Invariants for representations of Weyl groups and two-sided cells

Akihiko Gyoja ^{1}, Kyo Nishiyama ^{2} and Hiroyuki Shimura ^{3}

The notion of two-sided cell, which was originally introduced by A. Joseph and reformulated by D. Kazhdan and G. Lusztig, has played imporatnt role in the representation theory. Results concerning them have been obtained by very deep and sometimes ad hoc argument. Here we introduce certain polynomial invariants for irreducible representations of Weyl groups. Our invariants are easily calculated, and the calculational results show that they almost determine the two-sided cells. Moreover, the factorization pattern of our polynomial invariants seems to be controlled by the natural parameter set M(G) of each two-sided cell.

1Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-01, Japan. (gyoja@math.h.kyotou.ac.jp)

The first author is partly supported by Grant-in-Aid for Scientific Research, Ministry of Education, Japanese Government, No. 07640035.

2Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-01, Japan. (kyo@math.h.kyotou.ac.jp)

The second author is partly supported by Grant-in-Aid for Scientific Research, Ministry of Education, Japanese Government, No. 07740019.

3Faculty of Science, Kyoto University, Kyoto 606-01, Japan. (shimura@kusm.kyoto-u.ac.jp)

0 Introduction

0.1 Motivation.

Let W be a Weyl group and S the totality of simple reflections. For w ∈ W, let l(w) be the minimum of the length n of various expressions w = s_{1}s_{2} ⋯ s_{n} (s_{i} ∈ S). In [Gy1], a ‘q-analogue’ of the following matrix valued function was studied;

∑_{w∈W} ρ(w)t^{l(w)},  (0.1)

where ρ : W → GL_{N}(ℂ) is an irreducible representation. In particular, it was shown that these functions have similar properties as the congruence zeta functions of algebraic varieties.

As a modification of these functions, N. Iwahori proposed to replace the length function l(·) in (0.1) with another length function defined as follows. Let S′ be the totality of (not necessarily simple) reflections in W. Define l′(w) to be the minimum of the length m of the various expressions w = r_{1}r_{2} ⋯ r_{m} (r_{i} ∈ S′). Such length function l′(·) is studied by R. Carter [C1] in his study of conjugacy classes of Weyl groups. Since l′ is a class function, (0.1) becomes the scalar function

c(χ; t) := χ(e)^{−1} ∑_{w∈W} χ(w)t^{l′(w)}  (0.2)

after such modification, where χ = trace ρ.

Concerning the function (0.2), we can make the following observation by a case study. Let χ, χ′ ∈ W^{∨}, where W^{∨} is the set of irreducible characters of W. Then the following equivalence is almost true:

c(χ; t) = c(χ′; t) ⇔ χ ∼_{LR} (χ′)^{ι} for some ι ∈ Aut(W, S).  (0.3)

Here χ∼_{LR}χ′ means that χ and χ′ belong to the same family (= two-sided cell) in the sense

of G. Lusztig [L1, 4.2] (cf. (b) below). More precisely, (0.3) is true if W is of type A_{l}, B_{l} (= C_{l}) or G_{2}. In the other cases, some deviation occurs.

Let us explain why we are interested in (0.3). The partition of W as a set into left (resp. right, two-sided) cells was first introduced by A. Joseph [J] and then reformulated by D. Kazhdan and G. Lusztig [KL]. Concerning cells, we know the following, summing up works of Joseph, Kazhdan, Lusztig, Brylinski, Kashiwara, Beilinson, Bernstein, Barbasch, Vogan, · · ·.

(a) Let 𝔤 be a complex semisimple Lie algebra, and W its Weyl group. The primitive ideals of U(𝔤) with trivial central character are divided into several families parametrized by the set of two-sided cells of W. To each family F_{𝔤} there associates a special representation of W [L3] (= the Goldie rank representation in the sense of Joseph). The set of primitive ideals in a fixed family F_{𝔤} is in one-to-one correspondence with the set of left cells in the associated two-sided cell, and its cardinality is equal to the dimension of the corresponding special representation.

(b) Let G be a connected reductive group over a finite field 𝔽_{q} satisfying a certain mild assumption and whose Weyl group is the same W as in (a). A crucial step of the classification of the irreducible representations of G(𝔽_{q}) is the classification of the unipotent representations [L1]. The set of unipotent representations G(𝔽_{q})_{uni}^{∨} is divided into several families; G(𝔽_{q})_{uni}^{∨} = ∐ F_{G}. The set of these families is in one-to-one correspondence with the set of two-sided cells. For each family F_{G}, there associates a finite group G (≃ 𝔖_{2}^{k} (k ≥ 0), 𝔖_{3}, 𝔖_{4} or 𝔖_{5} if W is irreducible). The set of unipotent representations in a fixed family F_{G} is in one-to-one correspondence with M(G) := {(x, ρ) | x ∈ G, ρ ∈ Z_{G}(x)^{∨}}/Gconjugacy.

(c) Two-sided cells determine two-sided ideals of the group ring ℂW, and give a partition of W^{∨}. The latter partition gives the equivalence relation ∼_{LR} appeared in (0.3), and

is explicitly determined (cf. [L1, §4]). There is a natural inclusion W^{∨} → G(𝔽_{q})_{uni}^{∨}. Each F_{W} ∈ W^{∨}/∼_{LR} is contained in the (unique) family F_{G} as in (b), and the correspondence

F_{W} ↦ F_{G} gives a bijection W^{∨}/∼_{LR} → {F_{G}}. (For χ ∈ W^{∨}, let F_{W} (χ) ∈ W^{∨}/∼_{LR} be the class of χ, F_{G}(χ) the corresponding family of G(𝔽_{q})_{uni}^{∨}, and G(χ) the associated finite group.)

In spite of these beautiful and important roles played by the left/two-sided cells, the proof is often very deep and sometimes quite ad hoc. Thus it is surprising that such a simple formula like (0.3) characterizes the two-sided cells even if some deviation occurs.

0.2 Modification of c(χ; t).

Motivated by the reason explained at the end of §0.1, we want to reduce the deviation tainting (0.3). For this purpose, we modify the invariant c(χ; t). We start with introducing certain invariants τ̃(χ; q, y) ∈ ℚ(q)[y] and τ^{∗}(χ; t) ∈ ℤ[t]. The latter is essentially the same as c(χ; t), and can be obtained from the former by a certain specialization. By a case study, we observe that τ̃(χ; q, y) has a lot of factors of the form 1 + yq^{c} (c ∈ ℤ), and that these factors are often the same for χ’s in a fixed family F_{W} ∈ W^{∨}/∼_{LR}. Thus it is natural to consider the largest divisor of the form ∏_{i=1}^{κ}(1 + yq^{c_{i}}) of τ̃(χ; q, y) ∈ ℚ(q)[y]. For χ ∈ W^{∨}, let

c̃′(χ; q, y) = ∏_{i=1}^{κ′(χ)}(1 + yq^{c_{i}(χ)})  (0.4)

be such a largest divisor. As an invariant for F_{W} ∈ W^{∨}/∼_{LR}, we consider

c̃(χ; q, y) = GCD{c̃′(χ′; q, y) | χ′ ∼_{LR}χ} = ∏_{i=1}^{κ(χ)}(1 + yq^{c_{i}(χ)}).  (0.5)

There are a few χ’s such that c̃(χ; q, y) ≠ c̃′(χ; q, y). Calculational results indicate that this discrepancy is controled by M(G) in the cases of A_{l}, D_{4} and E_{l} (cf. Theorem 4.1). To control the same phenomenon in the cases of B_{l}, F_{4} and G_{2}, we need some modular representation theory (cf. §4.2). Define τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] and τ_{prim}^{∗}(χ; t) ∈ ℤ[t] so that

τ̃(χ; q, y) = (1 − q)^{−κ(χ)} ∏_{i=1}^{κ(χ)}(1 + yq^{c_{i}(χ)}) · τ̃_{prim}(χ; q, y),  (0.6)

τ^{∗}(χ; t) = ∏_{i=1}^{κ(χ)}(t + c_{i}(χ)) · τ_{prim}^{∗}(χ; t).  (0.7)

(As we shall see in Proposition 1.3, τ_{prim}^{∗}(χ; t) is a polynomial.) Our first main result is the following theorem which says that the invariant c̃(χ; q, y) separates the families.

Theorem 0.1 Let W be an irreducible Weyl group not of type D_{l} (l ≥ 5). For χ, χ′ ∈ W^{∨},

c̃(χ; q, y) = c̃(χ′; q, y) ⇔ χ ∼_{LR} (χ′)^{ι} for some ι ∈ Aut(W, S).

0.3 Primitive factors of the invariants.

Let K be a field. For a polynomial p(t) ∈ K[t], let p(t) = ∏_{i=1}^{n} p_{i}(t) be the irreducible factorization in K[t]. Then π_{K}(p(t)) := {deg p_{i}}_{1≤i≤n} is a partition of deg p, which we shall call the factorization pattern of p(t) over K.

Let W be an irreducible Weyl group of type A_{l}, D_{l}, or E_{l}. Then up to a little deviation, the factorization patterns π_{ℚ(q)}(τ̃_{prim}(χ; q, y)) and π_{ℚ}(τ_{prim}^{∗}(χ; t)) depend only on the element of M(G) associated to χ ∈ W^{∨}, if G = G(χ) is 𝔖_{m} (m = 1, 2, 3, 5). (Cf. §0.1.) More precisely, we have the following.

Theorem 0.2 Let W_{i} (i = 1, 2) be two irreducible Weyl groups of type A_{l}, D_{l}, or E_{l}, χ_{i} ∈ W_{i}^{∨}, G_{i} := G(χ_{i}) ∈ {𝔖_{m}| m = 1, 2, 3, 5}, and (x_{i}, ρ_{i}) ∈ M(G_{i}) the element associated to χ_{i}.

(1) The two factorization patterns π_{ℚ(q)}(τ̃_{prim}(χ_{i}; q, y)) for i = 1, 2 are the same, whenever G_{1} = G_{2} and (x_{1}, ρ_{1}) = (x_{2}, ρ_{2}).

(2) If (W_{i}, χ_{i}) (i = 1, 2) are not among the following list of (W, χ), the same result holds for π_{ℚ}(τ_{prim}^{∗}(χ; t)).

\begin{matrix}(a) & W = D_{l} (l ≥ 5) \\ (b) & W = W (E_{6}), χ ∈ \{30_{p}, 30′_{p}\}, \\ (c) & W = W (E_{8}), χ ∈ \{210_{x}, 210′_{x}\}, \\ (d) & W = W (E_{7}), χ ∈ \{512′_{a}, 512_{a}\}, \\ (e) & W = W (E_{8}), χ ∈ \{4096_{z}, 4096_{x}, 4096′_{z}, 4096′_{x}\}. \end{matrix}

Concerning the irreducible representations of Weyl groups, we follow the notation in [L1, §4].

In the above list, (2d) and (2e) seem to be real exceptions, but (2a) – (2c) do not. In fact, in the case of (2a) – (2c), τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] is an irreducible polynomial of degree 2 as is predicted by the general rule (cf. Theorem 4.1 (1b)). On the other hand, the extrapolation of the factorization patterns of τ_{prim}^{∗}(χ; t)’s except for χ’s in (2a) – (2c) might predict that they would be an irreducible polynomial of degree 2. But incidentally it decomposes into two linear factors, and thus we are obliged to exclude these χ’s (cf. Theorem 4.2 (1b)).

In §4.1, we shall generalize the above result on the factorization pattern so that any irreducible Weyl group not of type D_{l} (l ≥ 5) will be included. Further, in §4.2, we reformulate the result in terms of the modular representation theory of the (generic) Iwahori-Hecke algebras.

0.4 Product formula for τ^{∗}(χ; t) and τ̃(χ; q, y).

As we have seen, our invariants do not completely decompose into linear factors in general. However, τ^{∗}(χ_{1}; q)+τ^{∗}(χ_{2}; q) decomposes into linear factors for arbitrary χ_{1} ≠ χ_{2} in a family consisting of 3-elements. This seems to be a part of more general phenomenon. We have obtained some results in this direction. See Proposition 5.1 and (5.8) in Theorem 5.2. As for τ̃(χ; q, y) ∈ ℚ(q)[y], see the formula (5.9).

0.5 Convention and Notation.

We denote the rational integer ring by ℤ, the rational number field by ℚ, the real number field by ℝ, and the complex number field by ℂ. For a set X, #X denotes its cardinality, and 𝔖(X) the symmetric group on X. Put 𝔖_{n} = 𝔖(1, 2, ⋯, n) := 𝔖({1, 2, ⋯, n}). The identity element of a finite group is denoted by e, not by 1. For two complex valued functions f_{1} and f_{2} on a finite group Γ, put ⟨f_{1}, f_{2}⟩_{Γ} := (#Γ)^{−1} ∑_{g∈Γ} f_{1}(g)f_{2}(g^{−1}). The set

of isomorphism classes of irreducible (complex linear) representations of Γ is denoted by Γ^{∨}. For a character χ of Γ, we denote by dim χ the value χ(e) at the identity element, which is equal to the dimension of the corresponding representation of Γ. Concerning the irreducible representations of Weyl groups, we follow the notation in [L1, §4].

1 Definition of invariants.

Let V be a complex vector space of dimension l on which the Weyl group W acts faithfully as a reflection group. For w ∈ W, let V (w) denote the space of w-fixed vectors in V. We define τ̃(q, y), τ(t) and σ(q) as follows;

τ̃(q, y)(w) := \frac{det(1 + yw|V)}{det(1 − qw|V)},  (1.1)
τ(t)(w) := t^{dim V (w)},  (1.2)
σ(q)(w) := \frac{trace(w|V)}{det(1 − qw|V)},  (1.3)

where t, q and y are indeterminates. Note that σ(q)(w) is the coefficient of the linear term of τ̃(q, y)(w) with respect to the variable y. For an irreducible character χ of W, put

τ̃(χ; q, y) := ⟨χ, τ̃(q, y)⟩_{W} ,  (1.4)
τ(χ; t) := ⟨χ, τ(t)⟩_{W} ,  (1.5)
σ(χ; q) := ⟨χ, σ(q)⟩_{W} ,  (1.6)
τ^{∗}(χ; t) := (#W/ dim χ)τ(χ; t),  (1.7)
σ^{∗}(χ; q) := σ(χ; q)/τ̃(χ; q, 0),  (1.8)

where dim χ = χ(e) is the dimension of the representation corresponding to χ. Let l′(w) (w ∈ W) be the minimum of the length m of the various expressions w = r_{1}r_{2} ⋯ r_{m}, where r_{i}’s are arbitrary reflections in W. Then l′(w) = l − dim V (w) [C1, §2], and the function c(χ; t) in Introduction is given by

c(χ; t) = t^{l}τ^{∗}(χ; t^{−1}).  (1.9)

Therefore, we shall study the functions (1.4)–(1.8) instead of c(χ; t).

Let {m_{1}, ⋯, m_{l}} be the exponents of W (See [B, Chapter 5, §5, Exercise 3]). Here we understand the exponents so that {m_{i} + 1} are the degrees of the basic W -invariant polynomials on V. In particular, if we allow W to have non-zero fixed vectors in V, some of the exponents may be zero.

Proposition 1.1 (1) We have

∏_{i=1}^{l}(1 − q^{m_{i}+1})τ̃(χ; q, y) ∈ ℤ[q, y], ∏_{i=1}^{l}(1 − q^{m_{i}+1})σ(χ; q) ∈ ℤ[q].

(2) We have an expansion

\frac{τ̃(χ; q, y)}{τ̃(χ; q, 0)} = 1 + σ^{∗}(χ; q)y + ∑_{i=2}^{l} σ_{i}^{∗}(χ; q)y^{i}  (1.10)

with some σ_{i}^{∗}(χ; q) ∈ ℚ(q).

(3) We obtain τ^{∗}(χ; t) from τ̃(χ; q, y) by the following specialization:

τ^{∗}(χ; t) = \frac{#W}{dim χ} lim_{q→1} τ̃(χ; q, −1 + t(1 − q)) ∈ ℤ[t].  (1.11)

This proposition is an immediate consequence of the following lemma.

Lemma 1.2 (1) The functions τ and σ are obtained from τ̃(q, y) by the following specialization:

lim_{q→1} τ̃(q, −1 + t(1 − q))(w) = τ(t)(w), \frac{∂}{∂y} τ̃(q, y)(w)|_{y=0} = σ(q)(w).  (1.12)

(2) Let S^{n}(V) (resp. ⋀^{m}(V)) denote the n-th (resp. m-th) homogeneous part of the symmetric algebra (resp. the exterior algebra) of V. We denote the space of W -harmonic polynomials of degree n by H^{n}(V) ⊂ S^{n}(V) (cf. [St2]). Then we get

τ̃(χ; q, y) = \frac{1}{∏_{i=1}^{l}(1 − q^{m_{i}+1})} ∑_{n,m≥0} [χ : H^{n}(V) ⊗ ⋀^{m}(V)] q^{n}y^{m},  (1.13)
σ(χ; q) = \frac{1}{∏_{i=1}^{l}(1 − q^{m_{i}+1})} ∑_{n≥0} [χ : H^{n}(V) ⊗ V] q^{n},  (1.14)
σ^{∗}(χ; q) = \frac{∑_{n≥0} [χ : H^{n}(V) ⊗ V] q^{n}}{∑_{n≥0} [χ : H^{n}(V)] q^{n}} .  (1.15)

Here, we denote by [U_{1} : U_{2}] = Hom_{W} (U_{1}, U_{2}) the intertwining number of W -modules U_{i} (i = 1, 2).

Proof of Lemma. (1) is an easy consequence of the definition.

(2) By definition we can show that

τ̃(q, y)(w) = ∑_{n,m≥0} trace(w|S^{n}(V) ⊗ ⋀^{m}(V))q^{n}y^{m},  (1.16)
σ(q)(w) = ∑_{n≥0} trace(w|S^{n}(V) ⊗ V)q^{n}.  (1.17)

Note that the symmetric algebra is decomposed as S(V) = H(V) ⊗ I(V), where I(V) is the subalgebra of W -invariants in S(V). Since

∑_{n≥0} I^{n}(V)q^{n} = \frac{1}{∏_{i=1}^{l}(1 − q^{m_{i}+1})} (I^{n}(V) := I(V) ∩ S^{n}(V)),

we get the formulas (1.13) – (1.15) by the orthogonality relation of irreducible characters.

The polynomial τ̃(χ; q, y) contains a lot of factors of the form 1 + yq^{c} (c ∈ ℤ) (see §3 for examples). So we consider the largest divisor c̃′(χ; q, y) of τ̃(χ; q, y) ∈ ℚ(q)[y] which is of the form ∏_{i=1}^{κ}(1 + yq^{c_{i}}) (c_{i} ∈ ℤ). Put

c̃(χ; q, y) = GCD{c̃′(χ′; q, y) | χ′ ∼_{LR} χ} = ∏_{i=1}^{κ(χ)}(1 + yq^{c_{i}(χ)}),  (1.18)
τ̃_{prim}(χ; q, y) := τ̃(χ; q, y)\frac{(1 − q)^{κ(χ)}}{∏_{i=1}^{κ(χ)}(1 + yq^{c_{i}(χ)})},  (1.19)
τ_{prim}^{∗}(χ; t) := \frac{τ^{∗}(χ; t)}{∏_{i=1}^{κ(χ)}(t + c_{i}(χ))},  (1.20)

σ_{prim}^{∗}(χ; q) := σ^{∗}(χ; q) − ∑_{i=1}^{κ(χ)} q^{c_{i}(χ)}.  (1.21)

Proposition 1.3 (1) τ_{prim}^{∗}(χ; t) ∈ ℤ[t].

(2) The following formula holds with some σ_{i,prim}^{∗}(χ; q) ∈ ℚ(q);

\frac{τ̃_{prim}(χ; q, y)}{τ̃_{prim}(χ; q, 0)} = 1 + σ_{prim}^{∗}(χ; q)y + ∑_{i=2}^{l−κ(χ)} σ_{i,prim}^{∗}(χ; q)y^{i}.  (1.22)

Proof. Abbreviating κ = κ(χ) and c_{i} = c_{i}(χ), we get

ℚ(q)[t] ∋ \frac{#W}{dim χ}τ̃_{prim}(χ; q, −1 + t(1 − q))

= \frac{#W}{dim χ}τ̃(χ; q, −1 + t(1 − q))/ ∏_{i=1}^{κ} (tq^{c_{i}} + \frac{1 − q^{c_{i}}}{1 − q} )
→ τ^{∗}(χ; t)/ ∏_{i=1}^{κ}(t + c_{i}) (q → 1)  (1.23)

by (1.11). Therefore τ_{prim}^{∗}(χ; t) is a polynomial in t. Since τ^{∗}(χ; t) ∈ ℤ[t], we get (1). (2) immediately follows from the definition.

Proposition 1.4 Let sgn be the sign representation of W. Then we have

τ̃(χ; q, y) = (−q)^{−l}τ̃(χ ⊗ sgn; q^{−1}, y) = y^{l}τ̃(χ ⊗ sgn; q, y^{−1}),  (1.24)
τ^{∗}(χ; t) = (−1)^{l}τ^{∗}(χ ⊗ sgn; −t),  (1.25)
σ^{∗}(χ; q) = (−q)^{−l}σ^{∗}(χ ⊗ sgn; q^{−1}),  (1.26)
c̃′(χ; q, y) = c̃′(χ ⊗ sgn; q^{−1}, y), c̃(χ; q, y) = c̃(χ ⊗ sgn; q^{−1}, y),  (1.27)
κ(χ) = κ(χ ⊗ sgn), {c_{i}(χ)}_{i} = {−c_{i}(χ ⊗ sgn)}_{i},  (1.28)

τ̃_{prim}(χ; q, y) = (−q)^{κ(χ)−l}τ̃_{prim}(χ ⊗ sgn; q^{−1}, y)

= y^{l−κ(χ)}q^{− ∑_{i} c_{i}(χ)}τ̃_{prim}(χ ⊗ sgn; q, y^{−1}),  (1.29)
τ_{prim}^{∗}(χ; t) = (−1)^{l−κ(χ)}τ_{prim}^{∗}(χ ⊗ sgn; −t).  (1.30)

Proof. Note that w and w^{−1} are conjugate in W. Then we can show that

τ̃(q, y)(w) = (−q)^{−l}sgn(w) · τ̃(q^{−1}, y)(w) = y^{l}sgn(w) · τ̃(q, y^{−1})(w), (1.31)
τ(t)(w) = (−1)^{l}sgn(w) · τ(−t)(w),  (1.32)
σ(q)(w) = (−q)^{−l}sgn(w) · σ(q^{−1})(w).  (1.33)

The proposition immediately follows from the above three formulas.

2 Some specialization.

2.1 Specialization.

Let {m_{1}, ⋯, m_{l}} be the exponents of W. Put f_{q}(w) := ∏_{i=1}^{l}(1−q^{m_{i}+1})/ det(1−qw|V). Then

f(χ; q) := ⟨χ, f_{q}⟩_{W} = ∏_{i=1}^{l}(1 − q^{m_{i}+1})τ̃(χ; q, 0)  (2.1)

is called the fake degree of χ, whose explicit form is given in [St1] (for type A_{l}), [L2] (for types B_{l}(= C_{l}) and D_{l}), and [BL] (for exceptional types).

Proposition 2.1 If χ is the trivial character 1, then χ forms an equivalence class with respect to ∼_{LR} by itself, and we have

τ̃(1; q, y) = ∏_{i=1}^{l} \frac{1 + yq^{m_{i}}}{1 − q^{m_{i}+1}} , τ^{∗}(1; t) = ∏_{i=1}^{l}(t + m_{i}), σ^{∗}(1; q) = ∑_{i=1}^{l} q^{m_{i}},  (2.2)

c̃(1; q, y) = c̃′(1; q, y) = ∏_{i=1}^{l}(1 + yq^{m_{i}}).  (2.3)

Proof. For τ̃(1; q, y), see [B, Chapter 5, §5, Exercise 3]. The other formulas are easy consequences of it after appropriate specialization.

Lemma 2.2 The following special values of τ̃, τ^{∗} and σ^{∗} are obtained:

τ̃(χ; q, −q) = \{ \begin{matrix}1 & if χ = 1, \\ 0 & if χ ≠ 1, \end{matrix}  (2.4)
τ̃(χ; q, −q^{−1}) = \{ \begin{matrix}(−q)^{−l} & if χ = sgn, \\ 0 & if χ ≠ sgn, \end{matrix}  (2.5)
τ^{∗}(χ; 1) = \{ \begin{matrix}#W & if χ = 1, \\ 0 & if χ ≠ 1, \end{matrix}  (2.6)
τ^{∗}(χ; −1) = \{ \begin{matrix}(−1)^{l} · (#W) & if χ = sgn, \\ 0 & if χ ≠ sgn, \end{matrix}  (2.7)
σ^{∗}(χ; 1) = l (= dim V).  (2.8)
lim_{q→1} ∏_{i=1}^{l}(1 − q^{m_{i}+1}) · τ̃(χ; q, y)/ dim χ = (1 + y)^{l}  (2.9)

Proof. By the definition of τ̃ and τ^{∗} (2.4) and (2.6) are obvious. Thus (1.24) (resp. (1.25)) yields (2.5) (resp. (2.7)). To prove (2.8), let us put q = 1 in (1.15). Then the numerator becomes

∑_{n≥0} [χ : H^{n}(V) ⊗ V] = [χ : ℂW ⊗ V] = dim V · [χ : ℂW] = dim V · dim χ,

where ℂW denotes the left regular representation of W. Similarly the denominator becomes dim χ and we get the formula. For the last formula, we calculate as follows:

τ̃(χ; q, y) = \frac{1}{#W} ∑_{w∈W} χ(w)\frac{∑_{m} trace (w| ∧^{m} V)y^{m}}{det(1 − qw)}

= ∑_{m}⟨χ ⊗ ∧^{m}V, f_{q}⟩_{W} y^{m}/ ∏_{i=1}^{l}(1 − q^{m_{i}+1}),

hence we get

lim_{q→1} ∏_{i=1}^{l}(1 − q^{m_{i}+1})τ̃(χ; q, y) = dim χ ∑_{m} (\begin{matrix}l \\ m \end{matrix}) y^{m} = (dim χ)(1 + y)^{l},  (2.10)

using f(χ; 1) = dim χ.

2.2 Coefficients of τ̃(χ; q, y).

Consider τ̃(χ; q, y) as a polynomial in y with coefficients in ℚ(q). By (2.1) and (1.24),

τ̃(χ; q, y) = \frac{1}{∏_{i=1}^{l}(1 − q^{m_{i}+1})} (f(χ; q) + ⋯ + f(χ ⊗ sgn; q)y^{l})
= \frac{(−q)^{−l}}{∏_{i=1}^{l}(1 − q^{−m_{i}−1})} (f(χ ⊗ sgn; q^{−1}) + ⋯ + f(χ; q^{−1})y^{l}) . (2.11)

Here only the lowest and the highest terms are written. In the sequel, we use the similar abbreviation. In order to rewrite (2.11), we review some results of Lusztig. Assume that W is an irreducible Weyl group, and let

W_{ex.}^{∨} = \{ \begin{matrix}\{χ ∈ W^{∨} | dim χ = 512\}, & if W = W (E_{7}), \\ \{χ ∈ W^{∨} | dim χ = 4096\}, & if W = W (E_{8}), \\ ∅, & otherwise. \end{matrix}  (2.12)

Then W_{ex.}^{∨} is the set of exceptional representations [L3]. (Note that this notation is not compatible with that of [L1].) Let d(χ; q) be the generic degree of χ (cf. [L1, p.61, ↑ ℓℓ.5–1], where it is called the ‘formal dimension’). Express f(χ; q) and d(χ; q) as

f(χ; q) = α′(χ)q^{a′(χ)} + ⋯ + β′(χ)q^{b′(χ)},  (2.13)
d(χ; q) = α(χ)q^{a(χ)} + ⋯ + β(χ)q^{b(χ)},  (2.14)

where α′(χ) ≠ 0, β′(χ) ≠ 0, α(χ) ≠ 0, β(χ) ≠ 0, a′(χ) ≤ b′(χ), a(χ) ≤ b(χ), and the omitted terms are of the degrees between them. Then it is known that

χ_{1} ∼_{LR} χ_{2} ⇒ a(χ_{1}) = a(χ_{2}) [L1, 5.27]  (2.15)

for any χ_{1}, χ_{2} ∈ W^{∨}. Hence we may write a(F) := a(χ) for χ ∈ F ∈ W^{∨}/∼_{LR}. Further, if χ ∉ W_{ex.}^{∨}, it is known that

q^{−a′(χ⊗sgn)}f(χ ⊗ sgn; q) = q^{−a′(χ)}f(χ; q) [BL],  (2.16)
−a′(χ) + a′(χ ⊗ sgn) = −a(χ) + a(χ ⊗ sgn) [L3, (2.2) and (2.3)], (2.17)
a′(χ ⊗ sgn) + b′(χ) = a(χ ⊗ sgn) + b(χ) = ν [L3, (2.3)],  (2.18)

where ν is the number of reflections in W. Put A = A(F) := −a(F) + a(F ⊗ sgn). By (2.11), (2.16), (1.10) and (1.24), we get

τ̃(χ; q, y) = \frac{f(χ; q)}{∏_{i=1}^{l}(1 − q^{m_{i}+1})}(1 + σ^{∗}(χ; q)y + ⋯ + σ^{∗}(χ; q^{−1})q^{A}y^{l−1} + q^{A}y^{l}) (2.19)

for χ ∉ W_{ex.}^{∨}.

2.3 Coefficients of τ^{∗}(χ; t).

If we put

τ^{∗}(χ; t) = t^{l} + τ_{1}t^{l−1} + ⋯ + τ_{l} (τ_{i} = τ_{i}(χ) ∈ ℤ),  (2.20)

then as is easily seen

τ_{1}(χ) = \frac{1}{dim χ} ∑_{s} χ(s),  (2.21)

where s runs over the totality of the reflections in W.

Lemma 2.3 For any χ ∈ W^{∨}, the following two formulas hold;

τ_{1}(χ) = −a(F) + a(F ⊗ sgn) = A(F),  (2.22)
\frac{d}{dq} σ^{∗}(χ; q)|_{q=1} = −a(F) + a(F ⊗ sgn).  (2.23)

Proof. For χ ∉ W_{ex.}^{∨}, we have ν − τ_{1}(χ) = a′(χ) + b′(χ) by [BL, Proposition B]. Hence by (2.17) and (2.18), we get (2.22) if χ ∈ F ∈ W^{∨}/∼_{LR} and χ ∉ W_{ex.}^{∨}. Since we can directly

verify the same equality also for χ ∈ W_{ex.}^{∨}, (2.22) holds for any χ ∈ W^{∨}. Now consider the limit of

(1 − q)^{−l}(1 + σ^{∗}(χ; q)y + ⋯ + σ^{∗}(χ; q^{−1})q^{A}y^{l−1} + q^{A}y^{l})|_{y=−1+t(1−q)}  (2.24)

when q → 1, assuming χ ∉ W_{ex.}^{∨}. By (1.11), this limit is τ^{∗}(χ; t). On the other hand, the coefficient of t^{l−1} in (2.24) is

\frac{σ^{∗}(χ; q^{−1}) − l}{1 − q}q^{A}.

Hence

\frac{d}{dq} σ^{∗}(χ; q)|_{q=1} = lim_{q→1} \frac{σ^{∗}(χ; q) − l}{q − 1} by (2.8)
= τ_{1}(χ) = −a(F) + a(F ⊗ sgn) by (2.22)  (2.25)

if χ ∉ W_{ex.}^{∨}. Moreover a direct calculation shows that (2.23) holds even if χ ∈ W_{ex.}^{∨}.

3 Explicit form of invariants.

In this section, we calculate our functions τ̃, τ^{∗}, σ^{∗}, c̃′ and c̃ explicitly for each type of the Weyl groups, and see whether the invariant c̃(χ; q, y) separates the two-sided cells. Summing up, we get Theorem 0.1.

3.1 Type A_{l}.

We realize the reflection group W = W (A_{l}) (= 𝔖_{l+1}) as the group of permutation matrices of size l+1. (By an obvious modification of the argument below, we can calculate the invariants τ̃ etc. for the irreducible reflection group W of type A_{l}.) Then for w ∈ W of cycle type ρ = (ρ_{1} ≥ ⋯ ≥ ρ_{l+1−l′(w)} > 0) [M, p.60],

τ̃(q, y)(w) = ∏_{i} \frac{1 − (−y)^{ρ_{i}}}{1 − q^{ρ_{i}}} .  (3.1)

Let α be a partition of l + 1 and χ^{α} the corresponding irreducible representation of W (A_{l}) = 𝔖_{l+1}. We identify the partition α with the corresponding Young diagram, which we understand as a subset of ℤ^{2} as in [M]. For the sake of convenience, we recollect some of the notations in [M]: for α, |α| := ∑_{i} α_{i} = l + 1, n(α) := ∑_{i≥1}(i − 1)α_{i} and α^{∗} = (α_{1}^{∗} ≥ α_{2}^{∗} ≥ ⋯) the dual partition of α. For x = (i, j) ∈ α, c(x) := j − i (the content), and h(x) := (α_{i} − i) + (α_{j}^{∗} − j) + 1 (the hook length).

Proposition 3.1 (1) For the Weyl group W (A_{l}) of type A_{l}, we get

τ̃(χ^{α}; q, y) = q^{n(α)} ∏_{x∈α} \frac{1 + yq^{c(x)}}{1 − q^{h(x)}} ,  (3.2)
τ^{∗}(χ^{α}; t) = ∏_{x∈α}(t + c(x)),  (3.3)
σ^{∗}(χ^{α}; q) = ∑_{x∈α} q^{c(x)},  (3.4)
c̃(χ^{α}; q, y) = c̃′(χ^{α}; q, y) = ∏_{x∈α}(1 + yq^{c(x)}).  (3.5)

(2) The invariant c̃(χ^{α}; q, y) separates the two-sided cells in W (A_{l}).

Proof. (1) follows from [M, §I.2, Ex. 5 and §I.3, Ex. 3] (with a = 1, b = −y, q = q, and τ̃ being p_{ρ}). As for (2), see the proof of Proposition 3.3 below.

3.2 Type B_{l}(= C_{l}).

Let W = W_{l} = W (B_{l}) be the centralizer of (1, ¯1)(2, ¯2) ⋯ (l, l̄) in 𝔖(1, 2, ⋯, l, l̄, ⋯, ¯2, ¯1), θ : W → W/⟨(1, ¯1), (2, ¯2), ⋯, (l, l̄)⟩ = 𝔖_{l} the projection, s_{1} := (1, ¯1), s_{i} := (i − 1, i)(\overline{i − 1}, \overline{i}) (1 < i ≤ l), and S := \{s_{1}, ⋯, s_{l}\}. (Here ī should be understood as one symbol, or as −i.) Then (W, S) is a Coxeter system of type B_{l}. We realize W as a reflection group acting on V = ⨁_{i=1}^{l} ℂe_{i} by

s_{1}e_{j} = \{ \begin{matrix}−e_{1}, & for j = 1 \\ e_{j}, & for j ≠ 1, \end{matrix}  and s_{i}e_{j} = \{ \begin{matrix}e_{i}, & for j = i − 1 \\ e_{i−1}, & for j = i \\ e_{j}, & for j ≠ i − 1, i, \end{matrix}  (3.6)

for 1 < i ≤ l. Define ε ∈ Hom(W, ℂ^{×}) by ε(s_{1}) = −1 and ε(s_{i}) = 1 (i ≠ 1). Let l = l′ + l′′ (l′, l′′ ≥ 0), and (α, β) be an ordered pair of partitions of l′ and l′′. Let W′ (resp. W′′) be the pointwise stabilizer of {l′′ + 1, ⋯, l} (resp. {1, ⋯, l′′}) in W. Then W′ and W′′ are naturally isomorphic to W_{l′} and W_{l′′}, respectively. Define θ′ : W′ → 𝔖_{l′} and θ′′ : W′′ → 𝔖_{l′′} in the same way as θ. Put ε′′ := ε|_{W′′} and define χ^{α,β} as the induced character

χ^{α,β} := ind_{W′×W′′}^{W}((χ^{α} ∘ θ′) ⊗ (χ^{β} ∘ θ′′ ⊗ ε′′)).  (3.7)

Then χ^{α,β} are irreducible characters of W, and all the irreducible characters of W can be obtained uniquely in this way.

Express the partitions α and β as α = (α_{1} ≥ α_{2} ≥ ⋯ ≥ α_{m+1} ≥ 0) and β = (β_{1} ≥ β_{2} ≥ ⋯ ≥ β_{m} ≥ 0) for sufficiently large m, by adding some parts which are equal to zero. Let λ_{i} := α_{i} +m+1−i, μ_{i} := β_{i} +m−i, and {ν_{1}, ⋯, ν_{2m+1}} = {λ_{1}, ⋯, λ_{m+1}, μ_{1}, ⋯, μ_{m}} (including multiplicity).

Lemma 3.2 ([L1, (4.5.6)]) Let χ = χ^{α,β} and {ν_{1}, ⋯, ν_{2m+1}} as above. If {ν′_{1}, ⋯, ν′_{2m+1}} is associated to χ′ = χ^{α′,β′} by the same manner, then

χ ∼_{LR} χ′ ⇔ {ν_{1}, ⋯, ν_{2m+1}} = {ν′_{1}, ⋯, ν′_{2m+1}} up to permutation.  (3.8)

Proposition 3.3 (1) For the Weyl group W (B_{l}) of type B_{l}, we get

τ̃(χ^{α,β}; q, y) = q^{2n(α)+2n(β)+|β|} ∏_{x′∈α} \frac{1 + yq^{2c(x′)+1}}{1 − q^{2h(x′)}} ∏_{x′′∈β} \frac{1 + yq^{2c(x′′)−1}}{1 − q^{2h(x′′)}} ,  (3.9)
τ^{∗}(χ^{α,β}; t) = ∏_{x′∈α}(t + 2c(x′) + 1) · ∏_{x′′∈β}(t + 2c(x′′) − 1),  (3.10)
σ^{∗}(χ^{α,β}; q) = ∑_{x′∈α} q^{2c(x′)+1} + ∑_{x′′∈β} q^{2c(x′′)−1},  (3.11)
c̃′(χ^{α,β}; q, y) = ∏_{x′∈α}(1 + yq^{2c(x′)+1}) · ∏_{x′′∈β}(1 + yq^{2c(x′′)−1}).  (3.12)

(2) The invariant c̃(χ^{α,β}; q, y) coincides with c̃′(χ^{α,β}; q, y), and it separates the two-sided cells in W (B_{l}).

Proof. Put V′ := ⨁_{i>l′′} ℂe_{i}, V′′ := ⨁_{i≤l′′} ℂe_{i}, and define τ̃′ and τ̃′′ in the same way as τ̃ for W (A_{l}) in the former subsection, replacing V with V′ and V′′ respectively. Then

τ̃(χ^{α,β}; q, y) = ⟨χ^{α,β}, τ̃(q, y)⟩_{W}
= ⟨χ^{α} ∘ θ′, τ̃′(q, y)⟩_{W′} × ⟨(χ^{β} ∘ θ′′) ⊗ ε′′, τ̃′′(q, y)⟩_{W′′}
= ⟨χ^{α}, τ̃′(q^{2}, qy)⟩_{𝔖_{l′}} × ⟨χ^{β}, q^{l′′}τ̃′′(q^{2}, q^{−1}y)⟩_{𝔖_{l′′}}
= q^{2n(α)+2n(β)+|β|} ∏_{x′∈α} \frac{1 + yq^{2c(x′)+1}}{1 − q^{2h(x′)}} · ∏_{x′′∈β} \frac{1 + yq^{2c(x′′)−1}}{1 − q^{2h(x′′)}} , (3.13)

which proves (3.9). The rest of the formulas in (1) easily follows from (3.9).

(2) Let {ν_{1}, ⋯, ν_{2m+1}} = {λ_{1}, ⋯, λ_{m+1}, μ_{1}, ⋯, μ_{m}} be as above. Then as is easily seen from Example 3.4 below,

∏_{i=1}^{m}(1 + yq^{−2i+1})^{2i−1} · c̃′ (\binom{λ_{1} ⋯ λ_{m+1}}{μ_{1} ⋯ μ_{m}}; q, y) = ∏_{i=1}^{2m+1} ∏_{k=1}^{ν_{i}}(1 + yq^{2k−2m−1}).  (3.14)

(Here \binom{λ_{1}···λ_{m+1}}{μ_{1}···μ_{m}} is the symbol of χ^{α,β}. We often identify these two objects.) Hence for χ, χ′ ∈ W (B_{l})^{∨}, we have

c̃′(χ; q, y) = c̃′(χ′; q, y) ⇔ χ ∼_{LR} χ′,

by Proposition 3.2. Thus we get c̃(χ; q, y) = c̃′(χ; q, y) and clearly it separates the cells. (Note that in the B_{2} -case, Aut(W, S) acts on W^{∨}/ ∼_{LR} trivially.)

Example 3.4 If α = (5, 4, 4, 2) and β = (3, 2), we can take, for example, m = 3 and express β = (3, 2, 0). For the diagram α (resp. β), we put the content 2c(x) + 1 (x ∈ α) (resp. 2c(x) − 1 (x ∈ β)) in each box. Then we get the following tableaux.

α : \begin{matrix}1 & 3 & 5 & 7 & 9 \\ −1 & 1 & 3 & 5 \\ −3 & −1 & 1 & 3 \\ −5 & −3 \end{matrix} β : \begin{matrix}−1 & 1 & 3 \\ −3 & −1 \end{matrix}

Now we add m + 1 − i = 4 − i (resp. m − i = 3 − i) nodes to the left of the i-th row for α (resp. β).

\begin{aligned}−5 & −3 & −1 & 1 & 3 & 5 & 7 & 9 \\ & −5 & −3 & −1 & 1 & 3 & 5 \\ & & −5 & −3 & −1 & 1 & 3 \\ & & & −5 & −3 \end{aligned}  \begin{aligned}−5 & −3 & −1 & 1 & 3 \\ & −5 & −3 & −1 \end{aligned}

Then λ_{i} (resp. μ_{i}) is the number of nodes in the i-th row of the left (resp. right) tableau above. Note that we have added 2i − 1 nodes labeled by −2i + 1 (cf. (3.14)).

3.3 Type D_{l}.

We keep the notations in the preceding subsection. Let W = W̃_{l} := ker ε (⊂ W (B_{l})). Then W̃_{l} is the reflection group of type D_{l}. Let (α, β) be an unordered pair of partitions. If α ≠ β, put χ̃^{α,β} := χ^{α,β}|W̃_{l}. If α = β =: α, χ^{α,α}|W̃_{l} is a sum of two irreducible characters χ_{I}^{α} ≠ χ_{II}^{α}. Note that χ̃^{α,β} = χ̃^{β,α} because χ^{α,β} ⊗ ε = χ^{β,α} holds. Then {χ̃^{α,β}, χ_{I}^{α}, χ_{II}^{α}} are irreducible characters of W̃_{l}, and all the irreducible characters of W̃_{l} can be obtained uniquely in this way.

For α and β above, let λ_{i} := α_{m+1−i} + i − 1, μ_{i} := β_{m+1−i} + i − 1 for 1 ≤ i ≤ m, and |λ| := ∑_{i} λ_{i} etc. Here we assume that α_{m+1} = β_{m+1} = 0. Then Λ = \binom{λ_{1} ··· λ_{m}}{μ_{1} ··· μ_{m}} is the symbol associated to the irreducible character χ of W (D_{l}) ([L1]). We identify an irreducible character with the associated symbol.

Lemma 3.5 ([L1, (4.6.10)]) The equivalence relation ∼_{LR} for W (D_{l})^{∨} can be described as follows. Each irreducible character χ_{I}^{α} (resp. χ_{II}^{α}) forms an equivalence class by itself. If χ = χ̃^{α,β} = \binom{λ_{1} ··· λ_{m}}{μ_{1} ··· μ_{m}} with m ≫ 0, let \{ν_{1}, ⋯, ν_{2m}\} = \{λ_{1}, ⋯, λ_{m}, μ_{1}, ⋯, μ_{m}\} (including

multiplicity). If \{ν′_{1}, ⋯, ν′_{2m}\} is associated to χ′ = χ̃^{α′,β′} = \binom{λ′_{1} ··· λ′_{m}}{μ′_{1} ··· μ′_{m}} similarly, then

χ ∼_{LR} χ′ ⇔ {ν_{1}, ⋯, ν_{2m}} = {ν′_{1}, ⋯, ν′_{2m}} up to permutation.

Proposition 3.6 (1) For the representation χ̃^{α,β} (α ≠ β) we get

τ̃(χ̃^{α,β}; q, y) = \frac{q^{2n(α)+2n(β)}}{2 ∏_{x∈α⊔β}(1 − q^{2h(x)})} \{q^{|β|} ∏_{x′∈α}(1 + yq^{2c(x′)+1}) ∏_{x′′∈β}(1 + yq^{2c(x′′)−1})

+q^{|α|} ∏_{x′∈α}(1 + yq^{2c(x′)−1}) ∏_{x′′∈β}(1 + yq^{2c(x′′)+1})} ,  (3.15)

τ^{∗}(χ̃^{α,β}; t) = \frac{1}{2} \{ ∏_{x′∈α}(t + 2c(x′) + 1) ∏_{x′′∈β}(t + 2c(x′′) − 1)

+ ∏_{x′∈α}(t + 2c(x′) − 1) ∏_{x′′∈β}(t + 2c(x′′) + 1)} ,  (3.16)

σ^{∗}(χ̃^{α,β}; q) = \frac{1}{2(q^{|α|} + q^{|β|})} · \{q^{|β|} ( ∑_{x′∈α} q^{2c(x′)+1} + ∑_{x′′∈β} q^{2c(x′′)−1})

+q^{|α|} ( ∑_{x′∈α} q^{2c(x′)−1} + ∑_{x′′∈β} q^{2c(x′′)+1})} .  (3.17)

(2) For the representations χ_{I}^{α} and χ_{II}^{α}, we get

τ̃(χ_{I}^{α}; q, y) = τ̃(χ_{II}^{α}; q, y)
= q^{4n(α)+|α|} ∏_{x∈α} \frac{(1 + yq^{2c(x)+1})(1 + yq^{2c(x)−1)})}{(1 − q^{2h(x)})^{2}},  (3.18)
τ^{∗}(χ_{I}^{α}; t) = τ^{∗}(χ_{II}^{α}; t) = ∏_{x∈α}(t + 2c(x) + 1)(t + 2c(x) − 1),  (3.19)
σ^{∗}(χ_{I}^{α}; q) = σ^{∗}(χ_{II}^{α}; q) = ∑_{x∈α}(q^{2c(x)+1} + q^{2c(x)−1}).  (3.20)

Proof. (1) Since χ^{α,β} ⊗ ε = χ^{β,α}, we have

2τ̃(χ̃^{α,β}; q, y) = τ̃(χ^{α,β}; q, y) + τ̃(χ^{β,α}; q, y),

which proves (3.15). The rest of the formulas in (1) easily follows from (3.15).

(2) Since the inner automorphism by s_{1} in W (B_{l}) induces an outer automorphism ι of W (D_{l}) which interchanges χ_{I}^{α} and χ_{II}^{α}, we have

τ̃(χ^{α,α}; q, y) = 2τ̃(χ_{I}^{α}; q, y) = 2τ̃(χ_{II}^{α}; q, y).

From this we get (3.18), hence the rest of the formulas.

Next we calculate c̃(χ; q, y) for type D_{l}.

First, we consider the representations χ_{I}^{α} and χ_{II}^{α}. Since each χ_{I}^{α} (resp. χ_{II}^{α} ) forms an equivalence class with respect to ∼_{LR} by itself by Lemma 3.5, we have c̃(χ; q, y) = c̃′(χ; q, y) for χ = χ_{I}^{α} or χ_{II}^{α}, and

c̃(χ_{I}^{α}; q, y) = c̃(χ_{II}^{α}; q, y) = ∏_{x∈α}(1 + yq^{2c(x)+1})(1 + yq^{2c(x)−1}).  (3.21)

Secondly, we consider c̃ for χ̃^{α,β} (α ≠ β). Put [i] := 1 + yq^{i}, [j, k] := ∏ _{\begin{matrix}j≤i≤k \\ i is odd \end{matrix}} [i] if

{i | j ≤ i ≤ k, i is odd } ≠ ∅, and [j, k] := 1 if otherwise. Then the inside of { } of (3.15) multiplied by ∏_{i=1}^{m}[−2i + 1]^{2i−1} (m ≫ 0) is equal to

q^{−\binom{m}{2}} ∏_{i=1}^{m}[−2m + 1, 2λ_{i} − 2m − 1] · [−2m + 1, 2μ_{i} − 2m − 1]

× {q^{|μ|} ∏_{i=1}^{m}[2λ_{i} − 2m + 1] + q^{|λ|} ∏_{i=1}^{m}[2μ_{i} − 2m + 1]} ,  (3.22)

where λ_{i} := α_{m+1−i} + i − 1, μ_{i} := β_{m+1−i} + i − 1 for 1 ≤ i ≤ m as before. Let Λ = \binom{λ_{1} ··· λ_{m}}{μ_{1} ··· μ_{m}} = \binom{A⊔C}{B⊔C} with A ∩ B = ∅. From the assumption α ≠ β, A and B are nonempty. Then (3.22) is equal to

∏_{x∈A⊔B⊔C⊔(C+1)}[−2m + 1, 2x − 2m − 1]

×q^{−\binom{m}{2}} \{q^{|μ|} ∏_{a∈A}[2a − 2m + 1] + q^{|λ|} ∏_{b∈B}[2b − 2m + 1]\} ,  (3.23)

where C + 1 := {c + 1 | c ∈ C}.

Lemma 3.7 Put χ = χ̃^{α,β} (α ≠ β).

(1) If there exists d ∈ ℤ such that

A = {a_{1}, ⋯, a_{r}}, B = {d − a_{1}, ⋯, d − a_{r}}, and r is odd,  (3.24)

then c̃′(χ; q, y) is given by

∏_{i=1}^{m}[−2i + 1]^{2i−1} · c̃′(χ; q, y) = [d − 2m + 1] · ∏_{x∈A⊔B⊔C⊔(C+1)}[−2m + 1, 2x − 2m − 1]. (3.25)

(2) If the condition (3.24) does not hold, then c̃′(χ; q, y) is given by

∏_{i=1}^{m}[−2i + 1]^{2i−1} · c̃′(χ; q, y) = ∏_{x∈A⊔B⊔C⊔(C+1)}[−2m + 1, 2x − 2m − 1].  (3.26)

Proof. We can see that { } in (3.23) vanishes for y = −q^{−d+2m−1} (d ∈ ℤ), if and only if the condition (3.24) holds. Let us prove it. The condition which d should satisfy is

q^{∑_{i} b_{i}} ∏_{i=1}^{r}(1 − q^{2a_{i}−d}) + q^{∑_{i} a_{i}} ∏_{i=1}^{r}(1 − q^{2b_{i}−d}) = 0,

which implies {2a_{i} − d}_{i} = {ε_{i}(2b_{i} − d)}_{i} with some ε_{i} = ±1. Since A ∩ B = ∅, all the ε_{i}’s should be −1, hence we may assume that 2a_{i} − d = d − 2b_{i}, i.e., b_{i} = d − a_{i}. Thus we can easily see that (3.24) is a necessary and sufficient condition.

Next, let us show that y = −q^{−d+2m−1} is a simple root. By a change of variable y = xq^{−d+2m−1}, it suffices to show that x = −1 is a simple root of

φ(x) := ∏_{i=1}^{r}(q^{a_{i}} + xq^{b_{i}}) + ∏_{i=1}^{r}(q^{b_{i}} + xq^{a_{i}}),

assuming (3.24). Note that

φ′(−1) = (∑_{j=1}^{r} \frac{q^{a_{j}} + q^{b_{j}}}{q^{a_{j}} − q^{b_{j}}} ) · ∏_{i}(q^{a_{i}} − q^{b_{i}}).  (3.27)

Since A ∩ B = ∅, the first factor of (3.27) tends to ∑_{j=1}^{r}(±1) as q → 0. Since r is odd, this limit is non-zero, and hence the first factor is non-zero. Since A ∩ B = ∅, the second factor is also non-zero.

Next proposition immediately follows from the above lemma.

Proposition 3.8 Take χ = χ̃^{α,β} ∈ W (D_{l})^{∨} (α ≠ β), and let λ_{i} := α_{m+1−i} + i − 1, μ_{i} := β_{m+1−i} + i − 1 for 1 ≤ i ≤ m. Here we assume α_{m+1} = β_{m+1} = 0. Put C = {λ_{i}}_{i} ∩ {μ_{i}}_{i}, A = {λ_{i}}_{i} − C, and B = {μ_{i}}_{i} − C.

(1) If #A > 1 (hence #B = #A > 1 also), then c̃(χ; q, y) is given by

∏_{i=1}^{m}[−2i + 1]^{2i−1} · c̃(χ; q, y) = ∏_{x∈A⊔B⊔C⊔(C+1)}[−2m + 1, 2x − 2m − 1],  (3.28)

where C + 1 = {c + 1| c ∈ C}.

(2) If A = {a} and B = {b}, then c̃(χ; q, y) is given by

∏_{i=1}^{m}[−2i+1]^{2i−1} ·c̃(χ; q, y) = [a+b−2m+1]·  ∏_{x∈A⊔B⊔C⊔(C+1)}[−2m+1, 2x−2m−1]. (3.29)

As the following example shows, the invariant c̃(χ; q, y) does not separate the two-sided cells in W (D_{l})^{∨}.

Example 3.9 Let Λ = \binom{0 1 3}{1 4 6} and Λ′ = \binom{0 1 3}{2 3 6}, and identify them with the corresponding irreducible characters of W (D_{9}). Then

{c_{i}(Λ)}_{i} = {c_{i}(Λ′)}_{i} = {−3, −1, −1, 1, 1, 3, 5}.

Thus Λ ≁_{LR} Λ′ but c̃(Λ; q, y) = c̃(Λ′; q, y).

For a later use, we reformulate (3.16) as follows.

Lemma 3.10 Take χ ∈ W (D_{l})^{∨} corresponding to Λ = \binom{λ_{1} ··· λ_{m}}{μ_{1} ··· μ_{m}} and let the sets A, B and C be as in Proposition 3.8. Then we have

∏_{i=1}^{m}(t − 2i + 1)^{2i−1} × 2τ^{∗}(Λ; t) = ∏_{x∈A⊔B⊔C⊔(C+1)}[−2m + 1, 2x − 2m − 1]^{∗}

× { ∏_{a∈A}(t + 2a − 2m + 1) + ∏_{b∈B}(t + 2b − 2m + 1)} ,  (3.30)

where [j, k]^{∗} = ∏ _{\begin{matrix}j≤i≤k \\ i is odd \end{matrix}} (t + i).

3.4 Type G_{2}.

If χ = 1 (resp. χ = sgn), then c̃′(1; q, y) = (1 + yq)(1 + yq^{5}) (resp. c̃′(sgn; q, y) = (1 + yq^{−1})(1 + yq^{−5})) by (2.3) and (1.27). The trivial (resp. sign) representation forms a family by itself. The remaining four representations form one family. In this case, c̃′(χ; q, y) = (1 + yq^{−1})(1 + yq) by (2.4) and (2.5). Hence always τ̃_{prim}(χ; q, y) = 1 and c̃(χ; q, y) = c̃′(χ; q, y). See Table 2.

3.5 Types F_{4} and E_{l} ( l = 6, 7, 8 ).

We give c_{i}(χ) (1 ≤ i ≤ κ(χ)) and τ_{prim}^{∗}(χ; t) in Tables 3 – 6. Note that the set {c_{i}(χ) (1 ≤ i ≤ κ(χ))} completely determines the invariant c̃(χ; q, y), so the tables describe both c̃(χ; q, y) and τ^{∗}(χ; t). The explicit form of τ_{prim}(χ; q, y) is too complicated to include here.

Besides, we shall give the explicit form of c̃(χ; q, y) = ∏_{i=1}^{κ(χ)}(1 + q^{c_{i}(χ)}y), τ̃(χ; q, y) and τ^{∗}(χ; t) for χ ∈ W_{ex.}^{∨} in this subsection.

(1) If W = W (E_{7}) and dim χ = 512, then κ = κ(χ) = 2 and {c_{i}}_{1≤i≤κ} = {c_{i}(χ)}_{1≤i≤κ} = {−1, 1}. Put {c_{i}}_{κ<i≤l} = {−4, −2, 0, 2, 4} (l = 7). Then τ^{∗}(χ; t) = ∏_{i=1}^{l}(t + c_{i}),

τ̃(χ; q, y) = \frac{f(χ ⊗ sgn; q)}{∏_{i=1}^{l}(1 − q^{m_{i}+1})} ∏_{i=1}^{l}(1 + q^{c_{i}}y) + εq^{5} \frac{q^{2}ϕ_{5}ϕ_{10}y^{4} + ϕ_{5}ϕ_{8}ϕ_{10}y^{2} + q^{6}}{ϕ_{2}^{7}ϕ_{6}^{3}ϕ_{10}ϕ_{14}ϕ_{18}} ∏_{i=1}^{κ}(1 + q^{c_{i}}y),

where ϕ_{i} = ϕ_{i}(q) is the i -th cyclotomic polynomial (e.g., ϕ_{3} = q^{2} + q + 1), and ε = −1 (resp. +1) if χ = 512′_{a} (resp. 512_{a}).

(2) If W = W (E_{8}) and χ ∈ {4096_{z}, 4096_{x}}, then κ = κ(χ) = 3, and {c_{i}}_{1≤i≤κ} = {c_{i}(χ)}_{1≤i≤κ} = {−1, 1, 7}. Put {c_{i}}_{κ<i≤l} = {−4, −2, 2, 4, 8} (l = 8). Then τ^{∗}(χ; t) = ∏_{i=1}^{l}(t + c_{i}), and

τ̃(χ; q, y) = q^{−15} \frac{f(χ ⊗ sgn; q)}{∏_{i=1}^{l}(1 − q^{m_{i}+1})} ∏_{i=1}^{l}(1 + q^{c_{i}}y)

+εq^{5} \frac{(q^{12} + q^{10} + q^{6} + q^{2} + 1)q^{6}y^{4} + (q^{14} + q^{12} − q^{10} + q^{8} + 2q^{6} + 1)ϕ_{8}y^{2} + q^{6}}{ϕ_{1}ϕ_{2}^{8}ϕ_{6}^{4}ϕ_{10}^{2}ϕ_{14}ϕ_{18}ϕ_{30}} ∏_{i=1}^{κ}(1 + q^{c_{i}}y),

where ε = −1 (resp. +1) if χ = 4096_{z} (resp. 4096_{x}).

(3) If W = W (E_{8}) and χ ∈ {4096′_{z}, 4096′_{x}}, then κ(χ), c_{i}(χ) (1 ≤ i ≤ κ(χ)), τ^{∗}(χ; t) and τ̃(χ; q, y) can be explicitly determined by (2) and the duality (see Proposition 1.4).

3.6 The main theorem.

We sum up the results in the former subsections into the following theorem.

Theorem 3.11 Let W be an irreducible Weyl group not of type D_{l} (l ≥ 5). For χ, χ′ ∈ W^{∨}, we have

c̃(χ; q, y) = c̃(χ′; q, y) ⇔ χ ∼_{LR} (χ′)^{ι} for some ι ∈ Aut(W, S).

4 Primitive factors and the relation with the modular representation theory.

We extend the notion of exceptional representations in the case of type G_{2}; namely, we define W_{ex.m}^{∨} ⊂ W^{∨} by

W_{ex.m}^{∨} = \{ \begin{matrix}\{χ ∈ W^{∨} | dim χ = 2\}, & if W = W (G_{2}), \\ W_{ex.}^{∨}, & otherwise. \end{matrix}

We call them exceptional representations in a modified sense. For the motivation of this modification, see §4.2 below.

4.1 Factorization patterns of τ̃.

If W is of type A_{l}, D_{l} or E_{l}, the factorization patterns of τ̃_{prim}(χ; q, y) and τ_{prim}^{∗}(χ; t) seem to be controlled by M(G(χ)) (cf. §0.1 (b) and §0.3). Theorem 0.2 in the introduction can be refined, and generalized as follows.

Theorem 4.1 (1) Consider the Weyl group of type A_{l}, D_{l} or E_{l}, excluding χ’s in W_{ex.m}^{∨}.

(a) If G = 𝔖_{1}, then τ̃_{prim}(χ; q, y) = 1.

(b) If G = 𝔖_{2}, then τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] is an irreducible polynomial of degree 2.

(c) If G = 𝔖_{3} (so that W = W (E_{l}) (l = 6, 7, 8)), then the factorization pattern of τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] is as follows:

\begin{matrix}M(𝔖_{3}) & (1, 1) & (g_{2}, 1) & (1, r) & (g_{3}, 1) & (1, ϵ) \\ π_{ℚ(q)}(τ̃_{prim}(χ; q, y)) & \{21^{2}\} & \{4\} & \{4\} & \{4\} & \{21^{2}\} \end{matrix}

(See [L1, 4.3] for the notation (g_{2}, 1) etc. See §0.3 for the definition of factorization pattern.)

(d) If G = 𝔖_{5} (so that W = W (E_{8})), the factorization pattern of τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] is as follows:

\begin{matrix}M(𝔖_{5}) & (1, 1) & (g_{3}, 1) & (g′_{2}, 1) & (1, ν) & (1, λ′) & (g_{5}, 1) & (g_{3}, ε) & (1, ν′) \\ π_{ℚ(q)}(τ̃_{prim}(χ; q, y)) & \{6\} & \{1^{2}4\} & \{1^{2}4\} & \{6\} & \{6\} & \{6\} & \{6\} & \{1^{2}4\} \end{matrix}

\begin{matrix}(g′_{2}, ε′′) & (1, λ^{2}) & (g′_{2}, ε′) & (1, λ^{3}) & (g_{2}, 1) & (g_{4}, 1) & (g_{6}, 1) & (g_{2}, r) & (g_{2}, ε) \\ \{6\} & \{1^{2}4\} & \{6\} & \{1^{2}4\} & \{6\} & \{1^{2}2^{2}\} & \{6\} & \{1^{4}2\} & \{1^{2}4\} \end{matrix}

(2) Consider the case of B_{l} (= C_{l}), F_{4} or G_{2}, excluding χ’s in W_{ex.m}^{∨}. Then every τ̃_{prim}(χ; q, y) decomposes into linear factors in ℤ[t] except for {12_{1}, 4_{1}, 6_{1}, 6_{2}, 16_{1}} in the F_{4}-case. For these excepted χ’s, the factorization pattern of τ̃_{prim}(χ; q, y) is {2}.

(3) For χ ∈ W_{ex.m}^{∨}, τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] is irreducible.

Theorem 4.2 The factorization pattern of τ^{∗}(χ; t) ∈ ℤ[t] is always finer than that of τ̃(χ; q, y) ∈ ℚ(q)[y].

(1) Consider the Weyl group of type A_{l}, D_{l} or E_{l}, excluding χ’s in W_{ex.m}^{∨}.

(a) If G = 𝔖_{1}, then τ_{prim}^{∗}(χ; t) = 1.

(b) If G = 𝔖_{2} and W ≠ W (D_{l}) (l ≥ 5), then τ_{prim}^{∗}(χ; t) ∈ ℤ[t] is an irreducible polynomial of degree 2 except the following 4 cases:

W = W (E_{6}) and χ ∈ {30_{p}, 30′_{p}}, W = W (E_{8}) and χ ∈ {210_{x}, 210′_{x}}.

In these cases, the factorization pattern of τ^{∗}(χ; t) is {1^{2}}.

(c) If G = 𝔖_{3} (so that W = W (E_{l}) (l = 6, 7, 8)), the factorization pattern of τ_{prim}^{∗}(χ; t) ∈ ℤ[t] is as follows:

\begin{matrix}M(𝔖_{3}) & (1, 1) & (g_{2}, 1) & (1, r) & (g_{3}, 1) & (1, ϵ) \\ π_{ℚ}(τ_{prim}^{∗}(χ; t)) & \{21^{2}\} & \{4\} & \{2^{2}\} & \{2^{2}\} & \{21^{2}\} \end{matrix}

(d) If G = 𝔖_{5} (so that W = W (E_{8})), the factorization pattern of τ_{prim}^{∗}(χ; t) ∈ ℤ[t] is as follows:

\begin{matrix}M(𝔖_{5}) & (1, 1) & (g_{3}, 1) & (g′_{2}, 1) & (1, ν) & (1, λ′) & (g_{5}, 1) & (g_{3}, ε) & (1, ν′) \\ π_{ℚ}(τ_{prim}^{∗}(χ; t)) & \{2^{3}\} & \{1^{2}4\} & \{1^{2}2^{2}\} & \{1^{2}2^{2}\} & \{1^{2}2^{2}\} & \{24\} & \{1^{4}2\} & \{1^{2}2^{2}\} \end{matrix}

\begin{matrix}(g′_{2}, ε′′) & (1, λ^{2}) & (g′_{2}, ε′) & (1, λ^{3}) & (g_{2}, 1) & (g_{4}, 1) & (g_{6}, 1) & (g_{2}, r) & (g_{2}, ε) \\ \{2^{3}\} & \{1^{2}2^{2}\} & \{24\} & \{1^{2}2^{2}\} & \{24\} & \{1^{2}2^{2}\} & \{6\} & \{1^{4}2\} & \{1^{2}2^{2}\} \end{matrix}

(2) Consider the case of B_{l} (= C_{l}), F_{4} or G_{2}, excluding χ’s in W_{ex.m}^{∨}. Then the factorization patter of τ_{prim}^{∗}(χ; t) is the same as that of τ̃_{prim}(χ; q, y).

(3) For χ ∈ W_{ex.m}^{∨}, τ_{prim}^{∗}(χ; t) decomposes into linear factors in ℤ[t].

Remark. For type D_{l}, it is known that G = 𝔖_{2}^{m} (m ≥ 0). We do not know anything about the case for type D_{l} with G = 𝔖_{2}^{m} (m ≥ 2) yet.

4.2 Relation with the modular representation theory of (generic) Iwahori-Hecke algebra.

Here we try to find a unified picture for Theorem 4.1 from the viewpoint of the modular representation theory.

Let (W, S) be a Coxeter system, {q_{s}}_{s∈S} a family of indeterminates such that q_{s} = q_{s′} if and only if s and s′ are conjugate in W and such that different q_{s}’s are algebraically independent. Put R = ℤ[q_{s}^{1/2}, q_{s}^{−1/2}; s ∈ S] and K := ℚ(q_{s}^{1/2}; s ∈ S). Define an associative algebra structure in H_{0}(W) = ⨁_{s∈W} RT_{w} so that

T_{w}T_{w′} = T_{ww′} if l(w) + l(w′) = l(ww′), and
(T_{s} + 1)(T_{s} − q_{s}) = 0 for s ∈ S.

Put H(W) := H_{0}(W) ⊗_{R} K. For a ring A, let A^{∨} denote the set of simple A-modules (up to isomorphism). Then naturally we can identify H(W)^{∨} = (ℂW)^{∨} = W^{∨}. For χ ∈ W^{∨}, denote the corresponding simple H(W)-module by χ̃. For a fixed prime number p, we define the equivalence relation ∼_{p} in W^{∨} as follows: Let us consider ‘the reduction modulo pR’ of elements of H(W)^{∨}. For χ, χ′ ∈ W^{∨}, define χ∼_{p}χ′ if χ̃ and χ̃′ belong to the same

block (in the usual sense of the modular representation theory). Define χ∼_{∗}χ′ if there exists χ_{0}, χ_{1}, ⋯, χ_{n} ∈ W^{∨} and prime numbers p_{1}, ⋯, p_{n} such that

χ = χ_{0} ∼_{p_{1}} χ_{1} ∼_{p_{2}} ⋯ ∼_{p_{n}} χ_{n} = χ′.

This equivalence relation can be explicitly described as follows.

(1) If W is of type A_{l}, D_{l} or E_{l} (l = 6, 7), then χ ∼_{∗} χ′ ⇔ χ ∼_{LR} χ′ by [Gy2]. It is plausible

that the same holds also for E_{8} (see [Gy2]).

(2) If W is of type B_{l} (= C_{l}), then χ∼_{∗}χ′ ⇔ χ = χ′ by [Gy3] and [H].

(3) If W is of type G_{2}, then the equivalence classes are {1}, {sgn}, {ϵ_{1}}, {ϵ_{2}}, {V, V′} in the notation of [L1, 4.8].

(4) If W is of type F_{4}, then {12_{1}, 4_{1}, 6_{1}, 6_{2}, 16_{1}} forms an equivalence class with respect to ∼_{∗}, and each of the remaining χ’s forms an equivalence class by itself. (Proof. For a prime number p, let R_{p} be the localization of R by pR, and H_{p}(W) := H_{0}(W) ⊗_{R} R_{p}. First consider the case p = 2. By [Gy3, 2.17] and [C2, p.451], χ ∉ {12_{1}, 16_{1}, 4_{1}} = {ϕ_{12,4}, ϕ_{16,5}, ϕ_{4,8}} can be obtained from a principal indecomposable module by a scalar extension. In particular, each such χ forms a block by itself. Then by the argument of [Ge, p.2965, ↑ ℓℓ.14–7] and by the information given in [L1, (4.10.4)], we can see that 12_{1} + 16_{1} and 12_{1} + 4_{1} come from some direct summand, say U_{1} and U_{2}, of the left regular module H_{p}(W). By [Gy3, (2.16.1)] and by [C2, p.451], U_{i}’s are indecomposable. Hence if p = 2, {12_{1}, 16_{1}, 4_{1}} forms a single block, and its Brauer tree is 4_{1} 16_{1} 12_{1}. If p = 3, the same argument yields that {12_{1}, 6_{1}, 6_{2}} = {ϕ_{12,4}, ϕ′_{6,6}, ϕ′′_{6,6}} forms a single block, its Brauer tree is 6_{1} 12_{1} 6_{2}, and each of the remaining χ’s forms a block by itself.)

Using or assuming (1)–(4) above, we can restate Theorem 4.1 as follows.

Theorem 4.3 Consider an irreducible Weyl group W.

(1) If φ(χ) := #{χ′ ∈ W^{∨} | χ′∼_{∗}χ} = 1, then τ̃_{prim}(χ; q, y) = 1.

(2) If φ(χ) = 3 (i.e., W = W (D_{l}) or W (E_{l}), and G = 𝔖_{2}, but χ ∉ W_{ex.}^{∨}), then τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] is an irreducible polynomial of degree 2.

(3) If φ(χ) = 5 and G = 𝔖_{3} (i.e., W = W (E_{l}) and G = 𝔖_{3}), then the factorization pattern of τ̃_{prim}(χ; q, y) is given by (1c) of Theorem 4.1.

(4) If φ(χ) = 17 (i.e., W = W (E_{8}) and G = 𝔖_{5}), then the factorization pattern of τ̃_{prim}(χ; q, y) is given by (1d) of Theorem 4.1.

(5) If φ(χ) = 5 and G = 𝔖_{4} (i.e., W = W (F_{4}) and χ ∈ {12_{1}, 4_{1}, 6_{1}, 6_{2}, 16_{1}}), then the factorization pattern of τ̃_{prim}(χ; q, y) is {2}.

(6) If φ(χ) = 2 (i.e., χ ∈ W_{ex.m}^{∨}), then τ̃_{prim}(χ; q, y) ∈ ℚ(q)[y] is irreducible.

Remark 1. Theorem 4.2 can be also reformulated in a similar way.

Remark 2. The block structure of H_{0}(W) ⊗_{R} R_{p} can be determined if we know (1) the center of H_{0}(W) ⊗_{R} R_{p}, and

(2) the central characters of all the irreducible representations of H_{0}(W)⊗_{R} K_{p}, where K_{p} is the fractional field of R_{p}.

If W is of type A_{l} or B_{l} (= C_{l}), (1′) the center of H_{0}(W) ⊗_{R} K_{p}, and (2), are determined in [AK]. (See from p.239, ℓ.23 to the end.) In this connection, it would be worth noting that σ^{∗}(χ; q) (with a slight modification in the B_{l}-case) appears in [AK, Theorem 3.20].

5 Product formula.

Let W be of type A_{l}, D_{l} or E_{l}, χ ∈ W^{∨}, F the family containing χ, and G the group associated to F (cf. §0.1 (b)). If χ ∉ W_{ex.m}^{∨} and G ≠ 1, then τ^{∗}(χ; t) does not decompose into linear factors in ℤ[t] (with a small number of exceptions). However, if G = 𝔖_{2}^{r} (r ≥ 0), then certain linear combinations of τ^{∗}(χ; t) (χ ∈ F) completely decompose into linear factors. In §5.1, we give such a product formula. Moreover, if G = 𝔖_{2}, we can refine this observation. See §5.2, and also §0.4.

5.1 D_{l}-case.

Let χ = χ̃^{α,β}, λ_{i}, μ_{i}, Λ, A, B and C be as in §3.3. In the sequel, we identify χ with Λ. Take any bijection φ : A → B. Let Φ be the totality of permutations σ ∈ 𝔖(A⊔B⊔C)

such that σ({a, φ(a)}) = {a, φ(a)} (a ∈ A), and σ(c) = c (c ∈ C). Fix a_{0} ∈ A and put Φ_{0} := {a ∈ Φ | σ(a_{0}) = a_{0}}. Put

σ(Λ) = \binom{σ(λ_{1}), ⋯, σ(λ_{m})}{σ(μ_{1}), ⋯, σ(μ_{m})}.

Note that Λ∼_{LR}σ(Λ) for any σ ∈ Φ.

Proposition 5.1

∏_{i=1}^{m}(t − 2i + 1)^{2i−1} × ∑_{σ∈Φ_{0}} τ^{∗}(σ(Λ); t) =

2^{|A|−1} ∏_{x∈A⊔B⊔C⊔(C+1)}[−2m + 1, 2x − 2m − 1]^{∗} · ∏_{a∈A}(t + a + φ(a) − 2m + 1), (5.1)

where [j, k]^{∗} = ∏ _{\begin{matrix}j≤i≤k \\ i : odd \end{matrix}} (t + i).

Proof. From (3.30), we get

2 ∏_{i=1}^{m}(t − 2i + 1)^{2i−1} · ∑_{σ∈Φ_{0}} τ^{∗}(σ(Λ); t) = ∏_{x∈A⊔B⊔C⊔(C+1)}[−2m + 1, 2x − 2m − 1]^{∗}

× ∑_{σ∈Φ} ∏_{a∈σ(A)}(t + 2a − 2m + 1).  (5.2)

The second factor in the right hand side is equal to

∏_{a∈A} {(t + 2a − 2m + 1) + (t + 2φ(a) − 2m + 1)} = 2^{|A|} ∏_{a∈A}(t + a + φ(a) − 2m + 1). (5.3)

5.2 Families consisting of 3 elements.

Let W be of type D_{l} or E_{l}, and F a family consisting of three elements χ_{1}, χ_{2}, χ_{3}. We assume that χ_{1}, χ_{2}, χ_{3} correspond to (1, 1), (g_{2}, 1), (1, ϵ) in this order, and regard the suffix i of χ_{i} as an element of ℤ/3ℤ. (For example χ_{3+1} = χ_{1}.) Put

e = e(F) = e(χ) := −a(χ) + a(χ ⊗ sgn) − ∑_{i=1}^{κ(χ)} c_{i}(χ) for χ ∈ F.  (5.4)

By Theorem 4.1 (1b), (1.10), (1.19) and (2.19), we get

\frac{τ̃_{prim}(χ_{i}; q, y)}{τ̃_{prim}(χ_{i}; q, 0)} = 1 + σ_{prim}^{∗}(χ_{i}; q)y + q^{e(F)}y^{2}.  (5.5)

Theorem 5.2 To each family F ∈ W^{∨}/ ∼_{LR} consisting of 3 elements, there associate integers d_{i} = d_{i}(F) and d′_{i} = d′_{i}(F) (i ∈ ℤ/3ℤ) such that

d_{1} + d′_{1} = d_{2} + d′_{2} = d_{3} + d′_{3}  (5.6)

and

σ_{prim}^{∗}(χ_{i}; q) = (q^{d_{i}} + q^{d′_{i}})^{−1}(q^{d_{i+1}} + q^{d′_{i+1}})(q^{d_{i+2}} + q^{d′_{i+2}}),  (5.7)
\frac{1}{2}(τ_{prim}^{∗}(χ_{i}; t) + τ_{prim}^{∗}(χ_{i+1}; t)) = (t + d_{i+2})(t + d′_{i+2}),  (5.8)

\frac{∏_{i=1}^{l}(1 − q^{m_{i}+1})}{f(χ_{i}; q)}τ̃(χ_{i}; q, y)

= \frac{q^{d_{i}}}{q^{d_{i}} + q^{d′_{i}}} (1 + yq^{−d_{i}+d_{i+1}+d′_{i+2}})(1 + yq^{−d_{i}+d′_{i+1}+d_{i+2}}) ∏_{1≤j≤κ(χ_{i})}(1 + yq^{c(χ_{i})})

+ \frac{q^{d′_{i}}}{q^{d_{i}} + q^{d′_{i}}} (1 + yq^{−d′_{i}+d_{i+1}+d_{i+2}})(1 + yq^{−d′_{i}+d′_{i+1}+d′_{i+2}}) ∏_{1≤j≤κ(χ_{i})}(1 + yq^{c(χ_{i})}). (5.9)

Proof. First, let us consider the D_{l}-case. Let notation be as in §3.3, and assume that #A = #B = 2. For Λ := \binom{A⊔C}{B⊔C}, put

P (Λ; q) := q^{|A|−2m+1} + q^{|B|−2m+1}.  (5.10)

Now assume that A = {a_{1}, a_{2}}, B = {b_{1}, b_{2}} and a_{1} < b_{1} < a_{2} < b_{2}. Put A′ := {a_{1}, b_{1}}, B′ := \{a_{2}, b_{2}\}, A′′ := \{a_{1}, b_{2}\}, B′′ := \{a_{2}, b_{1}\}, Λ_{1} := Λ, Λ_{2} := \binom{A′⊔C}{B′⊔C} and Λ_{3} := \binom{A′′⊔C}{B′′⊔C}. Then Λ_{1}, Λ_{2}, Λ_{3} form a family, the associated elements in M(𝔖_{2}) are (1, 1), (g_{2}, 1) and (1, ϵ) in this order [L1, 4.6]. By Proposition 3.8 (1), (1.21) and (3.17), we obtain

Since (3.25) (resp. (3.17)) gives an explicit form of c̃(χ; q, y) (resp. σ^{∗}(Λ_{i}; q)), we can calculate σ_{prim}^{∗}(Λ_{i}; q), and we get

σ_{prim}^{∗}(Λ_{i}; q) = P (Λ_{i}; q)^{−1}P (Λ_{i+1}; q)P (Λ_{i+2}; q).  (5.11)

Then we get (5.6) and (5.7) in the D_{l}-case. In the E_{l}-case, d_{i} and d′_{i} can be obtained by a direct calculation. See Table 7 for their explicit values.

Now (5.9) follows from (2.1) and (5.5) – (5.7). Specializing (5.9) as in (1.23), we get (5.8).

References.

[AK] S.Ariki, K.Koike, A Hecke algebra of (ℤ/rℤ) ≀ 𝔖_{n} and construction of its representations. Advances in Math., 106(1994), 216–243.

[BL] W. M. Beynon, G. Lusztig, Some numerical results on the characters of exceptional Weyl groups. Math. Proc. Camb. Phil. Soc., 84(1978), 417–426.

[B] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4–6. Masson, 1981.

[C1] R. W. Carter, Conjugacy classes in the Weyl group. Compositio Math., 25(1972), 1–59.

[C2] R.W.Carter, Finite Groups of Lie Type. John Wiley and Sons, 1985.

[Ge] M.Geck, Brauer trees of Hecke algebras. Comm. in Algebra, 20(10)(1992), 2937– 2973.

[GP] M.Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras. Adv. in Math., 102(1993), 79–94.

[Gy1] A. Gyoja, A generalized Poincaré series associated to a Hecke algebra of finite or p-adic Chevalley group. Japan J. Math., 9(1983), 87–111.

[Gy2] A. Gyoja, Cells and modular representations of Hecke algebra. To appear in Osaka J. Math.

[Gy3] A. Gyoja, Modular representation theory over a ring of higher dimension with applications to Hecke algebras. J. Algebra, 174(1995), 553 – 572.

[H] N.Hoefsmit, Representations of Hecke algebras of finite groups with BN pairs of classical type. Ph.D. dissertation, Vancouver, 1974.

[J] A. Joseph, W -module structure in the primitive spectrum of the enveloping algebra of a semisimple Lie algebra. Lecture Notes in Math., 728(1979), 116 – 135, Springer.

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

[L1] G. Lusztig, Characters of Reductive Groups over a Finite Field. Princeton University Press, Princeton, 1984.

[L2] G. Lusztig, Irreducible representations of finite classical groups. Invent. Math., 43(1977), 125–175.

[L3] G. Lusztig, A class of irreducible representations of a Weyl group. Proc. Kon. Nederl. Akad., 82(1979), 323-335.

[M] I. G. Macdonald, Symmetric functions and Hall polynomials. Clarendon Press, Oxford, 1979.

[St1] R. Steinberg, A geometric approach to the representations of the full linear group over a Galois field. Trans. AMS, 71(1951), 274–282.

[St2] R. Steinberg, Differential equations invariant under finite reflection groups. Trans. AMS, 112(1964), 392 – 400.

Tables.

For each irreducible character χ of W = W (D_{4}), W (G_{2}), W (F_{4}) and W (E_{l}) (l = 6, 7, 8), we shall write down c_{i}(χ) (1 ≤ i ≤ κ(χ)) and τ_{prim}^{∗}(χ; t) in Tables 1 – 6. In Table 7, we shall list the integers d_{i} and d′_{i} (1 ≤ i ≤ 3) appeared in Theorem 5.2. Let us give some explanation on these tables.

On Tables 1 – 6. We shall give all the families belonging to W^{∨}/ ∼_{LR} and describe c_{i}(χ) (1 ≤ i ≤ κ(χ)) and τ_{prim}^{∗}(χ; t) in these tables. Note that

τ^{∗}(χ; t) = ∏_{i=1}^{κ(χ)}(t + c_{i}(χ)) · τ_{prim}^{∗}(χ; t).

So the tables describe τ^{∗}(χ; t) completely.

For simplicity, the families which consists of only one element (i.e., 1-element families) are collected together. For these families, the associated group is G = 𝔖_{1} = {e} and M(𝔖_{1}) = {(1, 1)}. We put the parameters of each irreducible representation χ in the first and the second columns. The first one is used in [C2] and the second is used in [L1, §4].

For the families which have more than 2 irreducible representations, we also list the associated parameters (1, 1), (g_{2}, 1), etc. ∈ M(G) in the third column. We call the family which have n elements n-element family.

For 1-element families, since τ_{prim}^{∗}(χ; t) = 1 always, we only list {c_{i}(χ)| 1 ≤ i ≤ l = κ(χ)} for each representation χ.

For n-element families (n ≥ 2), in the first row, we list {c_{i}(χ)| 1 ≤ i ≤ κ(χ)} with the associated group G = 𝔖_{2}, 𝔖_{3}, · · ·. The following rows contain two kinds of parameters of χ, by Carter [C2] and Lusztig [L1, §4], in the first and the second columns respectively; in the third column the associated parameter belonging to M(G) is given. In the last column, τ_{prim}^{∗}(χ; t) is displayed.

On Table 7. Here we list d_{i} and d′_{i} (1 ≤ i ≤ 3) of Theorem 5.2. In the first and the second columns, we list the special representations (i.e., χ ∈ W^{∨} corresponding to (1, 1) ∈ M(𝔖_{2})) of the families F ∈ W^{∨}/ ∼_{LR} consisting of three elements. The third column contains d_{1}, d′_{1}, d_{2}, d′_{2}, d_{3}, d′_{3} in this order.

Table 1 : Type D_{4}. 1-element families (𝔖_{1} = {e}).

(4, −) (\begin{matrix}4 \\ 0 \end{matrix}) 1 3 3 5
(1^{4}, −) (\begin{matrix}1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \end{matrix}) −5 −3 −3 −1
(3, 1)  (\begin{matrix}3 \\ 1 \end{matrix}) −1 1 3 3
(1^{3}, 1)  (\begin{matrix}1 & 2 & 3 \\ 0 & 1 & 3 \end{matrix}) −3 −3 −1 1
(31, −)  (\begin{matrix}1 & 4 \\ 0 & 1 \end{matrix}) −1 1 1 3
(2, 2)_{I}  (\begin{matrix}2 \\ 2 \end{matrix})_{I} −1 1 1 3
(2, 2)_{II}  (\begin{matrix}2 \\ 2 \end{matrix})_{II} −1 1 1 3
(21^{2}, −) (\begin{matrix}1 & 2 & 4 \\ 0 & 1 & 2 \end{matrix}) −3 −1 −1 1
(1^{2}, 1^{2})_{I} (\begin{matrix}1 & 2 \\ 1 & 2 \end{matrix})_{I} −3 −1 −1 1
(1^{2}, 1^{2})_{II} (\begin{matrix}1 & 2 \\ 1 & 2 \end{matrix})_{II} −3 −1 −1 1

3-element family (𝔖_{2}) : −1 1

(21, 1) (\begin{matrix}1 & 3 \\ 0 & 2 \end{matrix}) (1, 1) t^{2} − 3
(2^{2}, −) (\begin{matrix}2 & 3 \\ 0 & 1 \end{matrix}) (g_{2}, 1) t^{2} + 3
(2, 1^{2}) (\begin{matrix}0 & 3 \\ 1 & 2 \end{matrix}) (1, ε) t^{2} − 5

Table 2 : Type G_{2}. 1-element families (𝔖_{1} = {e}).

[1, 0] trivial 1 5
[1, 6] sign −5 −1

4-element family (𝔖_{3}) : −1 1

[1, 3]′ ε_{1} (1, r) 1
[1, 3]′′ ε_{2} (g_{3}, 1) 1
[2, 1] V (1, 1) 1
[2, 2] V′ (g_{2}, 1) 1

Table 3 : Type F_{4}.

1-element families (𝔖_{1} = {e}).

[1, 0] 1_{1}  1 5 7 11
[1, 24] 1_{4} −11 −7 −5 −1
[9, 2] 9_{1} −1 1 3 5
[9, 10] 9_{4} −5 −3 −1 1
[8, 3]′′ 8_{1} −1 1 1 5
[8, 9]′ 8_{2} −5 −1 −1 1
[8, 3]′ 8_{3} −1 1 1 5
[8, 9]′′ 8_{4} −5 −1 −1 1

3-element family (𝔖_{2}) : −1 1 5 7

[4, 1] 4_{2} (1, 1) 1
[2, 4]′′ 2_{1} (g_{2}, 1) 1
[2, 4]′ 2_{3} (1, ε) 1

3-element family (𝔖_{2}) : −7 −5 −1 1

[4, 13] 4_{5} (1, 1) 1
[2, 16]′′ 2_{4} (g_{2}, 1) 1
[2, 16]′ 2_{2} (1, ε) 1

11-element family (𝔖_{4}) : −1 1

[12, 4] 12_{1} (1, 1) t^{2} − 5
[9, 6]′′ 9_{2} (g′_{2}, 1) (t + 3) (t − 3)
[9, 6]′ 9_{3} (1, λ^{1}) (t + 3) (t − 3)
[1, 12]′′ 1_{2} (g′_{2}, ε′) (t − 5) (t + 5)
[1, 12]′ 1_{3} (1, λ^{2}) (t − 5) (t + 5)
[4, 8] 4_{1} (g′_{2}, ε′′) t^{2} − 13
[4, 7]′′ 4_{3} (g_{4}, 1) (t − 1) (t + 1)
[4, 7]′ 4_{4} (g_{2}, ε′′) (t − 1) (t + 1)
[6, 6]′ 6_{1} (g_{3}, 1) t^{2} + 7
[6, 6]′′ 6_{2} (1, σ) t^{2} − 17
[16, 5] 16_{1} (g_{2}, 1) t^{2} − 7

Table 4 : Type E_{6}. 1-element families (𝔖_{1} = {e}).

\begin{matrix}[1, 0] & 1_{p} & 1 & 4 & 5 & 7 & 8 & 11 \\ [1, 36] & 1′_{p} & −11 & −8 & −7 & −5 & −4 & −1 \\ [6, 1] & 6_{p} & −1 & 1 & 4 & 5 & 7 & 8 \\ [6, 25] & 6′_{p} & −8 & −7 & −5 & −4 & −1 & 1 \\ [20, 2] & 20_{p} & −1 & 1 & 2 & 4 & 5 & 7 \\ [20, 20] & 20′_{p} & −7 & −5 & −4 & −2 & −1 & 1 \\ [64, 4] & 64_{p} & −2 & −1 & 1 & 2 & 4 & 5 \\ [64, 13] & 64′_{p} & −5 & −4 & −2 & −1 & 1 & 2 \\ [60, 5] & 60_{p} & −1 & −1 & 1 & 1 & 2 & 4 \\ [60, 11] & 60′_{p} & −4 & −2 & −1 & −1 & 1 & 1 \\ [81, 6] & 81_{p} & −3 & −1 & 0 & 1 & 3 & 4 \\ [81, 10] & 81′_{p} & −4 & −3 & −1 & 0 & 1 & 3 \\ [24, 6] & 24_{p} & −2 & −1 & 1 & 1 & 2 & 5 \\ [24, 12] & 24′_{p} & −5 & −2 & −1 & −1 & 1 & 2 \end{matrix}

3-element family (𝔖_{2}) : −1 1 4 5

[30, 3] 30_{p} (1, 1) (t + 4) (t − 1)
[15, 4] 15_{q} (g_{2}, 1) t^{2} + 3 t + 8
[15, 5] 15_{p} (1, ε) t^{2} + 3 t − 16

3-element family (𝔖_{2}) : −5 −4 −1 1

[30, 15] 30′_{p} (1, 1) (t + 1) (t − 4)
[15, 16] 15′_{q} (g_{2}, 1) t^{2} − 3 t + 8
[15, 17] 15′_{p} (1, ε) t^{2} − 3 t − 16

5-element family (𝔖_{3}) : −1 1

[80, 7] 80_{s} (1, 1) (t − 2) (t + 2) (t^{2} − 7)
[60, 8] 60_{s} (g_{2}, 1) t^{4} − 5 t^{2} − 32
[90, 8] 90_{s} (1, r) (t^{2} + t − 8) (t^{2} − t − 8)
[10, 9] 10_{s} (g_{3}, 1) (t^{2} − 3 t + 8) (t^{2} + 3 t + 8)
[20, 10] 20_{s} (1, ε) (t − 4) (t + 4) (t^{2} − 13)

\begin{matrix}[1, 0] & 1_{a} & 1 & 5 & 7 & 9 & 11 & 13 & 17 \\ [1, 63] & 1′_{a} & −17 & −13 & −11 & −9 & −7 & −5 & −1 \\ [7, 1] & 7′_{a} & −1 & 1 & 5 & 7 & 9 & 11 & 13 \\ [7, 46] & 7_{a} & −13 & −11 & −9 & −7 & −5 & −1 & 1 \\ [27, 2] & 27_{a} & −1 & 1 & 3 & 5 & 7 & 9 & 11 \\ [27, 37] & 27′_{a} & −11 & −9 & −7 & −5 & −3 & −1 & 1 \\ [21, 3] & 21′_{b} & −1 & 1 & 3 & 5 & 7 & 7 & 11 \\ [21, 36] & 21_{b} & −11 & −7 & −7 & −5 & −3 & −1 & 1 \\ [189, 5] & 189′_{b} & −1 & −1 & 1 & 3 & 3 & 5 & 7 \\ [189, 22] & 189_{b} & −7 & −5 & −3 & −3 & −1 & 1 & 1 \\ [210, 6] & 210_{a} & −3 & −1 & 1 & 1 & 5 & 5 & 7 \\ [210, 21] & 210′_{a} & −7 & −5 & −5 & −1 & −1 & 1 & 3 \\ [105, 6] & 105_{b} & −1 & 1 & 1 & 1 & 3 & 5 & 5 \\ [105, 21] & 105′_{b} & −5 & −5 & −3 & −1 & −1 & −1 & 1 \\ [168, 6] & 168_{a} & −1 & −1 & 1 & 1 & 3 & 5 & 7 \\ [168, 21] & 168′_{a} & −7 & −5 & −3 & −1 & −1 & 1 & 1 \\ [189, 7] & 189′_{c} & −3 & −1 & 1 & 1 & 3 & 5 & 7 \\ [189, 20] & 189_{c} & −7 & −5 & −3 & −1 & −1 & 1 & 3 \\ [378, 9] & 378′_{a} & −3 & −1 & −1 & 1 & 1 & 3 & 5 \\ [378, 14] & 378_{a} & −5 & −3 & −1 & −1 & 1 & 1 & 3 \\ [210, 10] & 210_{b} & −1 & −1 & −1 & 1 & 1 & 1 & 3 \\ [210, 13] & 210′_{b} & −3 & −1 & −1 & −1 & 1 & 1 & 1 \\ [105, 12] & 105_{c} & −5 & −1 & −1 & 1 & 1 & 3 & 5 \\ [105, 15] & 105′_{c} & −5 & −3 & −1 & −1 & 1 & 1 & 5 \end{matrix}

2-element family (𝔖_{2}) : −1 1

[512, 11] 512′_{a} (1, 1) (t − 2) (t − 4) (t + 4) (t + 2) t
[512, 12] 512_{a} (1, ε) (t − 2) (t − 4) (t + 4) (t + 2) t

3-element family (𝔖_{2}) : −1 1 5 7 9

[56, 3] 56′_{a} (1, 1) t^{2} + 6 t − 1
[35, 4] 35_{b} (g_{2}, 1) t^{2} + 6 t + 17
[21, 6] 21_{a} (1, ε) t^{2} + 6 t − 31

3-element family (𝔖_{2}) : −9 −7 −5 −1 1

[56, 30] 56_{a} (1, 1) t^{2} − 6 t − 1
[35, 31] 35′_{b} (g_{2}, 1) t^{2} − 6 t + 17
[21, 33] 21′_{a} (1, ε) t^{2} − 6 t − 31

3-element family (𝔖_{2}) : −1 1 3 5 7

[120, 4] 120_{a} (1, 1) t^{2} + 6 t − 13
[15, 7] 15′_{a} (g_{2}, 1) t^{2} + 6 t + 29
[105, 5] 105′_{a} (1, ε) t^{2} + 6 t − 19

3-element family (𝔖_{2}) : −7 −5 −3 −1 1

[120, 25] 120′_{a} (1, 1) t^{2} − 6 t − 13
[15, 28] 15_{a} (g_{2}, 1) t^{2} − 6 t + 29
[105, 26] 105_{a} (1, ε) t^{2} − 6 t − 19

3-element family (𝔖_{2}) : −3 −1 1 3 5

[405, 8] 405_{a} (1, 1) t^{2} + 2 t − 7
[216, 9] 216′_{a} (g_{2}, 1) t^{2} + 2 t + 7
[189, 10] 189_{a} (1, ε) t^{2} + 2 t − 23

3-element family (𝔖_{2}) : −5 −3 −1 1 3

[405, 15] 405′_{a} (1, 1) t^{2} − 2 t − 7
[216, 16] 216_{a} (g_{2}, 1) t^{2} − 2 t + 7
[189, 17] 189′_{a} (1, ε) t^{2} − 2 t − 23

3-element family (𝔖_{2}) : −3 −1 1 1 5

[420, 10] 420_{a} (1, 1) t^{2} − 13
[84, 12] 84_{a} (g_{2}, 1) t^{2} + 11
[336, 11] 336′_{a} (1, ε) t^{2} − 19

3-element family (𝔖_{2}) : −5 −1 −1 1 3

[420, 13] 420′_{a} (1, 1) t^{2} − 13
[84, 15] 84′_{a} (g_{2}, 1) t^{2} + 11
[336, 14] 336_{a} (1, ε) t^{2} − 19

5-element family (𝔖_{3}) : −1 1 5

[315, 7] 315′_{a} (1, 1) (t + 3) (t^{2} + 2 t − 11) (t − 1)
[280, 8] 280_{b} (g_{2}, 1) t^{4} + 4 t^{3} − 4 t^{2} − 16 t − 81
[280, 9] 280′_{a} (1, r) (t^{2} − 13) (t^{2} + 4 t − 9)
[70, 9] 70′_{a} (g_{3}, 1) (t^{2} − 2 t + 9) (t^{2} + 6 t + 17)
[35, 13] 35′_{a} (1, ε) (t + 7) (t − 5) (t^{2} + 2 t − 27)

5-element family (𝔖_{3}) : −5 −1 1

[315, 16] 315_{a} (1, 1) (t − 3) (t^{2} − 2 t − 11) (t + 1)
[280, 17] 280′_{b} (g_{2}, 1) t^{4} − 4 t^{3} − 4 t^{2} + 16 t − 81
[280, 18] 280_{a} (1, r) (t^{2} − 4 t − 9) (t^{2} − 13)
[70, 18] 70_{a} (g_{3}, 1) (t^{2} − 6 t + 17) (t^{2} + 2 t + 9)
[35, 22] 35_{a} (1, ε) (t + 5) (t − 7) (t^{2} − 2 t − 27)

\begin{matrix}[1, 0] & 1_{x} & 1 & 7 & 11 & 13 & 17 & 19 & 23 & 29 \\ [1, 120] & 1′_{x} & −29 & −23 & −19 & −17 & −13 & −11 & −7 & −1 \\ [8, 1] & 8_{z} & −1 & 1 & 7 & 11 & 13 & 17 & 19 & 23 \\ [8, 91] & 8′_{z} & −23 & −19 & −17 & −13 & −11 & −7 & −1 & 1 \\ [35, 2] & 35_{x} & −1 & 1 & 5 & 7 & 11 & 13 & 17 & 19 \\ [35, 74] & 35′_{x} & −19 & −17 & −13 & −11 & −7 & −5 & −1 & 1 \\ [560, 5] & 560_{z} & −1 & −1 & 1 & 5 & 7 & 7 & 11 & 13 \\ [560, 47] & 560′_{z} & −13 & −11 & −7 & −7 & −5 & −1 & 1 & 1 \\ [567, 6] & 567_{x} & −3 & −1 & 1 & 3 & 7 & 9 & 11 & 13 \\ [567, 46] & 567′_{x} & −13 & −11 & −9 & −7 & −3 & −1 & 1 & 3 \\ [3240, 9] & 3240_{z} & −3 & −1 & 1 & 1 & 3 & 5 & 7 & 9 \\ [3240, 31] & 3240′_{z} & −9 & −7 & −5 & −3 & −1 & −1 & 1 & 3 \\ [525, 12] & 525_{x} & −5 & −1 & 1 & 1 & 5 & 5 & 7 & 11 \\ [525, 36] & 525′_{x} & −11 & −7 & −5 & −5 & −1 & −1 & 1 & 5 \\ [4536, 13] & 4536_{z} & −3 & −1 & −1 & 1 & 1 & 3 & 3 & 7 \\ [4536, 23] & 4536′_{z} & −7 & −3 & −3 & −1 & −1 & 1 & 1 & 3 \\ [2835, 14] & 2835_{x} & −3 & −1 & −1 & 1 & 1 & 3 & 3 & 5 \\ [2835, 22] & 2835′_{x} & −5 & −3 & −3 & −1 & −1 & 1 & 1 & 3 \\ [6075, 14] & 6075_{x} & −5 & −3 & −1 & 1 & 1 & 3 & 5 & 7 \\ [6075, 22] & 6075′_{x} & −7 & −5 & −3 & −1 & −1 & 1 & 3 & 5 \\ [4200, 15] & 4200_{z} & −5 & −1 & −1 & 1 & 1 & 1 & 5 & 5 \\ [4200, 21] & 4200′_{z} & −5 & −5 & −1 & −1 & −1 & 1 & 1 & 5 \\ [2100, 20] & 2100_{y} & −7 & −5 & −1 & −1 & 1 & 1 & 5 & 7 \end{matrix}

2-element family (𝔖_{2}) : −1 1 7

[4096, 11] 4096_{z} (1, 1) (t − 2) (t − 4) (t + 8) (t + 4) (t + 2)
[4096, 12] 4096_{x} (1, ε) (t − 2) (t − 4) (t + 8) (t + 4) (t + 2)

2-element family (𝔖_{2}) : −7 −1 1

[4096, 26] 4096′_{x} (1, 1) (t − 2) (t − 4) (t − 8) (t + 4) (t + 2)
[4096, 27] 4096′_{z} (1, ε) (t − 2) (t − 4) (t − 8) (t + 4) (t + 2)

3-element family (𝔖_{2}) : −1 1 7 11 13 17

[112, 3] 112_{z} (1, 1) t^{2} + 12 t + 17
[84, 4] 84_{x} (g_{2}, 1) t^{2} + 12 t + 47
[28, 8] 28_{x} (1, ε) t^{2} + 12 t − 73

3-element family (𝔖_{2}) : −17 −13 −11 −7 −1 1

[112, 63] 112′_{z} (1, 1) t^{2} − 12 t + 17
[84, 64] 84′_{x} (g_{2}, 1) t^{2} − 12 t + 47
[28, 68] 28′_{x} (1, ε) t^{2} − 12 t − 73

3-element family (𝔖_{2}) : −1 1 5 7 11 13

[210, 4] 210_{x} (1, 1) (t + 13) (t − 1)
[50, 8] 50_{x} (g_{2}, 1) t^{2} + 12 t + 83
[160, 7] 160_{z} (1, ε) t^{2} + 12 t − 43

3-element family (𝔖_{2}) : −13 −11 −7 −5 −1 1

[210, 52] 210′_{x} (1, 1) (t + 1) (t − 13)
[50, 56] 50′_{x} (g_{2}, 1) t^{2} − 12 t + 83
[160, 55] 160′_{z} (1, ε) t^{2} − 12 t − 43

3-element family (𝔖_{2}) : −1 1 1 5 7 11

[700, 6] 700_{x} (1, 1) t^{2} + 12 t + 23
[400, 7] 400_{z} (g_{2}, 1) t^{2} + 12 t + 41
[300, 8] 300_{x} (1, ε) t^{2} + 12 t − 1

3-element family (𝔖_{2}) : −11 −7 −5 −1 −1 1

[700, 42] 700′_{x} (1, 1) t^{2} − 12 t + 23
[400, 43] 400′_{z} (g_{2}, 1) t^{2} − 12 t + 41
[300, 44] 300′_{x} (1, ε) t^{2} − 12 t − 1

3-element family (𝔖_{2}) : −3 −1 1 3 7 9

[2268, 10] 2268_{x} (1, 1) t^{2} + 4 t − 17
[972, 12] 972_{x} (g_{2}, 1) t^{2} + 4 t + 23
[1296, 13] 1296_{z} (1, ε) t^{2} + 4 t − 47

3-element family (𝔖_{2}) : −9 −7 −3 −1 1 3

[2268, 30] 2268′_{x} (1, 1) t^{2} − 4 t − 17
[972, 32] 972′_{x} (g_{2}, 1) t^{2} − 4 t + 23
[1296, 33] 1296′_{z} (1, ε) t^{2} − 4 t − 47

3-element family (𝔖_{2}) : −1 −1 1 1 5 7

[2240, 10] 2240_{x} (1, 1) t^{2} + 6 t − 1
[1400, 11] 1400_{zz} (g_{2}, 1) t^{2} + 6 t + 17
[840, 13] 840_{z} (1, ε) t^{2} + 6 t − 31

3-element family (𝔖_{2}) : −7 −5 −1 −1 1 1

[2240, 28] 2240′_{x} (1, 1) t^{2} − 6 t − 1
[1400, 29] 1400′_{zz} (g_{2}, 1) t^{2} − 6 t + 17
[840, 31] 840′_{z} (1, ε) t^{2} − 6 t − 31

3-element family (𝔖_{2}) : −1 −1 1 1 5 7

[4200, 12] 4200_{x} (1, 1) t^{2} − 13
[840, 14] 840_{x} (g_{2}, 1) t^{2} + 11
[3360, 13] 3360_{z} (1, ε) t^{2} − 19

3-element family (𝔖_{2}) : −7 −5 −1 −1 1 1

[4200, 24] 4200′_{x} (1, 1) t^{2} − 13
[840, 26] 840′_{x} (g_{2}, 1) t^{2} + 11
[3360, 25] 3360′_{z} (1, ε) t^{2} − 19

3-element family (𝔖_{2}) : −5 −1 1 1 5 7

[2800, 13] 2800_{z} (1, 1) t^{2} + 4 t − 23
[700, 16] 700_{xx} (g_{2}, 1) t^{2} + 4 t + 31
[2100, 16] 2100_{x} (1, ε) t^{2} + 4 t − 41

3-element family (𝔖_{2}) : −7 −5 −1 −1 1 5

[2800, 25] 2800′_{z} (1, 1) t^{2} − 4 t − 23
[700, 28] 700′_{xx} (g_{2}, 1) t^{2} − 4 t + 31
[2100, 28] 2100′_{x} (1, ε) t^{2} − 4 t − 41

3-element family (𝔖_{2}) : −5 −1 −1 1 5 7

[5600, 15] 5600_{z} (1, 1) t^{2} − 13
[3200, 16] 3200_{x} (g_{2}, 1) t^{2} + 5
[2400, 17] 2400_{z} (1, ε) t^{2} − 37

3-element family (𝔖_{2}) : −7 −5 −1 1 1 5

[5600, 21] 5600′_{z} (1, 1) t^{2} − 13
[3200, 22] 3200′_{x} (g_{2}, 1) t^{2} + 5
[2400, 23] 2400′_{z} (1, ε) t^{2} − 37

5-element family (𝔖_{3}) : −1 1 7 11

[1400, 7] 1400_{z} (1, 1) (t + 5) (t^{2} + 6 t − 19) (t + 1)
[1344, 8] 1344_{x} (g_{2}, 1) t^{4} + 12 t^{3} + 28 t^{2} − 48 t − 353
[1008, 9] 1008_{z} (1, r) (t^{2} + 10 t + 1) (t^{2} + 2 t − 23)
[448, 9] 448_{z} (g_{3}, 1) (t^{2} + 12 t + 47) (t^{2} + 11)
[56, 19] 56_{z} (1, ε) (t − 7) (t + 13) (t^{2} + 6 t − 67)

5-element family (𝔖_{3}) : −11 −7 −1 1

[1400, 37] 1400′_{z} (1, 1) (t − 5) (t^{2} − 6 t − 19) (t − 1)
[1344, 38] 1344′_{x} (g_{2}, 1) t^{4} − 12 t^{3} + 28 t^{2} + 48 t − 353
[1008, 39] 1008′_{z} (1, r) (t^{2} − 10 t + 1) (t^{2} − 2 t − 23)
[448, 39] 448′_{z} (g_{3}, 1) (t^{2} + 11) (t^{2} − 12 t + 47)
[56, 49]  56′_{z} (1, ε) (t − 13) (t + 7) (t^{2} − 6 t − 67)

5-element family (𝔖_{3}) : −1 1 5 7

[1400, 8] 1400_{x} (1, 1) (t^{2} + 6 t − 19) (t + 7) (t − 1)
[1050, 10] 1050_{x} (g_{2}, 1) t^{4} + 12 t^{3} + 34 t^{2} − 12 t − 611
[1575, 10] 1575_{x} (1, r) (t^{2} + 8 t − 17) (t^{2} + 4 t − 29)
[175, 12] 175_{x} (g_{3}, 1) (t^{2} + 23) (t^{2} + 12 t + 59)
[350, 14] 350_{x} (1, ε) (t + 11) (t − 5) (t^{2} + 6 t − 43)

5-element family (𝔖_{3}) : −7 −5 −1 1

[1400, 32] 1400′_{x} (1, 1) (t^{2} − 6 t − 19) (t + 1) (t − 7)
[1050, 34] 1050′_{x} (g_{2}, 1) t^{4} − 12 t^{3} + 34 t^{2} + 12 t − 611
[1575, 34] 1575′_{x} (1, r) (t^{2} − 4 t − 29) (t^{2} − 8 t − 17)
[175, 36] 175′_{x} (g_{3}, 1) (t^{2} + 23) (t^{2} − 12 t + 59)
[350, 38] 350′_{x} (1, ε) (t − 11) (t + 5) (t^{2} − 6 t − 43)

17-element family (𝔖_{5}) : −1 1

[4480, 16] 4480_{y} (1, 1) (t^{2} − 7) (t^{2} − 19) (t^{2} − 13)
[3150, 18] 3150_{y} (g_{3}, 1) (t − 5) (t + 5) (t^{4} − 2 t^{2} + 193)
[4200, 18] 4200_{y} (g′_{2}, 1) (t − 5) (t + 5) (t^{2} + 11) (t^{2} − 13)
[4536, 18] 4536_{y} (1, ν) (t − 3) (t + 3) (t^{2} − 29) (t^{2} − 21)
[5670, 18] 5670_{y} (1, λ′) (t + 3) (t − 3) (t^{2} − 17) (t^{2} − 33)
[420, 20] 420_{y} (g_{5}, 1) (t^{2} + 23) (t^{4} − 2 t^{2} + 577)
[1134, 20] 1134_{y} (g_{3}, ε) (t^{2} − 41) (t − 3)^{2} (t + 3)^{2}
[1400, 20] 1400_{y} (1, ν′) (t − 5) (t + 5) (t^{2} − 13) (t^{2} − 37)
[2688, 20] 2688_{y} (g′_{2}, ε′′) (t^{2} + 11) (t^{2} − 31) (t^{2} − 19)
[1680, 22] 1680_{y} (1, λ^{2}) (t − 7) (t + 7) (t^{2} − 13) (t^{2} − 37)
[168, 24] 168_{y} (g′_{2}, ε′) (t^{2} + 11) (t^{4} + 10 t^{2} − 1451)
[70, 32]  70_{y} (1, λ^{3}) (t − 7) (t + 7) (t^{2} − 73) (t^{2} − 97)
[7168, 17] 7168_{w} (g_{2}, 1) (t^{2} − 19) (t^{4} − 20 t^{2} − 71)
[1344, 19] 1344_{w} (g_{4}, 1) (t^{2} − 19) (t^{2} + 11) (t − 1) (t + 1)
[2016, 19] 2016_{w} (g_{6}, 1) t^{6} − 9 t^{4} − 81 t^{2} − 5671
[5600, 19] 5600_{w} (g_{2}, r) (t − 5) (t + 5) (t^{2} − 31) (t − 1) (t + 1)
[448, 25] 448_{w} (g_{2}, ε) (t + 7) (t − 7) (t^{2} − 19) (t^{2} − 61)

Table 6. \begin{matrix}E_{6} : \\ [30, 3] & 30_{p} ( & 4, & −1) ( & 5, & −2) ( & 2, & 1) \\ [30, 15] & 30′_{p} ( & 1, & −4) ( & 2, & −5) ( & −1, & −2) \\ E_{7} : \\ [56, 3] & 56′_{a} ( & 7, & −1) ( & 8, & −2) ( & 4, & 2) \\ [56, 30] & 56_{a} ( & 1, & −7) ( & 2, & −8) ( & −2, & −4) \\ [120, 4] & 120_{a} ( & 5, & 1) ( & 8, & −2) ( & 4, & 2) \\ [120, 25] & 120′_{a} ( & −1, & −5) ( & 2, & −8) ( & −2, & −4) \\ [405, 8] & 405_{a} ( & 4, & −2) ( & 5, & −3) ( & 2, & 0) \\ [405, 15] & 405′_{a} ( & 2, & −4) ( & 3, & −5) ( & 0, & −2) \\ [420, 10] & 420_{a} ( & 2, & −2) ( & 4, & −4) ( & 1, & −1) \\ [420, 13] & 420′_{a} ( & 2, & −2) ( & 4, & −4) ( & 1, & −1) \\ E_{8} : \\ [112, 3] & 112_{z} ( & 13, & −1) ( & 14, & −2) ( & 8, & 4) \\ [112, 63] & 112′_{z} ( & 1, & −13) ( & 2, & −14) ( & −8, & −4) \\ [210, 4] & 210_{x} ( & 10, & 2) ( & 14, & −2) ( & 7, & 5) \\ [210, 52] & 210′_{x} ( & −2, & −10) ( & 2, & −14) ( & −7, & −5) \\ [700, 6] & 700_{x} ( & 10, & 2) ( & 11, & 1) ( & 8, & 4) \\ [700, 42] & 700′_{x} ( & −2, & −10) ( & −1, & −11) ( & −4, & −8) \\ [2268, 10] & 2268_{x} ( & 6, & −2) ( & 8, & −4) ( & 3, & 1) \\ [2268, 30] & 2268′_{x} ( & 2, & −6) ( & 4, & −8) ( & −1, & −3) \\ [2240, 10] & 2240_{x} ( & 7, & −1) ( & 8, & −2) ( & 4, & 2) \\ [2240, 28] & 2240′_{x} ( & 1, & −7) ( & 2, & −8) ( & −2, & −4) \\ [4200, 12] & 4200_{x} ( & 2, & −2) ( & 4, & −4) ( & 1, & −1) \\ [4200, 24] & 4200′_{x} ( & 2, & −2) ( & 4, & −4) ( & 1, & −1) \\ [2800, 13] & 2800_{z} ( & 5, & −1) ( & 8, & −4) ( & 2, & 2) \\ [2800, 25] & 2800′_{z} ( & 1, & −5) ( & 4, & −8) ( & −2, & −2) \\ [5600, 15] & 5600_{z} ( & 4, & −4) ( & 5, & −5) ( & 2, & −2) \\ [5600, 21] & 5600′_{z} ( & 4, & −4) ( & 5, & −5) ( & 2, & −2) \end{matrix}

37