

where 𝔖_{μ} is the stabilizer of μ in 𝔖_{m}. (Here we are considering finite variables x_{i}^{(k)}, x_{i}^{(k∓1)} (1 ≤ i ≤ m), and 𝔖_{m} acts on both variables).

1.10. The notion of symbols was introduced in [S1]. (Although a more general setting was discussed in [S2], we do not use it in the discussion below. We remark that similar symbols were also considered by G. Malle in [Ma]). Let m = (m_{0}, …, m_{e−1}) be as before. We denote by Z_{n}^{0,0} = Z_{n}^{0,0}(m) the set of e-partitions α = (α^{(0)}, …, α^{e−1)}) ∈ P_{n,e} such that each α^{(k)} is written ( as an element in ℤ^{m_{k}}) in the form α^{(k)} : α_{1}^{(k)} ≥ ⋯ ≥ α_{m_{k}}^{(k)} ≥ 0. We express α as α = (α_{j}^{(k)}) in matrix form. Let us fix integers r ≥ s ≥ 0 and define an e-partition Λ^{0} = Λ^{0}(m, s, r) = (Λ^{(0)}, …, Λ^{(e−1)}) as follows.

(1.10.1)  Λ^{(0)} : (m_{0} − 1)r ≥ ⋯ 2r ≥ r ≥ 0,
Λ^{(i)} : s + (m_{i} − 1)r ≥ ⋯ ≥ s + 2r ≥ s + r ≥ s

for i = 0, …, e − 1. We denote by Z_{n}^{r,s} = Z_{n}^{r,s}(m) the set of e-partitions of the form Λ = α + Λ^{0}, where α ∈ Z_{n}^{0,0} and the sum is taken entry-wise. We write Λ = Λ(α) if Λ = α+Λ^{0}, and call it the e-symbol of type (r, s) corresponding to α. We often denote the symbol Λ = (Λ^{(0)}, …, Λ^{(e−1)}) in the form Λ = (Λ_{j}^{(k)}) with Λ^{(k)} : Λ_{1}^{(k)} > ⋯ > Λ_{m_{k}}^{(k)} for k = 0, …, e − 1.

Put m′ = (m_{0} + 1, …, m_{e−1} + 1), and define a shift operation Z_{n}^{r,s}(m) → Z_{n}^{r,s}(m′) by associating Λ′ = (Λ′_{0}, …, Λ′_{e−1}) ∈ Z_{n}^{r,s}(m) to Λ = (Λ_{0}, …, Λ_{e−1}) ∈ Z_{n}^{r,s}(m), where Λ′_{0} = (Λ_{0} + r) ∪ {0}, and Λ′_{k} = (Λ_{k} + r) ∪ {s} for k = 0, …, e − 1. In other words, for Λ = Λ(α), Λ′ is obtained as Λ′ = α+Λ^{0}(m′, s, r), where α is regarded as an element of Z_{n}^{0,0}(m′) by adding 0 in the entries of α. We denote by Z̄_{n}^{r,s} the set of classes in ∐_{m′} Z_{n}^{r,s}(m′) under the equivalence relation generated by shift operations. Note that P_{n,e} coincides with the set Z̄_{n}^{0,0}. Also note that Λ^{0} is regarded as a symbol in Z_{n}^{r,s} with n = 0.

Two elements Λ and Λ′ in Z̄_{n}^{r,s} are said to be similar, and are written as Λ ∼ Λ′, if there exist representatives in Z_{n}^{r,s}(m) such that all the entries of them coincide with multiplicities. The set of symbols which are similar to a fixed symbol is called a family in Z_{n}^{r,s}.

We define a function a : Z_{n}^{r,s} → ℤ_{≥0}, for Λ ∈ Z_{n}^{r,s}, by
(1.10.2)  a(Λ) = ∑_{λ,λ′∈Λ}min{λ, λ′} − ∑_{μ,μ′∈Λ^{0}} min{μ, μ′}.

The function a on Z_{n}^{r,s} is invariant under the shift operation, and it induces a function a on Z̄_{n}^{r,s}. Clearly, the a-function takes a constant value on each family in Z_{n}^{r,s}. We regard the a-function as a function on Z_{n}^{0,0} by using the bijection Z_{n}^{0,0} ≃ Z_{n}^{r,s}.

1.11. Hall-Littlewood functions P_{Λ}^{±}(x; t) and Q_{Λ}^{±}(x; t) attached to symbols Λ were introduced in [S1]. We shall now construct a two parameter version of Hall-Littlewood functions. Let us introduce a total order α ≺ β on Z_{n}^{0,0} such that a(α) ≥ a(β) whenever α ≺ β and that each family in Z_{n}^{0,0} forms an interval.

The following proposition is easily obtained by a similar argument as in Remark 4.9 in [S1] (i.e., a generalization of Gram-Schmidt orthogonalization process) if one notices that Π(x, y; 0, t) coincides with Ω(x, y; t) in [S1, 2.5].

Proposition 1.12. There exists a unique function P_{Λ}^{±}(x; q, t) ∈ Ξ_{F} for Λ ∈ Z_{n}^{r,s} satisfying the following two properties.

(i) P_{Λ}^{±}(x; q, t) for Λ = Λ(α) can be expressed in terms of s_{β}(x) as

P_{Λ}^{±} = s_{α} + ∑_{β}u_{α,β}^{±}s_{β},

with u_{α,β}^{±} ∈ F, where u_{α,β}^{±} = 0 unless β ≺ α and β ≁ α. (ii) ⟨P_{Λ}^{+}, P_{Λ′}^{−}⟩ = 0 unless Λ ∼ Λ′.

We then define Q_{Λ}^{±}(x; q, t) as the dual of P_{Λ}^{∓}(x; q, t), i.e., by the property that

⟨P_{Λ}^{+}, Q_{Λ′}^{−}⟩ = ⟨Q_{Λ}^{+}, P_{Λ′}^{−}⟩ = δ_{Λ,Λ′}.

P_{Λ}^{±}(x; q, t), Q_{Λ}^{±}(x; q, t) are called the Macdonald functions associated to complex reflection groups W (with respect to symbols in Z_{n}^{r,s}).

Remark 1.13. (i) The orthogonality relations of Macdonald functions given above imply, by [M1, VI, 2.7], that

(1.13.1)  Π(x, y; q, t) = ∑_{Λ}P_{Λ}^{+}(x; q, t)Q_{Λ}^{−}(y; q, t)

= ∑_{Λ}Q_{Λ}^{+}(x; q, t)P_{Λ}^{−}(y; q, t).

(ii) In the case where q = 0, the scalar product given in (1.3.2) coincides with the one given in [S1, 4.7]. Then by Proposition 4.8 in [S1], one sees that

(1.13.2)  P_{Λ}^{±}(x; 0, t) = P_{Λ}^{±}(x, t), Q_{Λ}^{±}(x; 0, t) = Q_{Λ}^{±}(x; t),

where the right hand sides are the Hall-Littlewood functions defined in [S1].

(iii) In the case where q = t, the scalar product in (1.3.2) coincides with the usual scalar product on the space Ξ_{ℚ}, where the Schur functions form an orthonormal basis of it. Hence Proposition 1.12 implies that

(1.13.3)  P_{Λ(α)}^{±}(x; t, t) = Q_{Λ(α)}^{±}(x; t, t) = s_{α}(x).

2. Macdonald operators

2.1. The original Macdonald functions related to symmetric groups are characterized as the simultaneous eigenfunctions of Macdonald operators (see [M1, VI]). In this section, we shall construct certain operators which can be viewed as a generalization of Macdonald operators to the case of W. Here we restrict ourselves to the case where symbols are of the type (r, s) with r = 1 and s = 0. (We note that the arguments in this section can not be applied to other types of symbols. See Remark 2.9.) In particular, Λ^{0} = δ, where δ = (δ^{(0)}, …, δ^{(e−1)}) with δ^{(k)} = (m_{k} − 1, …, 1, 0). Hence, in the case where e = 2 (i.e., W is the Weyl group of type B_{n}) with m_{1} = m_{0} +1, these symbols are exactly the ones used to parameterize unipotent characters of finite classical groups Sp_{2n}(𝔽_{q}) or SO_{2n+1}(𝔽_{q}) by Lusztig [L1]. Each family F in Z_{n}^{1,0} contains a unique element Λ_{F} = (Λ_{j}^{(k)}) with the property

(2.1.1)  Λ_{1}^{(0)} ≥ Λ_{1}^{(1)} ≥ ⋯ ≥ Λ_{1}^{(e−1)} ≥ Λ_{2}^{(0)} ≥ Λ_{2}^{(1)} ≥ · · · .

Such an element is called a special symbol associated to the family F. The set of families is in bijection with the set of special symbols. Special symbols are regarded as partitions of N = ∑ Λ_{j}^{(k)} by (2.1.1).

We shall define a partial order on the set of families in Z_{n}^{1,0}. Let F and F′ be families in Z_{n}^{1,0} and Λ_{F}, Λ_{F′} be special symbols corresponding to them. We put F < F′ if Λ_{F} < Λ_{F′} with respect to the dominance order on P_{N}. Recall that for λ = (λ_{i}), μ = (μ_{i}) ∈ P_{N}, the dominance order λ < μ is defined by the condition that

∑_{i=1}^{k} λ_{i} ≤ ∑_{i=1}^{k} μ_{i}, (1 ≤ k ≤ ∑_{j} m_{j}).

We define a partial order on Z_{n}^{1,0} by inheriting the partial order on the set of families. For λ = (λ_{i}) ∈ P_{N}, put

n(λ) = ∑_{i≥1}(i − 1)λ_{i}.

Then, for each Λ in a family F, the value a(Λ) is given as α(Λ) = n(Λ_{F}) − n(Λ^{0}), where Λ_{F} is regarded as an element in P_{M}. In particular, we have a(Λ) > a(Λ′) if Λ < Λ′. As before, by using the bijection Z_{n}^{0,0} ≃ Z_{n}^{1,0}, we consider the partial order on Z_{n}^{0,0}, which will be denoted by the same symbol.

2.2. In order to construct Macdonald operators, we shall start with finitely many variables x = x_{m}. We consider the expansion of Π(x, y; q, t) in the case of finitely many variables. Assume that x = (x_{j}^{(k)}) = x_{m} is as in 1.2 and put Ξ_{m,F} = F ⊗_{ℤ} Ξ_{m}. Let us denote by P_{m} the set of e-partitions α = (α_{j}^{(k)}) such that l(α^{(k)}) ≤ m_{k}. Then m_{α}(x) = 0 unless α ∈ P_{m}, and those non-zero m_{α}(x) form a basis of Ξ_{m,F}. The same is true for Schur functions. Also by a similar argument as in [M1] we see that g_{α}(x; q, t) such that α ∈ P_{m} form an F -basis of Ξ_{m,F}.

Now by substituting x_{j}^{(k)} = y_{j}^{(k)} = 0 for j > m_{k}, we have a finite version of (1.5.2), i.e., for x = x_{m}, y = y_{m}, we have

(2.2.1)  Π(x, y; q, t) = ∑_{α∈P_{m}} g_{α,+}(x; q, t)m_{α}(y) = ∑_{α∈P_{m}} m_{α}(x)g_{α,−}(y; q, t).

This enables us to define a scalar product on Ξ_{m,F} by

(2.2.2)  ⟨g_{α,+}, m_{β}⟩ = ⟨m_{α}, g_{β,−}⟩ = δ_{α,β}.

We now consider the restriction of the functions P_{Λ}^{±} to Ξ_{m,F}. By Proposition 1.12, one sees that {P_{Λ(α)}^{±} | α ∈ P_{m}} form a basis of Ξ_{m,F}. Moreover, the finite variables version of Proposition 1.12 holds, and P_{Λ}^{±} ∈ Ξ_{F} is obtained as the limit of P_{Λ}^{±} ∈ Ξ_{m,F} determined by these properties.

2.3. Let

I = I(m) = {i = ( i_{0} ; ⋮ ; i_{e−1}) ∈ ℤ^{e} | 1 ≤ i_{k} ≤ m_{k}}.

For each i ∈ I and u ∈ F, we define an F -linear operator T_{u,i}^{±} : F [x] → F [x] by T_{u,i}^{±}f = f′, where f′ is a polynomial obtained from f by replacing the variables x_{i_{k}}^{(k)} by ux_{i_{k∓1}}^{(k∓1)} for k = 0, …, e − 1. (Here we understand that i_{e} = i_{o}). More generally, for each r such that 1 ≤ r ≤ M_{1} = min_{k}{m_{k}}, we define I_{r} as the set of J = {i_{1}, …, i_{r}} consisting of i_{k} ∈ I such that any two i_{k} have no common entries (i.e., i_{j} − i_{k} does not contain 0 entries for each j ≠ k as vectors in ℤ^{e}). Then we define, for each J ∈ I_{r}, an operator T_{u,J}^{±} : F [x] → F [x] by T_{u,J}^{±} = ∏_{k=1}^{r} T_{u,i_{k}}^{±}. Note that T_{u,i_{k}}^{±} in the product commute with each other, and so T_{u,J}^{±} does not depend on the order of the product.

Let Z = Z(m) be the set of sequences β = (β_{j}^{(k)}) (0 ≤ k ≤ e − 1, 1 ≤ j ≤ m_{k}) with β_{i}^{(k)} ∈ ℤ_{≥0}. For each β ∈ Z, we denote by [β] the element in Z obtained from β by permuting the entries inside each row, so that each row is arranged in decreasing order. We often regard β as a matrix, and denote its k-th row by β^{(k)}. Then the set J is regarded as a subset of the set of indices {(k, j) | 0 ≤ k ≤ e − 1, 1 ≤ j ≤ m_{k}} of (β_{j}^{(k)})

We put, for each β ∈ Z and J ∈ I_{r},

⟨β, J⟩ = ∑_{(k,j)∈J} β_{j}^{(k)}.

𝔖_{m} acts naturally on Z and on I_{r}, respectively, and this pairing is 𝔖_{m}-invariant. We also note that the action of 𝔖_{m} on I_{r} is transitive.

The operation of T_{q,J}^{±} on x = (x_{i}^{(k)}) also induces an action β ↦ β_{J±} on Z by permuting the entries of β so that T_{q,J}^{±}x^{β} = q^{⟨β,J⟩}x^{β^{J±}}, where, as usual, x^{β} denotes

the monomial ∏_{i,k}(x_{i}^{(k)})^{β_{i}^{(k)}}. For each β ∈ Z, we define a function a_{β}(x) by

a_{β}(x) = ∑_{w∈𝔖_{m}} ε(w)w(x^{β})

We now define, for each 1 ≤ r ≤ M_{1}, an F -linear operator D_{±}^{r}(q, t) on F [x] by

(2.3.1)  D_{±}^{r}(q, t) = a_{δ}(x)^{−1} ∑_{w∈𝔖_{m}} ε(w) ∑_{J∈I_{r}} x^{w(δ)_{J±}}t^{⟨w(δ),J⟩}T_{q,J}^{±}.

Then D_{±}^{r}(q, t) can be written as

(2.3.2)  D_{±}^{r}(q, t) = ∑_{J∈I_{r}} A_{J}^{±}(x; t)T_{q,J}^{±},

with

A_{J}^{±}(x; t) = a_{δ}(x)^{−1} ∑_{w∈𝔖_{m}} ε(w)x^{w(δ)_{J±}}t^{⟨w(δ),J⟩}
= a_{δ}(x)^{−1}T_{t,J}^{±}a_{δ}(x)

= t^{r(r−1)/2} ∏_{k=0}^{e−1} ∏_{\begin{matrix}(k,i)∉J \\ (k,j)∈J \end{matrix}} \frac{x_{i}^{(k)} − tx_{j′}^{(k∓1)}}{x_{i}^{(k)} − x_{j}^{(k)}} , (2.3.3)

where we write J = {i_{1}, …, i_{r}}, and take (k ∓ 1, j′) ∈ i_{a} if (k, j) ∈ i_{a} for a = 1, …, r. The formulas (2.3.2) and (2.3.3) are the analogue of (3.4)_{r} and (3.5)_{r} in [M, VI, 3].

First we show that

Lemma 2.4. (i) For α ∈ P_{m}, we have

D_{±}^{r}(q, t)m_{α}(x) = ∑_{β} ∑_{J∈I_{r}} t^{⟨δ,J⟩}q^{⟨β,J⟩}s_{(β+δ)_{J±}−δ}(x),

where β ∈ Z runs over all the row permutations of α. (ii) For α ∈ P_{m}, we have,

D_{±}^{r}(q, t)m_{α}(x) = ∑_{β∈Z_{n}^{0,0}} b_{α,β}^{r,±}(q, t)m_{β}(x)

with b_{α,β}^{r,±}(q, t) ∈ F, where b_{α,β}^{r,±}(q, t) = 0 unless β ∼ α or β < α. In particular,

D_{±}^{r} is an operator on the space Ξ_{m,F}.

Proof. For each β ∈ Z_{n}^{0,0}, we have

D_{±}^{r}(q, t)x^{β} = a_{δ}(x)^{−1} ∑_{w_{1}∈𝔖_{m}} ε(w_{1}) ∑_{J∈I_{r}} t^{⟨w_{1}(δ),J⟩}q^{⟨β,J⟩}x^{(β+w_{1}(δ))_{J±}}.

If we replace β by w_{2}(β) for w_{2} ∈ 𝔖_{m} and put w_{2} = w_{1}w, the term (β + w_{1}(δ))_{J±} (resp. ⟨β, J⟩ ) is replaced by w_{1}((w(β) + δ)_{J′±}) (resp. ⟨w(β), J′⟩ ), respectively, with J′ = w_{1}^{−1}(J). It follows that for α ∈ Z_{n}^{0,0}, we have

D_{±}^{r}(q, t)m_{α}(x) = |𝔖_{α}|^{−1}a_{δ}(x)^{−1} ∑_{w,w_{1}∈𝔖_{m}} ε(w_{1})

× ∑_{J′∈I_{r}} t^{⟨δ,J′⟩}q^{⟨w(α),J′⟩}x^{w_{1}((w(α)+δ)_{J′±})}

= |𝔖_{α}|^{−1} ∑_{w∈𝔖_{m}} ∑_{J∈I_{r}} t^{⟨δ,J⟩}q^{⟨w(α),J⟩}s_{(w(α)+δ)_{J±}−δ}(x).

This proves (i).

Next we show (ii). Since δ = Λ^{0} by our assumption, α + δ coincides with the symbol Λ(α) for α ∈ Z_{n}^{0,0}. We note that if w ≠ 1, then the symbol w(α) + δ belongs to a family strictly smaller than the family containing α. On the other hand, if w = 1, (α+δ)_{J±} is obtained from the symbol α+δ by permuting some entries, and s_{(α+δ)_{J±}−δ} coincides with ±s_{γ}, where γ + δ is obtained from (α + δ)_{J±} by rearranging the rows in decreasing order. It follows that γ + δ is contained in the same family as α + δ. Hence, for β ∈ Z_{n}^{0,0} in the expression of (ii), we see that β < α if w ≠ 1 and β ∼ α if w = 1. □

Next we show

Lemma 2.5. The operators D_{±}^{r} are adjoint each other, i.e., we have

(2.5.1)  ⟨D_{+}^{r}f, g⟩ = ⟨f, D_{−}^{r}g⟩ , (f, g ∈ Ξ_{m,F}).

Proof. By Lemma 1.6, (2.5.1) is equivalent to the formula

(2.5.2)  Π^{−1}(D_{+}^{r})_{x}Π = Π^{−1}(D_{−}^{r})_{y}Π.

But for J ∈ I_{r}, we have

Π^{−1}(T_{q,J}^{+})_{x}Π = ∏_{k=0}^{e−1} ∏_{j≥1} ∏_{(k,i)∈J} \frac{1 − x_{i}^{(k)}y_{j}^{(k)}}{1 − tx_{i}^{(k)}y_{j}^{(k+1)}} ,

Π^{−1}(T_{q,J}^{−})_{y}Π = ∏_{k=0}^{e−1} ∏_{i≥1} ∏_{(k,j)∈J} \frac{1 − x_{i}^{(k)}y_{j}^{(k)}}{1 − tx_{i}^{(k−1)}y_{j}^{(k)}} .

It follows that both of Π^{−1}(D_{+}^{r})_{x}Π and Π^{−1}(D_{−}^{r})_{y}Π are independent of q. Hence in the proof of (2.5.2), we may assume that q = t. In other words, we have only to prove (2.5.1) under the assumption that q = t.

Now assume that q = t. Since

T_{t,J}^{±}(x^{w(δ)}f) = x^{w(δ)_{J±}}t^{⟨w(δ),J⟩}T_{t,J}^{±}f

for any polynomial f ∈ F [x], we have

D_{±}^{r}(t, t)f = a_{δ}^{−1} ∑_{J∈I_{r}} T_{t,J}^{±}(a_{δ}f).

It follows that for any α ∈ Z_{n}^{0,0},

D_{±}^{r}(t, t)s_{α} = a_{δ}^{−1} ∑_{J∈I_{r}} T_{t,J}^{±}(a_{α+δ})

= ∑_{J∈I_{r}} t^{⟨α+δ,J⟩}s_{(α+δ)_{J±}−δ}.

As before, s_{(α+δ)_{J±}−δ} coincides with ±s_{β}, where β + δ = [(α + δ)_{J±}] (under the notation in 2.3). Now in the case where q = t, {s_{α}(x) | α ∈ P_{m}} is an orthonormal basis of Ξ_{m,F}. It follows that

(2.5.3)  ⟨D_{+}^{r}(t, t)s_{α}, s_{β}⟩ = ∑_{J} ε_{J+}t^{⟨α+δ,J⟩},

where J runs over all the elements in I_{r} such that [(α + δ)_{J+}] coincides with β + δ, and ε_{J} = (−1)^{l(w)} with w ∈ 𝔖_{m} such that [(α + δ)_{J+}] = w((α + δ)_{J+}). But if β + δ = w((α + δ)_{J+}), we have α + δ = w^{−1}((β + δ)_{J′−}) = [(β + δ)_{J′−}] with J′ = w(J). Also in this case,

⟨β + δ, J′⟩ = ⟨ w(α + δ)_{J′+}, J′⟩ = ⟨w(α + δ), J′⟩ = ⟨α + δ, J⟩ .

Since ε_{J+} = ε_{J′−}, we see that the right hand side of (2.5.3) is equal to

∑_{J′} ε_{J′−}t^{⟨β+δ,J′⟩},

where J′ ∈ I_{r} runs over all the elements such that [(β + δ)_{J′−}] = α + δ. Clearly this coincides with ⟨s_{α}, D_{−}^{r}(t, t)s_{β}⟩. So the lemma is proved. □

2.6. We fix a total order ≺ on Z_{n}^{0,0} as in 1.11, so that it is compatible with the partial order <. Let B_{±}^{r} = (b_{α,β}^{r,±}) be the matrix consisting of the coefficients in the formula in Lemma 2.4 (ii) with respect to the total order ≺. We consider B_{±}^{r} as a block matrix with respect to the equivalence relation α ∼ β, and denote it as (B_{F,F′}^{r,±}), where B_{F,F′}^{r,±} is the submatrix of B_{±}^{r} corresponding to the families F, F′. Then Lemma 2.4 (ii) implies that B_{±}^{r} is lower triangular as a block matrix. We consider the Macdonald functions P_{Λ}^{±} constructed via ≺. The following result shows that the set of Macdonald functions attached to symbols in a fixed family behaves as an eigenfunction for Macdonald operators, where the eigenvalues should be replaced by the diagonal blocks B_{F,F}^{r,±} of B_{±}.

Proposition 2.7. Let P_{Λ}^{±}(x; q, t) ∈ Ξ_{m,F} be Macdonald functions attached to Λ = Λ(α). Then we have

D_{±}^{r}P_{Λ}^{±} = ∑_{β∼α}b_{α,β}^{r,±}P_{Λ(β)}^{±},

where the coefficients b_{α,β}^{r,±} are the same as in Lemma 2.4 (ii).

Proof. By Proposition 1.12, P_{Λ}^{±} can be written as

P_{Λ}^{±} = m_{α} + ∑_{β≺α,β≁α}u′_{α,β}m_{β}

with u′_{α,β} ∈ F. It follows, by Lemma 2.4 (ii), that one can write as

D_{±}^{r}P_{Λ}^{±} = ∑_{β∼α}b_{α,β}^{r,±}P_{Λ(β)}^{±} + ∑_{Λ′≺Λ,Λ′≁Λ}c_{Λ,Λ′}^{±}P_{Λ′}^{±}.

Hence, for each Λ′ such that Λ′ ≺ Λ and that Λ′ ≁ Λ, we have

⟨D_{+}^{r}P_{Λ}^{+}, P_{Λ′}^{−}⟩ = c_{Λ,Λ′}^{+}.

On the other hand thanks to Lemma 2.5, we have

⟨D_{+}^{r}P_{Λ}^{+}, P_{Λ′}^{−}⟩ = ⟨P_{Λ}^{+}, D_{−}^{r}P_{Λ′}^{−}⟩ = 0

since D_{−}^{r}P_{Λ′}^{−} is a linear combination of P_{Λ′′}^{−}, where Λ′′ ∼ Λ′ or Λ′′ ≺ Λ′. It follows that c_{Λ,Λ′}^{+} = 0 and the proposition holds for the + case. The − case is similar. □

In view of Proposition 2.7, it is important to know the diagonal part of B_{±}. By lemma 2.4, the matrix B_{F,F}^{±} is described as follows.

Lemma 2.8. For α, β ∈ Z_{n}^{0,0} such that β ∼ α, we have

b_{α,β}^{r,±}(q, t) = ∑_{\begin{matrix}J∈I_{r} \\ [Λ_{J±}]=Λ(β) \end{matrix}} ε_{Λ,J±}(tq^{−1})^{⟨δ.J⟩}q^{⟨Λ,J⟩},

where Λ = α + δ ∈ Z_{n}^{1,0} and ε_{Λ,J±} = (−1)^{l(w)} for w ∈ 𝔖_{m} such that [Λ_{J±}] = w(Λ_{J±}).

Remark 2.9. The results in this section work only for a special type of symbols. It seems to be difficult to extend the definition of Macdonald operators in (2.3.1) directly to a more general case. A naive idea for the general situation is to replace δ by Λ^{0} in the definition (2.3.1). Then one gets some operator related to the symbols associated with Λ^{0}. However, the thus obtained operator does not preserve the set of polynomials in x, since it involves the factor a_{Λ}^{0}(x)^{−1}. If one leaves the denominator a_{δ}(x) unchanged, and replaces δ in all other places, then the operator preserves polynomials, but it does not preserve the degree of them, and is not so useful.

3. A characterization of Macdonald functions

3.1. We write the operator D_{±}^{r} as D_{m,±}^{r} to indicate the dependence on m. The operators D_{m,±}^{r} are not compatible with the restriction homomorphisms ρ_{m′,m}. In the case where r = 1, one can modify D_{m,±}^{1} as discussed in [M1], so that they are compatible with ρ_{m′,m}. Let us define an operator E_{m}^{±} = E_{m}^{±}(q, t) on Ξ_{m,F} by

(3.1.1)  E_{m}^{±} = t^{−M}D_{m,±}^{1} − ∑_{i∈I} t^{⟨δ,i⟩ −M}.

We show the following lemma.

Lemma 3.2. The operators E_{m}^{±} are compatible with ρ_{m′,m}, i.e., we have

ρ_{m′,m} ∘ E_{m′}^{±} = E_{m}^{±} ∘ ρ_{m′,m}

for m′ = (m_{0} + 1, …, m_{e−1} + 1).

Proof. Let us define another operator Ẽ_{m}^{±} : Ξ_{m,F} → Ξ_{m,F} by

Ẽ_{m}^{±} = t^{−M}D_{m,±}^{1} − ∑_{i∈I} t^{⟨δ,i⟩ −M}a_{δ}^{−1}T_{1,i}^{±}a_{δ}.

Then for each α ∈ P_{m}, we have

(3.2.1)  Ẽ_{m}^{±}m_{α} = ∑_{β} ∑_{i∈I}(q^{⟨β,i⟩} − 1)t^{− ∑ i_{k}}s_{(β+δ)_{i±}−δ},

where β ∈ Z runs over all the row permutations of α, and i = (i_{0}, …, i_{e−1}). In fact, by applying [M1, VI, 4] one can write m_{α} = ∑_{β} s_{β} with β as above. Since a_{δ}^{−1}T_{1,i}^{±}a_{δ}s_{β} = s_{(β+δ)_{i±}−δ}, we obtain (3.2.1).

We claim that Ẽ_{m}^{±} is compatible with ρ_{m′,m}, i.e.,

(3.2.2)  ρ_{m′,m} ∘ Ẽ_{m′}^{±} = Ẽ_{m}^{±} ∘ ρ_{m′,m}.

Recall that the map ρ_{m′,m} : Ξ_{m′,F} → Ξ_{m,F} is defined by substituting x_{m_{k}+1}^{(k)} = 0 for k = 0, …, e − 1. Take α ∈ Z_{n}^{0,0}(m) and let α′ ∈ Z_{n}^{0,0}(m′) be the element obtained by adding 0’s to the last part of α. We consider the expression of Ẽ_{m′}^{±}m_{α}′ as given in (3.2.1). Let δ′ be the element for m′ corresponding to δ for m, and take β′ ∈ Z(m′). We note that s_{(β′+δ′)_{i±}−δ′} goes to zero under ρ_{m′,m} if there exists some k such that β_{m_{k}+1}^{(k)} ≠ 0 for β′ = (β_{j}^{(k)}). In fact, if we write β′ + δ′ = (γ_{j}^{(k)}), then γ_{j}^{(k)} > 0 for j ≤ m_{k}. Hence if β_{m_{k}+1}^{(k)} ≠ 0 for some k, then (β′ + δ′)_{i±} = (γ′_{j}^{(k)}) contains a row γ′^{(k′)} whose entries are all non-zero, and so s_{(β′+δ′)_{i±}−δ′} goes to zero.

It follows that in the expression of Ẽ_{m′}^{±}m_{α}′ in (3.2.1), we may only consider β′ (a row permutation of α′ ) such that the last column consists of zeros. One can embed Z(m) into Z(m′) by adding 0 to the last part of β ∈ Z(m). Then those β′ are identified, under the embedding Z_{n}^{0,0}(m) ↪ Z_{n}^{0,0}(m′), with a row permutation β of α. Now I(m) is also embedded into I(m′) in the same way. Take i ∈ I(m′) such that i ∉ I(m). If i is not equal to i_{1} = ^{t}(m_{0} + 1, …, m_{e−1} + 1), then s_{(β′+δ′)_{i±}−δ′} goes to zero under ρ_{m′,m} by the same reason as before. If i = i_{1}, then ⟨β′, i⟩ = 0. In any case, such i does not give a contribution, and we may only consider i ∈ I(m) in (3.2.1). Hence (3.2.2) holds.

Let H̃_{±} be the coefficient matrix of Ẽ_{m}^{±}m_{α} in terms of m_{β}. Then by Lemma 2.4 (ii), H̃_{±} can be expressed as a block matrix H̃_{±} = (H̃_{F,F′}^{±}), where H̃_{F,F′}^{±} = 0 unless F = F′ or F′ < F. Then by comparing (3.2.1) with Lemma 2.4 (i) on the diagonal parts, we see that

H̃_{F,F}^{±} = t^{−M}B_{F,F}^{±} − ∑_{i∈I} t^{⟨δ,i⟩ −M}.

On the other hand, if we write H̃_{F,F}^{±} = (h_{α,β}^{±})_{α,β∈F}, Proposition 2.7 together with (3.1.1) implies that

E_{m}^{±}P_{Λ(α)}^{±} = ∑_{β∼α}h_{α,β}^{±}P_{Λ(β)}^{±}.

But by (3.2.2), the matrix H̃_{F,F}^{±} does not depend on the shift operation under m′ → m. This shows that the operator E_{m}^{±} is compatible with ρ_{m′,m}. The lemma is proved. □

3.3. By Lemma 3.2, one can define an operator E^{±} = E^{±}(q, t) on Ξ_{F} as the limit of E_{m}^{±}. Since E_{m}^{±} satisfies a similar formula as given in Lemma 2.5, the operator E^{±} also satisfies the adjointness property, i.e., we have

(3.3.1)  ⟨E^{+}f, g⟩ = ⟨f, E^{−}g⟩, (f, g ∈ Ξ_{F}).

By Lemma 2.4 (ii), one can write, for each α ∈ P_{n,e},

(3.3.2)  E^{±}(q, t)m_{α}(x) = ∑_{β∈Z_{n}^{0,0}} h_{α,β}^{±}(q, t)m_{β}(x)

with h_{α,β}^{±}(q, t) ∈ F, where h_{α,β}^{±}(q, t) = 0 unless β ∼ α or β < α. Let H_{±} = (h_{α,β}^{±}), and write it as a block matrix H_{±} = (H_{F,F′}^{±}) as in 2.6.

In the case of symmetric groups, the matrix H_{±} is a triangular matrix, with distinct eigenvalues. This property was used to characterize Macdonald functions as eigenfunctions of Macdonald operators. As an analogy, it is likely that the following property holds for the diagonal parts of the block matrix B_{±}.

Conjecture A. Let F, F′ be any distinct families in Z_{n}^{0,0}. Then the matrices H_{F,F}^{±} and H_{F′,F′}^{±} have no common eigenvalues (according to the sign + or −, respectively).

We have verified the conjecture in the case where e = 2, and n ≤ 5.

3.4. Before giving a characterization of Macdonald functions in terms of Macdonald operators, we prepare an easy lemma. Let A = (a_{ij}) (resp. B = (b_{ij})) be a square matrix of degree m (resp. n), and let C = (C_{α,β})_{1≤α,β≤n} be a block matrix of size mn, consisting of blocks C_{α,β} of size m, defined by

C_{α,β} = \begin{cases}A − b_{α,α}I_{m} & if β = α, \\ −b_{α,β}I_{m} & otherwise. \end{cases}

We consider a matrix equation AX = XB, where X = (x_{ij}) is a m × n matrix of unknown variables. Then this equation can be regarded as a system of linear equations with respect to the mn variables {x_{ij}}, whose coefficient matrix is given by the matrix C. Moreover, if B is a triangular matrix, then C is block wise triangular, and so det C ≠ 0 if and only if det(A − b_{α,α}I_{m}) ≠ 0 for 1 ≤ α ≤ n. Hence we have the following lemma. Lemma 3.5. Under the above notation, the following are equivalent.

(i) The matrix equation AX = XB has a unique solution X = 0.

(ii) det C ≠ 0.

(iii) The matrices A and B have no common eigenvalues.

We now show the following. Theorem 3.6. Suppose that Conjecture A holds. Then the Macdonald functions P_{Λ}^{±} ∈ Ξ_{F} are characterized by the following two properties.

(3.6.1)  P_{Λ}^{±} = m_{α} + ∑_{β<α}w_{α,β}^{±}m_{β},
(3.6.2)  E^{±}P_{Λ}^{±} = ∑_{β∼α}h_{α,β}^{±}P_{Λ(β)}^{±}.

In particular, P_{Λ}^{±} are determined independently from the choice of the total order ≺. Proof. Let {F_{i} | i ∈ I} be the set of families in Z_{n}^{0,0}. We give a total order ≺ on the index set I, according to the total order ≺ on Z_{n}^{0,0}, and write H_{±} as H_{±} = (H_{ij})_{i,j∈I}, where H_{ij} = H_{F_{i},F_{j}}^{±}. (Since the following discussion is independent of the sign {±}, we omit them). Let X = (w_{α,β}^{±}) be the transition matrix between basis {m_{α}} and {P_{Λ}^{±}} of Ξ_{F}, and write it as X = (X_{ij}). By Proposition 1.12, X is block wise lower triangular, with identity diagonal blocks. Moreover, by Proposition 2.7, we see that XHX^{−1} = G, where G = (G_{ij}) is a block diagonal matrix with diagonal blocks G_{ii} = H_{ii}. In order to prove the theorem, we have only to show the following.

(3.6.3) Let X = (X_{ij}) be a block wise lower triangular matrix, with identity diagonal blocks, such that XHX^{−1} = G. Then X is determined uniquely, and X_{ij} = 0 unless i = j or F_{j} < F_{i}.

We show (3.6.3). The equation XHX^{−1} = G can be written as

(3.6.4)  H_{ii}X_{ij} − X_{ij}H_{jj} = ∑_{j≺k≺i} X_{ik}H_{kj}

for any pair j ≺ i. By backwards induction on j, we may assume that X_{ik} are already determined for j ≺ k ≺ i. Then (3.6.4) determines X_{ij} uniquely, by Conjecture A and Lemma 3.5. Now suppose that F_{j} ≮ F_{i}. Again by induction, we may assume that X_{ik} = 0 unless F_{k} < F_{i}. Since H_{kj} = 0 unless F_{j} < F_{k} by (3.3.2), we must have H_{ii}X_{ij} − X_{ij}H_{jj} = 0. This implies that X_{ij} = 0 by Lemma 3.5, and we obtain (3.6.3). Thus the theorem is proved. □

Remark 3.7. The Hall-Littlewood function P_{Λ}^{±}(x; t) given in [S1] coincides with P_{Λ}^{±}(x; 0, t). Hence it satisfies similar formulas as (3.6.1), (3.6.2). In particular, P_{Λ}^{±}(x; t), and so the Kostka functions K_{α,β}^{±}(x; t) do not depend on the choice of the total order. This answers the questions posed in [S1, Remark 4.5, (ii)] and in [GM, Remark 2.4], modulo the truth of the conjecture.

3.8. We give here some examples of the matrices B_{F,F}^{±} and H_{F,F}^{±} for some small rank cases with e = 2. Assume that W is the Weyl group of type C_{2}. Then the symbols and families are given as follows.

F_{1} = \{\binom{3 0}{0}\}, F_{2} = \{\binom{2 1 0}{2 1}\},
F_{3} = \{\binom{2 0}{1}, \binom{2 1}{0}, \binom{1 0}{2}\},

which correspond, in this order, to the double partitions of 2,

(2; −), (−; 11), (1; 1), (11; −), (−; 2)

respectively. In the tables below, we express the matrices B_{F,F}^{±}, H_{F,F}^{±} as B_{F}, H_{F}, since they are independent of the sign ±. We have

B_{F_{1}} = (1), B_{F_{2}} = (t^{3}q + tq), B_{F_{3}} = (\begin{matrix}0 & q & tq^{2} \\ q & 0 & −tq \\ tq^{2} & −q^{2} & 0 \end{matrix}) .

Up to C_{5}, only 3-element or 1-element families occur. The Weyl group of type C_{6} contains a unique 10-element family, which is given as follows,

\begin{matrix}F_{4} = \{ & \binom{4 2 0}{3 1}, \binom{4 2 1}{3 0}, \binom{4 3 0}{2 1}, \binom{4 3 1}{2 0}, \binom{4 3 2}{1 0}, \\ \binom{4 1 0}{3 2}, \binom{3 2 0}{4 1}, \binom{3 2 1}{4 0}, \binom{3 1 0}{4 2}, \binom{2 1 0}{4 3}\}. \end{matrix}

The corresponding double partitions of 6 are, in this order, as follows.

(21; 21), (211; 2), (22; 11), (221; 1), (222; −),
(2; 22), (11; 31), (111; 3), (1; 32), (−; 33).

The matrix B_{F} is given by

B_{F} = (\begin{matrix}0 & q & t^{2}q^{3} & 0 & tq^{2} & tq^{2} & t^{3}q^{4} & 0 & 0 & t^{2}q^{3} \\ q & 0 & 0 & t^{2}q^{3} & −tq^{3} & −tq & 0 & t^{3}q^{4} & 0 & −t^{2}q^{2} \\ t^{2}q^{3} & 0 & 0 & q & −tq & −tq^{3} & −t^{3}q^{3} & 0 & t^{2}q^{3} & 0 \\ 0 & t^{2}q^{3} & q & 0 & tq^{2} & tq^{2} & 0 & −t^{3}q^{3} & −t^{2}q^{2} & 0 \\ tq^{2} & −t^{2}q^{2} & −q^{2} & tq^{2} & 0 & 0 & −t^{2}q^{2} & t^{3}q^{2} & 0 & 0 \\ tq^{2} & −q^{2} & −t^{2}q^{2} & tq^{2} & 0 & 0 & 0 & 0 & t^{3}q^{4} & −t^{2}q^{4} \\ t^{3}q^{4} & 0 & −t^{2}q^{4} & 0 & −tq^{3} & 0 & 0 & q & tq^{2} & −t^{2}q^{2} \\ 0 & t^{3}q^{4} & 0 & −t^{2}q^{4} & tq^{4} & 0 & q & 0 & −tq & t^{2}q \\ 0 & 0 & t^{2}q^{3} & −tq^{3} & 0 & t^{3}q^{4} & tq^{2} & −q^{2} & 0 & t^{2}q^{3} \\ t^{2}q^{3} & −tq^{3} & 0 & 0 & 0 & −t^{3}q^{3} & −tq^{3} & q^{3} & t^{2}q^{3} & 0 \end{matrix}) .

Throughout the above examples, H_{F} is given by

H_{F} = t^{−(2m+1)}B_{F} − t^{−(2m+1)} \frac{(1 − t^{m})(1 − t^{m+1})}{(1 − t)^{2}}

for symbols of the shape m = (m + 1, m), (so, m = 1 for F_{1}, F_{3}, and m = 2 for F_{2}, F_{4}, respectively). Note that B_{F} is not necessarily symmetric. However, if we put q = t, it turns out to be symmetric since D^{1} is a self adjoint operator, and the representation matrix with respect to the orthonormal basis of Schur functions coincides with the diagonal blocks (B_{F}).

3.9. In the case of symmetric groups, Macdonald operators are commuting with each other since they are simultaneously diagonalizable. In our case, Proposition 2.7 shows that D_{±}^{r} are simultaneously diagonalizable in the sense of block matrices. So they are commuting with each other if and only if the matrices B_{F,F}^{r,±} are commuting with for r = 1, …, M_{1} (for a fixed F). As the following examples show, one might expect that B_{F,F}^{r,±} are commuting with each other in general.

First consider a simple example. Let F_{3} = {Λ_{1}, Λ_{2}, Λ_{3}} be the 3-element family of C_{2} as in 3.8. We consider its m times shifts F_{3}^{(m)} = {Λ′_{1}, Λ′_{2}, Λ′_{3}}, where

Λ′_{i} = \binom{x + m, y + m, m − 1, …, 0}{z + m, m − 1, …, 0}

for Λ_{i} = \binom{x y}{z}. Then the matrices B_{F_{3}}^{r} are given as

B_{F_{3}}^{r} = aB_{F_{3}}^{1} + b

with

a = t^{2m} ∑_{J′} t^{⟨δ,J′⟩}, b = ∑_{J} t^{⟨δ,J⟩},

where J ∈ I_{r} runs over the elements of the form J = \{i_{j_{1}}, …, i_{j_{r}}\} for i_{j} = \binom{j+1}{j} with 2 ≤ j ≤ m + 1, and J′ ∈ I_{r−1} runs over the elements having similar properties. This implies that B_{F_{3}}^{r} are all commuting with for 1 ≤ r ≤ m.

In the following, we discuss some related results, i.e., we show that when e = 2, and q = t, then the operators D^{r} are commuting with each other. First we prepare a lemma. (Since we deal with the case where e = 2, we omit the sign ± in the discussion below.)

Lemma 3.10. Assume that e = 2. Then, for each r, r′, there exists a bijective map φ : I_{r} × I_{r′} → I_{r} × I_{r′} satisfying the following properties: let J ∈ I_{r}, J′ ∈ I_{r′} and put φ(J, J′) = (K, K′). For each α ∈ Z, we have (i) α_{JJ′} = α_{K′K}, (ii) ⟨α, J⟩ + ⟨α_{J}, J′⟩ = ⟨α, K′⟩ + ⟨α_{K′}, K⟩.

Proof. Take J ∈ I_{r}, J′ ∈ I_{r′}. Let J_{0} be the subset of J consisting of i = \binom{a}{b} such that there exists i′ = \binom{x}{y} ∈ J′ with x = a or y = b, and let J′_{0} be the subset of J′ having similar properties. We define an equivalence relation on J_{0} by connecting i = \binom{a}{b}, i′ = \binom{c}{d} ∈ J_{0} when there exists \binom{x}{y} ∈ J′ such that (x, y) = (a, d) or (x, y) = (c, b). We denote by {J_{C} | C ∈ C} the set of equivalence classes in J_{0}. Then the class J_{C} has the following form.

J_{C} = \{\binom{a_{1}}{c}, \binom{a_{2}}{b_{1}}, …, \binom{a_{k−1}}{b_{k−2}}, \binom{a_{k}}{b_{k−1}}, \binom{d}{b_{k}}\},

where \binom{a_{i}}{b_{i}} ∈ J′ for 1 ≤ i ≤ k. We put

J′_{C} = \{\binom{a_{i}}{b_{i}} | p ≤ i ≤ q\},

where p = 0 (resp. q = k + 1) if there exists \binom{a_{0}}{b_{0}} ∈ J′ (resp. \binom{a_{k+1}}{b_{k+1}} ∈ J′) such that b_{0} = c (resp. a_{k+1} = d), and p = 1, q = k otherwise.

For each C ∈ C, we define the sets K_{C}, K′_{C} as follows:

K_{C} = \begin{cases}J_{C} & if p = 1, q = k, \\ J′_{C} & if p = 1, q = k + 1, \\ J′_{C} & if p = 0, q = k, \\ \{\binom{a_{0}}{b_{1}}, …, \binom{a_{k}}{b_{k+1}}\} & if p = 0, q = k + 1. \end{cases}

K′_{C} = \begin{cases}\{\binom{a_{2}}{c}, \binom{a_{3}}{b_{1}}, …, \binom{a_{k}}{b_{k−2}}, \binom{d}{b_{k−1}}\} & if p = 1, q = k, \\ J_{C} & if p = 1, q = k + 1, \\ J_{C} & if p = 0, q = k, \\ J′_{C} & if p = 0, q = k + 1. \end{cases}

Since J, J′ are subsets of the index set of elements in Z, J_{C}, etc. induce permutations on the entries of elements in Z. We denote by x_{C}, x′_{C} (resp. y_{C}, y′_{C}) the permutations in 𝔖_{M} corresponding to J_{C}, J′_{C} (resp. K_{C}, K′_{C}), respectively. Then it is easy to check that

(3.10.1)  x′_{C} ∘ x_{C} = y_{C} ∘ y′_{C}.

We put

K = (J − J_{0}) ∪ ⋃_{C∈C} K_{C}, K′ = (J′ − J′_{0}) ∪ ⋃_{C∈C} K′_{C}.

Components of K (resp. K′) are mutually disjoint, and we see that K ∈ I_{r}, K′ ∈ I_{r′}. We now define the map φ : I_{r} × I_{r′} → I_{r} × I_{r′} by φ(J, J′) = (K, K′). Then one can check that φ^{2} = id, and so φ is a bijection. The assertion (i) follows from (3.10.1). To show (ii), it is enough to verify the formula in the case where J = J_{C}, J′ = J′_{C}, K = K_{C}, K′ = K′_{C}. The assertion is clear when K_{C} = J′_{C}, K′_{C} = J_{C}. Assume that p = 1, q = k or p = 0, q = k + 1. Then one can check by a direct computation that

⟨α, J_{C}⟩ = ⟨α_{K′_{C}}, K_{C}⟩, ⟨α_{J_{C}}, J′_{C}⟩ = ⟨α, K′_{C}⟩ .

Hence the formula holds in these cases also, and the lemma follows. □

Proposition 3.11. Assume that e = 2, and q = t. Then the operators D^{r}(t, t) are commuting with each other for r = 1, …, M_{1}.

Proof. As remarked in 3.9, it is enough to show that the matrices B_{F}^{r} = B_{F,F}^{r,±} are commuting with each other for r = 1, …, M_{1}. By Lemma 2.4 (i), the part corresponding to the diagonal block in the expression of D^{r}(t, t)s_{α} is given as

∑_{J∈I_{r}} t^{⟨Λ,J⟩}s_{Λ_{J} −δ},

where Λ = α + δ. Let X be the part of D^{r′}(t, t)D^{r}(t, t)s_{α} corresponding to the diagonal blocks. Then we have

X = ∑_{J∈I_{r}} ε_{J}t^{⟨Λ,J⟩} ∑_{J′′∈I_{r′}} t^{⟨[Λ_{J}],J′′⟩}s_{[Λ_{J}]_{J′′}−δ}.

For each J, there exists w ∈ 𝔖_{m} such that [Λ_{J}] = w(Λ_{J}) and ε_{J} is given by ε_{J} = (−1)^{l(w)}. Then if we put J′ = w^{−1}(J′′), we have ⟨[Λ_{J}], J′′⟩ = ⟨Λ_{J}, J′⟩, and [Λ_{J}]_{J′′} = w(Λ_{JJ′}). Hence one can write

X = ∑_{J∈I_{r}} ∑_{J′∈I_{r′}} ε_{J,J′}t^{⟨Λ,J⟩ + ⟨Λ_{J} ,J′⟩}s_{[Λ_{JJ′}]−δ},

where ε_{J,J′} = (−1)^{l(w′)} for w′ ∈ 𝔖_{m} such that [Λ_{JJ′}] = w′(Λ_{JJ′}). Now by applying Lemma 3.10 for α = Λ, we see that X coincides with the diagonal part for D^{r}(t, t)D^{r′}(t, t)s_{α}. The proposition is proved. □

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