REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 00, 1995

ON THE GENERIC DEGREES OF CYCLOTOMIC ALGEBRAS

GUNTER MALLE

We determine the generic degrees of cyclotomic Hecke algebras attached to exceptional finite complex reflection groups. The results are used to introduce the notion of spetsial reflection group, which in a certain sense is a generalization of finite Weyl group. In particular to spetsial W there is attached a set of unipotent degrees which in the case of a Weyl group is just the set of degrees of unipotent characters of finite reductive groups with Weyl group W, and in general enjoys many of their combinatorial properties.

1. Introduction

Cyclotomic algebras are certain deformations of the group algebras of finite complex reflection groups, defined analogously to Iwahori-Hecke algebras for Weyl groups. These cyclotomic algebras seem to play an important role in the understanding of the representation theory of finite groups of Lie type (see for example [3]). At the moment, the properties of cyclotomic algebras are not yet fully understood. This paper is devoted to the study of certain numerical invariants attached to them, the generic degrees.

It is conjectured (and known in all but finitely many cases [16]) that the cyclotomic algebra H = H(W, u) of an irreducible finite complex reflection group W is symmetric over the ground ring A = ℤ[u, u^{−1}], such that moreover the corresponding trace form t vanishes on all basis elements T_{w} for w ∈ W, w ≠ 1 reduced. Since t is a trace form, it can be expressed as a sum t = ∑_{ϕ} 1/c_{ϕ} ϕ of irreducible characters of H with nonvanishing coefficients. The c_{ϕ} appearing as coefficients are called the Schur elements of H (with respect to t). In this paper we compute the Schur elements with respect to any trace form as above for the exceptional complex reflection groups. The method is an extension of [13] where the 2-dimensional case was considered.

In the final section we use the results obtained so far to introduce so-called spetsial reflection groups (as announced in [14]). This is a subclass of all finite complex reflection groups, which includes in particular all those which can already be defined over the real numbers. It can be defined by a variety of equivalent characterizations all related to the Schur elements (see Prop. 8.1). At present there is no conceptual understanding of the equivalence of these properties, even in the case of finite Coxeter groups (where they are always satisfied). It would be very interesting to find such an

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

I’m grateful to J. Michel for spotting some inaccuracies in an earlier version of this paper.

I thank the Science University of Tokyo for its hospitality and the Deutsche Forschungsgemeinschaft for financial support

explanation, since this property is a necessary and sufficient condition for the existence of unipotent degrees attached to a complex reflection group W, i.e., for W behaving like the Weyl group of some as yet mysterious object generalizing the concept of an algebraic group, see Section 8.3.

2. Cyclotomic Hecke algebras

We recall the definition and some properties of cyclotomic Hecke algebras, some facts about character values, and some basic properties of symmetric algebras.

2A. The cyclotomic algebra of a complex reflection group.

Let W be a finite irreducible complex reflection group on the vector space V. The ring of invariants of W in the symmetric algebra S(V) of V is a polynomial ring, generated by homogeneous invariants of degrees d_{1}, …, d_{n}, with n = dim(V). The Poincaré polynomial P_{W} of W is given by P_{W} := ∏_{j=1}^{n}(x^{d_{j}} − 1)/(x − 1). For an irreducible character ϕ ∈ Irr(W) the fake degree is defined as

(2.1)  R_{ϕ} := (x − 1)^{n}P_{W} \frac{1}{|W |} ∑_{w∈W} \frac{det_{V} (w)ϕ(w)}{det_{V} (x − w)} ∈ ℤ[x] ,

where det_{V} denotes the determinant on V. The b-value of ϕ ∈ Irr(W) is the order of zero at x = 0 of the fake degree R_{ϕ}.

Let D be the diagram associated to W in [5]. This defines a presentation of W on a set of generators S with order relations s^{d_{s}} = 1 for s ∈ S, together with certain homogeneous relations, the so-called braid relations. The braid group B = B(W) associated to W is by definition the group generated by a set {s | s ∈ S} in bijection with S, subject to the braid relations of D. Let u = (u_{s,j} | s ∈ S, 0 ≤ j ≤ d_{s} − 1) be transcendentals over ℤ, such that u_{s,j} = u_{t,j} whenever s and t are conjugate in W. The generic cyclotomic Hecke algebra H(W, u) of W with parameter set u is defined to be the quotient

(2.2)  H(W, u) := ℤ[u, u^{−1}]B/I,  with I = ( ∏_{j=0}^{d_{s}−1}(s − u_{s,j}) | s ∈ S)

of the group algebra of B over A := ℤ[u, u^{−1}] by the ideal I generated by certain deformed order relations. We will write T_{w} for the image in H(W, u) of an element w = s_{1} … s_{k} ∈ B. Any ring homomorphism f : ℤ[u, u^{−1}] → R endows R with an A-module structure, and we write

H_{R}(W, u) := H(W, u) ⊗_{A} R

for the corresponding specialization. Note that such a homomorphism f is uniquely determined by the images f(u_{s,j}), s ∈ S, 0 ≤ j ≤ d_{s} − 1. A specialization of H will be called admissible if the specialization

(2.3)  u_{s,j} ↦ exp(2πij/d_{s})  for s ∈ S, 0 ≤ j ≤ d_{s} − 1,

to the group algebra of the complex reflection group W factors through it. One particularly important example is the 1-parameter specialization H(W, x) of H(W, u) induced by the map

(2.4)  f_{x} : u_{s,j} ↦ \{ \begin{matrix}x & j = 0, \\ exp(2πij/d_{s}) & j > 0, \end{matrix}

where x is an indeterminate. This is the analogue of the usual 1-parameter Iwahori-Hecke algebra for real W.

Henceforth we will make the following assumption:

Assumption 2.5. The cyclotomic algebra H(W, u) is free over A = ℤ[u, u^{−1}] of rank |W |.

This assumption is known to hold for all infinite families of irreducible reflection groups and for some of the remaining 34 exceptional groups (see the references in [15]). We conjecture it to be true in all cases [5, Sect. 4].

2B. Character values.

Let k be the character field of the reflection representation of W. So k is a finite cyclotomic extension of ℚ and we write μ(k) for the group of roots of unity in k. For s ∈ S, 0 ≤ j ≤ d_{s} − 1, let v_{s,j} be such that v_{s,j}^{|μ(k)|} = exp(−2πij/d_{s})u_{s,j} and set K_{W} := k(v). We extend the specialization (2.3) to ℤ[v, v^{−1}] by

(2.6)  v_{s,j} ↦ 1  for s ∈ S, 0 ≤ j ≤ d_{s} − 1 .

By [15, Thm. 5.2] it is known that K_{W} is a splitting field for H. In particular, the values of all irreducible characters on an A-basis of H are contained in K_{W}. Furthermore, it follows from Assumption 2.5 and Tits’ deformation theorem that H_{K_{W}} is isomorphic to the group algebra K_{W} W, and the specialization (2.6) induces a bijection

(2.7)  Irr(W) ⟶^{∼} Irr(H_{K_{W}}),  ϕ ↦ ϕ_{v} ,

between Irr(H_{K_{W}}) and Irr(W) which furthermore carries over to any admissible specialization of H_{k[v,v^{−1}]}.

In [5] we defined a certain central element π ∈ Z(B) which maps to the identity under the canonical epimorphism B → W. Clearly, its image T_{π} in H is also central. Thus, it acts as a scalar in any (absolutely) irreducible representation of H_{K_{W}}. Extending an idea of Springer, Broué and Michel have observed that this allows to evaluate irreducible characters on roots of T_{π} without knowing the corresponding representation explicitly. Let us describe this method.

Let S′ be a system of representatives of the generators in S up to conjugation in W. Write π = s_{1} … s_{l} in B. For s ∈ S′ let N_{s} = |{j | s_{j} ∼ s}| denote the number of factors in the decomposition of π conjugate to s. (This does not depend on the chosen expression for π, as can be seen by evaluating linear characters of B.) For an irreducible character ϕ of W let m_{s,j}^{ϕ} denote the multiplicity of the eigenvalue exp(2πij/d_{s}) of s in a representation affording ϕ.

Proposition 2.8. (Broué-Michel [6]) Let w ∈ B such that w^{d} = π for some d ≥ 1. Then the value of the irreducible character ϕ_{v} ∈ Irr(H_{K_{W}}) on T_{w}^{l} is given by

ϕ_{v}(T_{w}^{l}) = ϕ(w^{l}) ∏_{s∈S′} ∏_{j=0}^{d_{s}−1} v_{s,j}^{l|μ(k)|m_{s,j}^{ϕ}N_{s}/dϕ(1)}

It can be shown by a general argument that m_{s,j}^{ϕ}N_{s}/ϕ(1) is always an integer (see [6, 4.17]). As a consequence of [15, Thm. 5.2], which relies on a case-by-case analysis, the exponent |μ(k)|m_{s,j}^{ϕ}N_{s}/dϕ(1) in Proposition 2.8 has to be integral unless ϕ(w) = 0.

2C. Symmetrizing forms.

By what we saw in 2A the cyclotomic algebra H is isomorphic to the group algebra of W over the splitting field K_{W}. Since K_{W} W is a symmetric algebra, the same is true for H_{K_{W}}. Thus there exists a symmetric form ⟨ , ⟩ : H ⊗ H → K_{W} on H . Moreover this can be normalized such that the associated trace form t_{u} : H(W, u) → K_{W} defined by t_{u}(h) := ⟨1, h⟩ under (2.3) specializes to the canonical trace form on the group algebra of W. Over the splitting field K_{W} of H(W, u), we may write t_{u} as a sum over the irreducible characters of H_{K_{W}} with non-vanishing coefficients:

t_{u} = ∑_{ϕ∈Irr(W)} \frac{1}{c_{ϕ}} ϕ_{v} .

The c_{ϕ} are called Schur elements of H(W, u) (with respect to t_{u}). A basis C of H with 1 ∈ C is called quasi-symmetric (with respect to t_{u}) if t_{u}(T) = δ_{T,1} for T ∈ C. Thus for any quasi-symmetric basis, the (inverses of the) Schur elements are uniquely determined by the linear system

(2.9)  ∑_{ϕ∈Irr(W)} c_{ϕ}^{−1} ϕ_{v}(T) = δ_{T,1}  (for T ∈ C) .

Here, already the equations on a subset of C whose image under specialization to W covers all conjugacy classes yield a system of maximal rank.

It is well-known that for Coxeter groups W the Iwahori-Hecke algebra H(W) carries a canonical symmetrizing form endowing it with a structure of symmetric algebra over A. This form is characterized by the fact that the standard basis elements form a quasi-symmetric basis (see [8, Prop. 8.1.1]). The Schur elements with respect to this form are explicitly known in all cases.

It was shown in [16] for all but finitely many irreducible complex reflection groups W that there exists a symmetrizing form t_{u} on H(W, u) making it to a symmetric algebra over A (i.e., the form has Gram matrix invertible over A). Moreover there exists a quasi-symmetric basis with respect to t_{u} consisting of monomials in the generators T_{s}, s ∈ S. Furthermore, restriction of t_{u} to parabolic subalgebras gives the corresponding symmetrizing form there. The Schur elements of this symmetrizing form were determined explicitly in [7].

On the other hand in [13] we dealt with the 2-dimensional primitive groups. It is the purpose of the present paper to analyze the remaining exceptional cases. More precisely we will determine explicitly the Schur elements c_{ϕ} with respect to any symmetrizing form which behaves nicely under restriction to parabolic subalgebras and which vanishes on some suitable monomials in the generators.

2D. Symmetries and action on Irr(W).

It will be convenient to introduce and study the action of several groups of automorphisms on cyclotomic algebras and associated objects. Let k̃ := k(exp(2πi/|μ(k)|^{2})), and K̃_{W} := k̃(v) the composite with K_{W}. Then K̃_{W} is a Galois extension of ℚ(u). Its Galois group G := Gal(K̃_{W} /ℚ(u)) acts naturally on HK̃_{W}, hence on Irr(H). Via the bijection (2.7) we thus obtain a G-action on Irr(W) where g(ϕ) is defined by the condition g(ϕ)_{v} = g(ϕ_{v}).

On the other hand, the factor group Gal(k̃(u)/ℚ(u)) of G acts naturally on Irr(W). Now restriction yields an isomorphism between the subgroup G_{0} := Gal(K̃_{W} /ℚ(v)) of G and Gal(k̃/ℚ). The definition of the bijection (2.7) via the specialization (2.6) shows that the natural action of Gal(k/ℚ) on Irr(W) and its action induced by the identification with G_{0} do coincide. If H is symmetric over A, the linear independence of the irreducible characters together with (2.9) shows that the corresponding Schur elements satisfy:

Lemma 2.10. If H is symmetric with corresponding Schur elements c_{ϕ} (ϕ ∈ Irr(W)), and G := Gal(K̃_{W} /ℚ(u)), then we have

(2.11)  c_{g(ϕ)} = g(c_{ϕ})  for all ϕ ∈ Irr(W), g ∈ G.

Next observe that for s ∈ S′ = S/W the symmetric group 𝔖_{d_{s}} acts (from the left) on the set of indeterminates {u_{s,j} | 0 ≤ j ≤ d_{s} − 1} by

σ(u_{s,j}) := u_{s,σ(j)} .

Let 𝔖 = 𝔖(W) := ∏_{s∈S′} 𝔖_{d_{s}}, the symmetry group of W. By (2.2) H(W, u) is already defined over the ring of invariants ℤ[u, u^{−1}]^{𝔖} under this symmetry group. By trivial action on the constants k̃ the 𝔖-action on ℤ[u, u^{−1}] extends to an action on k̃[u, u^{−1}]. Setting

σ(v_{s,j}) := exp(2πi(σ(j) − j)/d_{s}|μ(k)|)v_{s,σ(j)}

this further extends to an action on k̃[v, v^{−1}]. This induces an 𝔖-action on H and then also on Irr(H) by

σ(ϕ)(T) := σ(ϕ(σ^{−1}(T))) for ϕ ∈ Irr(H), T ∈ H_{K_{W}} , σ ∈ 𝔖.

Via (2.7) this defines an 𝔖-action on Irr(W).

Now assume that H has a quasi-symmetric 𝔖-invariant basis, for example one consisting of images of group elements in the braid group under the natural epimorphism ℤ[u, u^{−1}]B → H, i.e., of monomials in the generators T_{s}, s ∈ S. Then H is symmetric already over ℤ[u, u^{−1}]^{𝔖} and (2.9) shows:

Lemma 2.12. If H is symmetric with respect to an 𝔖-invariant quasi-symmetric basis, with corresponding Schur elements c_{ϕ} (ϕ ∈ Irr(W)), then we have

(2.13)  c_{σ(ϕ)} = σ(c_{ϕ})  for all ϕ ∈ Irr(W), σ ∈ 𝔖.

This observation on symmetries together with (2.11) will be used later on in order to reduce the number of Schur elements to be printed explicitly.

By (2.9) the Schur elements are rational functions in v. But moreover we have:

Lemma 2.14. Assume that H has a quasi-symmetric basis C consisting of monomials in the T_{s}. Then for any s ∈ S the Schur elements are homogeneous of degree 0 in {v_{s,j} | 0 ≤ j ≤ d_{s} − 1}.

Proof. Indeed, let (α_{s} | s ∈ S) be invertible elements α_{s} ∈ k^{×} such that α_{s} = α_{t} whenever s ∼ t. Then the elements {α_{s}T_{s} | s ∈ S} of H(W, u) satisfy the relations of the cyclotomic algebra H(W, u′) with u′ = (α_{s}u_{s,j} | s ∈ S, 0 ≤ j ≤ d_{s} − 1). Any monomial in the T_{s} is a non-zero scalar multiple of a monomial in the α_{s}T_{s}. But if C is quasi-symmetric then any basis C′ containing 1 and consisting of non-zero scalar multiples of the elements of C is quasi-symmetric with respect to the same symmetrizing form. Thus the Schur elements of H remain invariant under replacing the v_{s,j} by non-zero scalar multiples (which only depend on s) as claimed. □

3. Generic degrees for G_{24}, G_{27}, G_{29}

In this and the next section we compute the Schur elements for those exceptional non-real reflection groups of dimension n ≥ 3, which are generated by n involutive reflections. In the classification of Shephard and Todd, these are the groups G_{i}, i ∈ {24, 27, 29, 33, 34}. In these cases, all reflections are conjugate, so the cyclotomic algebra has just two parameters (u_{1}, u_{2}). Since the Schur elements are homogeneous in the parameters, they can be recovered from the Schur elements for the (admissible) specialization with parameters (x, −1). So it suffices to consider this latter case.

It turns out that in all cases the Schur elements (with the above specialization) divide the Poincaré polynomial P (W) of W, considered as elements of the ring k[x]. We call δ_{ϕ} := P (W)/c_{ϕ} the generic degree of ϕ. Clearly, it is enough to know the generic degrees in order to recover the Schur elements.

We denote the irreducible characters ϕ ∈ Irr(W) as ϕ_{d,b}, where d = ϕ(1) is the degree and b is the b-invariant, that is, the order of zero at x = 0 of the fake degree R_{ϕ} (see 2A). In most cases, this distinguishes the characters unambiguously. In the few remaining cases, we give more detailed information below.

In the tables, we only list one character from each pair of complex conjugate ones, since the generic degree of the other on can be obtained by applying complex conjugation, according to (2.11). Moreover, if ϵ denotes the sign character of W, we only list one of ϕ, ϵ ⊗ ϕ, since, as it turns out,

(3.1)  δ_{ϕ}(x) = x^{N} δ_{ϵ⊗ϕ}(x^{−1})

in all cases, where N is the number of reflections in W. Combining this with (2.11) this can be rephrased to saying that δ_{ϕ} is semi-palindromic in the sense of [15, 6B].

3A. The complex reflection group G_{24}.

The 3-dimensional primitive complex reflection group W = G_{24}, abstractly isomorphic to L_{2}(7)×2, has twelve irreducible characters. It is generated by three conjugate reflections s_{1}, s_{2}, s_{3} of order 2. It has two classes of maximal parabolic subgroups, one of type B_{2} and one of type A_{2}, hence both are Coxeter groups.

Let c := s_{1}s_{2}s_{3} be a Coxeter element in the braid group B(W) of W. Then the central element π ∈ B(W) is given by π = c^{14} (see [5, Table 4]).

Theorem 3.2. Let t be a symmetrizing form on H(G_{24}; (x, −1)) restricting to the canonical symmetrizing form on the maximal parabolic subalgebras and such that there exists a quasi-symmetric basis containing T_{c}^{l} (1 ≤ l ≤ 7). Then the generic degrees with respect to t are given in Table 3.3.

Proof. Assume that t is a symmetrizing form on H(W) which restricts to the canonical symmetrizing forms of the Iwahori-Hecke algebras of these parabolic subgroups. Since the generic degrees of finite Coxeter groups are known, restriction of characters to these subgroups yields conditions on the generic degrees of H(W) with respect to t. More precisely, since five of the conjugacy classes of W have representatives in one of these parabolic subgroups, we obtain five equations.

Since π is a power of c, Proposition 2.8 allows to compute the values of all irreducible characters on T_{c}^{l}. Plugging these into (2.9) gives linear equations for the unknown generic degrees, which together with the equations from restriction to parabolic subalgebras yield a system of full rank. □

Table 3.3. Generic degrees for G_{24}.

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,21}
ϕ_{3,1} \frac{\sqrt{−7}}{14}xΦ_{3}Φ_{4}Φ_{6}Φ′_{7}Φ′′_{14} ϕ_{3,3} ϕ_{3,8} ϕ_{3,10}
ϕ_{6,2}  \frac{1}{2}xΦ_{2}^{2}Φ_{3}Φ_{6}Φ_{14}  ϕ_{6,9}
ϕ_{7,3}  x^{3}Φ_{7}Φ_{14}  ϕ_{7,6}
ϕ_{8,4}  \frac{1}{2}x^{4}Φ_{2}^{3}Φ_{4}Φ_{6}Φ_{14}  ϕ_{8,5}

Here, as in the subsequent tables, Φ_{n} denotes the n-th cyclotomic polynomial over ℚ. Moreover, with μ = (−1 + \sqrt{−7})/2 = ζ_{7} + ζ_{7}^{2} + ζ_{7}^{4}, \overline{μ} = (−1 − \sqrt{−7})/2, where ζ_{7} := exp(2πi/7), we let Φ′_{7} = x^{3} − μx^{2} + \overline{μ}x − 1, Φ′_{14} = x^{3} + μx^{2} + \overline{μ}x + 1.

3B. The complex reflection group G_{27}.

The 3-dimensional primitive complex reflection group W = G_{27}, abstractly isomorphic to 3.A_{7} × 2, has 34 irreducible characters. It is generated by three conjugate reflections s_{1}, s_{2}, s_{3} of order 2. It has four conjugacy classes of maximal parabolic subgroups, of types H_{2}, B_{2} and A_{2} (two classes). All of these are Coxeter groups, hence their Iwahori-Hecke algebras have a canonical symmetrizing form. Restriction to these subgroups gives eight equations for the generic degrees. The Coxeter element c := s_{1}s_{2}s_{3} satisfies c^{30} = π [5, Table 4]. Its fifth power c^{5} is central, thus more generally we may compute the value of any irreducible character on products T_{w}T_{c}^{5l} for all T_{w} lying in proper parabolic subalgebras of H. This yields enough independent equations, and as above we obtain:

Theorem 3.4. Let t be a symmetrizing form on H(G_{27}; (x, −1)) such that there exists a quasi-symmetric basis containing T_{w}T_{c}^{5l} (0 ≤ l ≤ 5) for w running through

Table 3.5. Generic degrees for G_{27}.

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,45}
ϕ_{3,1,25} −^{\frac{\sqrt{−15}ζ_{3}^{2}}{30}} xΦ′′_{3}^{3}Φ_{4}Φ′_{5}Φ′_{6}^{3}Φ′_{10}Φ_{12}Φ′_{15}Φ_{15}^{ba}Φ′_{30}Φ_{30}^{ab} ϕ_{3,5,29} ϕ_{3,16,40} ϕ_{3,20,44}
ϕ_{3,5,23} −\frac{\sqrt{−15}ζ_{3}}{30} xΦ′_{3}^{3}Φ_{4}Φ′′_{5}Φ′′_{6}^{3}Φ′′_{10}Φ_{12}Φ′′_{15}Φ_{15}^{aa}Φ′_{30}Φ_{30}^{ba} ϕ_{3,7,25} ϕ_{3,20,38} ϕ_{3,22,40}
ϕ′_{5,6}  \frac{1}{2}x^{3}Φ_{4}Φ_{5}Φ_{10}Φ_{12}Φ_{15}Φ_{30}  ϕ′′_{5,15}
ϕ′′_{5,6}  \frac{1}{2}x^{3}Φ_{4}Φ_{5}Φ_{10}Φ_{12}Φ_{15}Φ_{30}  ϕ′_{5,15}
ϕ_{6,2}  \frac{1−ζ_{3}^{2}}{6} xΦ_{2}^{2}Φ′_{3}^{3}Φ_{6}^{2}Φ′′_{6}Φ_{10}Φ′_{12}Φ′′_{15}Φ_{30} ϕ_{6,4}  ϕ_{6,17}  ϕ_{6,19}
ϕ_{8,6,36}  \frac{(3−λ)}{10}x^{6}Φ_{2}^{3}Φ_{4}Φ′_{5}Φ_{6}^{3}Φ_{10}Φ_{12}Φ_{15}^{B}Φ_{30}  ϕ_{8,9,39}
ϕ_{8,9,33}  \frac{(2+λ)}{10}x^{6}Φ_{2}^{3}Φ_{4}Φ′′_{5}Φ_{6}^{3}Φ_{10}Φ_{12}Φ_{15}^{A}Φ_{30}  ϕ_{8,12,36}
ϕ_{9,4}  \frac{1}{3}x^{4}Φ_{3}^{3}Φ_{6}^{3}Φ_{12}Φ_{15}Φ_{30} ϕ_{9,8}  ϕ_{9,13}  ϕ_{9,11}
ϕ_{9,6}  \frac{1}{3}x^{4}Φ_{3}^{3}Φ_{6}^{3}Φ_{12}Φ_{15}Φ_{30}  ϕ_{9,9}
ϕ_{10,3}  \frac{1}{2}x^{3}Φ_{2}^{2}Φ_{5}Φ_{6}^{2}Φ_{10}Φ_{15}Φ_{30}  ϕ_{10,12}
ϕ_{15,5}  \frac{1−ζ_{3}^{2}}{3} x^{5}Φ′′_{3}^{3}Φ_{5}Φ′_{6}^{3}Φ_{10}Φ′′_{12}Φ_{15}Φ_{30} ϕ_{15,7} ϕ_{15,8}  ϕ_{15,10}

Here, Φ′_{3} = x−ζ_{3}, Φ′_{6} = x+ζ_{3}^{2}, Φ′_{12} = x^{2} +ζ_{3}^{2}, Φ′′_{5} = x^{2} +λx+1, Φ′_{10} = x^{2} −λx+1, Φ′_{15} = x^{4} + ζ_{3}^{2}x^{3} + ζ_{3}x^{2} + x + ζ_{3}^{2}, Φ_{15}^{A} = x^{4} − λx^{3} + λx^{2} − λx + 1, Φ_{15}^{aa} = x^{2} + λζ_{3}^{2}x + ζ_{3}, Φ_{15}^{ab} = x^{2} + λζ_{3}x + ζ_{3}^{2}, Φ′′_{30} = Φ′_{15}(−x), Φ_{30}^{aa} = Φ_{15}^{aa}(−x), with λ = (1 + \sqrt{5})/2 = 1 + ζ_{5} + ζ_{5}^{4}.

representatives of minimal length of the conjugacy classes of the maximal parabolic subgroups of W. Then the generic degrees with respect to t are given in Table 3.5.

Here, in order to distinguish the characters of degree 3 and of degree 8 among themselves, we have given as a third index the degree in x of the fake degree R_{ϕ}. We denote by ϕ′_{5,6}, ϕ′_{5,15} the characters of degree 5 which occur in the permutation character of W on the parabolic subgroup of type A_{2}, and by ϕ′′_{5,6}, ϕ′′_{5,15} the other two. (It turns out that the two characters ϕ_{5,6}, respectively ϕ_{5,15} have identical generic degrees.)

3C. The complex reflection group G_{29}.

The 4-dimensional primitive complex reflection group W = G_{29} has 37 irreducible characters. It is generated by four conjugate reflections s_{1}, …, s_{4} of order 2, satisfying the relations implied by the Coxeter diagram

◯_{s_{1}}_{2} ◯_{s_{2}}_{2} ◯_{s_{3}}_{2} ⧸^{◯ 2 ^{s_{4}}} ⧹

together with (s_{2}s_{3}s_{4})^{2} = (s_{4}s_{2}s_{3})^{2}. It has five conjugacy classes of maximal parabolic subgroups, of types B_{3}, A_{3} (2 classes), A_{2} + A_{1}, G(4, 4, 3). While the first four of these are Coxeter groups, the last one is an imprimitive non-real reflection group.

In order to use induction we first determine generic degrees for G(4, 4, 3). Now all maximal parabolic subgroups of G(4, 4, 3) are Coxeter groups, and restriction of characters to these yields five equations. Furthermore, evaluation of characters on the first five powers of the Coxeter element gives five additional equations, thus determining the generic degrees with respect to any quasi-symmetric basis containing these elements. We give the results in Table 3.6.

Table 3.6. Generic degrees for G(4, 4, 3).

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,12}
ϕ_{2,4}  x^{4}Φ_{8}
ϕ_{3,1} \frac{1+i}{4}xΦ_{3}Φ′_{4}^{2}Φ′′_{8} ϕ_{3,2} ϕ_{3,5} ϕ_{3,6}
ϕ_{3,2} \frac{1}{2}xΦ_{3}Φ_{8} ϕ_{3,6}
ϕ_{6,3}  x^{3}Φ_{3}Φ_{8}

3.7. Generic degrees for G_{29}.

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,40}
ϕ_{4,1}  \frac{1+i}{4}xΦ_{2}^{2}Φ′′_{4}^{2}Φ_{6}Φ′_{8}Φ_{10}Φ_{12}^{a}Φ′′_{20} ϕ_{4,3} ϕ_{4,21} ϕ_{4,23}
ϕ_{4,4}  \frac{1}{2}xΦ_{4}^{3}Φ_{12}Φ_{20}  ϕ_{4,24}
ϕ_{5,8}  \frac{1}{2}x^{4}Φ_{5}Φ_{8}Φ_{10}Φ_{12}Φ_{20}  ϕ_{5,16}
ϕ′_{6,10}  \frac{1}{5}x^{6}Φ_{3}Φ_{5}Φ_{6}Φ_{8}Φ_{10}Φ_{12}Φ_{20}  ϕ′′_{6,10}
ϕ′′′_{6,10} −\frac{1}{20}x^{6}Φ_{3}Φ′_{4}^{4}Φ_{5}Φ_{6}Φ_{8}Φ_{10}Φ_{12}Φ′′_{20} ϕ′′′′_{6,10}
ϕ_{6,12}  \frac{1}{4}x^{6}Φ_{3}Φ_{4}^{2}Φ_{6}Φ_{8}Φ_{10}Φ_{12}Φ_{20}
ϕ_{10,2} x^{2}Φ_{5}Φ_{8}Φ_{10}Φ_{20} ϕ_{10,18}
ϕ_{10,6}  \frac{1}{2}x^{4}Φ_{4}^{2}Φ_{5}Φ_{10}Φ_{12}Φ_{20} ϕ_{10,14}
ϕ′_{15,4}  x^{4}Φ_{3}Φ_{5}Φ_{6}Φ_{10}Φ_{12}Φ_{20}  ϕ′_{15,12}
ϕ′′_{15,4}  \frac{1}{2}x^{4}Φ_{3}Φ_{5}Φ_{6}Φ_{8}Φ_{10}Φ_{20}  ϕ′′_{15,12}
ϕ_{16,3}  \frac{1}{2}x^{3}Φ_{4}^{4}Φ_{8}Φ_{12}Φ_{20} ϕ_{16,5} ϕ_{16,15} ϕ_{16,13}
ϕ_{20,5} \frac{1−i}{4}x^{5}Φ_{2}^{2}Φ′′_{4}^{2}Φ_{5}Φ_{6}Φ′′_{8}Φ_{10}Φ_{12}^{a}Φ_{20} ϕ_{20,7} ϕ_{20,9} ϕ_{20,11}
ϕ_{20,6} \frac{1}{2}x^{5}Φ_{4}^{3}Φ_{5}Φ_{10}Φ_{12}Φ_{20} ϕ_{20,10}
ϕ_{24,6}  \frac{1}{20}x^{6}Φ_{2}^{4}Φ_{3}Φ_{5}Φ_{6}Φ_{8}Φ_{12}Φ_{20}
ϕ_{24,7}  \frac{1}{4}x^{6}Φ_{2}^{2}Φ_{3}Φ_{4}^{3}Φ_{6}Φ_{10}Φ_{12}Φ_{20} ϕ_{24,9}
ϕ_{30,8}  \frac{1}{4}x^{6}Φ_{3}Φ_{4}^{2}Φ_{5}Φ_{6}Φ_{8}Φ_{12}Φ_{20}

Here Φ′_{4} = x − i, Φ′_{8} = x^{2} − i, Φ_{12}^{a} = x^{2} − ix − 1, Φ′′_{20} = x^{4} − ix^{3} − x^{2} + ix + 1.

Let us return to W = G_{29}. Restriction to the maximal parabolic subgroups gives 16 equations. The Coxeter element c := s_{1}s_{2}s_{3}s_{4} in the braid group B(W) of W satisfies c^{20} = π. The equations given by evaluating the irreducible characters of H on T_{c}^{l}, 1 ≤ l ≤ 18, are linearly independent from the equations obtained by restriction. The character tables of all maximal parabolic subalgebras of H(G_{29}; (x, −1)) are known, respectively can easily be computed for the non-Coxeter group G(4, 4, 3). Since c^{5} is central in B(W), the values of irreducible characters on elements T_{w}T_{c}^{5} with T_{w} in some parabolic subalgebra of H are known. The corresponding equations determine the generic degrees uniquely.

Theorem 3.8. Let t be a symmetrizing form on H(G_{29}; (x, −1)) restricting to the canonical symmetrizing form on the maximal parabolic subalgebras of Coxeter type (respectively to the form determined above for G(4, 4, 3)) and such that there exists a quasi-symmetric basis containing T_{c}^{l} (1 ≤ l ≤ 18) and elements T_{w}T_{c}^{5} for w running through representatives of minimal length of the conjugacy classes of parabolic subgroups of W. Then the generic degrees with respect to t are given in Table 3.7.

All characters of W are distinguished by their degree and their b-invariant, except for four characters of degree 6 with b = 10 and for two pairs of characters of degree 15. We denote by ϕ′′_{15,4} the character of degree 15 occurring in the tensor product of the reflection character ϕ_{4,1} with ϕ_{4,3}, and by ϕ′′_{15,12} its tensor product with the sign character ϕ_{1,40}. We denote by ϕ′_{6,10} the real character of type (6, 10) occurring in the permutation character of W on the parabolic subgroup of type A_{3}, and by ϕ′′_{6,10} the other real character of degree 6 with b = 10 (these two have identical generic degrees). By ϕ′′′_{6,10} we denote the character of degree 6 appearing in the tensor product of the reflection character with itself, and by ϕ′′′′_{6,10} its complex conjugate.

4. Generic degrees for G_{33} and G_{34}

Here we conclude the determination of Schur elements for exceptional non-real reflection groups of dimension n ≥ 3 generated by n involutive reflections by considering the cases G_{33} and G_{34} in the Shephard-Todd classification.

4A. Some imprimitive groups.

We first compute the generic degrees for some imprimitive complex reflection groups which occur as parabolic subgroups in the primitive groups G_{33} and G_{34}. More precisely, these are the groups G(3, 3, 3), G(3, 3, 4) and G(3, 3, 5). Again, the results can easily be obtained by the methods presented so far, and we just give the lists in Tables 4.1–4.3.

Table 4.1. Generic degrees for G(3, 3, 3).

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,9}
ϕ_{2,3}  x^{3}Φ_{2}Φ_{6}
ϕ_{3,1} \frac{1−ζ_{3}^{2}}{3} xΦ′_{3}^{3}Φ′′_{6} ϕ_{3,2} ϕ_{3,4} ϕ_{3,5}

Table 4.2. Generic degrees for G(3, 3, 4).

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,18}
ϕ_{2,6}  \frac{1}{3}x^{4}Φ_{4}Φ_{6}Φ_{9}
ϕ_{3,3} x^{3}Φ_{9} ϕ_{3,9}
ϕ_{4,1}  \frac{1−ζ_{3}^{2}}{3} xΦ_{2}Φ_{4}Φ′_{6}Φ′′_{9} ϕ_{4,2} ϕ_{4,10} ϕ_{4,11}
ϕ_{6,2} x^{2}Φ_{4}Φ_{9} ϕ_{6,8}
ϕ′_{6,5} −\frac{ζ_{3}^{2}}{3} x^{4}Φ′_{3}^{3}Φ_{4}Φ_{6}Φ′′_{9} ϕ′′_{6,5}
ϕ_{8,4}  \frac{1}{3}x^{4}Φ_{2}^{2}Φ_{4}Φ_{9} ϕ_{8,5}
ϕ_{12,3}  x^{3}Φ_{2}Φ_{4}Φ_{9}  ϕ_{12,6}

Here, we have denoted by ϕ′_{6,5} the character of degree 6 with b-value 5 appearing in the tensor square of the reflection character ϕ_{4,1}, by ϕ′′_{6,5} its complex conjugate.

Table 4.3. Generic degrees for G(3, 3, 5).

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,30}
ϕ_{4,3}  x^{3}Φ_{2}Φ_{4}Φ_{6}Φ_{12}  ϕ_{4,18}
ϕ_{5,1}  \frac{1−ζ_{3}^{2}}{3} xΦ_{5}Φ′_{6}Φ′_{9}Φ′′_{12} ϕ_{5,2} ϕ_{5,19} ϕ_{5,20}
ϕ_{5,6} \frac{1}{3}x^{4}Φ_{5}Φ_{6}Φ_{9}Φ_{12} ϕ_{5,12}
ϕ_{6,9}  x^{9}Φ_{4}Φ_{9}Φ_{12}
ϕ_{10,2}  \frac{1−ζ_{3}^{2}}{3} x^{2}Φ_{4}Φ_{5}Φ′_{6}Φ′′_{9}Φ_{12} ϕ_{10,3} ϕ_{10,14} ϕ_{10,15}
ϕ_{10,5}  −\frac{ζ_{3}^{2}}{3} x^{4}Φ_{4}Φ_{5}Φ′_{6}^{2}Φ_{9}Φ′′_{12} ϕ_{10,6} ϕ_{10,11} ϕ_{10,12}
ϕ_{10,7}  \frac{1}{3}x^{6}Φ_{4}Φ_{5}Φ_{6}Φ_{9}Φ_{12} ϕ_{10,8}
ϕ_{15,4}  \frac{1}{3}x^{4}Φ′′_{3}^{3}Φ_{5}Φ′_{6}^{2}Φ′′_{9}Φ_{12} ϕ_{15,5} ϕ_{15,10} ϕ_{15,11}
ϕ_{20,3} x^{3}Φ_{2}Φ_{4}Φ_{5}Φ_{6}^{2}Φ_{12} ϕ_{20,12}
ϕ_{20,5} \frac{1−ζ_{3}}{3} x^{5}Φ_{2}Φ_{4}Φ_{5}Φ_{6}Φ′′_{6}Φ′_{9}Φ_{12} ϕ_{20,6} ϕ_{20,8} ϕ_{20,9}
ϕ_{30,4} x^{4}Φ_{4}Φ_{5}Φ_{9}Φ_{12} ϕ_{30,10}
ϕ′_{30,7}  \frac{1}{3}x^{6}Φ′_{3}^{3}Φ_{4}Φ_{5}Φ′′_{6}^{2}Φ′_{9}Φ_{12} ϕ′′_{30,7}
ϕ_{40,6}  \frac{1}{3}x^{6}Φ_{2}^{2}Φ_{4}Φ_{5}Φ_{9}Φ_{12}

Here, ϕ′′_{30,7} denotes the character of degree 30 occurring in the fourth tensor power of the reflection character ϕ_{5,1}, and ϕ′_{30,7} its complex conjugate.

In all tables of this section the notation for cyclotomic polynomials is as follows. Let ζ_{3} := exp(2πi/3) be a third root of unity. Then for any j multiple of 3 we have a factorization Φ_{j} = Φ′_{j}Φ′′_{j} over ℚ(ζ_{3}) and we choose notation such that Φ′_{j}(exp(2πi/j)) = 0.

More specifically, this means that Φ′_{3} = x−ζ_{3}, Φ′_{6} = x+ζ_{3}^{2}, Φ′_{9} = x^{3}−ζ_{3}, Φ′_{12} = x^{2}+ζ_{3}^{2}, Φ′_{18} = x^{3} + ζ_{3}^{2}, ...

4B. The complex reflection group G_{33}.

The 5-dimensional primitive complex reflection group W = G_{33}, abstractly isomorphic to 2 × O_{5}(3), has 40 irreducible characters. It is generated by five conjugate reflections s_{1}, …, s_{5} of order 2 satisfying the relations implied by the Coxeter diagram

◯_{s_{1}}_{2} ◯_{s_{2}}_{2} △ ◯_{s_{4}}_{2} ⧸^{◯ 2 ^{s_{3}}} ⧹ ◯_{s_{5}}_{2}

together with (s_{2}s_{3}s_{4})^{2} = (s_{3}s_{4}s_{2})^{2} = (s_{4}s_{2}s_{3})^{2}. It has four conjugacy classes of maximal parabolic subgroups, of types D_{4}, A_{4}, A_{3} +A_{1}, G(3, 3, 4). The first three are Coxeter groups, and the last one was considered in 4A. Restriction to these subgroups gives 22 equations. The Coxeter element c := s_{1}s_{2}s_{3}s_{4}s_{5} satisfies c^{18} = π. The equations given by evaluating the irreducible characters of H on T_{c}^{m}, 1 ≤ m ≤ 17, are linearly independent from the equations obtained by restriction. Furthermore, the defining relations of B(W) (i.e., the braid relations of W) show that w^{5} = c^{9} for the element w := cs_{1}s_{2}s_{3}s_{4} ∈ B(W) of length 9, hence it is a 10-th root of π. Evaluating characters on T_{w} gives a further independent equation, to yield a system of full rank.

Theorem 4.4. Let t be a symmetrizing form on H(G_{33}; (x, −1)) restricting to the canonical symmetrizing form on the maximal parabolic subalgebras of Coxeter type (respectively to the form determined above for G(3, 3, 4)) and such that there exists a quasi-symmetric basis containing T_{c}^{l} (1 ≤ l ≤ 17) and T_{w}. Then the generic degrees with respect to t are given in Table 4.5.

All irreducible characters of W are determined by their degree and their b-value, except for two pairs of characters of degree 10 and two pairs of characters of degree 40. We denote by ϕ′_{10,8} the character of degree 10 whose fake degree is of the form x^{28} + x^{26}+ lower powers of x, and by ϕ′_{40,5} the one of degree 40 with fake degree x^{31} + x^{29}+ lower powers of x. The notation for the other characters is now fixed by the information given in Table 4.5.

4C. The complex reflection group G_{34}.

The 6-dimensional primitive complex reflection group W = G_{34}, abstractly isomorphic to 6.U_{4}(2).2, has 169 irreducible characters. It is generated by a set S = {s_{1}, …, s_{6}} of six conjugate reflections of order 2, satisfying the relations implied by the Coxeter diagram

◯_{s_{1}}_{2} ◯_{s_{2}}_{2} △ ◯_{s_{4}}_{2} ⧸^{◯ 2 ^{s_{3}}} ⧹ ◯_{s_{5}}_{2} ◯_{s_{6}}_{2}

together with (s_{2}s_{3}s_{4})^{2} = (s_{3}s_{4}s_{2})^{2} = (s_{4}s_{2}s_{3})^{2}. It has eight conjugacy classes of maximal parabolic subgroups, of types G_{33}, G(3, 3, 5), D_{5}, A_{5} (2 classes), G(3, 3, 4) + A_{1}, A_{4} + A_{1}, A_{3} + A_{2}. All of these only involve real groups, one of the imprimitive groups considered above, or G_{33}. Thus for all maximal parabolic subgroups we have a well defined symmetrizing form. Restriction to these subgroups gives 71 equations.

4.5. Generic degrees for G_{33}.

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,45}
ϕ_{5,1}  \frac{1−ζ_{3}^{2}}{3} xΦ_{5}Φ′_{9}Φ_{10}Φ′_{12}Φ′′_{18} ϕ_{5,3} ϕ_{5,28} ϕ_{5,30}
ϕ_{6,5}  \frac{1}{2}x^{3}Φ_{4}^{2}Φ_{9}Φ_{10}Φ_{12}Φ_{18}  ϕ_{6,20}
ϕ′_{10,8}  \frac{ζ_{3}^{2}}{6} x^{4}Φ_{4}^{2}Φ_{5}Φ′′_{6}^{3}Φ_{9}Φ_{10}Φ_{12}Φ′_{18} ϕ′′_{10,8} ϕ′_{10,17} ϕ′′_{10,17}
ϕ_{15,2}  x^{2}Φ_{5}Φ_{9}Φ_{10}Φ_{18}  ϕ_{15,23}
ϕ_{15,9}  x^{9}Φ_{5}Φ_{9}Φ_{10}Φ_{12}Φ_{18}  ϕ_{15,12}
ϕ_{20,6}  \frac{1}{3}x^{4}Φ_{4}^{2}Φ_{5}Φ_{9}Φ_{10}Φ_{12}Φ_{18}  ϕ_{20,15}
ϕ_{24,4}  \frac{1}{2}x^{3}Φ_{2}^{4}Φ_{6}^{2}Φ_{9}Φ_{10}Φ_{18}  ϕ_{24,19}
ϕ_{30,3}  \frac{1}{2}x^{3}Φ_{4}^{2}Φ_{5}Φ_{9}Φ_{12}Φ_{18}  ϕ_{30,18}
ϕ_{30,4}  −\frac{ζ_{3}^{2}}{6} x^{4}Φ′_{3}^{3}Φ_{4}^{2}Φ_{5}Φ′′_{9}Φ_{10}Φ_{12}Φ_{18} ϕ_{30,6} ϕ_{30,13} ϕ_{30,15}
ϕ′_{40,5} \frac{1−ζ_{3}}{6} x^{4}Φ_{2}^{4}Φ_{5}Φ_{6}^{2}Φ′′_{6}Φ′′_{9}Φ_{10}Φ′_{12}Φ_{18} ϕ′′_{40,5} ϕ′_{40,14} ϕ′′_{40,14}
ϕ_{45,7}  \frac{1−ζ_{3}^{2}}{3} x^{7}Φ′′_{3}^{3}Φ_{5}Φ′_{6}^{3}Φ_{9}Φ_{10}Φ′′_{12}Φ_{18} ϕ_{45,9} ϕ_{45,10} ϕ_{45,12}
ϕ_{60,7}  x^{7}Φ_{4}^{2}Φ_{5}Φ_{9}Φ_{10}Φ_{12}Φ_{18}  ϕ_{60,10}
ϕ_{64,8}  \frac{1}{2}x^{8}Φ_{2}^{5}Φ_{4}^{2}Φ_{6}^{3}Φ_{10}Φ_{12}Φ_{18}  ϕ_{64,9}
ϕ_{81,6}  x^{6}Φ_{3}^{3}Φ_{6}^{3}Φ_{9}Φ_{12}Φ_{18}  ϕ_{81,11}

As usual we denote by s_{j} the canonical preimage in B(W) of the generator s_{j} ∈ S. The Coxeter element c := s_{1} … s_{6} satisfies c^{42} = π in B(W) [5, Table 4]. Its seventh power is central in B(W).

Theorem 4.7. The generic degrees of the cyclotomic Hecke algebra H(G_{34}; (x, −1)) with respect to any quasi-symmetric basis containing only T_{c}^{m}, 1 ≤ m ≤ 41, and elements of the form T_{v}T_{c}^{7l} (1 ≤ l ≤ 5) for v running through representatives of minimal length of the conjugacy classes of the maximal parabolic subgroups of W are given in Table 4.6.

Proof. The character values of H on the powers of T_{c} can be evaluated with Proposition 2.8. Moreover, character values on a basis of any proper parabolic subalgebra are known by the preceding section, respectively by [3]. Since T_{c}^{7l} is central in H, the values on any product T_{v}T_{c}^{5l} can be determined, once those on T_{v} are known. Thus the values of all irreducible characters on all the elements occurring in the statement can be computed. Using the computer algebra system GAP it was checked that this system has maximal rank, and thus determines the generic degrees uniquely. □

All irreducible characters of W are determined by their degree and their b-value, except for seven pairs and three triples of characters. The notation for the characters of degree 105 and 840 is fixed by Table 4.6, since for each ambiguous character, its complex conjugate is determined by its degree and b-value. In the three triples, we

14 GUNTER MALLE

4.6. Generic degrees for G_{34}.

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{1,0}  1  ϕ_{1,126}
ϕ_{6,1}  −\frac{\sqrt{−3}}{3}ζ_{3}xΦ′_{3}^{3}Φ′′_{6}^{3}Φ_{8}Φ′_{12}Φ_{24}Φ′′_{15}Φ′_{21}Φ′_{30}Φ′′_{42} ϕ_{6,5} ϕ_{6,85} ϕ_{6,89}
ϕ_{15,14}  \frac{ζ_{3}}{6} x^{4}Φ′_{3}^{3}Φ_{5}Φ′′_{6}^{3}Φ_{8}Φ′_{9}Φ_{10}Φ′_{12}^{2}Φ_{14}Φ_{15}Φ′_{18}Φ_{21}Φ_{24}Φ_{30}Φ′′_{42} ϕ_{15,16} ϕ_{15,56} ϕ_{15,58}
ϕ′_{20,33}  \frac{1}{42}x^{15}Φ_{4}^{2}Φ_{5}Φ′_{6}^{6}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ′′_{42} ϕ′′_{20,33}
ϕ_{21,2}  −\frac{\sqrt{−3}}{3}ζ_{3}x^{2}Φ′′_{3}^{3}Φ′_{6}^{3}Φ_{7}Φ′′_{12}Φ_{14}Φ′_{15}Φ_{21}Φ′_{24}Φ′′_{30}Φ_{42} ϕ_{21,4} ϕ_{21,68} ϕ_{21,70}
ϕ_{21,6}  \frac{1}{2}x^{3}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{14}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{21,60}
ϕ_{35,6}  \frac{1}{2}x^{3}Φ_{5}Φ_{7}Φ_{8}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{42}  ϕ_{35,60}
ϕ_{35,18}  \frac{1}{3}x^{10}Φ_{5}Φ_{7}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{35,36}
ϕ_{56,3}  \frac{1}{2}x^{3}Φ_{2}^{4}Φ_{6}^{4}Φ_{7}Φ_{10}Φ_{14}Φ_{18}Φ_{30}Φ_{21}Φ_{42}  ϕ_{56,57}
ϕ_{56,9}  \frac{1}{3}x^{5}Φ_{4}^{2}Φ_{7}Φ_{8}Φ_{9}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{56,45}
ϕ′_{70,9}  x^{9}Φ_{5}Φ_{7}Φ_{8}Φ_{10}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ′_{70,45}
ϕ′′_{70,9}  −\frac{ζ_{3}^{2}}{3} x^{5}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ′_{12}^{2}Φ_{14}Φ′_{15}Φ_{18}Φ_{21}Φ_{24}Φ′′_{30}Φ_{42} ϕ′′′_{70,9} ϕ′′_{70,45} ϕ′′′_{70,45}
ϕ_{84,13}  −\frac{ζ_{3}^{2}}{6} x^{7}Φ′′_{3}^{3}Φ_{4}^{2}Φ′_{6}^{3}Φ_{7}Φ_{8}Φ′_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ′_{18}Φ_{21}Φ_{24}Φ′′_{30}Φ_{42} ϕ_{84,17} ϕ_{84,37} ϕ_{84,41}
ϕ_{90,6} \frac{1}{3}x^{4}Φ_{3}^{3}Φ_{5}Φ_{6}^{3}Φ_{8}Φ_{10}Φ_{12}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{90,48}
ϕ_{105,4}  −\frac{ζ_{3}^{2}}{6} x^{4}Φ′′_{3}^{3}Φ_{5}Φ′_{6}^{3}Φ_{7}Φ_{8}Φ′_{9}Φ_{10}Φ′′_{12}^{2}Φ_{15}Φ′_{18}Φ′′_{21}Φ_{24}Φ_{30}Φ_{42} ϕ′_{105,8} ϕ_{105,46} ϕ_{105,50}
ϕ′′_{105,8}  \frac{1}{3}x^{6}Φ′′_{3}^{3}Φ_{5}Φ′_{6}^{3}Φ_{7}Φ′′_{9}Φ_{10}Φ′′_{12}^{2}Φ_{14}Φ_{15}Φ′_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{105,10} ϕ_{105,38} ϕ_{105,40}
ϕ_{105,20}  −\frac{1}{6}x^{13}Φ′_{3}^{3}Φ_{5}Φ′′_{6}^{6}Φ_{7}Φ_{8}Φ′′_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ′_{30}Φ_{42} ϕ_{105,22} ϕ_{105,26} ϕ_{105,28}
ϕ_{120,5}  −\frac{\sqrt{−3}}{6}ζ_{3}x^{4}Φ_{2}^{4}Φ′_{3}^{3}Φ_{5}Φ_{6}^{4}Φ′′_{6}Φ_{10}Φ′_{12}Φ_{14}Φ_{15}Φ_{18}Φ′_{21}Φ′′_{24}Φ_{30}Φ_{42} ϕ_{120,7} ϕ_{120,47} ϕ_{120,49}
ϕ′_{120,21} \frac{1}{7}x^{15}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ′′_{120,21}
ϕ_{126,5}  \frac{1}{3}x^{5}Φ_{3}^{3}Φ_{6}^{3}Φ_{7}Φ_{8}Φ_{12}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{126,7} ϕ_{126,41} ϕ_{126,43}
ϕ_{140,12} \frac{1}{2}x^{9}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{140,30}
ϕ_{140,21}  \frac{1}{2}x^{18}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}
ϕ_{189,18} \frac{1}{6}x^{13}Φ_{3}^{3}Φ_{6}^{3}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{189,24}
ϕ_{210,8}  \frac{1}{3}x^{6}Φ′′_{3}^{3}Φ_{5}Φ′_{6}^{3}Φ_{7}Φ_{8}Φ′_{9}Φ_{10}Φ′′_{12}^{2}Φ_{14}Φ_{15}Φ′′_{18}Φ_{21}Φ′_{24}Φ_{30}Φ_{42} ϕ_{210,10} ϕ_{210,38} ϕ_{210,40}
ϕ_{210,12} x^{12}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{210,30}
ϕ_{210,13} −\frac{ζ_{3}}{3} x^{11}Φ′_{3}^{3}Φ_{5}Φ′′_{6}^{3}Φ_{7}Φ_{8}Φ′′_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ′_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{210,17} ϕ_{210,25} ϕ_{210,29}
ϕ′_{280,12}  −\frac{ζ_{3}^{2}}{3} x^{10}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ′_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ′′_{24}Φ_{30}Φ_{42} ϕ′′_{280,12} ϕ′_{280,30} ϕ′′_{280,30}
ϕ_{315,6} \frac{1}{3}x^{6}Φ_{3}^{3}Φ_{5}Φ_{6}^{3}Φ_{7}Φ_{10}Φ_{12}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{315,36}
ϕ_{315,10}  \frac{1}{3}x^{10}Φ_{3}^{3}Φ_{5}Φ_{6}^{3}Φ_{7}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{315,14} ϕ_{315,28} ϕ_{315,32}
ϕ_{315,18} \frac{1}{6}x^{13}Φ_{3}^{3}Φ_{5}Φ_{6}^{3}Φ_{7}Φ_{8}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{315,24}
ϕ_{336,8} −\frac{\sqrt{−3}}{6}ζ_{3}x^{7}Φ_{2}^{4}Φ′_{3}^{3}Φ_{6}^{4}Φ′′_{6}Φ_{7}Φ_{8}Φ_{10}Φ′_{12}Φ_{14}Φ′′_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{336,10} ϕ_{336,32} ϕ_{336,34}
ϕ_{336,17} −\frac{ζ_{3}^{2}}{6} x^{13}Φ_{2}^{4}Φ′_{3}^{3}Φ_{6}^{4}Φ′′_{6}^{2}Φ_{7}Φ_{8}Φ′_{9}Φ_{10}Φ′_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ′′_{24}Φ_{30}Φ_{42} ϕ_{336,19} ϕ_{336,23} ϕ_{336,25}

ϕ  δ_{ϕ}  ϕ̄  ϵ ⊗ ϕ ϵ ⊗ ϕ̄
ϕ_{384,8} −\frac{\sqrt{−3}}{6}ζ_{3}x^{8}Φ_{2}^{5}Φ′′_{3}^{3}Φ_{4}^{2}Φ_{6}^{5}Φ′_{6}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{14}Φ′_{15}Φ_{18}Φ′′_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{384,10} ϕ_{384,32} ϕ_{384,34}
ϕ_{384,11} −\frac{\sqrt{−3}}{6}ζ_{3}x^{8}Φ_{2}^{5}Φ′′_{3}^{3}Φ_{4}^{2}Φ_{6}^{5}Φ′_{6}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{14}Φ′_{15}Φ_{18}Φ′′_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{384,13} ϕ_{384,29} ϕ_{384,31}
ϕ_{420,7}  −\frac{ζ_{3}^{2}}{6} x^{7}Φ′′_{3}^{3}Φ_{4}^{2}Φ_{5}Φ′_{6}^{3}Φ_{7}Φ_{8}Φ′′_{9}Φ_{12}^{2}Φ_{14}Φ′_{15}Φ′′_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{420,11} ϕ_{420,31} ϕ_{420,35}
ϕ_{420,12}  \frac{1}{2}x^{9}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{420,30}
ϕ_{420,14}  \frac{ζ_{3}^{2}}{6} x^{13}Φ′_{3}^{3}Φ_{4}^{2}Φ_{5}Φ′′_{6}^{6}Φ_{7}Φ_{8}Φ′_{9}Φ′_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{420,16} ϕ_{420,20} ϕ_{420,22}
ϕ_{420,21}  \frac{1}{2}x^{18}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42}
ϕ_{504,9}  \frac{1}{3}x^{7}Φ_{3}^{3}Φ_{4}^{2}Φ_{6}^{3}Φ_{7}Φ_{8}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{504,33}
ϕ_{504,15}  \frac{1}{6}x^{13}Φ_{2}^{4}Φ_{3}^{3}Φ_{6}^{5}Φ_{7}Φ_{10}Φ_{12}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{504,21}
ϕ_{540,17}  \frac{1}{6}x^{15}Φ_{3}^{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{3}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{540,19}
ϕ′_{540,21}  −\frac{1}{42}x^{15}Φ′′_{3}^{6}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ′_{21}Φ_{24}Φ_{30}Φ_{42} ϕ′′_{540,21}
ϕ_{560,9}  \frac{1}{2}x^{9}Φ_{2}^{4}Φ_{5}Φ_{6}^{4}Φ_{7}Φ_{8}Φ_{10}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{560,27}
ϕ′_{560,18}  \frac{1}{2}x^{18}Φ_{2}^{4}Φ_{5}Φ_{6}^{4}Φ_{7}Φ_{8}Φ_{10}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}
ϕ′′_{560,18}  \frac{1}{6}x^{15}Φ_{2}^{4}Φ_{5}Φ_{6}^{4}Φ′_{6}^{2}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ′′_{12}^{2}Φ_{14}Φ_{15}Φ_{18}Φ′_{21}Φ_{24}Φ_{30}Φ_{42} ϕ′′′_{560,18}
ϕ_{630,11} −\frac{\sqrt{−3}}{3}ζ_{3}x^{11}Φ′′_{3}^{3}Φ_{5}Φ′_{6}^{3}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ′′_{12}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{630,13} ϕ_{630,23} ϕ_{630,25}
ϕ_{630,14} \frac{\sqrt{−3}}{3}ζ_{3}^{2}x^{14}Φ′_{3}^{3}Φ_{5}Φ′′_{6}^{3}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ′_{12}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{630,16} ϕ_{630,20} ϕ_{630,22}
ϕ_{630,15}  \frac{1}{3}x^{11}Φ_{3}^{3}Φ_{5}Φ_{6}^{3}Φ_{7}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{15}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{630,27}
ϕ_{720,16}  \frac{1}{6}x^{15}Φ_{2}^{4}Φ_{3}^{3}Φ_{5}Φ_{6}^{5}Φ_{8}Φ_{10}Φ_{12}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{720,20}
ϕ_{729,10}  \frac{1}{3}x^{10}Φ_{3}^{6}Φ_{6}^{6}Φ_{9}Φ_{12}^{2}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{729,14} ϕ_{729,28} ϕ_{729,26}
ϕ_{729,12}  \frac{1}{3}x^{10}Φ_{3}^{6}Φ_{6}^{6}Φ_{9}Φ_{12}^{2}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{729,24}
ϕ_{756,14} −\frac{ζ_{3}^{2}}{6} x^{13}Φ′_{3}^{6}Φ_{4}^{2}Φ′′_{6}^{3}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ′_{12}^{2}Φ_{14}Φ_{15}Φ′′_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{756,16} ϕ_{756,20} ϕ_{756,22}
ϕ_{840,11}  \frac{1}{3}x^{11}Φ′′_{3}^{3}Φ_{4}^{2}Φ_{5}Φ′_{6}^{3}Φ_{7}Φ_{8}Φ′′_{9}Φ_{10}Φ′′_{12}^{2}Φ_{14}Φ_{15}Φ′_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ′_{840,13} ϕ′′_{840,23} ϕ_{840,25}
ϕ′′_{840,13}  \frac{1}{6}x^{13}Φ_{2}^{4}Φ′′_{3}^{3}Φ_{5}Φ_{6}^{4}Φ′_{6}^{2}Φ_{7}Φ′_{9}Φ_{10}Φ′′_{12}^{2}Φ_{14}Φ′_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{840,17} ϕ_{840,19} ϕ′_{840,23}
ϕ_{896,12}  \frac{1}{2}x^{12}Φ_{2}^{5}Φ_{4}^{2}Φ_{6}^{5}Φ_{7}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{896,24}
ϕ_{896,15}  \frac{1}{2}x^{12}Φ_{2}^{5}Φ_{4}^{2}Φ_{6}^{5}Φ_{7}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{14}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}  ϕ_{896,21}
ϕ_{945,14}  \frac{1}{6}x^{13}Φ′_{3}^{6}Φ_{5}Φ′′_{6}^{3}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{14}Φ′′_{15}Φ′_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{945,16} ϕ_{945,20} ϕ_{945,22}
ϕ_{1260,17}  \frac{1}{6}x^{15}Φ_{3}^{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{3}Φ_{7}Φ_{8}Φ_{10}Φ_{12}^{2}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42} ϕ_{1260,19}
ϕ_{1280,15}  \frac{1}{42}x^{15}Φ_{2}^{6}Φ_{4}^{2}Φ_{5}Φ_{7}Φ_{8}Φ_{9}Φ_{10}Φ_{12}^{2}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}
ϕ_{1280,18}  \frac{1}{6}x^{15}Φ_{2}^{6}Φ_{4}^{2}Φ_{5}Φ_{6}^{4}Φ_{8}Φ_{9}Φ_{10}Φ_{14}Φ_{15}Φ_{18}Φ_{21}Φ_{24}Φ_{30}Φ_{42}

choose ϕ′_{70,9}, ϕ′_{70,45}, ϕ′_{560,18} to be the real characters. We denote by ϕ′′_{20,33}, ϕ′′′_{70,9} the characters of degree 20 respectively 70 occurring in the tensor product of the reflection character ϕ_{6,1} with ϕ_{15,14}, and by ϕ′_{20,33}, ϕ′′_{70,9} their complex conjugates. The tensor product of ϕ′′_{70,9} with the sign character is ϕ′′_{70,45}, its complex conjugate ϕ′′′_{70,45}. We write ϕ′_{120,21} for the character of type (120, 21) occurring in the permutation character on the parabolic subgroup of type A_{5}. Finally, we call ϕ′_{280,12}, ϕ′′_{280,30}, ϕ′′′_{560,18} the characters of the respective types which occur in ϕ_{6,1} ⊗ ϕ_{336,17}, and ϕ′_{540,21} the one occurring in ϕ_{6,1} ⊗ ϕ_{105,20}. This fixes the notation for all elements of Irr(W).

5. Schur elements for G_{31}

The 4-dimensional primitive complex reflection group W = G_{31} has a single class of reflections. It differs from all other exceptional reflection groups of dimension n at least three in that it cannot be generated by n of its reflections. It has 59 irreducible characters and 3 conjugacy classes of maximal parabolic subgroups, of types G(4, 2, 3), A_{3}, A_{2} + A_{1}. In order to be able to use induction on parabolic subalgebras, we first have to compute the Schur elements of the imprimitive group G(4, 2, 3).

5A. The imprimitive group G(4, 2, 3).

There exists a unique symmetrizing form on the cyclotomic Hecke algebra of the imprimitive group G(4, 2, 3) restricting to the canonical symmetrizing form on the maximal parabolic subalgebras (of Coxeter type) and vanishing on T_{c}^{l} (1 ≤ l ≤ 5). The corresponding Schur elements can then be computed from (2.9). It turns out that they agree with the values conjectured in [12, Satz 5.13].

They can also be described in a different way as follows. The group G(4, 2, 3) occurs as the relative Weyl group of the Φ_{4}-Sylow torus in finite reductive groups of type D_{6}. Conjecturally, this should imply that the Schur elements of H(G(4, 2, 3); (x, −1)) are related to the degree polynomials of the unipotent characters of ^{2}D_{6} in the principal Φ_{4}-block (see [3]). It turns out that this is in fact true. For this, let U_{4}(^{2}D_{6}) denote the set of unipotent characters of ^{2}D_{6} whose degree polynomial is not divisible by Φ_{4}, and write deg(γ) ∈ ℚ[x] for the degree polynomial of γ ∈ U_{4}(^{2}D_{6}).

Theorem 5.1. There is a bijection

Irr(G(4, 2, 3)) ⟶ U_{4}(^{2}D_{6}), ϕ ↦ γ_{ϕ},

between the irreducible characters of the cyclotomic algebra H(G(4, 2, 3); (x, −1)) and U_{4}(^{2}D_{6}) (given by Table 5.2) such that the Schur element c_{ϕ} of ϕ ∈ Irr(G(4, 2, 3)) is given by

c_{ϕ}(x) = \frac{(x + 1)(x^{2} − 1)(x^{4} − 1)(x^{5} + 1)(x^{6} − 1)}{(x − 1)^{3} deg(γ_{ϕ})(i\sqrt{x})} .

Here, we denote the unipotent characters of ^{2}D_{6} by symbols of rank 6 as in [10].

5B. The complex reflection group G_{31}.

We return to the primitive reflection group W = G_{31}. Twenty of its conjugacy classes have non-empty intersection with proper parabolic subgroups, so by restriction we obtain 20 independent equations for the Schur elements. Let {s_{1}, …, s_{5}} denote the set of standard generators of W, with the relations implied by the Coxeter diagram

◯_{s_{4}}_{2}⧸◯_{s_{1}}_{2} ◯_{s_{2}}_{2}  ◯_{s_{5}}_{2} ◯_{s_{3}}_{2}⧹

5.2. A bijection Irr(G(4, 2, 3)) ⟷ U_{4}(^{2}D_{6}).

ϕ  γ_{ϕ}  ϕ̄  ϵ ⊗ ϕ  ϵ ⊗ ϕ̄
ϕ_{1,0}  (06|−)  ϕ_{1,15}
ϕ_{1,3}  (016|1)  ϕ_{1,12}
ϕ_{2,4}  (0126|12)  ϕ_{2,7}
ϕ_{3,2,6}  (24|−)  ϕ_{3,9}
ϕ_{3,5,9}  (124|1)  ϕ_{3,6,10}
ϕ_{3,1}  (012|5)  ϕ_{3,3}  ϕ_{3,8}  ϕ_{3,6,14}
ϕ_{3,2,10}  (013|4)  ϕ_{3,4}  ϕ_{3,7}  ϕ_{3,5,13}
ϕ_{6,3}  (023|3)  ϕ_{6,4}

together with s_{1}s_{2}s_{3} = s_{2}s_{3}s_{1} = s_{3}s_{1}s_{2}. The Coxeter element c := s_{1}s_{2}s_{3}s_{4}s_{5} of the braid group B(W) satisfies c^{24} = π (see [5]). The equations given by evaluating the irreducible characters of H on T_{c}^{m}, 1 ≤ m ≤ 23, are linearly independent from the equations obtained by restriction.

Furthermore, let w = s_{1}s_{2}s_{3}s_{4}s_{5}s_{1} = cs_{1}. From the defining relations of B(W) it follows that w^{5} = c^{6}, hence w is a 20-th root of π in B(W). (In fact, the image of w in W is a Coxeter element in the maximal rank reflection subgroup G_{29} < G_{31}.) Evaluating characters on powers of T_{w} gives further 14 independent equations. The remaining two equations are obtained by evaluating the irreducible characters on products of central elements by elements in proper parabolic subalgebras.

The group G_{31} occurs as the relative Weyl group of the Φ_{4}-Sylow torus in finite reductive groups of type E_{8}. Conjecturally, this should imply that the Schur elements of H(W, x) are related to the degree polynomials of the unipotent characters of E_{8} in the principal Φ_{4}-block. It turns out that this is in fact true. For this, let U_{4}(E_{8}) denote the set of unipotent characters of E_{8} whose degree polynomial is not divisible by Φ_{4}, and write deg(γ) ∈ ℚ[x] for the degree polynomial of γ ∈ U_{4}(E_{8}). We follow the naming convention in [10] for the unipotent characters of E_{8}.

Theorem 5.4. There is a bijection

Irr(G_{31}) ⟶ U_{4}(E_{8}), ϕ ↦ γ_{ϕ},

between the irreducible characters of the cyclotomic Hecke algebra H(G_{31}; (x, −1)) and U_{4}(E_{8}) (given by Table 5.3). This bijection is such that the Schur element c_{ϕ} of ϕ ∈ Irr(G_{31}) with respect to any quasi-symmetric basis containing T_{c}^{m} (1 ≤ m ≤ 23) and T_{w}^{m} (1 ≤ m ≤ 16) and elements of the form T_{v}T_{c}^{6l} (0 ≤ l ≤ 3) with v contained in a proper parabolic subgroups of B(W) is given by

c_{ϕ}(x) = \frac{(x + 1)(x^{4} − 1)(x^{6} − 1)(x^{7} + 1)(x^{9} + 1)(x^{10} − 1)(x^{12} − 1)(x^{15} + 1)}{(x − 1)^{4} deg(γ_{ϕ})(i\sqrt{x})} .

Except for five pairs the irreducible characters of G_{31} are determined by their degree and their b-invariant. We denote by ϕ′_{15,8}, ϕ′_{15,20}, ϕ′′_{45,8} the irreducible characters of

5.3. A bijection Irr(G_{31}) ⟷ U_{4}(E_{8}).

ϕ  γ_{ϕ}  ϕ̄  ϵ ⊗ ϕ  ϵ ⊗ ϕ̄
ϕ_{1,0}  ϕ_{1,0}  ϕ_{1,60}
ϕ_{4,1}  D_{4}, ϕ_{1,0}  ϕ_{4,7}  ϕ_{4,37}  ϕ_{4,31}
ϕ_{5,4}  ϕ_{35,2}  ϕ_{5,40}
ϕ_{5,12}  ϕ_{525,12}  ϕ_{5,24}
ϕ_{6,14}  ϕ_{70,32}  ϕ_{6,18}
ϕ_{9,8}  ϕ_{567,6}  ϕ_{9,28}
ϕ_{10,2}  ϕ_{50,8}  ϕ_{10,6}  ϕ_{10,30}  ϕ_{10,26}
ϕ_{10,12}  ϕ_{1050,10}  ϕ_{10,24}
ϕ′_{15,8}  ϕ_{1575,10}  ϕ′_{15,20}
ϕ′′_{15,8}  ϕ_{175,12}  ϕ′′_{15,20}
ϕ_{16,16}  D_{4}, ϕ_{16,5}
ϕ_{20,3}  D_{4}, ϕ′_{1,12}  ϕ_{20,5}  ϕ_{20,23}  ϕ_{20,21}
ϕ_{20,7}  ϕ_{2800,13}  ϕ′_{20,13}  ϕ_{20,19}  ϕ′′_{20,13}
ϕ_{20,14}  ϕ_{2100,20}
ϕ_{24,6}  E_{8}^{II}[1]  ϕ_{24,14}
ϕ_{30,4}  ϕ_{350,14}  ϕ_{30,16}
ϕ′_{30,10}  ϕ_{1134,20}  ϕ′′_{30,10}
ϕ_{36,5}  ϕ_{1296,13}  ϕ_{36,7}  ϕ_{36,17}  ϕ_{36,15}
ϕ_{36,10}  ϕ_{420,20}
ϕ_{40,6}  D_{4}, ϕ′_{8,3}  ϕ_{40,18}
ϕ_{40,7}  ϕ_{3360,13}  ϕ_{40,9}  ϕ_{40,15}  ϕ_{40,13}
ϕ_{40,10}  ϕ_{2688,20}  ϕ_{40,14}
ϕ′_{45,8}  ϕ_{2835,14}  ϕ′_{45,12}
ϕ′′_{45,8}  ϕ_{6072,14}  ϕ′′_{45,12}
ϕ_{64,9}  E_{8}[i]  ϕ_{64,11}

the respective types occurring in the tensor product ϕ_{4,1} ⊗ ϕ_{20,7}, and we let ϕ′_{30,10} be the character of degree 30 with fake degree x^{50} + x^{46}+ lower powers of x. The remaining notation is now fixed by the information in Table 5.3.

6. Schur elements for G_{25}, G_{26} and G_{32}

The three exceptional reflection groups G_{25}, G_{26} and G_{32} do contain reflections of order 3. Thus their Schur elements cannot be recovered from a 1-parameter specialization.

6A. The complex reflection group G_{26}.

The 3-dimensional primitive complex reflection group W = G_{26} is the direct product of G_{25} with a central element of order 2. It can be shown (see [3, Bem. 4.14(b)]) that the generic cyclotomic algebra for G_{25} is a subalgebra of index 2 of a suitable specialization of the cyclotomic algebra for G_{26}. This allows to study G_{25} using results from G_{26}. So we will first consider the latter group.

The complex reflection group W = G_{26} is generated by three reflections s_{1}, s_{2}, s_{3} of orders 3,3,2 respectively, which satisfy the defining relations indicated by this Coxeter type diagram:

◯_{s_{1}}_{3}  ◯_{s_{2}}_{3}  ◯_{s_{3}}_{2}

We denote by T_{s_{1}}, T_{s_{2}}, T_{s_{3}} the corresponding generators of the generic cyclotomic algebra H = H(W, (x, y)) of type G_{26}, hence

(T_{s_{j}} − y_{1})(T_{s_{j}} − y_{2})(T_{s_{j}} − y_{3}) = 0 for j = 1, 2, (T_{s_{3}} − x_{1})(T_{s_{3}} − x_{2}) = 0,

with the five parameters x = (x_{1}, x_{2}), y = (y_{1}, y_{2}, y_{3}). The group W has three conjugacy classes of maximal parabolic subgroups, of types G(3, 1, 2), G(3, 1, 1) + A_{1} and G_{4}. Fifteen of the 48 conjugacy classes of G_{26} have a representative in one of these parabolic subgroups. The Coxeter element c := s_{1}s_{2}s_{3} of the braid group B(W) satisfies c^{18} = π (see [5]). Its third power c^{3} is central. Since the irreducible representations of all proper parabolic subalgebras are explicitly known by [13, 3A] and [3, 5B] we may evaluate the irreducible characters of H on all elements of the form T_{c}^{l} or T_{w}T_{c}^{3l} where T_{w} lies in some maximal parabolic subalgebra. This gives enough linear independent equations for the determination of Schur elements.

For 1 ≤ j ≤ 3 let z := (z_{1}, z_{2}, z_{3}) with z_{j} := \sqrt{−x_{1}x_{2}y_{1}y_{2}y_{3}/y_{j}} in an algebraic closure of ℚ(x, y). We consider the specialization

x_{j} ↦ (−1)^{j−1},  y_{j} ↦ ζ_{3}^{j−1},  z_{j} ↦ (−ζ_{3}^{2})^{j},

of the generic cyclotomic algebra H to the group algebra of W. By [15, Table 8.2] (or [3, 6A]) the field ℚ(ζ_{3})[x, y, z] is a splitting field for H(W), so by Tits’ deformation theorem this identifies Irr(H(W)) with Irr(W). According to 2D the symmetry group 𝔖 = 𝔖_{2} × 𝔖_{3} of W acts on Irr(W) by permuting the first two and the last three parameters among themselves. In order to minimize the number of Schur elements to be printed, we record in Table 6.1 the action of 𝔖 on Irr(W) induced by the (left) action on {x_{1}, x_{2}, y_{1}, y_{2}, y_{3}}. We let τ_{1} := (x_{1}, x_{2}), τ_{2} := (y_{1}, y_{2}) and σ := (y_{1}, y_{2}, y_{3}).

Table 6.1. 𝔖_{2} × 𝔖_{3}-action on Irr(G_{26}).
ι  σ  σ^{2} τ_{2} στ_{2} τ_{2}σ τ_{1} τ_{1}σ τ_{1}σ^{2} τ_{1}τ_{2} τ_{1}στ_{2} τ_{1}τ_{2}σ
ϕ_{1,0}  ϕ_{1,12} ϕ_{1,24} ϕ_{1,12} ϕ_{1,24} ϕ_{1,0} ϕ_{1,9} ϕ_{1,21} ϕ_{1,33} ϕ_{1,21} ϕ_{1,33} ϕ_{1,9}
ϕ_{2,3}  ϕ_{2,15} ϕ_{2,9} ϕ_{2,3} ϕ_{2,15} ϕ_{2,9} ϕ_{2,12} ϕ_{2,24} ϕ_{2,18} ϕ_{2,12} ϕ_{2,24} ϕ_{2,18}
ϕ_{3,1}  ϕ′_{3,13} ϕ′′_{3,13} ϕ′′_{3,5} ϕ_{3,17} ϕ′_{3,5} ϕ_{3,4} ϕ′_{3,16} ϕ′′_{3,16} ϕ′′_{3,8} ϕ_{3,20} ϕ′_{3,8}
ϕ_{3,6}  ϕ_{3,15}
ϕ_{6,2}  ϕ′_{6,8} ϕ′′_{6,8} ϕ′′_{6,4} ϕ_{6,10} ϕ′_{6,4} ϕ_{6,5} ϕ′_{6,11} ϕ′′_{6,11} ϕ′′_{6,7} ϕ_{6,13} ϕ′_{6,7}
ϕ_{8,3}  ϕ′′_{8,6} ϕ′′_{8,9} ϕ′′_{8,6} ϕ′′_{8,9} ϕ_{8,3}
ϕ′_{8,6}  ϕ′_{8,9} ϕ_{8,12} ϕ′_{8,9} ϕ_{8,12} ϕ′_{8,6}
ϕ_{9,5}  ϕ_{9,8}
ϕ_{9,7}  ϕ_{9,10}

We denote the elements of Irr(W) by ϕ_{d,b}, where d = ϕ_{d,b}(1) and b is the b-value of ϕ_{d,b}. This distinguishes the characters except for ten pairs. The eight pairs with d = 3 or d = 6 have the property that the complex conjugate of any character in a pair is already determined by d, b. We will see in Theorem 6.2 that the generic degrees for the relevant characters are ℚ(x, y)-rational. Thus interchanging y_{2} with y_{3} induces complex conjugation on Irr(W). So the complex conjugates can be read of from Table 6.1 and thus also the notation for the first eight ambiguous pairs is fixed. In each of the two pairs of characters of degree 8, although the b-values agree, the fake degrees themselves are different for the members of a pair. We choose notation such that the fake degrees of ϕ′_{8,6} and ϕ′_{8,9} are monic as polynomials in x. This fixes the notation for the irreducible characters of W.

Theorem 6.2. The Schur elements of the cyclotomic algebra H(G_{26}; (x, y)) with respect to any quasi-symmetric basis containing T_{c}^{l} (1 ≤ l ≤ 7) and T_{w}T_{c}^{3l} (0 ≤ l ≤ 5) for w running through representatives of minimal length of the conjugacy classes of the maximal parabolic subgroups of W are as given below.

ϕ_{1,0} :(x_{2} − x_{1})(y_{1} − y_{2})(y_{1} − y_{3})(x_{1}y_{1} + x_{2}y_{2})(x_{1}y_{1} + x_{2}y_{3})(x_{1}y_{1}^{2} − x_{2}y_{2}^{2})

(x_{1}y_{1}^{2} − x_{2}y_{3}^{2})(x_{1}y_{1}^{3} + x_{2}y_{2}^{2}y_{3})(x_{1}y_{1}^{3} + x_{2}y_{2}y_{3}^{2})(x_{1}^{2}y_{1}^{4} − x_{1}x_{2}y_{1}^{2}y_{2}y_{3} + x_{2}^{2}y_{2}^{2}y_{3}^{2})

(y_{1}^{2} + y_{2}y_{3})(y_{1}^{2} − y_{1}y_{2} + y_{2}^{2})(y_{1}^{2} − y_{1}y_{3} + y_{3}^{2})/x_{2}^{9}y_{2}^{12}y_{3}^{12}

ϕ_{2,3} :(x_{1} − x_{2})(y_{1} − y_{3})(y_{2} − y_{3})(x_{1}y_{1} − x_{2}y_{3})(x_{1}y_{2} − x_{2}y_{3})(x_{1}y_{1} + x_{2}y_{3})

(x_{1}y_{2} + x_{2}y_{3})(x_{1}y_{1} + x_{2}y_{2})(x_{1}y_{2} + x_{2}y_{1})(x_{1}^{2}y_{1}^{2}y_{2}^{2} − x_{1}x_{2}y_{1}y_{2}y_{3}^{2} + x_{2}^{2}y_{3}^{4})

(y_{1}y_{2} + y_{3}^{2})(y_{1}^{2} − y_{1}y_{2} + y_{2}^{2})/x_{2}^{9}y_{1}^{2}y_{2}^{2}y_{3}^{12}

ϕ_{3,1} :(x_{1} − x_{2})(y_{1} − y_{2})(y_{1}^{2} − y_{3}^{2})(y_{2} − y_{3})(y_{1}y_{2} + y_{3}^{2})(y_{1}^{2} + y_{2}y_{3})(y_{1}^{2} − y_{1}y_{3} + y_{3}^{2})

(x_{1}y_{1} + x_{2}y_{3})(x_{1}y_{2} + x_{2}y_{1})(x_{1}y_{1}^{2} − x_{2}y_{2}^{2})(y_{1}^{2}y_{2}x_{1} + y_{3}^{3}x_{2})/x_{1}x_{2}^{4}y_{1}y_{2}^{4}y_{3}^{12}

ϕ_{3,6} :(x_{2}^{3} − x_{1}^{3})(x_{1}y_{1} + x_{2}y_{2})(x_{1}y_{1} + x_{2}y_{3})(x_{1}y_{2} + x_{2}y_{1})(x_{1}y_{2} + x_{2}y_{3})(x_{1}y_{3} + x_{2}y_{1})

(x_{1}y_{3} + x_{2}y_{2})(y_{1}y_{2} + y_{3}^{2})(y_{1}y_{3} + y_{2}^{2})(y_{2}y_{3} + y_{1}^{2})/x_{2}^{9}y_{1}^{4}y_{2}^{4}y_{3}^{4}

ϕ_{6,2} :(x_{2} − x_{1})(y_{1} − y_{2})(y_{1} − y_{3})(y_{2} − y_{3})(y_{1}^{3} + y_{3}^{3})(y_{1}y_{3} + y_{2}^{2})(x_{1}y_{2} − x_{2}y_{3})

(x_{1}y_{3} + x_{2}y_{1})(x_{1}y_{2} + x_{2}y_{3})(x_{1}y_{1}^{3} + x_{2}y_{2}^{2}y_{3})/x_{1}x_{2}^{4}y_{1}^{2}y_{2}^{4}y_{3}^{8}

ϕ_{8,3} :2(y_{1} − y_{2})(y_{1} − y_{3})(x_{1}y_{2} + x_{2}y_{3})(x_{2}y_{2} + x_{1}y_{3})(z_{3}y_{1} + x_{2}y_{3}^{2})(x_{2}y_{1}y_{2} − z_{3}y_{3})

(z_{3}y_{1} + x_{2}y_{2}^{2})(x_{2}y_{1}y_{3} − z_{3}y_{2})(x_{1}y_{1}^{2} − z_{3}y_{1} − x_{2}y_{2}y_{3})

(x_{2}y_{1}^{2} − z_{3}y_{1} − x_{1}y_{2}y_{3})/x_{1}^{3}x_{2}^{5}y_{1}^{2}y_{2}^{7}y_{3}^{7}

ϕ_{9,5} :(ζ_{3}^{2} − 1)(y_{1}^{2} − y_{1}y_{2} + y_{2}^{2})(y_{1}^{2} − y_{1}y_{3} + y_{3}^{2})(y_{2}^{2} − y_{2}y_{3} + y_{3}^{2})(x_{2}y_{1}^{2} + ζ_{3}^{2}x_{1}y_{2}y_{3})

(x_{2}y_{2}^{2} + ζ_{3}^{2}x_{1}y_{1}y_{3})(x_{2}y_{3}^{2} + ζ_{3}^{2}x_{1}y_{1}y_{2})(x_{1} − x_{2})(x_{1} − ζ_{3}x_{2})/x_{1}x_{2}^{4}y_{1}^{4}y_{2}^{4}y_{3}^{4}

The Schur element of ϕ′_{8,6} is the image of the one for ϕ_{8,3} under the non-trivial automorphism of k(x, y, z_{3})/k(x, y), the Schur element of ϕ_{9,7} is the complex conjugate of the one for ϕ_{9,5}.

6B. The complex reflection group G_{25}.

In order to treat the 3-dimensional primitive complex reflection group W_{1} := G_{25} we consider the specialization

x_{1} ↦ 1, x_{2} ↦ −1,

of the generic cyclotomic algebra for G_{26}. By [3, Bem. 4.14] its generic cyclotomic algebra H_{1} := H(G_{25}; y) is a subalgebra of index 2 of the specialization H′ := H(G_{26}; ((1, −1), y)). It was shown in [3, Satz 4.7] that H_{1} satisfies Assumption 2.5. Furthermore, by [15, Table 8.2] (or [3, Bem. 6.5]) the field ℚ(ζ_{3})(y) is a splitting field for H_{1}. So by Tits’ deformation theorem the specialization y_{j} ↦ ζ_{3}^{j} induces a bijection between Irr(H_{1}) and Irr(W_{1}). On the other hand, W is isomorphic to the direct product of W_{1} with the cyclic group of order 2. Since W_{1} has no normal subgroup of order 2 this allows to identify Irr(W_{1}) with the subset of elements of Irr(W) having this direct factor in the kernel. It turns out that these are precisely the characters appearing in the first six columns of Table 6.1.

Having thus given a labeling for the irreducible characters of H(W_{1}) by a subset of Irr(W), we may state the following:

Theorem 6.3. The generic degree d_{ϕ} of an irreducible character ϕ of the cyclotomic Hecke algebra H_{1} = H(G_{25}; y) with respect to the restriction of a symmetric form on H′ = H(G_{26}; y, 1, −1) as in Theorem 6.2 is given by twice the corresponding generic degree for H′.

Proof. This follows from the above observations by an easy descent argument given for example in [12, Lemma 5.11]. □

These Schur elements were already computed in [3, Bem. 5.15], but without giving the labeling by irreducible characters of G_{25}, so we include them once more for the convenience of the reader:

ϕ_{1,0} :(y_{1} − y_{2})^{2}(y_{1} − y_{3})^{2}(y_{1}^{2} + y_{2}^{2})(y_{1}^{2} + y_{3}^{2})(y_{1}^{3} − y_{2}^{2}y_{3})(y_{1}^{3} − y_{2}y_{3}^{2})(y_{1}^{2} + y_{2}y_{3})
(y_{1}^{4} + y_{1}^{2}y_{2}y_{3} + y_{2}^{2}y_{3}^{2})(y_{1}^{2} − y_{1}y_{2} + y_{2}^{2})(y_{1}^{2} − y_{1}y_{3} + y_{3}^{2})/y_{2}^{12}y_{3}^{12}
ϕ_{2,3} :(y_{1} − y_{2})^{2}(y_{1} − y_{3})^{2}(y_{2} − y_{3})^{2}(y_{1} + y_{3})(y_{2} + y_{3})(y_{1}^{2}y_{2}^{2} + y_{1}y_{2}y_{3}^{2} + y_{3}^{4})
(y_{1}y_{2} + y_{3}^{2})(y_{1}^{2} − y_{1}y_{2} + y_{2}^{2})/y_{1}^{2}y_{2}^{2}y_{3}^{12}
ϕ_{3,1} : − (y_{1} − y_{2})^{2}(y_{1} − y_{3})^{2}(y_{2} − y_{3})(y_{1}y_{2} + y_{3}^{2})(y_{1}^{2} + y_{2}y_{3})(y_{1}^{3} + y_{3}^{3})(y_{1}^{2} + y_{2}^{2})
(y_{1}^{2}y_{2} − y_{3}^{3})/y_{1}y_{2}^{4}y_{3}^{12}
ϕ_{3,6} : − (y_{1} − y_{2})^{2}(y_{1} − y_{3})^{2}(y_{2} − y_{3})^{2}(y_{1}y_{2} + y_{3}^{2})(y_{1}y_{3} + y_{2}^{2})(y_{1}^{2} + y_{2}y_{3})/y_{1}^{4}y_{2}^{4}y_{3}^{4}
ϕ_{6,2} :(y_{1} − y_{2})(y_{1} − y_{3})^{2}(y_{2} − y_{3})^{2}(y_{1}^{3} + y_{3}^{3})(y_{1}y_{3} + y_{2}^{2})(y_{2} + y_{3})(y_{1}^{3} − y_{2}^{2}y_{3})/y_{1}^{2}y_{2}^{4}y_{3}^{8}
ϕ_{8,3} :(y_{1} − y_{3})(y_{1} − y_{2})(y_{2} − y_{3})^{2}(y_{2}^{3} − y_{1}^{2}y_{3})(y_{1}^{2}y_{2} − y_{3}^{3})(y_{2}^{2}y_{3}^{2} + y_{1}^{2}y_{2}y_{3} + y_{1}^{4})/y_{1}^{2}y_{2}^{6}y_{3}^{6}
ϕ_{9,5} :(ζ_{3} − ζ_{3}^{2})(y_{1}^{2} − y_{1}y_{2} + y_{2}^{2})(y_{1}^{2} − y_{1}y_{3} + y_{3}^{2})(y_{2}^{2} − y_{2}y_{3} + y_{3}^{2})(y_{1}^{2} − ζ_{3}^{2}y_{2}y_{3})
(y_{2}^{2} − ζ_{3}^{2}y_{1}y_{3})(y_{3}^{2} − ζ_{3}^{2}y_{1}y_{2})/y_{1}^{4}y_{2}^{4}y_{3}^{4}

The Schur element of ϕ_{9,7} is the complex conjugate of the one for ϕ_{9,5}.

6C. The complex reflection group G_{32}.

The 4-dimensional primitive complex reflection group W = G_{32} is generated by four reflections s_{1}, …, s_{4} of order 3, which satisfy the braid relations of the A_{4}- diagram. It has two conjugacy classes of maximal parabolic subgroups, of types G_{25} and G(3, 1, 1) + G_{4}. Out of the 102 conjugacy classes of G_{32}, 32 have a representative in one of these parabolic subgroups. For both of these, a symmetrizing form and the corresponding Schur elements are known by virtue of the previous section and by [3], respectively. Let T_{s_{1}}, …, T_{s_{4}} be the generators of the generic cyclotomic algebra H = H(W, y) of type G_{32} satisfying

(T_{s_{j}} − y_{1})(T_{s_{j}} − y_{2})(T_{s_{j}} − y_{3}) = 0 for j = 1, …, 4 ,

with the parameters y = (y_{1}, y_{2}, y_{3}).

Table 6.4. 𝔖_{3}-action on Irr(G_{32}).

\begin{matrix}(1) & (1, 2, 3) & (1, 3, 2) & (1, 2) & (1, 3) & (2, 3) \\ ϕ_{1,0} & ϕ_{1,40} & ϕ_{1,80} & ϕ_{1,40} & ϕ_{1,80} & ϕ_{1,0} \\ ϕ_{4,1} & ϕ_{4,41} & ϕ_{4,51} & ϕ_{4,21} & ϕ_{4,61} & ϕ_{4,11} \\ ϕ_{5,4} & ϕ_{5,44} & ϕ_{5,36} & ϕ_{5,12} & ϕ_{5,52} & ϕ_{5,20} \\ ϕ_{6,8} & ϕ_{6,48} & ϕ_{6,28} & ϕ_{6,8} & ϕ_{6,48} & ϕ_{6,28} \\ ϕ_{10,2} & ϕ_{10,30} & ϕ_{10,34} & ϕ_{10,14} & ϕ_{10,42} & ϕ_{10,10} \\ ϕ_{15,6} & ϕ_{15,22} & ϕ_{15,38} & ϕ_{15,22} & ϕ_{15,38} & ϕ_{15,6} \\ ϕ_{15,8} & ϕ_{15,24} & ϕ_{15,16} & ϕ_{15,8} & ϕ_{15,24} & ϕ_{15,16} \\ ϕ_{20,3} & ϕ_{20,25} & ϕ′_{20,29} & ϕ_{20,13} & ϕ_{20,35} & ϕ′_{20,9} \\ ϕ_{20,5} & ϕ_{20,33} & ϕ_{20,19} & ϕ_{20,5} & ϕ_{20,33} & ϕ_{20,19} \\ ϕ_{20,7} & ϕ′′_{20,29} & ϕ_{20,21} & ϕ′′_{20,9} & ϕ_{20,31} & ϕ_{20,17} \\ ϕ_{20,12} & ϕ_{20,16} & ϕ_{20,20} & ϕ_{20,16} & ϕ_{20,20} & ϕ_{20,12} \\ ϕ_{24,6} & ϕ_{24,16} & ϕ_{24,26} & ϕ_{24,16} & ϕ_{24,26} & ϕ_{24,6} \\ ϕ_{30,4} & ϕ′′_{30,20} & ϕ_{30,24} & ϕ′_{30,12} & ϕ_{30,28} & ϕ_{30,8} \\ ϕ′′_{30,12} & ϕ_{30,16} & ϕ′_{30,20} & ϕ_{30,16} & ϕ′_{30,20} & ϕ′′_{30,12} \\ ϕ_{36,5} & ϕ_{36,15} & ϕ_{36,25} & ϕ_{36,15} & ϕ_{36,25} & ϕ_{36,5} \\ ϕ_{36,7} & ϕ_{36,17} & ϕ_{36,27} & ϕ_{36,17} & ϕ_{36,27} & ϕ_{36,7} \\ ϕ_{40,8} & ϕ_{40,18} & ϕ_{40,22} & ϕ_{40,14} & ϕ_{40,24} & ϕ_{40,10} \\ ϕ_{45,6} & ϕ_{45,22} & ϕ_{45,14} & ϕ_{45,6} & ϕ_{45,22} & ϕ_{45,14} \\ ϕ_{45,10} & ϕ_{45,26} & ϕ_{45,18} & ϕ_{45,10} & ϕ_{45,26} & ϕ_{45,18} \\ ϕ_{60,7} & ϕ_{60,17} & ϕ′_{60,15} & ϕ_{60,9} & ϕ_{60,19} & ϕ′_{60,11} \\ ϕ′′_{60,11} & ϕ′′_{60,15} & ϕ_{60,13} & ϕ′′_{60,11} & ϕ′′_{60,15} & ϕ_{60,13} \\ ϕ_{60,12} & ϕ_{60,16} & ϕ_{60,20} & ϕ_{60,16} & ϕ_{60,20} & ϕ_{60,12} \\ ϕ_{64,8} & ϕ_{64,18} & ϕ_{64,13} & ϕ_{64,8} & ϕ_{64,18} & ϕ_{64,13} \\ ϕ_{64,11} & ϕ_{64,21} & ϕ_{64,16} & ϕ_{64,11} & ϕ_{64,21} & ϕ_{64,16} \\ ϕ_{80,9} & ϕ_{80,13} & ϕ_{80,17} & ϕ_{80,13} & ϕ_{80,17} & ϕ_{80,9} \end{matrix}

We consider the specialization y_{j} ↦ ζ_{3}^{j−1} of H to the group algebra of W. By [15, Table 8.2] (or [3, 6A]) the field K_{W} := ℚ(ζ_{3})[\sqrt[6]{y_{j}} | 1 ≤ j ≤ 3] is a splitting field for H(W; y). By 2D the symmetry group 𝔖 = 𝔖_{3} of W in its action on the parameters induces an action on Irr(W). As in the case of G_{26} we first give a table showing the action of 𝔖 on Irr(W) induced by the (left) action on {y_{1}, y_{2}, y_{3}}.

The three characters ϕ_{81,10}, ϕ_{81,12}, ϕ_{81,14} are invariant under the 𝔖_{3}-action.

We denote the elements of Irr(W) by ϕ_{d,b}, where d = ϕ_{d,b}(1) and b is the b-value of ϕ_{d,b}. This distinguishes the characters except for six pairs. We denote by ϕ′′_{30,12} and ϕ′′_{60,15} the only two rational characters in these pairs. The other characters in these pairs have the property that their complex conjugate is already determined by its d, b. Interchanging y_{2} with y_{3} induces complex conjugation on Irr(W). So the complex conjugates can be read of from Table 6.4 and thus the notation for the ambiguous pairs is fixed.

We denote by c := s_{1}s_{2}s_{3}s_{4} the Coxeter element of the braid group B(W) of W. It satisfies c^{30} = π and its fifth power is central in B(W). Let w := s_{1}^{2}s_{2}s_{3}s_{4} = s_{1}c. By direct computation with the defining relations of B(W) we find that w^{4} = c^{5}, hence w is a 24-th root of π in the braid group.

Theorem 6.5. The Schur elements of the cyclotomic Hecke algebra H(G_{32}; y) with respect to any quasi-symmetric basis only containing T_{c}^{l} (1 ≤ l ≤ 29), T_{w}^{l} (1 ≤ l ≤ 23) and elements of the form T_{v}T_{c}^{5l} (1 ≤ l ≤ 5) for v running through representatives of minimal length of the conjugacy classes of the maximal parabolic subgroups of B(W) are given in the table below.

Proof. The character values of H on the powers of T_{c} and of T_{w} can be evaluated with Proposition 2.8. Moreover, character values on all elements of proper parabolic subalgebras are known by the preceding section, respectively by [3]. Since T_{c}^{5l} is central in H, the values on any product T_{v}T_{c}^{5l} can be determined, once those on T_{v} are known. Thus the values of all irreducible characters on all the elements occurring in the statement can be computed. Using the computer algebra system GAP it was checked that this system has maximal rank, and thus determines the generic degrees uniquely. □

ϕ_{1,0} :(y^{2}z^{2} + yzx^{2} + x^{4})(y^{2} − yx + x^{2})(z^{2} − zx + x^{2})(y^{2}z − x^{3})(yz^{2} − x^{3})
(y^{4}z^{4} + y^{3}z^{3}x^{2} + y^{2}z^{2}x^{4} + yzx^{6} + x^{8})(y^{4} − y^{3}x + y^{2}x^{2} − yx^{3} + x^{4})
(z^{4} − z^{3}x + z^{2}x^{2} − zx^{3} + x^{4})(y^{3}z + x^{4})(yz^{3} + x^{4})(y^{4}z^{2} − y^{2}zx^{3} + x^{6})

(y^{2}z^{4} − yz^{2}x^{3} + x^{6})(y^{2}z^{3} − x^{5})(y^{3}z^{2} − x^{5})(y^{2}z^{2} + x^{4})(y^{3}z^{3} + x^{6})(y^{2} + x^{2})

(z^{2} + x^{2})(yz + x^{2})(y − x)^{2}(z − x)^{2}/y^{40}z^{40}
ϕ_{4,1} :(x^{8} − x^{7}z + z^{3}x^{5} − x^{4}z^{4} + z^{5}x^{3} − z^{7}x + z^{8})(y^{4} − y^{3}x + y^{2}x^{2} − yx^{3} + x^{4})

(z^{2}y^{2} + zyx^{2} + x^{4})(x^{3} + z^{3})(y^{2} − yx + x^{2})(z^{4}y + x^{5})(yx^{3} + z^{4})(z^{3}y − x^{4})

(z^{2}y − x^{3})(zy^{2} − x^{3})(yx^{2} − z^{3})(z^{2}y + x^{3})(x^{2} + z^{2})(yz + x^{2})(xy + z^{2})
(x − z)^{2}(x − y)(y − z)/xy^{15}z^{40}

ϕ_{5,4} :(y^{2}x^{2} + yxz^{2} + z^{4})(z^{4} − z^{2}x^{2} + x^{4})(x^{4}y + z^{5})(y^{2}x^{3} − z^{5})(z^{2} − zx + x^{2})^{2}

(y^{2} − yx + x^{2})(yx^{2} − z^{3})(z^{2}y + x^{3})(y^{2}x + z^{3})(y^{2} + x^{2})(zy + x^{2})(yx + z^{2})

(z − x)^{2}(y − x)^{2}(z − y)^{2}(z + y)(z + x)^{2}/x^{2}y^{6}z^{40}

ϕ_{6,8} :(x^{4}y^{4} + x^{3}y^{3}z^{2} + x^{2}y^{2}z^{4} + xyz^{6} + z^{8})(x^{4} − x^{3}y + x^{2}y^{2} − xy^{3} + y^{4})

(x^{3}y^{3} + z^{6})(z^{2} − zy + y^{2})(z^{2} − zx + x^{2})(z^{3} − x^{2}y)(xy^{2} − z^{3})(x^{2}y + z^{3})

(xy^{2} + z^{3})(zx + y^{2})(zy + x^{2})(xy + z^{2})(z − y)^{2}(x − y)^{2}(z − x)^{2}(z + x)

(z + y)/x^{4}y^{4}z^{40}

ϕ_{10,2} : − (x^{4}y^{2} − x^{2}yz^{3} + z^{6})(y^{2}z^{2} + yzx^{2} + x^{4})(x^{2} − xz + z^{2})(x^{2} − xy + y^{2})

(y^{3}z + x^{4})(x^{2}y − z^{3})(yz^{2} − x^{3})(yz^{2} + x^{3})(x^{2} + z^{2})(xy + z^{2})

(xz + y^{2})(yz + x^{2})(x − z)^{3}(y − z)^{2}(x − y)(x + z)^{2}(y + z)/x^{2}y^{12}z^{30}

ϕ_{15,6} :(y^{2}z^{2} − yzx^{2} + x^{4})(y^{2}z^{2} + x^{4})(x^{2} − xz + z^{2})(x^{2} − xy + y^{2})(x^{3}y + z^{4})

(x^{3}z + y^{4})(x^{3} − y^{2}z)(yz^{2} − x^{3})(xy + z^{2})(xz + y^{2})(yz + x^{2})^{2}(x − z)^{2}

(y − z)^{2}(x − y)^{2}(x + y)(x + z)/x^{2}y^{20}z^{20}

ϕ_{15,8} :(x^{2}y^{2} + z^{4})(z^{2} − zy + y^{2})(z^{2} − zx + x^{2})(zx^{2} − y^{3})(x^{3} − y^{2}z)(zx + y^{2})

(zy + x^{2})(xy + z^{2})^{2}(z − y)^{2}(x − y)^{2}(z − x)^{2}(z + x)^{2}(z + y)^{2}/x^{6}y^{6}z^{20}

ϕ_{20,3} :(y^{4}z^{2} − y^{2}zx^{3} + x^{6})(x^{2}y^{2} + xyz^{2} + z^{4})(z^{2} − zx + x^{2})(z^{3}y − x^{4})(zx^{2} − y^{3})

(yz^{2} − x^{3})(xy^{2} + z^{3})(z^{2} + x^{2})(zx + y^{2})(zy + x^{2})(xy + z^{2})(z − y)(x − y)^{2}

(z − x)^{2}(z + x)^{2}/x^{3}y^{12}z^{25}

ϕ_{20,5} :(y^{2}x^{2} + xyz^{2} + z^{4})(y^{2}x^{2} − xyz^{2} + z^{4})(y^{2} − yx + x^{2})(z^{2} − zx + x^{2})

(y^{2} − yz + z^{2})(x^{3} − y^{2}z)(x^{2}y − z^{3})(xy^{2} − z^{3})(zx^{2} − y^{3})(yz^{2} + x^{3})

(z^{2}x + y^{3})(yx + z^{2})^{2}(z − x)^{2}(y − z)^{2}/x^{5}y^{5}z^{30}

ϕ_{20,7} : − (y^{2}z^{2} + yzx^{2} + x^{4})(x^{3} + z^{3})^{2}(y^{2} − yz + z^{2})(x^{3}z + y^{4})(x^{3}y − z^{4})

(xy^{2} − z^{3})(xy^{2} + z^{3})(xy + z^{2})(xz + y^{2})(x − y)^{2}(x − z)^{2}(y − z)^{2}/x^{5}y^{6}z^{25}

ϕ_{20,12} :2 (z^{2}y^{2} + yzx^{2} + x^{4})(z^{2} − zy + y^{2})^{2}(z^{2}x + y^{3})(xy^{2} + z^{3})(zy + x^{2})^{2}(y − z)^{2}

(x − y)^{3}(z − x)^{3}(x + z)^{2}(x + y)^{2}/x^{6}y^{12}z^{12}

ϕ_{24,6} :(y^{4}z^{4} + y^{3}z^{3}x^{2} + y^{2}z^{2}x^{4} + yzx^{6} + x^{8})(x^{3} + z^{3})(x^{3} + y^{3})(x^{2}y − z^{3})(y^{3} − zx^{2})

(x^{2} + z^{2})(x^{2} + y^{2})(xy + z^{2})(xz + y^{2})(x − y)^{2}(x − z)^{2}(y − z)^{2}/x^{4}y^{15}z^{15}

ϕ_{30,4} :(x^{5} − z^{2}y^{3})(x^{3} + y^{3})(y^{3} + z^{3})(x^{3} + z^{3})(yz^{4} + x^{5})(xz^{2} + y^{3})(x^{4} − z^{4})

(xz + y^{2})(zy + x^{2})(x − y)^{2}(y − z)^{2}(x − z)^{2}/x^{4}y^{12}z^{20}

ϕ′_{30,12} :(y^{3} + z^{3})^{2}(x^{2} − xy + y^{2})(z^{2} − zx + x^{2})(zx^{2} − y^{3})(x^{2}y − z^{3})(zx + y^{2})

(xy + z^{2})(x − y)^{3}(x − z)^{3}(x + z)^{2}(x + y)^{2}/x^{6}y^{12}z^{12}

ϕ_{36,5} :(ζ_{3}^{2} − 1)(ζ_{3}z^{3} + x^{2}y)(ζ_{3}y^{3} + x^{2}z)(xy − ζ_{3}z^{2})(xz − ζ_{3}y^{2})(x^{2} − ζ_{3}^{2}yz)(x − y)

(ζ_{3}^{2}y^{4} + xy^{3} + ζ_{3}x^{2}y^{2} + ζ_{3}^{2}x^{3}y + x^{4})(ζ_{3}^{2}z^{4} + xz^{3} + ζ_{3}x^{2}z^{2} + ζ_{3}^{2}x^{3}z + x^{4})

(x^{6} + y^{3}z^{3})(x^{2} − xz + z^{2})(y^{2} − yz + z^{2})(x^{2} − xy + y^{2})(x − z)/x^{4}y^{15}z^{15}

ϕ_{40,8} :(x^{2}z^{2} + xzy^{2} + y^{4})(y^{2}x^{3} − z^{5})(x^{3} + y^{3})(z^{2} − zy + y^{2})(zy^{2} − x^{3})(x^{2} + y^{2})

(x^{2} + z^{2})(xy + z^{2})(zy + x^{2})(x − y)^{2}(z − y)(x − z)^{3}(x + z)/x^{5}y^{12}z^{15}

ϕ_{45,6} :(1 − ζ_{3}^{2})(ζ_{3}x^{3} + y^{2}z)(ζ_{3}y^{3} + zx^{2})(−ζ_{3}^{2}y^{2} + zx)(−ζ_{3}^{2}x^{2} + zy)(ζ_{3}xy − z^{2})

(ζ_{3}xy + z^{2})(ζ_{3}^{2}xy + z^{2})(ζ_{3}^{2}y^{2} + z^{2})(ζ_{3}^{2}x^{2} + z^{2})(ζ_{3}y + z)^{2}(ζ_{3}^{2}y + z)

(x^{2} − xy + y^{2})(ζ_{3}x + z)^{2}(ζ_{3}^{2}x + z)(xy + z^{2})(z − y)(z − x)/x^{6}y^{6}z^{20}

ϕ_{60,7} :(x^{3} + y^{3})(z^{2} − zy + y^{2})(yx^{4} + z^{5})(y^{2}x − z^{3})(z^{2}y − x^{3})(x^{2} + y^{2})(xz + y^{2})

(x^{2} + yz)(x − y)^{2}(y^{2} − z^{2})^{2}(x − z)^{2}/x^{6}y^{9}z^{15}

ϕ′′_{60,11} :(y^{2}x^{2} − xyz^{2} + z^{4})(y^{2} − yx + x^{2})(yz^{3} − x^{4})(z^{3}x − y^{4})(yz + x^{2})(yx + z^{2})^{2}

(zx + y^{2})(x^{2} − y^{2})^{2}(x − z)^{2}(y − z)^{2}/x^{9}y^{9}z^{12}

ϕ_{60,12} :2 (y^{2}z^{2} − yzx^{2} + x^{4})(y^{3} + z^{3})^{2}(xy^{2} − z^{3})(z^{2}x − y^{3})(x^{2} + z^{2})(x^{2} + y^{2})

(yz + x^{2})^{2}(x − z)^{3}(x − y)^{3}/x^{6}y^{12}z^{12}

ϕ_{64,8} :2 (x^{2}y^{2} − rxyz + xyz^{2} − rz^{3} + z^{4})(x^{2}y^{2} + xyz^{2} + z^{4})(zx^{2} + ry^{2})(rx^{2} + zy^{2})

(y^{4} − ry^{3} + rxy^{2} − x^{2}y^{2} + rx^{2}y − rx^{3} + x^{4})(xr − z^{2})(ry − z^{2})(y^{3} − z^{2}x)

(x^{3} − yz^{2})(y − z)(x − y)^{2}(x − z)/rx^{8}y^{8}z^{15}

ϕ_{80,9} :2 (y^{2}z^{2} + yzx^{2} + x^{4})(y^{4} − y^{2}z^{2} + z^{4})(y^{2}z^{2} + x^{4})(xy^{2} − z^{3})(z^{2}x − y^{3})

(x^{2} + z^{2})(x^{2} + y^{2})(x − z)^{2}(y − z)^{2}(x − y)^{2}(x + y)(x + z)/x^{6}y^{12}z^{12}

ϕ_{81,10} :3 (sx^{2} + syz + s^{2}x + yzx + x^{3})(sz^{2} + sxy + s^{2}z + yzx + z^{3})

(sy^{2} + sxz + s^{2}y + yzx + y^{3})(xy + sz)(xz + sy)(yz + sx)(sz + x^{2})

(sz + y^{2})(sx + z^{2})(sy + x^{2})(sx + y^{2})(sy + z^{2})(xy + z^{2})(xz + y^{2})

(x^{2} + yz)(s + z)(s + y)(s + x)/x^{12}y^{12}z^{12}

In the table, we do not print the Schur elements for ϕ_{36,7}, ϕ_{45,10}, since they are the complex conjugates of those for ϕ_{36,5}, ϕ_{45,6}, respectively. In order to avoid indices, we write (x, y, z) instead of (y_{1}, y_{2}, y_{3}). Finally, r, s denote elements of K_{W} with r^{2} = xy and s^{3} = xyz.

7. d-Howlett-Lehrer-Lusztig theories for unipotent blocks

It was first noted by Springer that non-real reflection groups naturally appear inside Weyl groups as what is now called relative Weyl groups, that is, normalizers modulo centralizers of subspaces in the natural reflection representation. By the standard dictionary between Weyl groups and finite reductive groups these relative Weyl groups can also be seen as normalizer modulo centralizer of (so-called d-split) Levi subgroups in groups of Lie type. The importance of this construction was revealed in the work of Broué, Michel and the author on the ℓ-blocks of characters of finite groups of Lie type. There it was shown that the unipotent characters inside an ℓ-block are parametrized by the irreducible characters of a relative Weyl group, which in general is a non-real reflection group.

A possible conceptual interpretation of this result was subsequently proposed by Broué and the author [3]. Namely, it was conjectured there that the cyclotomic Hecke algebras attached to relative Weyl groups govern the decomposition of Lusztiginduced d-cuspidal characters. If this does in fact hold, it implies that the degrees of the constituents of those Lusztig induced characters can be expressed in terms of the Schur elements of the corresponding cyclotomic algebras.

This latter fact was already verified in [12, Folg. 3.16 and 6.11] for all relative Weyl groups which are imprimitive in their natural reflection representation, in [13, Prop. 5.2] for the 2-dimensional ones, and in [3, Folg. 5.16] for G_{25}. With the Schur elements computed above, we can check the remaining cases and thus obtain (we refer to [3] or [13] for the notation):

Proposition 7.1. Let G be a group of Lie type, E(G, (L, λ)) a Φ_{d}-block of unipotent characters. Then the degrees of the unipotent characters γ ∈ E(G, (L, λ)) satisfy conjecture (d-HV6) in [3] with respect to the values of parameters given in [3, 2B and Table 8.1].

Proof. By the references cited above, it only remains to consider the cases

(G, d) ∈ {(E_{7}, 3), (E_{7}, 6), (E_{8}, 4), (E_{8}, 3), (E_{8}, 6)},

with relative Weyl groups G_{26}, G_{26}, G_{31}, G_{32}, G_{32} respectively. The Schur elements for the cyclotomic Hecke algebras of these latter groups have all been determined above, and the result follows by comparing the specialized Schur elements with the unipotent character degrees. (This was done explicitly in Theorem 5.4 for the case (E_{8}, 4).) □

8. Spetsial reflection groups

In this last section we observe some general properties of the Schur elements of complex reflection groups, using the explicit results derived here, respectively in [7] and [13]. These will bring to light an important property shared by certain finite complex reflection groups which seems to lie at the heart of the existence of so-called unipotent degrees (see for example [12] and Section 8.3). This result was announced in the ICM-report [14].

8A. A property of some complex reflection groups.

Let W be a finite irreducible complex reflection group defined over k, ℤ_{k} the ring of integers of k, H(W, x) the 1-parameter cyclotomic algebra (2.4) over ℤ[x, x^{−1}], k(y) with y^{|μ(k)|} = x a splitting field for H(W, x) (see 2A).

We now associate a certain rational function c_{ϕ} in y to each irreducible character of W. If W is a real reflection group, c_{ϕ} denotes the Schur element of ϕ with respect to the canonical symmetrizing form on the Iwahori-Hecke algebra of W (see [8, 9.4]). Similarly, if W = G(m, p, n), c_{ϕ} denotes the Schur element of ϕ with respect to the symmetrizing form on H(W, x) constructed by Bremke and the author. According to [7], Cor. 1.5, and [12], Satz 5.13, it is given as follows. The irreducible characters of W are indexed by so-called m-symbols S. For each symbol S let D_{S} be the rational function defined in [12, (5.12)]. We let c_{ϕ} = 1/D_{S}(x; w_{0}, …, w_{m−1}) if ϕ is indexed by S, where the w_{j} are given by

w_{j} = \{ \begin{matrix}x^{\frac{1}{p} −1} & if j = 0, \\ x^{\frac{1}{p}} ζ_{m}^{j} & if t|j, j ≠ 0, \\ ζ_{m}^{j} & else, \end{matrix}

with ζ_{m} := exp(2πi/m). If W is a 2-dimensional exceptional reflection group, we let c_{ϕ} be the Schur element corresponding to ϕ computed in [13]. Finally, if W is a nonreal exceptional reflection group in dimension at least 3, then c_{ϕ} is the 1-parameter specialization of the Schur element computed in the previous sections.

Thus, at least conjecturally, the c_{ϕ} are the Schur elements with respect to a nice (natural) symmetric form on the 1-parameter cyclotomic algebra H(W, x) of W.

For ϕ ∈ Irr(W) we now call δ_{ϕ} := P (W)/c_{ϕ} the generic degree of ϕ. It is a rational function in y. We write a(ϕ) for |μ(k)|^{−1} times the order of zero of δ_{ϕ} at y = 0, and b(ϕ) for the order of zero of the fake degree R_{ϕ} at x = 0. Following the notation of Lusztig [9] for the real case, a character ϕ with a(ϕ) = b(ϕ) is called special.

The explicit formulae for the generic degrees now allow to observe the following:

Proposition 8.1. Let W be a finite irreducible complex reflection group. Then the following are equivalent:

(i) for all ϕ ∈ Irr(W) there exists a special ψ ∈ Irr(W) with a(ϕ) = a(ψ); (ii) a(ϕ) ≤ b(ϕ) for all ϕ ∈ Irr(W); (iii) (rationality) δ_{ϕ} ∈ k(x) for all ϕ ∈ Irr(W); (iv) (integrality) δ_{ϕ} ∈ k[y] for all ϕ ∈ Irr(W);

(v) δ_{1} = 1 (i.e. c_{1} = P (W));

(vi) (representability) the k-subspaces ⟨δ_{ϕ} | ϕ⟩ and ⟨R_{ϕ} | ϕ⟩ of k(y) coincide.

Proof. First note that the fake degrees R_{ϕ} lie in ℤ[x], so statement (vi) implies (iii) and (iv). For the other implications, the proof is case by case. If W is a real reflection group, then it is known that all statements (i)–(vi) hold. For example, they can be checked from the explicitly known values of the c_{ϕ}. In the case of Weyl groups, it is possible to give general proofs for some of the statements by using the fact that the generic degrees specialize to the degrees of actual characters of groups of Lie type with this Weyl group. Properties (i) and (v) for all real types but H_{4} are contained in [9, 12.d) and e)], property (ii) in [9, (2.1)], (iii) and (iv) in [2, Th. 2.6], and (vi) in [10, (4.24)]. The case of H_{4} is treated in [1].

For the monomial groups G(m, 1, n), (i) and (ii) hold by [12, Bem. 2.24], (iii) by definition of the c_{ϕ} in this case (see [12, (2.21)]), (iv) by [12, Folg. 3.17], (v) by inspection and (vi) by [12, Satz 4.17]. For the imprimitive groups G(m, m, n), (i) and (ii) are shown in [12, Lemma 5.16], (iii) follows from [12, (5.14)], (iv) from [12, Folg. 6.12], (v) again by inspection and (vi) from [12, Satz 6.26].

Now assume that W = G(m, p, n) with n ≥ 2, p ≠ 1, m. We claim that none of the statements (i)–(vi) holds for W. For this we compute the generic degrees of the trivial character and of the reflection character. The trivial character is parametrized by the m-tuple of partitions (n, −, …, −). The formula in [12] yields

δ_{1} = \frac{x^{nt} − 1}{x^{n} − 1} ∏_{k=1}^{n} \frac{x^{(k−1)p+1} − 1}{x^{(k−1)m+t} − 1} ∏_{k=1}^{n−1} \frac{x^{mk} − 1}{x^{pk} − 1} ,

where t := m/p. For p ≠ 1, m the factor x^{(n−1)m+t} − 1 in the denominator does not cancel, hence δ_{1} is not integral and (iv),(v),(vi) fail. The reflection character ρ is parametrized by the multi-partition (n − 1, 1, −, …, −). Here, for p ≠ m we find

δ_{ρ} = \frac{x^{2− \frac{1}{p}} (x^{n} − 1)(x^{m(n−1)+t} − 1)(x^{p(n−1)} − 1)(x^{n−2+ \frac{1}{p}} − ζ)(x^{\frac{1}{p}} − ζ)}{(x^{n−1} − 1)(x^{p(n−1)+1} − 1)(x − 1)(x^{n−1+ \frac{1}{p}} − ζ)(x^{\frac{p−1}{p}} − ζ^{−1})(x − ζ^{p})} δ_{1} ,

with ζ = exp(2πi/m). Thus a(ρ) = 2 − \frac{1}{p} is not integral, and (i), (iii) cannot hold. Moreover, by definition the reflection character occurs in the first symmetric power of the natural representation of W, so b(ρ) = 1 < a(ρ), and (ii) is violated.

The generic degrees of the 2-dimensional primitive reflection groups were computed in [13]. From these, it can be checked that (i)–(vi) hold for the groups G_{4}, G_{6}, G_{8}, G_{14}.

For the remaining 2-dimensional groups, the explicit formulae in [13] show that δ_{1} is not integral, so (iv), (v) and (vi) fail to hold. Also, in all cases there exists a non-rational degree, so (iii) is violated. For all groups except G_{12} and G_{22} there even exists some ϕ with non-integral a(ϕ), which contradicts (i). For G_{12}, the reflection character ρ has a(ρ) = 2, but there is no ψ with b(ψ) = 2. Thus (i) and (ii) do not hold. The group G_{22} has a 3-dimensional character ϕ with a(ϕ) = 1, but a(ρ) = 3, so there is no special character ψ with a(ψ) = 1, again contradicting (i) and (ii). It can be checked that a(ρ) > b(ρ) for G_{7}, G_{11}, G_{13}, G_{15} and G_{19}. For the remaining groups, condition (ii) is violated for some other irreducible character.

It remains to consider the exceptional, non-real groups of dimension at least 3. The results of the previous sections show that for G_{31}, none of the statements holds, while the other groups satisfy (i)–(vi). □

A reflection group all of whose irreducible components satisfy the above (very special) equivalent conditions will be called spetsial. From the proof of Proposition 8.1 we see that the irreducible spetsial groups are

𝔖_{n}, G(m, 1, n), G(m, m, n), G_{i} with i ∈ {4, 6, 8, 14, 23, …, 30, 32, …, 37}.

With this list it is straightforward to verify that spetsiality is in a certain sense a local property:

Proposition 8.2. A reflection group is spetsial if and only if all of its 2-dimensional parabolic subgroups are spetsial.

Since taking parabolic subgroups is transitive, we obtain the immediate consequence that all parabolic subgroups of a spetsial reflection group are spetsial.

8B. Special characters of spetsial groups.

The explicit formulae for the generic degrees obtained in the previous sections or given in the above-mentioned references allow to compile the list of all special characters of irreducible spetsial reflection groups W. For real W, the special characters were listed in [9] and [1]. For W = G(m, 1, n) the special characters are described in [12, Bem. 2.24] in terms of the associated symbols, for W = G(m, m, n) in [12, Lemma 5.16]. For the exceptional spetsial reflection groups, the special characters can be read off from the explicit data in [13] and in the previous sections. Thus we obtain the following lists: G_{4}: ϕ_{1,0}, ϕ_{2,1}, ϕ_{3,2}, ϕ_{1,4}. G_{6}: ϕ_{1,0}, ϕ_{2,1}, ϕ_{3,4}, ϕ_{2,5,13}, ϕ_{1,10}. G_{8}: ϕ_{1,0}, ϕ_{2,1}, ϕ_{3,2}, ϕ_{4,3}, ϕ_{1,6}. G_{14}: ϕ_{1,0}, ϕ_{2,1}, ϕ_{4,5}, ϕ_{3,6}, ϕ_{2,9}, ϕ_{1,20}. G_{24}: ϕ_{1,0}, ϕ_{3,1}, ϕ_{7,3}, ϕ_{8,4}, ϕ_{7,6}, ϕ_{3,8}, ϕ_{1,21}. G_{25}: ϕ_{1,0}, ϕ_{3,1}, ϕ_{6,2}, ϕ′′_{6,4}, ϕ_{8,6}, ϕ′_{6,8}, ϕ_{1,12}. G_{26}: ϕ_{1,0}, ϕ_{3,1}, ϕ_{6,2}, ϕ_{8,3}, ϕ′_{6,4}, ϕ_{6,5}, ϕ′′_{8,6}, ϕ′′_{6,11}, ϕ′′_{3,16}, ϕ_{1,21}. G_{27}: ϕ_{1,0}, ϕ_{3,1}, ϕ_{10,3}, ϕ_{9,4}, ϕ_{15,5}, ϕ_{8,6}, ϕ_{15,8}, ϕ_{9,9}, ϕ_{10,12}, ϕ_{3,16}, ϕ_{1,45}. G_{29}: ϕ_{1,0}, ϕ_{4,1}, ϕ_{10,2}, ϕ_{16,3}, ϕ′_{15,4}, ϕ′′_{15,4}, ϕ_{20,5}, ϕ_{24,6}, ϕ_{20,9}, ϕ′′_{15,12}, ϕ′_{15,12}, ϕ_{16,13},

ϕ_{10,18}, ϕ_{4,21}, ϕ_{1,40}. G_{32}: ϕ_{1,0}, ϕ_{4,1}, ϕ_{10,2}, ϕ_{20,3}, ϕ_{30,4}, ϕ_{20,5}, ϕ_{45,6}, ϕ_{64,8}, ϕ_{60,9}, ϕ_{81,10}, ϕ_{30,12}, ϕ_{36,15},

ϕ_{30,20}, ϕ_{20,25}, ϕ_{10,30}, ϕ_{1,40}. G_{33}: ϕ_{1,0}, ϕ_{5,1}, ϕ_{15,2}, ϕ_{30,3}, ϕ_{30,4}, ϕ_{81,6}, ϕ_{60,7}, ϕ_{45,7}, ϕ_{64,8}, ϕ_{15,9}, ϕ_{45,10}, ϕ_{81,11},

ϕ_{60,10}, ϕ_{15,12}, ϕ_{30,13}, ϕ_{30,18}, ϕ_{15,23}, ϕ_{5,28}, ϕ_{1,45}. G_{34}: ϕ_{1,0}, ϕ_{6,1}, ϕ_{21,2}, ϕ_{56,3}, ϕ_{105,4}, ϕ_{126,5}, ϕ_{315,6}, ϕ_{420,7}, ϕ_{384,8}, ϕ′_{70,9}, ϕ_{560,9}, ϕ_{315,10},

ϕ_{729,10}, ϕ_{630,11}, ϕ_{840,11}, ϕ_{896,12}, ϕ_{210,12}, ϕ′_{840,13}, ϕ_{630,14}, ϕ_{1280,15}, ϕ′_{560,18}, ϕ_{630,20}, ϕ_{840,19}, ϕ_{210,30}, ϕ_{896,21}, ϕ′′_{840,23}, ϕ_{630,23}, ϕ_{729,24}, ϕ_{315,28}, ϕ_{560,27}, ϕ′_{70,45}, ϕ_{384,29}, ϕ_{420,31}, ϕ_{315,36}, ϕ_{126,41}, ϕ_{105,46}, ϕ_{56,57}, ϕ_{21,68}, ϕ_{6,85}, ϕ_{1,126}.

Recall that an n-dimensional irreducible finite complex reflection group W is called well-generated if it can be generated by n of its reflections. The proof of Proposition 8.1, together with results from [15], immediately yields the following characterization:

Corollary 8.3. Let W be irreducible. Then the following are equivalent: (i) W is well-generated; (ii) the reflection character ρ of W is special; (iii) the generic degree δ_{ρ} is rational.

Proof. We proceed case by case. The groups G(m, p, n), n ≥ 2, p ≠ 1, m, are not well-generated. In the proof of Proposition 8.1 we saw that a(ρ) = 2 − 1/p, so neither is ρ special nor δ_{ρ} rational. For the remaining non-well-generated groups G_{i}, i ∈ {7, 11, 12, 13, 15, 19, 22, 31} the assertion has either been verified in the course of proving Proposition 8.1, or it is a consequence of the explicit formulae for δ_{ρ} in Theorem 5.4 or in [13].

Now assume that W is well-generated. If W is moreover spetsial, then δ_{ρ} is rational by Proposition 8.1. The a-value can be checked from the lists of Schur elements determined here or in [12]. All non-spetsial well-generated groups are 2-dimensional, and for them the result follows from the formulae in loc. cit. □

8C. Unipotent degrees and spetses.

We end by giving some motivation for the study of spetsial reflection groups. In the course of his classification of the irreducible characters of finite reductive groups G(q) = G^{F} Lusztig [10] determined the important class of unipotent characters and showed that they can be indexed by a set which only depends on the Weyl group W of the associated algebraic group G, together with the action of the Frobenius endomorphism F on it. Moreover the degrees of these unipotent characters can naturally be written as polynomials with rational coefficients in the size q of the underlying field. We call this the (multi-) set of unipotent degrees attached to (W, F). Many properties of unipotent characters are already reflected by combinatorial properties of the unipotent degrees, for example the distribution into Harish-Chandra series, and the distribution into families (which restricted to principal series characters are just the Kazhdan-Lusztig cells).

In the course of this classification Lusztig observed that similar sets can formally also be attached to those finite real reflection groups W which are not Weyl groups [11]. We showed in [12] that in fact such sets of unipotent degrees exist for all imprimitive spetsial complex reflection groups, and in [4] corresponding sets are constructed for the primitive spetsial groups (using the results of the present paper).

More precisely we obtain sets E(W) together with a degree map

Deg : E(W) → \frac{1}{|W |}O_{K}[x],

where O_{K} denotes the ring of integers of the character field K of W (the field of definition of the reflection representation of W). The Galois group Gal(K/ℚ) acts on E(W) such that Deg is equivariant. The set E(W) falls into Φ-Harish-Chandra series for each cyclotomic polynomial Φ over K, and into families. The Φ-Harish-Chandra series are completely described by the Schur elements of cyclotomic Hecke algebras attached to certain relative Weyl groups. For Φ = x − 1 and W a Weyl group, these are just the ordinary Harish-Chandra series of unipotent characters. Each family contains a unique special character, and the degrees in the family are connected to the corresponding fake degrees by a Fourier transform matrix. To each element of E(W) is attached a root of unity (called Frobenius eigenvalue), such that for each family the diagonal matrix of Frobenius eigenvalues together with the Fourier matrix give a representation of SL_{2}(ℤ). For details and references we refer to [12] and [14]. It turns out that all (equivalent) conditions in Proposition 8.1 are necessary for the existence of such sets of unipotent degrees, so that we can state:

Theorem 8.4. A finite complex reflection group has unipotent degrees if and only if it is spetsial.

Thus in a certain sense spetsial complex reflection groups behave just as if they were the Weyl groups of an algebraic group. It is tempting to speculate about an underlying algebraic structure, baptized ‘spets‘ in [14], giving rise to the unipotent degrees attached to complex reflection groups. We don’t yet know what these spetses should be, but a lot of intriguing evidence for their existence has been collected.

References

[1] D. Alvis and G. Lusztig, The representations and generic degrees of the Hecke algebra of type H_{4}, J. reine angew. Math. 336 (1982), 201–212; correction, ibid. 449 (1994), 217–218.

[2] C. T. Benson and C. W. Curtis, On the degrees and rationality of certain characters of finite Chevalley groups, Trans. Amer. Math. Soc. 165 (1972), 251–273; corrections and additions, ibid. 202 (1975), 405–406.

[3] M. Broué and G. Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189.

[4] M. Broué, G. Malle and J. Michel, Towards spetses II, in preparation.

[5] M. Broué, G. Malle and R. Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. reine angew. Math. 500 (1998), 127–190.

[6] M. Broué and J. Michel, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Finite Reductive Groups : Related Structures and Representations (M. Cabanes, ed.), Progress in Mathematics, vol. 141, Birkhäuser, 1997, pp. 73–140.

[7] M. Geck, L. Iancu and G. Malle, Weights of Markov traces and generic degrees, Indag. Mathem. (2000) (to appear).

[8] M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Oxford University Press, Oxford, 2000.

[9] G. Lusztig, A class of irreducible representations of a Weyl group, Indag. Mathem. 41 (1979), 323–335; II, ibid. 44 (1982), 490–498.

[10] , Characters of reductive groups over a finite field, Ann. Math. Studies, vol. 107, Princeton Univ. Press, Princeton, 1984.

[11] , Coxeter groups and unipotent representations, Astérisque 212 (1993), 191–203.

[12] G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 (1995), 768–826.

[13] , Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, Finite Reductive Groups : Related Structures and Representations (M. Cabanes, ed.), Progress in Mathematics, vol. 141, Birkhäuser, 1997, pp. 311–332.

[14] , Spetses, Doc. Math. J. DMV Extra Volume ICM II (1998), 87–96.

[15] , On the rationality and fake degrees of characters of cyclotomic algebras, J. Math. Sci. Univ. Tokyo 6 (1999), 647–677.

[16] G. Malle and A. Mathas, Symmetric cyclotomic Hecke algebras, J. Algebra 205 (1998), 275–293.

FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany

   E-mail address: malle@mathematik.uni-kassel.de