GREEN FUNCTIONS ASSOCIATED TO COMPLEX REFLECTION GROUPS G(e, 1, n)

TOSHIAKI SHOJI

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

0. Introduction

Green polynomials Q_{λ}^{μ}(q) of GL_{n}(F_{q}), where λ, μ are partitions of n, were first introduced by J.A.Green [G] in 1955 in a combinatorial framework of symmetric functions. They are obtained as the transition matrix between Schur functions and Hall-Littlewood functions. Those Green functions play a crucial role in [G] in describing irreducible characters of GL_{n}(F_{q}). In 1976, P. Deligne and G. Lusztig constructed in [DL] Green functions Q_{T}^{G} for finite reductive groups G(F_{q}) in general, as the restriction to unipotent classes of G(F_{q}) of Deligne-Lusztig’s virtual character R_{T}^{G}(θ), associated to a maximal torus T of G defined over F_{q}. The Green function Q_{χ}, for an irreducible character χ of the Weyl group W of G, is defined as a certain linear combination of Q_{T}^{G} for various T. In the case of GL_{n}(F_{q}), Q_{χ^{λ}}(u_{μ}) coincides with Q_{λ}^{μ}(q), where χ^{λ} is the irreducible character of W = 𝔖_{n} corresponding to λ, and u_{μ} is the unipotent class corresponding to μ. Lusztig proved in [L3], generalizing the result in [S1], that there exists a simple algorithm of computing Green functions. In this context, Green functions arise as a unique solution of a certain type of matrix equation

(*)  P Λ^{t}P = Ω

with unknown P, Λ. (The entries of the matrix P give Green functions Q_{χ}(u)). Note that the matrix Ω is completely determined by the property of irreducible characters of W, while the shape of P and Λ are determined by the property of the Springer correspondence between unipotent classes of G and irreducible characters of W. It is known, by the geometric interpretation of Green functions in [L5], that the entries of the matrices P and Λ are in ℤ[q].

In [GM], Geck and Malle considered an analogy of the equation in (*) by making use of a-functions of irreducible characters of Weyl groups and families of unipotent characters, and showed that they have a unique solution with coefficients in ℚ(t). They verified in the case of exceptional groups that the matrices P, Λ have coefficients in ℤ[t], and also the matrix Λ provides the polynomials conjectured in [L6] expressing the cardinality of special pieces in G(F_{q}). They conjecture that similar facts will hold for reductive groups in general.

In the case of classical groups, the Springer correspondence is described in terms of certain combinatorial objects called “u-symbols” introduced by Lusztig [L2]. The remarkable fact is that the matrix equation (*) is completely described by the combinatorial property of such u-symbols. The notion of u-symbols was generalized by Malle [Ma1] so that it fits to the case of complex reflection groups G(e, 1, n) as a generalization of Weyl group of type C_{n}. Now the ingredients used to construct the equation (*) make sense even for this case, and one can define a matrix equation by using the combinatorics of symbols. By using a similar argument as in [GM], it is shown that this equation has a unique solution. So it would be natural to call the solution of this equation as the Green function associated to G(e, 1, n).

After Green [G], the character table of GL_{n}(F_{q}), especially the part corresponding to the values at unipotent elements of unipotent characters, is completely dominated by Green functions. Thanks to Lusztig [L3], similar facts hold in general; the character values at unipotent elements of finite reductive groups, say, Sp_{2n}(F_{q}) or SO_{2n+1}(F_{q}), of unipotent characters are dominated by Green functions (or rather by generalized Green functions). Lusztig showed that his method of classifying unipotent characters of finite reductive groups makes sense even for other Coxeter groups W such as H_{3}, H_{4}, and determined their ”degrees”, even there does not exist a finite group G(F_{q}) whose Weyl group is W. Inspired by this, Malle [Ma1] classified, in the case of G(e, 1, n), the “unipotent characters” and described their degrees. One might expect that Green functions associated to G(e, 1, n) will provide us the values of unipotent characters at other “unipotent elements”.

In this note, we try to find a combinatorial theory of symmetric functions which fits to the situation for G(e, 1, n). In our theory, the role of partitions is replaced by symbols. In particular, we construct Hall-Littlewood functions associated to G(e, 1, n), which are parametrized by symbols. We show that a large part of arguments used to construct Green polynomials of GL_{n}(F_{q}) can be generalized to our setting, and that the Green functions are obtained as the transition matrix between Schur functions and Hall-Littlewood functions. Thus obtained Green functions are rational functions in ℚ(t). As some examples show, it is very likely that these Green functions are actually polynomials in t with positive integral coefficients.

The details of this note will be found in [S3].

The author is grateful for G. Malle and F. Lübeck for some useful discussions, and for Malle for the computation of examples by using computers.

1. A background from finite reductive groups

1.1. Let G be a connected reductive algebraic group defined over a finite field F_{q}, with Frobenius map F : G → G. We assume that G is split over F_{q}. We are interested in the representation theory of the finite group G^{F} over Q̄_{l}, an algebraic closure of the l-adic number field Q_{l} with l ≠ ch F_{q}. For an F -stable maximal torus T and a character θ : T^{F} → Q̄_{l}^{∗}, Deligne-Lusztig’s virtual character R_{T}^{G}(θ) : G^{F} → Q̄_{l} is defined. The Green function Q_{T}^{G} is defined as the restriction of R_{T}^{G}(θ) to the set of unipotent elements in G^{F}, which is independent of the choice of θ. The functions R_{T}^{G}(θ) play a crucial role in the representation theory of G^{F}, and it is known that the computation of R_{T}^{G}(θ) is reduced to the computation of various Green functions of smaller reductive groups.

Let W be the Weyl group of G. Then the G^{F} -conjugacy classes of F -stable maximal tori are parametrized by the conjugacy classes of W. We denote by T_{w} the F -stable maximal torus corresponding to w ∈ W (up to conjugacy). Let V_{uni} be the space of G^{F} -invariant functions on the set of unipotent elements in G^{F}. Thus Q_{T}^{G} ∈ V_{uni}. Let W^{∧} be the set of irreducible characters of W. For χ ∈ W^{∧}, let us define a function Q_{χ} on V_{uni} by

Q_{χ} = |W |^{−1} ∑_{w∈W} χ(w)Q_{T_{w}}^{G}.

We refer Q_{χ} also as Green functions. Clearly, the determination of Q_{T_{w}}^{G} for any w ∈ W is equivalent to that of Q_{χ} for any χ ∈ W^{∧}.

1.2. Let I be the set of pairs (C, ρ), where C is a unipotent class in G, and ρ is an irreducible character of A_{G}(u). (Here C is F -stable, and u is an element in C^{F}. A_{G}(u) is the component group Z_{G}(u)/Z_{G}^{0}(u)). It is known, by a suitable choice of u ∈ C^{F}, that the G^{F} -conjugacy classes in C^{F} is in bijection with the set of conjugacy classes in A_{G}(u). We denote by u_{a} the class in C^{F} corresponding to a ∈ A_{G}(u). For each i = (C, ρ), we define a function Y_{i} ∈ V_{uni} by

Y_{i}(v) = \begin{cases}ρ(a) & if v ∼ u_{a} (G^{F} -conjugate), \\ 0 & if v ∉ C^{F} . \end{cases}

Then {Y_{i} | i ∈ I} forms a basis of the space V_{uni}.

There exists a natural injection f, the so-called “Springer correspondence” from W^{∧} to I. We denote by I_{0} the image of W^{∧} under the map f. If f(χ) = i, we write Q_{χ} also as Q_{i}. Then it is known that Q_{χ} can be written as

Q_{χ} = Q_{i} = ∑_{j∈I_{0}} p_{ij}Y_{j}

with coefficients p_{ij} ∈ Q̄_{l}. In fact, it is clear that Q_{i} is written as a linear combination of Y_{j} with j ∈ I, and it is a deep result due to Lusztig that p_{ij} = 0 if j ∉ I_{0}. We fix a total order on I_{0} which is compatible with the closure relations of unipotent classes, and consider the square matrix P = (p_{ij}). The determination of Q_{i} for various i ∈ I_{0} is equivalent to the determination of the matrix P. Lusztig showed that P is determined as a unique solution of the matrix equation of the form P Λ^{t}P = Ω as in (*), where Ω is a known matrix obtained from the data on W^{∧}. In fact, we define an equivalence relation ∼ on I_{0}, by the condition that i ∼ i′ if i = (C, ρ), i′ = (C′, ρ′) with C = C′, and consider the block matrix with respect to this equivalence relation. Then it is shown that P is a block lower triangular matrix with diagonal blocks of the form q^{d_{u}}I if the block corresponds to C ∋ u. Here d_{u} is the dimension of the variety B_{u} = {B_{1} ∈ G/B | u ∈ B_{1}}. Furthermore, the matrix Λ is a non-singular block diagonal matrix. By this condition, P and Λ are uniquely determined from the equation (*).

1.3. In the case where G = Sp_{2n} or SO_{2n+1}, all of the ingredients involved in the above arguments are described in a purely combinatorial way; the set I is in bijection with certain combinatorial objects, “u-symbols”. Then the subset I_{0} of I corresponds to the u-symbols of defect 1. Furthermore, the equivalence class in I with respect to ∼ corresponds to the similarity class of u-symbols. The assignment u → d_{u} is defined as a function a : I_{0} → ℤ_{≥0} combinatorially.

As explained in the introduction, these formalism can be extended to the case of complex reflection groups W = G(e, 1, n). In the next section, we formulate the equation corresponding to (*) for such W.

2. Green functions associated to G(e, 1, n)

2.1. Let W = 𝔖_{n} ⋉ (ℤ/eℤ)^{n} be the imprimitive complex reflection group G(e, 1, n), and V = ℂ^{n} the natural reflection representation of W. Let S(V) be the symmetric algebra of V, and I_{+} the ideal of S(V) generated by the homogeneous W -invariant vectors of strictly positive degrees. We denote by R = ⊕R_{i} the coinvariant algebra of W, which is a graded algebra defined as the the quotient of S(V) by I_{+}. Let t be an indeterminate. The Poincaré polynomial P_{W} (t) is defined as P_{W} (t) = ∑_{i≥0}(dim_{ℂ} R_{i})t^{i}, which is explicitly given as

P_{W} (t) = ∏_{i=1}^{n} \frac{t^{ei} − 1}{t − 1}.

For any class function f on W, we define R(f) by

(2.1.1)  R(f) = (t − 1)^{n}P_{W} (t)\frac{1}{|W |} ∑_{w∈W} \frac{det_{V} (w)f(w)}{det_{V} (t · id_{V} −w)},

where det_{V} denotes the determinant on V. If χ is an irreducible character of W, then R(χ) coincides with ∑_{i}⟨χ, R_{i}⟩_{W} t^{i}, where ⟨ , ⟩_{W} denotes the inner product of class functions. Hence R(χ) coincides with the fake degree of χ, and R(f) ∈ ℤ[t] for a generalized character f of W.

Let N^{∗} be the number of complex reflections in W. Then N^{∗} is the maximal degree in R, and the W -module R_{N^{∗}} coincides with \overline{det}_{V}.

2.2. An e-partition α = (α^{(0)}, …, α^{(e−1)}) is an e-tuple of partitions α^{(k)}. The size |α| of α is defined as |α| = ∑_{i=0}^{e−1} |α^{(k)}|. We denote by P_{n,e} the set of e-partitions of size n. Then the set W^{∧} of irreducible characters of W is in bijection with the set P_{n,e}. We denote by χ^{α} the irreducible character corresponding to α ∈ P_{n,e}. In particular, the unit character corresponds to α = ((n), −, …, −), and \overline{det}_{V} corresponds to α = (−…. , −, (1^{n})).

Let us fix m = (m_{0}, …, m_{e−1}), where m_{k} are positive integers such that m_{k} ≥ n for any k. We denote by Z_{n}^{0,0} = Z_{n}^{0,0}(m) the set of e-partitions α with size n, such that α^{(k)} is regarded as an element in ℤ^{m_{k}} written in the form, α^{(k)} : α_{1}^{(k)} ≥ ⋯ ≥ α_{m_{k}}^{(k)} ≥ 0. We fix integers r ≥ s ≥ 0, and consider an e-tuple of partitions Λ^{0} = Λ^{0}(m) = (Λ_{0}, …, Λ_{e−1}) defined by

(2.2.1)  Λ_{0} : (m_{0} − 1)r ≥ ⋯ ≥ 2r ≥ r ≥ 0,
Λ_{i} : s + (m_{i} − 1)r ≥ ⋯ ≥ s + 2r ≥ s + r ≥ s  for 1 ≤ i ≤ 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-wisely. We denote by Λ = Λ(α) if Λ is as above, and call it the e-symbol of type (r, s) corresponding to α. Let us 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 m′ = (m_{0} + 1, …, m_{e−1} + 1) with Λ′_{0} = (Λ_{0} + r) ∪ {0} and Λ′_{k} = (Λ_{k} + r) ∪ {s} for k = 1, …, e − 1.

Two elements Λ and Λ′ in Z_{n}^{r,s} is said to be similar if all the entries of them coincide each other with multiplicities. The set of symbols which are similar to a fixed symbol is called a similarity class in Z_{n}^{r,s}.

We shall define a function a : Z_{n}^{r,s} → ℤ_{≥0}. Take Λ ∈ Z_{n}^{r,s} and let Λ^{0} be as before. We put

(2.2.2)  a(Λ) = ∑_{λ,λ′∈Λ}min(λ, λ′) − ∑_{μ,μ′∈Λ^{0}} min(μ, μ′),

where in the first sum, we assume that λ ≠ λ′ if λ and λ′ are contained in the same Λ_{i}, and similarly for the second sum (this occurs only when r = 0). The function a on Z_{n}^{r,s} is invariant under the shift operation, and takes a constant value on each similarity class in Z_{n}^{r,s}.

Remark 2.3. The notion of symbols already appeared in several articles in various forms. If e = 1, Z_{n}^{r,s} is in bijection with the set of partitions of n. Next consider the case where e = 2. Then the set Z_{n}^{1,0} (up to shift operation, all the same below) was used in [L1] to parameterize unipotent characters of Sp_{2n}(F_{q}) or SO_{2n+1}(F_{q}). In that case, the a-function defined in (2.2.2) coincides with the original a-function, i.e., the exact power of q dividing the degree of the corresponding unipotent character. The set Z_{n}^{2,1} (resp. Z_{n}^{2,0}) was used in [L2] to describe the generalized Springer correspondence for G = Sp_{2n}(k) (resp. G = SO_{2n+1}(k)) subject to the condition that ch k ≠ 2. In this case, the similarity classes are in 1 to 1 correspondence with the unipotent classes of G, and the value a(Λ) for such a symbol Λ coincides with dim B_{u}, (see, for example [L4, 4.4]). While, the set Z_{n}^{4,2} was used in [LS] to describe the generalized Springer correspondence for Sp_{2n}(k) with ch k = 2. On the other hand, for arbitrary e, the set Z_{n}^{1,0} was used in [Ma1] to parameterize unipotent degrees associated to the complex reflection group G(e, 1, n).

2.4. Under the identification of Z_{n}^{0,0} with P_{n,e}, we consider the a function as the function on the set P_{n,e}. Also the similarity classes in P_{n,e} are defined by inheriting the classes in Z_{n}^{r,s}. We denote by α ∼ β if α, β ∈ P_{n,e} are in the same similarity class. We choose a total order ≻ on P_{n,e} such that a(α) ≤ a(β) whenever α ≻ β, and that each similarity class in P_{n,e} forms an interval. In the following, we consider the matrices of degree |P_{n,e}| with respect to this order. For each α, β ∈ P_{n,e}, we define a polynomial ω_{α,β} ∈ ℤ[t] by

(2.4.1)  ω_{α,β} = t^{N^{∗}}R(χ^{α} ⊗ χ^{β} ⊗ \overline{det}_{V}),

and put Ω = (ω_{α,β}), the matrix of degree |P_{n,e}|. Let P = (p_{α,β}) and Λ = (λ_{α,β}) be matrices of degree |P_{n,e}|. We consider the following system of equations with respect to the unknown variables λ_{α,β}, p_{α,β}.

λ_{α,β} = 0 unless α ∼ β,
(2.4.2)  p_{α,β} = 0 unless either α ≻ β and α ≁ β, or α = β,
p_{α,α} = t^{a(α)},
P Λ^{t}P = Ω.

2.5. There exists a system of equations which is closely related to (2.4.2). Let Ω′ = (ω′_{α,β}) be a matrix defined by

(2.5.1)  ω′_{α,β} = t^{N^{∗}}R(χ^{α} ⊗ \overline{χ}^{β} ⊗ \overline{det}_{V}),

where \overline{χ}^{β} is the complex conjugate of χ^{β}. We consider a matrix equation of the form

(2.5.2)  P′Λ′ ^{t}P′′ = Ω′,

where either of P′ = (p′_{α,β}) and P′′ = (p′′_{α,β}) satisfies similar conditions as in the second and the third one in (2.4.2), and Λ′ = (λ′_{α,β}) as in the first one. Let σ = (σ_{α,β}) be the permutation matrix realizing the complex conjugation of irreducible characters of W, i.e., σ_{α,β} = 1 if χ^{β} = \overline{χ}^{α} and σ_{α,β} = 0 otherwise. Then we have Ω′ = Ωσ, and the equation P Λ^{t}P = Ω implies that

P · Λσ · σ^{−1}^{t}P σ = Ω′.

It is easily checked that the equation (2.4.2) has a solution if and only if the equation (2.5.2) has a solution, and in that case, we have P′ = P, Λ′ = Λσ, P′′ = σ^{−1}^{t}P σ. In the remainder of this note, we report that the solution of (2.5.2) can be described in terms of certain symmetric functions as in the case of GL_{n}.

3. A combinatorial setting for G(e, 1, n)

3.1. For a given m = (m_{0}, …, m_{e−1}) as in 2.2, we introduce indeterminates x_{j}^{(k)} (0 ≤ k ≤ e − 1, 1 ≤ j ≤ m_{k}). We denote by x the whole variables (x_{j}^{(k)}), and also denote by x^{(k)} the variables x_{1}^{(k)}, …, x_{m_{k}}^{(k)}. Power sum symmetric functions and Schur functions are defined as in [M, Appendix B]. 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

(3.1.1)  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 the monomial symmetric function m_{α}(x) by

(3.1.2)  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)}.

3.2. Let t be an indeterminate. In what follows we regard the variables x_{i}^{(k)} defined for k ∈ ℤ/eℤ ≃ {0, 1, …, e − 1}. For each 0 ≤ k ≤ e − 1 and an integer r ≥ 0, we define a function q_{r,±}^{(k)}(x; t) by

(3.2.1)  q_{r,±}^{(k)}(x; t) = ∑_{i≥1}(x_{i}^{(k)})^{r+δ} \frac{∏_{j} x_{i}^{(k)} − tx_{j}^{(k±1)}}{∏_{j≠i} x_{i}^{(k)} − x_{j}^{(k)}}  (r ≥ 1),

where δ = m_{k} − 1 − m_{k±1}, and by q_{r,±}^{(k)}(x; t) = 1 for r = 0. In the product of the denominator, x_{j}^{(k)} runs over all the variables in x^{(k)} except x_{i}^{(k)}, while in the numerator, x_{j}^{(k±1)} runs over all the variables in x^{(k±1)}. Note that if x_{i}^{(k)} = x_{i}^{(k±1)} = x_{i} and m_{k} = m_{k±1}, q_{r,±}^{(k)}(x; t) coincide with the function q_{r}(x; t) given in [M, III, 2.9]. It can be shown that q_{r,±}^{(k)}(x; t) is a polynomial in x, t with integral coefficients, homogeneous of degree r with respect to the variable x.

For an e-partition α = (α^{(0)}, …, α^{(e−1)}) ∈ P_{n,e}, we define a function q_{α,±}(x) by

(3.2.2)  q_{α,±}(x; t) = ∏_{k=0}^{e−1} ∏_{j=1}^{m_{k}} q_{α_{j}^{(k)},±}^{(k)}(x; t).

Next, for a partition α^{(k)} : α_{1}^{(k)} ≥ ⋯ ≥ α_{l_{k}}^{(k)} > 0, (here l_{k} = l(α^{(k)}) is the number of parts of α^{(k)}), we define a function z_{α^{(k)}}(t) by

z_{α^{(k)}}(t) = ∏_{j=1}^{l_{k}}(1 − ζ^{k}t^{α_{j}^{(k)}})^{−1},

Then we define z_{α}(t) by

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

where z_{α} is the order of the centralizer of w_{α} in W. Explicitly, z_{α} is given as follows. For α ∈ P_{n,e}, put l(α) = ∑_{i=0}^{e−1} l(α^{(i)}). 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)}}.

The importance of the quantity z_{α}(t) lies in the following relations with R(f) in (2.1.1);

det_{V} (t · id_{V} −w_{α}) = ∏_{k=0}^{e−1} ∏_{j=1}^{l_{k}}(t^{α_{j}^{(k)}} − ζ^{k}).

In particular, we have z_{α}(t^{−1}) = z_{α}t^{n} det_{V} (t · id_{V} −w_{α})^{−1}.

We now introduce infinitely many variables x_{i}^{(k)}, y_{i}^{(k)} for i = 1, 2, … and for 0 ≤ k ≤ e − 1. We may regard the above functions as functions with infinitely many variables x_{1}^{(k)}, x_{2}^{(k)}, . . .. Under this setting, the following proposition holds.

Proposition 3.3. Let

Ω(x, y; t) = ∏_{k=0}^{e−1} ∏_{i,j} \frac{1 − tx_{i}^{(k+1)}y_{j}^{(k)}}{1 − x_{i}^{(k)}y_{j}^{(k)}} .

Then we have

(3.3.1)  Ω(x, y; t) = ∑_{α}q_{α,+}(x; t)m_{α}(y) = ∑_{α}m_{α}(x)q_{α,−}(y; t),
(3.3.2)  Ω(x, y; t) = ∑_{α}z_{α}(t)^{−1}p_{α}(x)p̄_{α}(y),

where α runs over all the e-partitions of any size. In (3.3.2), p̄_{α}(y) denotes the complex conjugate of p_{α}(y).

3.4. We now introduce functions Q_{Λ}^{±}(x; t) and P_{Λ}^{±}(x; t) associated to symbols Λ ∈ Z_{n}^{r,s}, which is in analogy with Q_{λ}(x; t) and P_{λ}(x; t) in [M, III]. Let A be the subring of ℚ(t) consisting of functions which have no pole at t = 0. Then A is a local ring with the unique maximal ideal tA. Hence A^{∗} = A − tA is the set of units in A. In what follows, we identify the set Z_{n}^{r,s} and Z_{n}^{0,0} by the natural map, and we fix a total order ≺ on Z_{n}^{0,0} as in 1.4. Then we have the following theorem.

Theorem 3.5. (i) For each Λ ∈ Z_{n}^{r,s}, there exists a unique function P_{Λ}^{±}(x; t) satisfying the following properties.

(a) P_{Λ}^{±}(x; t) can be expressed as

P_{Λ}^{±}(x; t) = ∑_{β∈Z^{0,0}} c_{α,β}(t)q_{β,±}(x; t),

where c_{α,β}(t) ∈ ℚ(t) and c_{α,β}(t) = 0 unless β ≻ α or β ∼ α.

(b) P_{Λ}^{±}(x; t) can be expressed as

P_{Λ}^{±}(x; t) = s_{α}(x) + ∑_{β∈Z^{0,0}} u_{α,β}(t)s_{β}(x),

where u_{α,β}(t) ∈ tA, and u_{α,β}(t) = 0 unless β ≺ α and β ≁ α. (ii) For each Λ ∈ Z_{n}^{r,s}, there exists a unique function Q_{Λ}^{±}(x; t) satisfying the fol-

lowing properties.

(a) Q_{Λ}^{±}(x; t) can be expressed as

Q_{Λ}^{±}(x; t) = q_{α,±}(x; t) + ∑_{β∈Z^{0,0}} d_{α,β}(t)q_{β,±}(x; t),

where d_{α,β}(t) ∈ ℚ(t) and d_{α,β}(t) = 0 unless β ≻ α and β ≁ α.

(b) Q_{Λ}^{±}(x; t) can be expressed as

Q_{Λ}^{±}(x; t) = ∑_{β∈Z^{0,0}} w_{α,β}(t)s_{β}(x),

where w_{α,β}(t) ∈ A, and w_{α,β}(t) = 0 unless β ≺ α or β ∼ α. Moreover,

w_{α,β} ∈ tA if β ≠ α, and w_{α,α} ∈ A^{∗}.

Remark 3.6. We call P_{Λ}^{±} the Hall-Littlewood function associated to Λ ∈ Z_{n}^{r,s}. It is likely that P_{Λ}^{±}(x; t) ∈ ℤ[x; t]. In fact, in the classical case, i.e., the case where e = 2, r = 2, s = 1, the assertion follows from the known properties of Green functions.

The functions P_{Λ}^{±}, Q_{Λ}^{±} have the stability property for the shift operation of symbols. Hence they can be regarded as functions with infinitely many variables. As a corollary to Theorem 3.5, we have

Corollary 3.7. Let Ω(x, y; t) be as in Proposition 3.3. Then we have

(3.7.1)  Ω(x, y; t) = ∑_{Λ,Λ′} b_{Λ,Λ′}(t)P_{Λ}^{+}(x; t)P_{Λ′}^{−}(y; t).
(3.7.2)  Ω(x, y; t) = ∑_{Λ}Q_{Λ}^{+}(x; t)P_{Λ}^{−}(y; t) = ∑_{Λ}P_{Λ}^{+}(x; t)Q_{Λ}^{−}(y; t).

where in (3.7.1), Λ, Λ′ run over all the elements in ⋃_{n=1}^{∞} Z_{n}^{r,s}, and b_{Λ,Λ′}(t) = 0 unless |Λ| = |Λ′| and Λ ∼ Λ′. In (3.7.2), Λ runs over all the elements in ⋃_{n=1}^{∞} Z_{n}^{r,s}.

3.8. Let 𝔖_{m} = 𝔖_{m_{0}} × ⋯ × 𝔖_{m_{e−1}} be a parabolic subgroup of 𝔖_{n} of type m. 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 other x_{j}^{(k)} invariant. Ξ = ⨁_{i≥0} Ξ^{i} is called the space of symmetric functions. Schur function s_{α}(x) with infinitely many variables x_{1}^{(k)}, … is regarded as an element in Ξ^{n} with n = |α|, and the set {s_{α}(x)} with α ∈ Z_{n}^{0,0} forms a ℤ-basis of Ξ^{n}. Put Ξ_{ℚ}^{n}[t] = ℚ(t) ⊗_{ℤ} Ξ^{n}. It can be shown that the set {P_{Λ}^{±} | Λ ∈ Z_{n}^{r,s}} (resp. the set {Q_{Λ}^{±} | Λ ∈ Z_{n}^{r,s}}, {q_{α,±} | α ∈ Z_{n}^{0,0}}) forms a ℚ(t)-basis of the space Ξ_{ℚ}^{n}[t]. A similar property holds if one replaces Ξ^{n} by Ξ_{m}^{n}

It follows from the above discussion that {s_{α}} gives rise to a basis of the ℤ[t]- module Ξ[t] = ℤ[t] ⊗_{ℤ} Ξ, hence so is {m_{α}}. Also, we see that {q_{α,±}}, {Q_{Λ}^{±}}, and {P_{Λ}^{±}} turn out to be bases of Ξ_{ℚ}[t]. Moreover, {p_{α}} gives a basis of ℂ(t)-space Ξ_{ℂ}[t] = ℂ(t) ⊗_{ℤ} Ξ. We now define a scalar product on Ξ_{ℚ}[t] by the condition that

(3.8.1)  ⟨q_{α,+}(x; t), m_{β}(x)⟩= δ_{α,β},

and extend it to a sesquilinear form on Ξ_{ℂ}[t]. By using a similar argument as in [M, Chap. I, 4], it follows from Corollary 3.7 and Proposition 3.3 that we have

⟨m_{α}(x), q_{β,−}(x; t)⟩= δ_{α,β},
(3.8.2)  ⟨P_{Λ}^{+}(x; t), Q_{Λ′}^{−}(x; t)⟩=⟨Q_{Λ}^{+}(x; t), P_{Λ′}^{−}(x; t)⟩= δ_{Λ,Λ′},
⟨p_{α}(x), p_{β}(x)⟩= z_{α}(t)δ_{α,β}.

In the special case where e = 2, the function q_{α,+} (resp. P_{Λ}^{+}, Q_{Λ}^{+}) coincides with q_{α,−} (resp. P_{Λ}^{−}, Q_{Λ}^{−}). We also have p_{α} = p̄_{α} and z_{α}(t) ∈ ℚ(t). Hence in this case, the scalar product ⟨ , ⟩ turns out to be symmetric.

The following result gives a characterization of P_{Λ}^{±} and Q_{Λ}^{±} in terms of the inner product. Proposition 3.9. The functions P_{Λ}^{±} are characterized by the following properties. (Then the functions Q_{Λ}^{±} is charactersized as the dual bases of P_{Λ}^{∓} in the sense of (3.8.2)).

(i) P_{Λ}^{±}(x; t) can be expressed in terms of m_{β}(x) as

P_{Λ}^{±} = m_{α} + ∑_{β}c_{α,β}^{±}m_{β},

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

3.10. Let K_{±}(t) = M(s, P^{±}) be the transition matrix between Schur functions and Hall-Littlewood functions, i.e.,

s_{β}(x) = ∑_{Λ(α)} K_{β,α}^{±}(t)P_{Λ(α)}^{±}(x; t).

We call K_{β,α}^{±}(t) Kostka functions associated to Z_{n}^{r,s}, in analogy to the case of GL_{n}(F_{q}). By Theorem 3.5, K_{±}(t) is a block lower triangular matrix with identity diagonal blocks, with entries K_{α,β}^{±}(t) in ℚ(t). We put K̃_{α,β}^{±}(t) = t^{a(β)}K_{α,β}^{±}(t^{−1}) and let K̃_{±}(t) = (K̃_{α,β}^{±}(t)). Then the matrix K̃_{±}(t) satisfies a similar condition as P given in (2.4.2). Next we consider the matrix D(t) = (b_{Λ,Λ′}(t)), indexed by symbols in Z_{n}^{r,s}, where b_{Λ,Λ′}(t) is as in Corollary 3.7. We write D(t)^{−1} = (b′_{α,β}(t)), and define a matrix Λ̃(t) by Λ̃(t) = (d(t)t^{a(β)−a(α)}b′_{α,β}(t^{−1})), where d(t) = t^{N^{∗}−n}(t − 1)^{n}P_{W} (t). Clearly, Λ̃(t) is a block diagonal matrix. Under this setting, we have the following theorem. Theorem 3.11. We have

K̃_{−}(t)Λ̃(t)^{t}K̃_{+}(t) = Ω′.

In particular P′ = K̃_{−}(t), P′′ = K̃_{+}(t) and Λ′ = Λ̃(t) gives a solution for the equation (2.5.2).

Theorem 3.11 shows that the solution of (2.5.2), hence that of (2.4.2) can be described in terms of Kostka functions, just as in the case of Green polynomials of GL_{n}(F_{q}). In particular, the matrix P coincides with K̃_{−}(t).

Remarks 3.12. (i) As some examples show one can conjecture the following. K_{α,β}^{±}(t) is a polynomial in ℤ[t] with positive coefficients. Moreover, deg K_{α,β}^{±}(t) ≤ a(β)−a(α). This implies that K̃_{α,β}(t) is also in ℤ[t]. The conjecture also implies that P_{Λ}^{±}(x; t) is a polynomial in ℤ[x; t].

(ii) From the discussion in 3.8, we see that {q_{α,±}(x; t)} and {m_{α}(x)}, and {Q_{Λ}^{±}(x; t)} and {P^{∓}(x; t)} are dual bases of each other. Hence we have

(3.12.1)  M(Q^{∓}, q_{∓}) = M(P^{±}, m)^{∗} = (K_{±}(t)^{−1}K)^{∗} = ^{t}K_{±}(t)K^{∗},

where K = M(s, m) is the Kostka matrix, and X^{∗} denotes the transposed inverse of the matrix X. Since K^{∗} is a matrix with entries in ℤ, we see that K_{±}(t) is a matrix with entries in ℤ[t] if and only if M(Q^{∓}, q_{∓}) is a matrix with entries in ℤ[t]. It follows that P_{Λ}^{±}(x; t) lie in ℤ[x; t] if an only if the coefficients d_{α,β}(t) in Theorem 3.5, (ii), (a) lie in ℤ[t] for any α, β ∈ Z_{n}^{0,0}.

3.13. Concerning the above conjecture, we have some partial result in the case where e = 2. Now assume that e = 2, r ≥ 1, and that m_{0} = m + 1, m_{1} = m for some m > 0. Let Λ = (Λ_{0}, Λ_{1}) ∈ Z_{n}^{r,s} be a symbol. We write Λ_{0} : ν_{1} > ν_{3} > ⋯ > ν_{2m+1} and Λ_{1} : ν_{2} > ν_{4} > ⋯ > ν_{2m}. We say that Λ is very special if ν_{i} ≥ ν_{i+1} +r for i = 1, 2, …. Then we have the following.

Proposition 3.14. Assume that e = 2 and that Λ = Λ(α) ∈ Z_{n}^{r,s} is very special. Then the coefficient d_{α,β}(t) of q_{β,±} in Q_{Λ}^{±} is in ℤ[t]. In particular, K_{β,α}^{±}(t) ∈ ℤ[t] for any β ∈ Z_{n}^{r,s}. Moreover, we have deg K_{β,α}^{±} ≤ a(α) − a(β). Hence K̃_{β,α}^{±}(t) ∈ ℤ[t].

3.15. In the remainder of this section, we shall give explicit examples of Green functions associated to various W = G(e, 1, n). In each case, Z_{n}^{0,0} are written as α_{1}, …, α_{h}, and the corresponding symbols in Z_{n}^{r,s} are written as Λ_{i} = Λ(α_{i}). In the matrix P of Green functions, the rows and columns are arranged along the order α_{1}, α_{2}, . . ..

(A) W = G(2, 1, 2). In this case W is the Weyl group of type B_{2}, and we give the table just for the sake of reference. Z_{n}^{0,0} is given as

α_{1} = (−; 1^{2}), α_{2} = (1^{2}; −), α_{3} = (1; 1), α_{4} = (−; 2), α_{5} = (2; −).

Assume that r = 2 and s = 1. Then the corresponding symbols and similarity classes are given as

{Λ_{1} = (420; 42)}, {Λ_{2} = (31; 1)},
{Λ_{3} = (30; 2), Λ_{4} = (20; 3)}, {Λ_{5} = (2; −)}.

The matrix P of Green functions is given in Table 1.

Table 1. G(2, 1, 2) (r = 2, s = 1) \begin{matrix}(−; 1^{2}) & q^{4} \\ (1^{2}; −) & q^{2} & q^{3} \\ (1; 1) & q^{3} + q & q & q \\ (−; 2) & q^{2} & & & q \\ (2; −) & 1 & 1 & 1 & & 1 \end{matrix}

(B) W = G(3, 1, 2). Then Z_{n}^{0,0} is given as
α_{1} = (−; 1^{2}; −), α_{2} = (−; −; 1^{2}), α_{3} = (1^{2}; −; −), α_{4} = (−; 1; 1),
α_{5} = (1; 1; −), α_{6} = (1; −; 1), α_{7} = (−; 2; −), α_{8} = (−; −; 2), α_{9} = (2; −; −).

Assume that r = 2, s = 1. Then the corresponding symbols and similarity classes are given as

{Λ_{1} = (420; 42; 31), Λ_{2} = (420; 31; 42)}, {Λ_{3} = (31; 1; 1)}, {Λ_{4} = (20; 2; 2)} {Λ_{5} = (30; 2; 1), Λ_{6} = (30; 1; 2), Λ_{7} = (20; 3; 1)Λ_{8} = (20; 1; 3)}, {Λ_{9} = (2; −; −)}.

The matrix P of Green functions is given in Table 2.

Table 2. G(3, 1, 2) (r = 2, s = 1) \begin{matrix}(−; 1^{2}; −) & q^{5} \\ (−; −; 1^{2}) & & q^{5} \\ (1^{2}; −; −) & q^{3} & q^{4} & q^{3} \\ (−; 1; 1) & q^{3} & & & q^{3} \\ (1; 1; −) & q^{4} + q & & q & q & q \\ (1; −; 1) & q^{2} & q^{3} & q^{2} & q^{2} & & q \\ (−; 2; −) & q^{2} & & q^{2} & & & & q \\ (−; −; 2) & & q^{2} & q & & & & & q \\ (2; −; −) & 1 & & 1 & 1 & 1 & & & & 1 \end{matrix}

(C) W = G(4, 1, 2). Then Z_{n}^{0,0} is given as
α_{1} = (−; 1^{2}; −; −), α_{2} = (−; −; 1^{2}; −; −), α_{3} = (−; −; −; 1^{2}),
α_{4} = (1^{2}; −; −; −), α_{5} = (−; 1; 1; −), α_{6} = (−; 1; −; 1), α_{7} = (−; −; 1; 1),
α_{8} = (1; 1; −; −), α_{9} = (1; −; 1; −), α_{10} = (−; 2; −; −), α_{11} = (1; −; −; 1),
α_{12} = (−; −; 2; −), α_{13} = (−; −; −; 2), α_{14} = (2; −; −; −).

Assume that r = 2 and s = 1. Then the corresponding symbols and similarity classes are given as

{Λ_{1} = (420; 42; 31; 31), Λ_{2} = (420; 31; 42; 31), Λ_{3} = (420; 31; 31; 42)},
{Λ_{4} = (31; 1; 1; 1)}, {Λ_{5} = (20; 2; 2; 1), Λ_{6} = (20; 2; 1; 2), Λ_{7} = (20; 1; 2; 2)},
{Λ_{8} = (30; 2; 1; 1), Λ_{9} = (30; 1; 2; 1), Λ_{10} = (20; 3; 1; 1),
Λ_{11} = (30; 1; 1; 2), Λ_{12} = (20; 1; 3; 1), Λ_{13} = (20; 1; 1; 3)}
{Λ_{14} = (2; −; −; −)}.

The matrix P of Green functions is given in Table 3.

Table 3. G(4, 1, 2) (r = 2, s = 1) \begin{matrix}(−; 1^{2}; −; −) & q^{6} \\ (−; −; 1^{2}; −) & & q^{6} \\ (−; −; −; 1^{2}) & & & q^{6} \\ (1^{2}; −; −; −) & q^{4} & & & q^{4} \\ (−; 1; 1; −) & q^{3} & q^{5} & & & q^{3} \\ (−; 1; −; 1) & q^{4} & & q^{4} & & & q^{3} \\ (−; −; 1; 1) & & q^{3} & q^{5} & & & & q^{3} \\ (1; 1; −; −) & q^{5} + q & & & q & q & & & q \\ (1; −; 1; −) & q^{2} & q^{4} & & q^{2} & q^{2} & & & & q \\ (−; 2; −; −) & q^{2} & & & & q^{2} & & & & & q \\ (1; −; −; 1) & q^{3} & & q^{3} & q^{3} & & q^{2} & & & & & q \\ (−; −; 2; −) & & q^{2} & & & & & q^{2} & & & & & q \\ (−; −; −; 2) & & & q^{2} & & & q & & & & & & & q \\ (2; −; −; −) & 1 & & & 1 & 1 & & & 1 & & & & & & 1 \end{matrix}

(D) W = G(3, 1, 3). Then Z_{n}^{0,0} is given as

α_{1} = (−; 1^{3}; −), α_{2} = (−; −; 1^{3}), α_{3} = (1^{3}; −; −), α_{4} = (−; 1^{2}; 1),

α_{5} = (−; 1; 1^{2}), α_{6} = (1; 1^{2}; −), α_{7} = (−; 21; −), α_{8} = (1; −; 1^{2}),

α_{9} = (−; −; 21), α_{10} = (1^{2}; 1; −), α_{11} = (1^{2}; −; 1), α_{12} = (21; −; −), α_{13} = (1; 1; 1), α_{14} = (−; 2; 1), α_{15} = (−; 1; 2), α_{16} = (1; 2; −), α_{17} = (1; −; 2), α_{18} = (2; 1; −), α_{19} = (2; −; 1), α_{20} = (−; 3; −), α_{21} = (−; −; 3), α_{22} = (3; −; −).

Assume that r = 2 and s = 1. Then the corresponding symbols and families are given as

{Λ_{1} = (6420; 642; 531), Λ_{2} = (6420; 531; 642)}, {Λ_{3} = (531; 31; 31)}, {Λ_{4} = (420; 42; 41), Λ_{5} = (420; 41; 42)}, {Λ_{6} = (520; 42; 31), Λ_{7} = (420; 52; 31), Λ_{8} = (520; 31; 42), Λ_{9} = (420; 31; 52)}, {Λ_{10} = (31; 2; 1), Λ_{11} = (31; 1; 2)}, {Λ_{12} = (41; 1; 1)}, {Λ_{13} = (30; 2; 2), Λ_{14} = (20; 3; 2), Λ_{15} = (20; 2; 3)}, {Λ_{16} = (30; 3; 1), Λ_{17} = (30; 1; 3)}, {Λ_{18} = 40; 2; 1), Λ_{19} = (40; 1; 2), Λ_{20} = (20; 4; 1), Λ_{21} = (20; 1; 4)}, {Λ_{22} = (3; −; −)}.

The matrix P of Green functions is given in Table 4.

Table 4. G(3, 1, 3) (r = 2, s = 1) ; \begin{matrix}(−; 1^{3}; −) & q^{12} \\ (−; −; 1^{3}) & & q^{12} \\ (1^{3}; −; −) & q^{9} & & q^{9} \\ (−; 1^{2}; 1) & q^{10} + q^{7} & q^{10} & & q^{7} \\ (−; 1; 1^{2}) & q^{8} & q^{11} + q^{8} & & & q^{7} \\ (1; 1^{2}; −) & q^{11} + q^{8} + q^{5} & & q^{5} & q^{5} & & q^{5} \\ (−; 21; −) & q^{9} + q^{6} & & & q^{6} & & & q^{5} \\ (1; −; 1^{2}) & q^{7} & q^{10} + q^{7} & q^{7} & & q^{6} & & & q^{5} \\ (−; −; 21) & & q^{9} + q^{6} & & & q^{5} & & & & q^{5} \\ (1^{2}; 1; −) & q^{10} + q^{7} + q^{4} & & q^{7} + q^{4} & q^{4} & & q^{4} & & & & q^{4} \\ (1^{2}; −; 1) & q^{8} + q^{5} & q^{8} & q^{8} + q^{5} & q^{5} & & & & q^{3} & & & q^{4} \\ (21; −; −) & q^{6} + q^{3} & & q^{6} + q^{3} & q^{3} & & q^{3} & & & & q^{3} & & q^{3} \\ (1; 1; 1) & q^{9} + 2q^{6} + q^{3} & q^{9} + q^{6} & q^{6} + q^{3} & q^{6} + q^{3} & q^{5} & q^{3} & & q^{4} & & q^{3} & & & q^{3} \\ (−; 2; 1) & q^{7} + q^{4} & q^{7} & & q^{4} & q^{6} & & q^{3} & & & & & & & q^{3} \\ (−; 1; 2) & q^{5} & q^{8} + q^{5} & & q^{5} & q^{4} & & & & q^{4} & & & & & & q^{3} \\ (1; 2; −) & q^{8} + q^{5} + q^{2} & & q^{2} & q^{5} + q^{2} & & q^{2} & q^{4} & & & q^{2} & & & q^{2} & & & q^{2} \\ (1; −; 2) & q^{4} & q^{7} + q^{4} & q^{4} & q^{4} & q^{3} & & & q^{2} & q^{3} & & q^{3} & & & & q^{2} & & q^{2} \\ (2; 1; −) & q^{7} + q^{4} + q & & q^{4} + q & q^{4} + q & & q^{4} + q & & & & q & & q & q & & & q & & q \\ (2; −; 1) & q^{5} + q^{2} & q^{5} & q^{5} + q^{2} & q^{2} & q^{4} & q^{2} & & q^{3} & & q^{2} & & q^{2} & q^{2} & & & & & & q \\ (−; 3; −) & q^{3} & & & q^{3} & & & q^{2} & & & & & & & q^{2} & & & & & & q \\ (−; −; 3) & & q^{3} & & & q^{2} & & & & q^{2} & & & & & & q & & q & & & & q \\ (3; −; −) & 1 & & 1 & 1 & & 1 & & & & 1 & & 1 & 1 & & & 1 & & 1 & & & & 1 \end{matrix}

14

References

[DL] P. Deligne and G. Lusztig; Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103–161.

[G] J.A. Green; The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402–447.

[GM] M. Geck and G. Malle; On special pieces in the unipotent variety, Experimental Math. 8 (1999).

[L1] G. Lusztig; Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125–175

[L2] G. Lusztig; Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), 205–272.

[L3] G. Lusztig; Character sheaves V, Advances in Math. 61 (1986), 103–155.

[L4] G. Lusztig; On the character values of finite Chevalley groups at unipotent elements, J. Algebra, 104 (1986), 146–194.

[L5] G. Lusztig; Green functions and character sheaves, Annals of Mathematics, 131 (1990), 355– 408.

[L6] G. Lusztig; Notes on unipotent classes, Asian J. Math. 1 (1997), 194–207.

[LS] G. Lusztig and N. Spaltenstein; On the generalized Springer correspondence for classical groups, in “Advanced Studies in Pure Math.” – Algebraic groups and related topics –, 6, 1985, pp. 289–316.

[M] I.G. Macdonald; Symmetric functions and Hall Polynomials, second edition. Clarendon Press. Oxford 1995.

[Ma1] G. Malle; Unipotent Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra, 177 (1995), 768–826.

[S1] T. Shoji; On the Green polynomials of classical groups. Inventiones Math., 74 (1983), 239 - 267.

[S2] T. Shoji; Character sheaves and almost characters of reductive groups, II, Adv. in Math. 111 (1995), 314 - 354.

[S3] T. Shoji; Green functions associated to complex reflection groups. Preprint.