SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS

Gunter Malle and Andrew Mathas

Universität Heidelberg and Universität Bielefeld

In this paper we prove that the generic cyclotomic Hecke algebras for imprimitive complex reflection groups are symmetric over any ring containing inverses of the parameters. For this we show that the determinant of the Gram matrix of a certain canonical symmetrizing form introduced in [3] is a unit in any such ring. On the way we show that the Ariki-Koike bases of these algebras are also quasi-symmetric.

0. Introduction and statement of the main result

Work of Broué, Michel and the first author on ℓ-blocks in finite groups of Lie type suggest that the decomposition of certain Lusztig induced characters should be governed by suitable complex reflection groups. This lead Broué and the first author in [4] to introduce the concept of the cyclotomic Hecke algebra H(W, u) associated to a complex reflection group W. Conjecturally specializations of this algebra should occur as endomorphism algebras of Lusztig induced characters, thus explaining the aforementioned observation (see [4]).

This general conjecture has been proved only in a small number of cases. Nevertheless, several structural and numerical consequences of this conjecture have been verified. One of the properties of cyclotomic algebras implied by the general conjecture is that H should carry a canonical symmetrizing form making it into a symmetric algebra over the ground ring A in which just the parameters are inverted. Such a canonical symmetrizing form over the field of fractions of A was constructed in [3] for all but finitely many of the cyclotomic algebras associated to irreducible complex reflection groups. In this paper we prove that pairs of dual bases with respect to this form are defined over A:

Theorem. The generic cyclotomic algebra H = H(W, u) of an irreducible imprimitive complex reflection group W is symmetric over ℤ[u, u^{−1}].

The proof of this result will be given in the subsequent sections. There we will also give more precise statements about properties of the corresponding symmetrizing form.

The assertion of the main theorem is well known in the case of Iwahori-Hecke algebras, i.e., for cyclotomic algebras for real reflection groups. In that case a dual basis can immediately be written down in terms of the standard basis of H and the

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The first author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft. The second author thanks the Universität Heidelberg for its hospitality during the writing of this paper and also the Sonderforschungsbereich 343 at the Universität Bielefeld for financial support.

assertion follows by inspection. It turns out that the situation for the complex case is more subtle.

We would like to point out that it is still open whether the cyclotomic algebras for all of the finitely many primitive complex reflection groups enjoy the properties stated in the theorem.

1. The symmetrizing form on H(W_{n}, u)

In this section we recall the definition of the cyclotomic Hecke algebra H(W_{n}, u) and the construction of a symmetrizing form on H(W_{n}, u) ⊗ ℚ(u) from [3].

Let V be the n-dimensional unitary space V = ⟨e_{1}, …, e_{n}⟩ with the standard unitary inner product. Fix r > 1, and let W_{n} = G(r, 1, n) ⊂ GL(V) be the imprimitive complex reflection group on V generated by the reflections t_{1}, …, t_{n−1} of order 2 with roots e_{2} −e_{1}, …, e_{n} −e_{n−1}, respectively, and a (complex) reflection t_{0} of order r with root e_{1}. On these generators, W_{n} has a presentation given by the diagram

(1.1)  B_{n}^{(r)}:  ◯_{t_{0}}_{r}  ◯_{t_{1}}  ◯_{t_{2}}  ⋯ ◯_{t_{n−1}}

The elements t_{1}, …, t_{n−1} generate a subgroup of W_{n} isomorphic to the symmetric group 𝔖_{n}, and form a set of standard generators of this Coxeter group. We will consider 𝔖_{n} as a subgroup of W_{n} via this embedding.

Let u = (u_{1}, …, u_{r}, q) be indeterminates and A := ℤ[u, u^{−1}], where we write u^{−1} := (u_{1}^{−1}, …, u_{r}^{−1}, q^{−1}). The generic cyclotomic algebra H := H(W_{n}, u) attached to W_{n} is the free unital associative A-algebra on generators T_{0}, T_{1}, …, T_{n−1}, subject to the braid relations implied by the diagram (1.1) and the deformed order relations

(T_{0} − u_{1}) … (T_{0} − u_{r}) = 0 = (T_{i} − q)(T_{i} + 1), 1 ≤ i ≤ n − 1,

(see [4, 4.1]). Under the specialization

(1.2)  u_{k} ↦ ζ_{r}^{k}, 1 ≤ k ≤ r,  q ↦ 1,

where ζ_{r} := exp(2πi/r), the cyclotomic algebra specializes to the group algebra of W_{n} over ℤ[ζ_{r}]. Further, it is known that H is a free A-algebra of rank |W_{n}| (see [2]). Thus the subalgebra generated by T_{1}, …, T_{n−1} can and will be identified with the generic Iwahori-Hecke algebra H(𝔖_{n}, q) of type A_{n−1}.

For x = (x_{1}, …, x_{l}), x_{i} ∈ {t_{0}, t_{1}, …, t_{n−1}}, a reduced expression of an element x = x_{1} … x_{l} in W_{n}, let T_{x} := T_{x_{1}} … T_{x_{l}}. Note that it is not in general true that two reduced expressions x, x′ of the same element x ∈ W_{n} give rise to identical elements T_{x}, T_{x′} in H. Nevertheless, this property does hold if x ∈ 𝔖_{n} ≤ W_{n}.

Let X = {x | x ∈ W } be an arbitrary set of reduced expressions for the elements of W. Then it is known that the set B_{X} := {T_{x} | x ∈ X} is an A-basis of H (see for example [3, Prop. 2.4]). Any such basis will be called a reduced basis of H. Choose one such reduced basis B_{X}. Let f_{H} : H → A be the linear form defined on B_{X} by

(1.3)  f_{H} : H → A,  f_{H}(T_{x}) = \{ \begin{matrix}1 & if x = 1, \\ 0 & otherwise. \end{matrix}

It was shown in [3, 2B] that the linear form f_{H} does not depend on the particular reduced expressions chosen in the definition of the reduced basis B_{X}. Hence f_{H} is defined unambiguously on H. Moreover, it was proved in [3, Prop. 2.7] that f_{H} is a trace form. Thus the bilinear form

(1.4)  ⟨ , ⟩ : H ⊗ H → A,  (h_{1}, h_{2}) ↦ f_{H}(h_{1}h_{2})

is symmetric. Clearly the specialization of ⟨ , ⟩ to the group algebra of W_{n} is the standard symmetrizing form which sends (w_{1}, w_{2}) to the coefficient of 1 in the product w_{1}w_{2}. Since the algebra H_{K} := H(W_{n}, u)⊗_{A}K over K = Quot(A) is split semisimple [2], this implies:

Theorem 1.5 ([3, Th. 2.8]). The linear form f_{H} defines a non-degenerate symmetric form

⟨h_{1}, h_{2}⟩ := f_{H}(h_{1}h_{2}) for h_{1}, h_{2} ∈ H_{K}

on H_{K}. This satisfies

(1.6) f_{H}(T_{x}) = 0 for all reduced expressions x of non-identity elements of W_{n}.

We will show that f_{H} actually defines a non-degenerate symmetric form on H over A. This seems to be important for a further study of this algebra.

Let O be a domain and H an unital associative O-algebra. Recall from [4, 1C] that an O-basis B of H with 1 ∈ B is called quasi-symmetric if the linear form f : H → O defined by setting f(b) = δ_{1b} for b ∈ B is a trace form and via (1.4) gives rise to a non-degenerate symmetric form on H. We then also say that B is quasi-symmetric with respect to f. In this terminology, the above theorem can be rephrased to saying that any reduced basis of H(W_{n}, u)_{K} is quasi-symmetric with respect to the form f_{H} defined above.

2. The Ariki-Koike basis of H is quasi-symmetric

It will be convenient to introduce a further quasi-symmetric basis of H = H(W_{n}, u). For this let L_{1} := T_{0} and inductively L_{i} := q^{−1}T_{i−1}L_{i−1}T_{i−1} for 2 ≤ i ≤ n. It was shown by Ariki and Koike [2, Th. 3.10] that

(2.1)  B := {L_{1}^{b_{1}} … L_{n}^{b_{n}}T_{w} | 0 ≤ b_{i} ≤ r − 1, w ∈ 𝔖_{n}}

is a basis of H(W_{n}, u). Here T_{w} denotes the standard basis element in the Iwahori-Hecke subalgebra H(𝔖_{n}, q) of H(W_{n}, u). Note that B is non-reduced if r > 1 and n > 1.

We can now prove the analogue of (1.6) for this basis.

Proposition 2.2. The Ariki-Koike basis B is quasi-symmetric with respect to f_{H}, i.e., it satisfies

(2.3)  f_{H}(b) = 0 for all 1 ≠ b ∈ B.

Proof. We compare B to a particular reduced basis B_{X} of H and show that any 1 ≠ b ∈ B is a linear combination of non-identity basis elements from B_{X}. The assertion then follows from Theorem 1.5.

To describe B_{X} we introduce the elements

L_{k,a} := \{ \begin{matrix}T_{k−1} … T_{1}T_{0}^{a} & for a > 0, \\ 1 & for a = 0, \end{matrix}

for 1 ≤ k ≤ n, 0 ≤ a ≤ r − 1. According to [3, Lemma 1.5] the set

B_{X} := {L_{1,a_{1}} … L_{n,a_{n}}T_{w} | 0 ≤ a_{i} ≤ r − 1, w ∈ 𝔖_{n}}

is a reduced basis of H (the corresponding set X of reduced words is clear). Furthermore, the following relations are proved in [3, Lemma 2.3]:

Lemma 2.4. We have:

(a)  T_{i}L_{k,a} = \{ \begin{matrix}L_{k,a}T_{i} & if i > k or a = 0, \\ L_{k+1,a} & if i = k, a ≠ 0, \\ qL_{k−1,a} + (q − 1)L_{k,a} & if i = k − 1, a ≠ 0, \\ L_{k,a}T_{i+1} & if i < k − 1, a ≠ 0, \end{matrix}

for 1 ≤ i ≤ n − 1, 1 ≤ k ≤ n.

(b) L_{k,a}L_{k−m,c} =

L_{k−m−1,c}L_{k,a}T_{1} + (q − 1) ∑_{i=1}^{c} (L_{k−m−1,a+c−i}L_{k,i} − L_{k−m−1,i}L_{k,a+c−i})

for a, c ≠ 0, k − m > 1.

Lemma 2.5. Let i ≥ 1.

(a) Let 1 ≤ k ≤ r−1. Then L_{i}^{k} is equal to an A-linear combination of basis elements L_{1,c_{1}} … L_{i,c_{i}}T_{w} ∈ B_{X} with 0 ≤ c_{j} ≤ r − 1, 1 ≤ c_{i} ≤ k, w ∈ ⟨t_{1}, …, t_{i−1}⟩.

(b) Let 1 ≤ b_{i} ≤ r − 1. Then L_{1}^{b_{1}} … L_{i}^{b_{i}} is equal to an A-linear combination of basis elements L_{1,c_{1}} … L_{i,c_{i}}T_{w} ∈ B_{X} with 0 ≤ c_{j} ≤ r − 1, c_{i} ≠ 0, w ∈ ⟨t_{1}, …, t_{i−1}⟩.

Proof. According to our definitions we have L_{i}^{k} = (q^{1−i}L_{i,1}T_{1} … T_{i−1})^{k}. Thus assertion (a) is certainly correct if k = 1. Now assume that (a) has been proved for all exponents strictly smaller than k. Hence L_{i}^{k} = L_{i}^{k−1}L_{i} is a linear combination of terms L_{1,c_{1}} … L_{i,c_{i}}T_{v}L_{i,1}T_{w} with 1 ≤ c_{i} ≤ k − 1 and v, w ∈ ⟨t_{1}, …, t_{i−1}⟩ = 𝔖_{i}. But Lemma 2.4(a) shows that T_{v}L_{i,1} is equal to an A-linear combination of terms L_{j,1}T_{w} with j ≤ i, w ∈ 𝔖_{i}. Thus we have that L_{i}^{k} is a linear combination of terms L_{1,c_{1}} … L_{i,c_{i}}L_{j,1}T_{w} with j ≤ i, w ∈ 𝔖_{i}. Now L_{i,c_{i}}L_{j,1} with 1 ≤ c_{i} ≤ k − 1, j ≤ i is a linear combination of terms L_{1,d_{1}} … L_{i,d_{i}} with 1 ≤ d_{i} ≤ k by Lemma 2.4(b). This shows that L_{i}^{k} is a linear combination of terms hL_{i,d_{i}}T_{w} with 1 ≤ d_{i} ≤ k and h in the subalgebra H(W_{i−1}, u).

Expanding h in the reduced basis of this subalgebra we obtain summands of the form L_{1,c_{1}} … L_{i−1,c_{i−1}}T_{v}L_{i,c_{i}}T_{w} with v ∈ ⟨t_{1}, …, t_{i−2}⟩. It remains to apply Lemma 2.4(a) again to move the T_{v} past L_{i,c_{i}}. This completes the induction step and hence the proof of assertion (a).

For part (b) we use induction on i. If i = 1 then the result is just part (a). So assume that the assertion is proved for all j < i. By induction we have that L_{1}^{b_{1}} … L_{i}^{b_{i}} is a linear combination of terms L_{1,c_{1}} … L_{i−1,c_{i−1}}T_{w}L_{i}^{b_{i}} with w ∈ ⟨t_{1}, …, t_{i−2}⟩. By the first part, L_{i}^{b_{i}} is an A-linear combination of terms L_{1,c_{1}} … L_{i,c_{i}}T_{w} with c_{i} ≠ 0. Again the result follows by Lemma 2.4. □

We are now in a position to complete the proof of the proposition. Let 1 ≠ b = L_{1}^{b_{1}} … L_{i}^{b_{i}}T_{v} ∈ B with a_{i} ≠ 0. If i = 0 then b = T_{v} is already contained in B_{X} so there is nothing to prove. If i > 0 then according to Lemma 2.5(b) we have that b is equal to a linear combination of basis elements L_{1,c_{1}} … L_{i,c_{i}}T_{w} ∈ B_{X} with c_{i} ≠ 0. In particular, f_{H}(b) = 0 by (1.6). □

3. The determinant of the Gram matrix of type G(r, 1, n)

Let ∗: H⟶H be the A–linear antiautomorphism of H determined by T_{i}^{∗} = T_{i} for 0 ≤ i < n. So, in particular, L_{i}^{∗} = L_{i} and T_{w}^{∗} = T_{w^{−1}} for all i with 1 ≤ i ≤ n and all w ∈ 𝔖_{n}. Given a set H ⊆ H let H^{∗} = { h^{∗} | h ∈ H }.

In order to show that f_{H} induces a symmetric form on H over A it suffices to exhibit two bases B, C of H over A such that the determinant of the Gram matrix G_{BC} := (⟨b, c⟩)_{B,C} = (f_{H}(bc))_{b∈B,c∈C} is a unit in A.

For the basis B we take the Ariki–Koike basis defined in (2.1) above. For the basis C we essentially take B^{∗}; however, for technical reasons (see Proposition 3.8 below), the definition is more complicated. Let ℓ_{1} = 1 and for i = 2, …, n let ℓ_{i} = q^{−1}T_{i−1}ℓ_{i−1}T_{i−1}. Then ℓ_{i} is the analogue of L_{i} in the subalgebra H(𝔖_{n}, q); indeed, L_{i} = u_{1}ℓ_{i} if r = 1. We define

C := {T_{v}ℓ_{1}^{δ(c_{1})} … ℓ_{n}^{δ(c_{n})}L_{1}^{c_{1}} … L_{n}^{c_{n}} | 0 ≤ c_{i} ≤ r − 1, v ∈ 𝔖_{n}}.

where δ(c) = −1 if c ≠ 0 and δ(0) = 0. For a given c = (c_{1}, …, c_{n}), the set {T_{v}ℓ_{1}^{δ(c_{1})} … ℓ_{n}^{δ(c_{n})}}_{v∈𝔖_{n}} is a basis of H(𝔖_{n}, q) because each ℓ_{i} is invertible. Consequently, C is a basis of H by (2.1).

Theorem 3.1. Let B and C be as above. Then det (G_{BC}) = ±q^{l}(u_{1} … u_{r})^{m} for some integers l and m.

By Proposition 2.2, and the above remarks, the main Theorem is a consequence of Theorem 3.1 when W is of type B_{n}^{(r)}; in the next section we deduce the general case from this.

The proof of Theorem 3.1 will occupy the whole of this section. We begin with a well–known Lemma which follows directly from the relations in H (see, for example, [2, Lemma 3.3]).

Lemma 3.2. Suppose that 1 ≤ i < n and that 1 ≤ j ≤ n. Then

(a) L_{i} commutes with L_{j}.

(b) T_{i} commutes with L_{j} if j ≠ i, i + 1.

(c) T_{i} commutes with L_{i}L_{i+1}.

The elements ℓ_{1}, …, ℓ_{n} also satisfy similar relations to those given in the Lemma; however, we shall not need this.

Lemma 3.3. Suppose that 1 ≤ i < n and let k and m be non–negative integers.

(a) If k ≥ m then L_{i}^{k}L_{i−1}^{m}T_{i−1} = T_{i−1}L_{i}^{m}L_{i−1}^{k} + (q − 1) ∑_{c=m+1}^{k} L_{i}^{c}L_{i−1}^{k+m−c}.

(b) If k ≤ m then L_{i}^{k}L_{i−1}^{m}T_{i−1} = T_{i−1}L_{i}^{m}L_{i−1}^{k} − (q − 1) ∑_{c=k+1}^{m} L_{i}^{c}L_{i−1}^{k+m−c}.

Proof. Part (a) is proved in [6, Lemma 5.10]; the argument is a routine induction on k −m (use Lemma 3.2(c) to start the induction). To prove (b), apply the involution ∗ to (a) and rearrange. □

Corollary 3.4. Suppose that 1 < i ≤ n and k ≥ 1. Then

L_{i}^{k} = q^{−1}T_{i−1}L_{i−1}^{k}T_{i−1} + q^{−1}(q − 1) ∑_{c=1}^{k−1} L_{i}^{c}L_{i−1}^{k−c}T_{i−1}.

Proof. When k = 1 this is just the definition of L_{i}. In general, taking m = 0 in part (a) of the Lemma, we see that

L_{i}^{k+1} = q^{−1}L_{i}^{k}T_{i−1}L_{i−1}T_{i−1}

= q^{−1} {T_{i−1}L_{i−1}^{k} + (q − 1) ∑_{c=1}^{k} L_{i}^{c}L_{i−1}^{k−c}} L_{i−1}T_{i−1}

= q^{−1}T_{i−1}L_{i−1}^{k+1}T_{i−1} + q^{−1}(q − 1) ∑_{c=1}^{k} L_{i}^{c}L_{i−1}^{k+1−c}T_{i−1}. □

Recall the trace form f_{H}: H ⟶ A from (1.3). Given h, h′ ∈ H we define an

equivalence relation =^{f_{H}} on H by saying that h =^{f_{H}} h′ whenever f_{H}(h) = f_{H}(h′). This equivalence relation does not respect multiplication in H; however, because f_{H} is a trace form, it does have the following crucial property.

(3.5) Let h, h′ ∈ H. Then hh′ =^{f_{H}} h′h.

For j = 0, …, n let H_{j} be the A–submodule of H with basis

{L_{j}^{a_{j}} … L_{1}^{a_{1}}T_{w} | 0 ≤ a_{i} < r, w ∈ 𝔖_{n}}.

Note that H_{n} = H and that H_{j} is not an H–module for j ≠ n. However, H_{j} is always a right H(𝔖_{n}, q)–module and, by Proposition 2.2, we also have the next result.

(3.6) Suppose that 1 ≤ j < n, 1 ≤ k ≤ r − 1 and h ∈ H_{j}. Then L_{j+1}^{k}h =^{f_{H}} 0.

What we really need to know, however, is how to compute f_{H}(L_{j+1}^{r}h).

Lemma 3.7. Suppose that i, j and k are integers such that 1 ≤ i ≤ j < n, and 1 ≤ k ≤ r and let h ∈ H_{j}. Then

L_{j+1}^{k}h =^{f_{H}} q^{i−j−1}T_{j} … T_{i}L_{i}^{k}T_{i} … T_{j}h.

Proof. For the duration of this proof let T_{j..i} = T_{j} … T_{i} and T_{i..j} = T_{i} … T_{j}. By declaring that T_{j..i+1} = 1 = T_{i+1..j} we can state the lemma when i = j + 1; in which case the lemma asserts that f_{H}(L_{j+1}^{k}h) = f_{H}(L_{j+1}^{k}h). Since this is certainly true, we may now proceed by downwards induction on i ≤ j.

Now, by induction and by Corollary 3.4,

\begin{aligned}q^{j−i+2}L_{j+1}^{k}h & =^{f_{H}} qT_{j..i}L_{i}^{k}T_{i..j}h \\ & = T_{j..i}T_{i−1}L_{i−1}^{k}T_{i−1}T_{i..j}h + (q − 1) ∑_{c=1}^{k−1} T_{j..i}L_{i}^{c}L_{i−1}^{k−c}T_{i−1}T_{i..j}h. \end{aligned}

Thus, it suffices to show that T_{j..i}L_{i}^{c}L_{i−1}^{k−c}T_{i−1}T_{i..j}h =^{f_{H}} 0 whenever 0 < c < k. Fix such an integer c. Then, using the definition of the elements L_{a} together with Lemma 3.2, we find

T_{j..i}L_{i}^{c}L_{i−1}^{k−c}T_{i−1}T_{i..j}h = q^{j−i+2}T_{j..i}L_{i}^{c}L_{i−1}^{k−c−1}(T_{i−1}^{−1}T_{j..i}^{−1}L_{j+1})h
= q^{j−i+2}T_{j..i}(L_{i−1}^{k−c−1}L_{i}^{c})T_{i−1}^{−1}T_{j..i}^{−1}L_{j+1}h
= q^{j−i+1}T_{j..i}L_{i−1}^{k−c−1}(L_{i}^{c−1}T_{i−1}L_{i−1})T_{j..i}^{−1}L_{j+1}h
= q^{j−i+1}T_{j..i}L_{i}^{c−1}L_{i−1}^{k−c−1}T_{i−1}T_{j..i}^{−1}L_{j+1}L_{i−1}h.

Now T_{a}^{−1} = q^{−1}(T_{a} + 1 − q) for 1 ≤ a < n. Therefore, by Lemma 3.3, together with Lemma 3.2(a), there exist scalars a_{L} ∈ A such that

L_{i}^{c−1}L_{i−1}^{k−c−1}T_{i−1}T_{j..i}^{−1} = ∑_{L=T_{w}L_{j+1}^{b_{j+1}} ...L_{1}^{b_{1}} ∈B^{∗}} a_{L}T_{w}L_{j+1}^{b_{j+1}} … L_{1}^{b_{1}}

where b_{j+1} ≤ max{c − 1, k − c − 1} ≤ k − 2 whenever a_{L} ≠ 0. Consequently,

\begin{aligned}T_{j..i}L_{i}^{c}L_{i−1}^{k−c}T_{i−1}T_{i..j}h & = ∑_{L=T_{w}L_{j+1}^{b_{j+1}} ...L_{1}^{b_{1}}} q^{j−i+1}a_{L}T_{j..i}T_{w}L_{j+1}^{b_{j+1}} … L_{1}^{b_{1}}L_{j+1}L_{i−1}h \\ & =^{f_{H}} ∑_{L=T_{w}L_{j+1}^{b_{j+1}} ...L_{1}^{b_{1}}} q^{j−i+1}a_{L}L_{j+1}^{1+b_{j+1}} … L_{1}^{b_{1}}L_{i−1}hT_{j..i}T_{w} \end{aligned}

by (3.5). However, 1 + b_{j+1} ≤ k − 1 < r; so T_{j..i}L_{i}^{c}L_{i−1}^{k−c}T_{i−1}T_{i..j}h =^{f_{H}} 0 by (3.6) as

required. □

Recall the element ℓ_{j+1} = q^{−j}T_{j} … T_{1}T_{1} … T_{j}.

Proposition 3.8. Let j be an integer such that 1 ≤ j < n and suppose that h ∈ H_{j}. Then

L_{j+1}^{r}h =^{f_{H}} (−1)^{r}u_{1} … u_{r}hℓ_{j+1}.

Proof. By Lemma 3.7, f_{H}(L_{j+1}^{r}h) = q^{−j}f_{H}(T_{j} … T_{1}T_{0}^{r}T_{1} … T_{j}h). Therefore, since (T_{0} − u_{1}) … (T_{0} − u_{r}) = 0, we may write f_{H}(L_{j+1}^{r}h) as an A–linear combination of terms f_{H}(T_{j} … T_{1}T_{0}^{k}T_{1} … T_{j}h) for k = 0, …, r − 1. However, by Lemma 3.7 and (3.6), if 0 < k < r then

T_{j} … T_{1}T_{0}^{k}T_{1} … T_{j}h =^{f_{H}} q^{j}L_{j+1}^{k}h =^{f_{H}} 0.

Therefore, L_{j+1}^{r}h =^{f_{H}} (−1)^{r}q^{−j}u_{1} … u_{r}hT_{j} … T_{1}T_{1} … T_{j} = (−1)^{r}u_{1} … u_{r}hℓ_{j+1} as re-

quired. □

We can now prove the main result of this section.

Proof of Theorem 3.1. Recall the bases B and C of H and the Gram matrix G_{BC} = (f_{H}(bc))_{b∈B,c∈C} from the beginning of this section.

An arbitrary element of B is of the form L_{n}^{b_{n}} … L_{1}^{b_{1}}T_{w} for some w ∈ 𝔖_{n} and some integers 0 ≤ b_{i} < r. In the matrix G_{BC} order the elements of B lexicographically according to the exponents (b_{n}, …, b_{1}), and similarly for the elements of C.

For j = 0, …, n let B_{j} = B ∩ H_{j} and C_{j} = C ∩ H_{j}^{∗} and define G_{j} to be the Gram matrix G_{B_{j}C_{j}} = (f_{H}(bc))_{b∈B_{j},c∈C_{j}}. Then G_{BC} = G_{n} and our ordering of the elements of B and C is such that the matrix G_{j} sits in the top left hand corner of the matrix G_{j+1} for each 0 ≤ j < n.

Suppose that 0 ≤ j < n and let α = (−1)^{r}u_{1} … u_{r}. Then we claim that, as an r × r–block matrix, G_{j+1} has the form

( \begin{matrix}G_{j} & 0 & … & 0 & 0 \\ 0 & 0 & … & 0 & αG_{j} \\ ⋮ & ⋮ & ... & ... & ∗ \\ 0 & 0 & ... & ∗ & ⋮ \\ 0 & αG_{j} & ∗ & ⋯ & ∗ \end{matrix} )

To see this let b = L_{j+1}^{b_{j+1}} … L_{1}^{b_{1}}T_{w} and c = T_{v}ℓ_{1}^{δ(c_{1})} … ℓ_{j+1}^{δ(c_{j+1})}L_{1}^{c_{1}} … L_{j+1}^{c_{j+1}} be arbitrary elements of B_{j+1} and C_{j+1} respectively. Then

\begin{aligned}bc & = L_{j+1}^{b_{j+1}} … L_{1}^{b_{1}}T_{w}T_{v}ℓ_{1}^{δ(c_{1})} … ℓ_{j+1}^{δ(c_{j+1})}L_{1}^{c_{1}} … L_{j+1}^{c_{j+1}} \\ & =^{f_{H}} L_{j+1}^{b_{j+1}+c_{j+1}}L_{j}^{b_{j}+c_{j}} … L_{1}^{b_{1}+c_{1}}T_{w}T_{v}ℓ_{1}^{δ(c_{1})} … ℓ_{j}^{δ(c_{j})}ℓ_{j+1}^{δ(c_{j+1})} \end{aligned}

by (3.5) and Lemma 3.2(a). If b_{j+1} = c_{j+1} = 0 the assertion is just the definition of G_{j} (note that δ(0) = 0). If 0 < b_{j+1} + c_{j+1} < r then f_{H}(bc) = 0 by (3.6). Finally, if b_{j+1} + c_{j+1} = r then by Proposition 3.8

\begin{aligned}bc & =^{f_{H}} (−1)^{r}u_{1} … u_{r}L_{j}^{b_{j}+c_{j}} … L_{1}^{b_{1}+c_{1}}T_{w}T_{v}ℓ_{1}^{δ(c_{1})} … ℓ_{j}^{δ(c_{j})}ℓ_{j+1}^{δ(c_{j+1})}ℓ_{j+1} \\ & = ( −1)^{r}u_{1} … u_{r}L_{j}^{b_{j}+c_{j}} … L_{1}^{b_{1}+c_{1}}T_{w}T_{v}ℓ_{1}^{δ(c_{1})} … ℓ_{j}^{δ(c_{j})}, \end{aligned}

since δ(c_{j+1}) = −1 because c_{j+1} > 0. This proves the claim.

Therefore, det G_{j+1} = ±(u_{1} … u_{r})^{r−1}( det G_{j})^{r} for all 0 ≤ j < n. Consequently, det G_{BC} = det G_{n} = ±(u_{1} … u_{r})^{m}( det G_{0})^{k} for some integers k and m. However, G_{0} = (f_{H}(T_{w}T_{v}))_{v,w∈𝔖_{n}} and it is well–known, and easy to prove, that f_{H}(T_{w}T_{v}) = q^{l}δ_{wv^{−1}}, where l is the length of a reduced expression for w. Hence, det G_{0} = ±q^{z} for some integer z. This completes the proof of Theorem 3.1. □

4. The determinant of the Gram matrix of type G(r, 2, 2)

For any divisor p of r let W_{n,p} be the normal subgroup of index p of W_{n} generated by the n + 1 reflections

{t_{0}^{p}, t_{0}^{−1}t_{1}t_{0}, t_{1}, …, t_{n−1}}

if p < r, respectively the n reflections

{t_{0}^{−1}t_{1}t_{0}, t_{1}, …, t_{n−1}}

if p = r. Thus W_{n,p} is the imprimitive complex reflection group G(r, p, n) in the notation of Shephard and Todd.

We would like to introduce a symmetrizing form on the generic cyclotomic algebra H(W_{n,p}, u) defined in [5]. This coincides with the algebra introduced by Ariki in [1] except for the case where n = 2, and both r and p are even, where the generic algebra has one additional parameter. In this latter case, no quasi-symmetric basis had been given in [3], so we need to treat this algebra separately.

So, suppose that r is even, n = p = 2, and set d := r/2. Then W_{2,2} has a presentation described by the diagram

(4.1)  D_{2}^{(d)} : ◯  ◯ ◯ d

Let H = H(W_{2,2}, u), where u = (u_{1}, …, u_{d}; q; q′), be the corresponding cyclotomic algebra, generated as associative unital ℤ[u, u^{−1}]-algebra by elements R, S, T subject to the relations

\begin{matrix}(T − u_{1}) … (T − u_{d}) = 0 = (R − q)(R + 1) = (S − q′)(S + 1), \\ RST = ST R = T RS \end{matrix}

(see [4, 4A]). We let U := RS and V := \frac{q′−1}{q′}U + q − 1. Note that T and U commute and the element T U is central in H. The following result is inspired by [1, Lemma 1.2].

Lemma 4.2. Let j, k be nonnegative integers. Then

(a) ST^{j} = (qq′)^{−j}T^{j}U^{2j}S − \frac{1}{q}T^{j}UV ∑_{i=0}^{j−1}(qq′)^{−i}U^{2i},

(b) SU^{k} = (qq′)^{k}U^{−k}S + \frac{1}{q}UV ∑_{i=0}^{k−1}(qq′)^{k−i}U^{2i−k},

(c) ST^{j}U^{k} = (qq′)^{k−j}T^{j}U^{2j−k}S + \{ \begin{matrix}−\frac{1}{q}T^{j}V ∑_{i=0}^{j−k−1}(qq′)^{−i}U^{2i+1+k} & if j ≥ k, \\ \frac{1}{q}T^{j}V ∑_{i=j−k}^{−1}(qq′)^{−i}U^{2i+1+k} & if j < k. \end{matrix}

Proof. We prove (a) by induction on j. The result trivially holds if j = 0. If j = 1 then

\begin{aligned}ST = ST RR^{−1} & = \frac{1}{q}T UR − \frac{q − 1}{q}T U = \frac{1}{q}T U^{2}S^{−1} − \frac{q − 1}{q}T U \\ & = (qq′)^{−1}T U^{2}S − \frac{1}{q}T UV, \end{aligned}

as claimed. The induction hypothesis and the case j = 1 now yield

\begin{aligned}ST^{j+1} & = (qq′)^{−j}T^{j}U^{2j}ST − \frac{1}{q}T^{j+1}U ∑_{i=0}^{j−1}(qq′)^{−i}U^{2i}V \\ & = (qq′)^{−j−1}T^{j+1}U^{2j+2}S − \frac{1}{q}T^{j+1}U ∑_{i=0}^{j}(qq′)^{−i}U^{2i}V, \end{aligned}

proving the first part.

In part (b) the case k = 0 is again obvious. If k = 1 then

SU = SRS = (q′S^{−1} + q′ − 1)(qR^{−1} + q − 1)S
= qq′U^{−1}S + (q − 1)q′ + (q′ − 1)(qR^{−1} + q − 1)S
= qq′U^{−1}S + (q − 1)q′ + (q′ − 1)U = qq′U^{−1}S + q′V.

The inductive step is verified by the same type of calculation as in the first part. The last part is an immediate consequence of parts (a) and (b). □

Note that T, U generate a commutative subalgebra of H. Since it is known that H is ℤ[u, u^{−1}]-free of rank |W_{2,2}| by [4, Satz 4.7] we conclude as there that

{T^{j}U^{k}, T^{j}U^{k}S | 0 ≤ 2j − k, k ≤ r − 1}

is a ℤ[u, u^{−1}]-basis of H (compare also with [1, Prop. 1.4] where the case q = q′ is considered).

We proceed to show that this basis has good properties.

Proposition 4.3. The basis

(4.4)  C = {T^{j}U^{k}, T^{j}U^{k}S | 0 ≤ 2j − k, k ≤ r − 1}

of H = H(W_{2,2}, u) is quasi-symmetric.

Proof. We have to show that the linear form f_{H} : H → ℤ[u, u^{−1}] defined by f_{H}(1) = 1, f_{H}(b) = 0 if 1 ≠ b ∈ C is a trace form. Since it specializes to the standard symmetrizing form of the complex reflection group W_{2,2} the induced symmetric form on H is then automatically non-degenerate over the quotient field of A.

We will have to consider elements of H of the form T^{j}U^{k}S^{l} with 0 ≤ k ≤ r − 1, l ∈ {0, 1} and j an arbitrary integer. First note that on such elements

(4.5)  f_{H}(T^{j}U^{k}S^{l}) = 0  if k ≠ 0 or l ≠ 0.

Indeed, if T^{j}U^{k}S^{l} ∉ C then by the relation on T we may replace T^{j} by a linear combination of elements T^{j+i} or T^{j−i}, with 1 ≤ i ≤ d. Induction and the definition of f_{H} now show the claim.

Since R and S are invertible in H it clearly suffices to prove that f_{H}(bL) = f_{H}(Lb) for b ∈ C and L ∈ {S, U, T U}. This is immediate for the central element L = T U. Now assume L = S. Inspection shows that if 0 ≤ 2j − k, k ≤ r − 1 then all monomials on the right hand side of Lemma 4.2(c) lie in C. Thus, Lemma 4.2(c) gives the decomposition in C of ST^{j}U^{k}. By the definition of f_{H} it follows that f_{H}(S · T^{j}U^{k}) = 0 = f_{H}(T^{j}U^{k} · S). Similarly, f_{H}(T^{j}U^{k}S · S) equals

f_{H}(S · T^{j}U^{k}S) = \{ \begin{matrix}q′ & if j = k = 0, \\ 0 & else. \end{matrix}

So finally assume that L = U. As U commutes with T, the assertion is again clear if b = T^{j}U^{k}. For b = T^{j}U^{k}S we have to compare the value of f_{H} on T^{j}U^{k+1}S and on

(4.6)  T^{j}U^{k}SU = qq′T^{j}U^{k−1}S + q′(q − 1)T^{j}U^{k} + (q′ − 1)T^{j}U^{k+1}

(by Lemma 4.2(b)). If k ≠ 0, r − 1 and 2j − k ≠ 0, r − 1 then all monomials in the above expressions lie in C and f_{H} vanishes on both sides. If 2j − k ∈ {0, r − 1} then f_{H} again vanishes by (4.5). We consider the two remaining cases separately.

First assume that k = 0. We have f_{H}(U · T^{j}S) = f_{H}(ST^{j}U) by what we proved for S, so the result is true if j = 0. For j > 0

ST^{j}U = T UST^{j−1} = (qq′)^{1−j}T^{j}U^{2j−1}S − \frac{1}{q}T^{j}U^{2}V ∑_{i=0}^{j−2}(qq′)^{−i}U^{2i}

by Lemma 4.2(a). All monomials appearing in the right-hand side are non-trivial basis elements, so f_{H} vanishes. On the other hand we have the equality

\begin{aligned}SU & = S ∑_{l=0}^{d−1} a_{l}T^{l+1}U = ∑_{l=0}^{d−1} a_{l}T UST^{l} \\ & = ∑_{l=0}^{d−1} a_{l} ((qq′)^{−l}T^{l+1}U^{2l+1}S − \frac{1}{q}T^{l+1}U^{2}V ∑_{i=0}^{l−1}(qq′)^{−i}U^{2i}) \end{aligned}

(with suitable constants a_{l}) since T U is central and by Lemma 4.2(a). Here, f_{H} is zero by (4.5).

It remains to consider k = r − 1. Lemma 4.2(a) gives

(4.7)  \begin{matrix}ST^{d} = & (qq′)^{−d}T^{d}U^{r}(S − (q′ − 1)) \\ − \frac{q − 1}{q}T^{d}U^{r−1}(qq′)^{1−d} − \frac{1}{q}T^{d}UV ∑_{i=0}^{d−2}(qq′)^{−i}U^{2i}. \end{matrix}

The last two terms on the right-hand side are of the type (4.5). On the other hand, replacing T^{d} by a linear combination of T^{i} with 0 ≤ i ≤ d − 1 and again applying Lemma 4.2(a) shows that the left-hand side is a sum of elements of the form considered in (4.5). Multiplication of both sides by T^{−(d−j)} does not change this fact, and thus f_{H}(T^{j}U^{r}S) = (q′ − 1)f_{H}(T^{j}U^{r}). But now (4.6), with k = r − 1, gives

f_{H}(T^{j}U^{r−1}S · U) = (q′ − 1)f_{H}(T^{j}U^{r}) = f_{H}(T^{j}U^{r}S) = f_{H}(U · T^{j}U^{r−1}S)

as required. □

We write G_{C} for the Gram matrix of f_{H} with respect to the pair of bases C, C^{∗}, where C^{∗} is obtained from C by reversing the order of the factors in each element.

Proposition 4.8. Let C be the basis of H defined in (4.4). Then det (G_{C}) is invertible in A = ℤ[u, u^{−1}]. In particular H is symmetric over A.

Proof. Elements of C are of the form T^{j}U^{k}S^{l} for some j, k, l with 0 ≤ k, 2j −k ≤ r−1 and l ∈ {0, 1}. We order C lexicographically with respect to (k, l, j). Then (4.5) shows that the Gram matrix has the shape

( \begin{matrix}G_{1} & 0 & … & 0 & 0 \\ 0 & 0 & … & 0 & G_{2} \\ ⋮ & ⋮ & ... & ... & ∗ \\ 0 & 0 & ... & ∗ & ⋮ \\ 0 & G_{r} & ∗ & ⋯ & ∗ \end{matrix} )

(as an r × r-block matrix). Again by (4.5) we see that G_{1} is a block diagonal 2 × 2matrix, and the blocks on the diagonal are the same as for the algebra H(W_{1}, u). Thus the determinant of G_{1} has the required form. The matrices G_{i}, 2 ≤ i ≤ r, can also be considered as 2 × 2-block matrices, with entries f_{H}(T^{j_{1}+j_{2}}U^{r}S^{l_{1}+l_{2}}) where l_{1}, l_{2} ∈ {0, 1}. Application of (4.7) shows that G_{i} has the block form

( \begin{matrix}G′_{i} & (q′ − 1)G′_{i} \\ (q′ − 1)G′_{i} & (q′ + (q′ − 1)^{2})G′_{i} \end{matrix} ) .

Here G′_{i} is the matrix with entries f_{H}(T^{j_{1}+j_{2}}U^{r}). Subtraction of (q′−1) times the first block of r columns from the second block of r columns thus yields det G_{i} = q′ det(G′_{i})^{2}. To evaluate the determinant of G′_{i} we need a further calculation. Application of (4.7) gives

T^{l}U^{r} = \frac{1}{q′} T^{l}U^{r}(S − (q′ − 1))S

= \frac{1}{q′} (qq′)^{d}T^{l−d} (ST^{d} + \frac{q − 1}{q}T^{d}U^{r−1}(qq′)^{1−d} − \frac{1}{q}T^{d}UV ∑_{i=0}^{d−2}(qq′)^{−i}U^{2i}) S

= q(qq′)^{d−1}T^{l−d}ST^{d}S + (q − 1)T^{l}U^{r−1}S − T^{l}UV ∑_{i=0}^{d−2}(qq′)^{−i−1}U^{2i}S.

According to (4.5) the form f_{H} vanishes on all summands on the right-hand side, except possibly the first. Hence, expanding T^{d} = ∑_{j=0}^{d−1} a_{j}T^{j}, we have

T^{l}U^{r} =^{f_{H}} q(qq′)^{d−1}T^{l−d}S ∑_{j=0}^{d−1} a_{j}T^{j}S

= q(qq′)^{d−1}T^{l−d} ∑_{j=0}^{d−1} a_{j} ((qq′)^{−j}T^{j}U^{2j}S − \frac{1}{q}T^{j}UV ∑_{i=0}^{j−1}(qq′)^{−i}U^{2i}) S

=^{f_{H}} q(qq′)^{d−1}T^{l−d}a_{0}S^{2} =^{f_{H}} (qq′)^{d}u_{1} … u_{d}T^{l−d}

by Lemma 4.2(a) and (4.5). Note that this is the analogue of Proposition 3.8 in this case. In particular this last equation shows that the determinant of G′_{i} is the same as the one occurring in H(W_{1}, u) (up to units in A). The assertion follows. □

Let f_{H} : H → A be the symmetrizing form induced by the quasi-symmetric basis C. Over the quotient field K := Quot(A) we may decompose f_{H} as a sum

f_{H} = ∑_{χ∈Irr(H_{K})} δ_{χ} χ

of irreducible characters of H_{K} := H ⊗ K, with δ_{χ} ∈ K. The coefficients δ_{χ} are called the relative degrees associated to f_{H}, respectively to the quasi-symmetric basis C. Clearly we have

∑_{χ∈Irr(H_{K})} δ_{χ} χ(c) = 0

for all elements c ≠ 1 of a quasi-symmetric basis with respect to f_{H}, and ∑ δ_{χ} χ(1) = 1. Using this property of the relative degrees the first author in [8] computed candidates for the relative degrees attached to a nice symmetrizing form on H. We can now verify:

Corollary 4.9. The relative degrees of the symmetric form induced by the quasisymmetric basis C are those given in [8, Th. 3.9]

Proof. According to the result in [8] it suffices to show that f_{H} vanishes on the elements

{RT^{j}U^{j}, ST^{j}U^{j} | 0 ≤ j ≤ d − 1} ∪ {T^{i+j}U^{j} | 0 < 2i + j ≤ r − 1}

of H. But RT^{j}U^{j} = T^{j}U^{j+1}S^{−1} = \frac{1}{q′} T^{j}U^{j+1}(S − (q′ − 1)), on which f_{H} vanishes by (4.7), ST^{j}U^{j} = T^{j}U^{j}S ∈ C \ {1}, and also T^{i+j}U^{j} ∈ C, so the result follows. □

5. The cyclotomic algebras H(G(r, p, n), u)

Finally, we consider the general case where W_{n,p} is an arbitrary imprimitive complex reflection group G(r, p, n). Let H(W_{n,p}, u) be the generic cyclotomic algebra associated to W_{n,p} in [5]. This coincides with the algebra defined by Ariki [1, Prop. 1.6] except in the case where n = 2 and r, p are both even.

Theorem 5.1. Let W = G(r, p, n) be an imprimitive complex reflection group. Then the associated generic cyclotomic algebra H = H(W, u) carries a canonical symmetrizing form f_{H} over ℤ[u, u^{−1}].

This form vanishes on all non-identity elements of the Ariki-Koike basis of H, while f_{H}(1) = 1. In the case p = 1 it also vanishes on all non-identity elements of any reduced basis of H.

Proof. We reduce this to the case p = 1 (respectively p = 2 if n = 2 and r, p are even) where such a symmetrizing form was found in Theorem 3.1 (resp. Proposition 4.8). Assume first that either n > 2 or one of r or p is odd. Let d := r/p, ζ_{p} := exp(2πi/p) and Ã := ℤ[ζ_{p}, u, u^{−1}]. Then according to Ariki [1, Prop. 1.6] the algebra H ⊗ Ã can be identified with a subalgebra of the specialized Ã–algebra H̃ = H(W_{n}, ũ), with parameters

(5.2)  ũ = (u_{1}, …, u_{d}, ζ_{p}u_{1}, …, ζ_{p}u_{d}, …, ζ_{p}^{p−1}u_{1}, …, ζ_{p}^{p−1}u_{d}; q) ,

generated by

{T_{0}^{p}, T_{0}^{−1}T_{1}T_{0}, T_{1}, …, T_{n−1}} .

Let f̃ be the linear form on H̃ obtained by specializing the symmetrizing form constructed in Section 3. Since the specialization to the group algebra factorizes through the specialization (5.2) the form f̃ gives rise to a non-degenerate symmetric form on H̃. In particular, all irreducible characters of H̃ ⊗ ℚ(u) occur with non-zero multiplicity in f̃. This implies that all irreducible characters of the subalgebra H ⊗ Ã of H̃ occur with non-zero multiplicity in the restriction f_{H} of f̃ to H ⊗ Ã. Thus, f_{H} is a symmetrizing form for H over the quotient field of Ã.

It is shown in [1, Prop. 1.6] that

{L_{n}^{a_{n}} … L_{1}^{a_{1}}T_{w} | 0 ≤ a_{i} ≤ r − 1, ∑ a_{i} ≡ 0 (mod p), w ∈ 𝔖_{n}}

is an A-basis of H. Tensoring with Ã, this gives a basis of H ⊗ Ã which is a subset of the Ariki-Koike basis of the cyclotomic algebra H̃; it follows that f_{H} is a symmetrizing form for H ⊗ Ã. However, the Gram determinants for the bases of H and H ⊗ Ã are equal and are elements of A, so f_{H} is a symmetrizing form for H (over A), as required.

In the case where n = 2 and r and p are even, we embed H instead into a suitable specialization H(W_{2,2}, ũ) of the generic algebra of W_{2,2} studied in 4.A (see [8, Prop. 3.8]). The remainder of the argument is then the same as in the general case. □

Remark 5.3. We expect that f_{H} should also vanish on non-identity reduced basis elements in the general case p ≠ 1. In this context note that a word which is reduced in W_{n,p} does not necessarily have to be reduced in W_{n}.

Remark 5.4. In [7, Verm. 2.20 and Satz 5.13] the first author put forward a conjecture on the values of the relative degrees δ_{χ} associated to a nice symmetrizing form on H(W_{n,p}). Using the calculations in [8, Th. 3.4] and Corollary 4.9 it follows that this conjecture is correct if n = 2 for the symmetrizing form studied in this paper. We expect that in general the relative degrees attached to the symmetrizing form f_{H} above are those predicted in [7].

Finally, suppose that O is an integral domain and let H_{O}(W_{n,p}, û) be the specialized cyclotomic Hecke algebra of W_{n,p} with parameters û ∈ O. Given a set of indeterminates u over ℤ we can consider O as a ℤ[u]–module by specifying that u acts as û. Then H_{O}(W_{n,p}, û) ≅ H_{ℤ[u]}(W_{n,p}, u) ⊗_{ℤ[u]} O. Therefore, by Theorem 5.1, we have the following.

Corollary 5.5. Let H_{O}(W_{n,p}, û) be a specialized cyclotomic Hecke algebra over O. Then H_{O}(W_{n,p}, û) is a symmetric algebra with respect to f_{H} if and only if the parameters û are invertible in O.

In [6] an analogue of the q̂–Schur algebra for a cyclotomic Hecke algebra H_{O} of type B_{n}^{(r)} was introduced. Key to the main results of that paper is the calculation of the “double annihilator” of certain elements of H (see [6, Thm. 5.16]). When the parameters u are invertible elements of O, the required result of [6] is an immediate consequence of Corollary 5.5 (significantly simplifying the argument there).

Interestingly, the results of [6] require only that q̂ be invertible in O. This suggests that H_{O} may be a quasi–Frobenius algebra so long as q̂ is an invertible element of O.

References

[1] S. Ariki, Representation theory of a Hecke algebra of G(r, p, n), J. Algebra 177 (1995), 164–185.

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

[3] K. Bremke and G. Malle, Reduced words and a length function for G(e, 1, n), Indag. Mathem. (1997) (to appear).

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

[5] M. Broué, G. Malle, and R. Rouquier, On complex reflection groups and their associated braid groups, Representations of groups (B.N. Allison and G.H. Cliff, eds.), CMS Conference Proceedings, vol. 16, AMS, Providence, 1995, pp. 1–13.

[6] R. Dipper, G. James, and A. Mathas, Cyclotomic q–Schur algebras, Math. Z. (to appear).

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

[8] G. Malle, Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, Finite Reductive Groups: Related Structures and Representations (M. Cabanes, ed.), Progr. Math, vol. 141, Birkhäuser, Boston, 1997, pp. 311–332.

G. M. : IWR, Im Neuenheimer Feld 368, Universität Heidelberg, D–69120 Heidelberg, Germany

E-mail address: malle@urania.iwr.uni-heidelberg.de

A. M. : Fakultät für Mathematik, SFB 343, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: sfb32@Mathematik.Uni-Bielefeld.de