COMPLEX REFLECTION GROUPS,

BRAID GROUPS, HECKE ALGEBRAS

Michel Broué, Gunter Malle, Raphaël Rouquier

Abstract. Presentations “à la Coxeter” are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their associated braid groups, such as the computation of the center of the braid group and the construction of deformations of the finite group algebra (Hecke algebras). We introduce monodromy representations of the braid groups which factorize through the Hecke algebras, extending results of Cherednik, Opdam, Kohno and others.

Summary

Introduction

1. Complex reflection groups and their presentations A. Background from complex reflection groups B. Presentations

2. Braid groups and their diagrams A. Generalities about hyperplane complements B. Generalities about the braid groups C. The braid diagrams D. Parabolic subgroups

3. Proofs of the main theorems for the braid groups B(de, e, r)

A. Notation and prerequisites

B. Computation of B(de, e, r) and of its center for d ≠ 1 C. Computation of B(e, e, r) and of its center

4. Hecke algebras A. Background from differential equations and monodromy B. A family of monodromy representations of the braid group C. Hecke algebras

5. Diagrams and tables Appendix 1 : Generators of the monodromy around a divisor Appendix 2 : tables 1 to 5

1991 Mathematics Subject Classification. Primary 20, 20G. We thank Jean Michel, Peter Orlik, Pierre Vogel for useful conversations, and the Isaac

Newton Institute for its hospitality while the last version of this manuscript was written up. The second named author gratefully acknowledges financial support by the Fondation Alexander von Humboldt for his stays in Paris.

Introduction

In [DeMo] (17.20), Deligne and Mostow raised the following question :

“Let W ⊂ GL(V) be an irreducible finite group generated by complex reflections, and let V′ be the complement in V of the fixed hyperplanes of the complex reflections in W. For H the fixed hyperplane of a complex reflection in W, let s_{H} be the generator of the monodromy around the image of H in V/W. It is well defined up to conjugacy in π := π_{1}(V′/W). The fundamental group π is an extension of W by the fundamental group of V′, and s_{H} projects in W to the inverse of the generator of the fixer of H, with non trivial eigenvalue of the form exp(2πi/n_{H}).

Question 3. For each conjugacy class of each hyperplane H fixed by a complex reflection, let q_{H}(t) be a path in ℂ^{×}, starting at exp(−2πi/n_{H}). Is it uniquely possible to deform with t a representation ρ_{t} of π, starting at t = 0 with the given representation of the quotient W of π, so that ρ_{t}(s_{H}) is a complex reflection with non trivial eigenvalue q_{H}(t) ?” As noticed by Deligne and Mostow, in the case where W is a Coxeter group, the

existence of the Hecke algebra as an image of the group algebra of the braid group π provides a positive answer to their question.

It is one of our purposes here to give a positive answer to the preceding question, at least for all infinite series of irreducible finite complex reflection groups, and for some exceptional ones (a more partial answer, without proofs, had been announced in [BMR]). We shall get this answer by exhibiting a generalized Hecke algebra for these groups again as an image of the group algebra of the associated “braid group” π.

Through recent work on representations of reductive finite groups and related topics (like representations of finite Coxeter groups and associated Hecke algebras)^{1} it has become clearer and clearer that finite “complex reflection groups” (i.e., linear groups generated by pseudo-reflections) behave very much like Coxeter groups, or even like Weyl groups.

• Many of them behave as if they were the Weyl group of a reductive algebraic group : in particular, they determine families of polynomials which share many properties of the set of generic degrees of the unipotent characters of a reductive algebraic group.

• Through suitable presentations by generators and relations, it has become possible to deform the complex group algebra of most complex reflection groups in a way which generalizes the construction of classical Hecke algebras of finite Coxeter groups.

Here we prove in particular that these presentations are naturally associated to presentations of the corresponding braid groups, thus providing a more intrinsic definition.

It should be noticed that some other nice properties of “Coxeter braid groups” extend to this more general setting.

For example, generalizing a result of Deligne and Brieskorn–Saito valid for Coxeter groups, we check here, in most cases, that the centers of braid groups associated to irreducible complex reflection groups are cyclic.

Also (at least in the case of the infinite series) the pure braid group of an r–dimensional irreducible complex reflection group has a natural structure as an r–fold iterated semidirect product of free groups (of finite rank).

1see for example [AlLu], [Ari], [ArKo], [BreMa], [BrMa], [BrMi], [BMM], [Lu], [Ma1], [Ma2].

Nevertheless, it should be emphasized that this must only be the beginning of a long story which is still to be discovered. Our results are – almost – general, but few of our proofs are. Moreover, new questions emerge now : how to characterize the distinguished generators and the diagrams representing the relations ? How to explain the natural “diagram invariants” like degrees, codegrees, zeta function (see §5 below) ?

1. Complex reflection groups and their presentations

A. Background from complex reflection groups.

For all the results quoted here, we refer the reader to the classical literature on complex reflections groups, such as [Bou], [Ch], [Co], [ShTo], [Sp], and also to the more recent fundamental work on the subject by Orlik, Solomon and Terao (see [OrSo1], [OrSo2], [OrTe]).

Let V be a complex vector space of dimension r. A pseudo–reflection of GL(V) is a non trivial element s of GL(V) which acts trivially on a hyperplane, called the reflecting hyperplane of s. Let W be a finite subgroup of GL(V) generated by pseudo–reflections. The pair (V, W) is called a complex reflection group.

A parabolic subgroup of W is by definition the subgroup of elements of W which act trivially on a subspace of V. The following result is due to Steinberg ([St], Theorem 1.5)

— cf. also exercises 7 and 8 in [Bou], Ch. v, §6.

1.1. Theorem. Let V′ be a subspace of V. Then the parabolic subgroup W_{V′}, consisting of all elements of W which fix V′ pointwise, is still generated by pseudo–reflections : W_{V′} is generated by those pseudo–reflections in W whose reflecting hyperplane contains V′.

We denote by A the set of reflecting hyperplanes of (V, W), and we set N := |A|. We denote by N^{∗} the number of pseudo-reflections in W (note that for real reflection groups we have N = N^{∗}).

For H ∈ A, we denote by W_{H} the pointwise stabilizer of H, and we set e_{H} := |W_{H}|. The group W_{H} is a minimal non trivial parabolic subgroup of W. All its non trivial elements are pseudo-reflections. The group W_{H} is cyclic : if s_{H} denotes the element of W_{H} with determinant exp(2iπ/e_{H}), we have W_{H} = ⟨s_{H}⟩, the group generated by s_{H}.

The centralizer C_{W} (W_{H}) of W_{H} in W is also its normalizer, as well as the normalizer (setwise stabilizer) of H.

For C ∈ A/W an orbit of hyperplanes, we denote by N_{C} its cardinality. We have N_{C} = |W : C_{W} (W_{H})| for H ∈ C. We also set e_{C} := e_{H} for H ∈ C.

We denote by S the symmetric algebra of V, by R = S^{W} the algebra of invariants of W, by R_{+} the ideal of R consisting of elements of positive degree, and we set S_{W} := S/R_{+}S.

The following facts are known (they are introduced here in an order which is convenient for the exposition, but not necessarily for their proof).

• Degrees.

There is a family of r integers d_{1}, d_{2}, …, d_{r} called the degrees of (V, W), defined as follows : the Poincaré polynomial of the graded module (V ⊗ S_{W})^{W} is

q^{d_{1}−1} + q^{d_{2}−1} + ⋯ + q^{d_{r}−1} .

We have

(q + d_{1} − 1)(q + d_{2} − 2) ⋯ (q + d_{r} − 1) = ∑_{w∈W} q^{dimV^{⟨w⟩}}

(where V^{⟨w⟩} denotes the space of fixed points of w). It follows that

∑_{j=1}^{j=r}(d_{j} − 1) = ∑_{H∈A}(e_{H} − 1) = ∑_{C∈A/W} N_{C}(e_{C} − 1) = N^{∗} .

• Codegrees.

There is a family of r integers d_{1}^{∗}, d_{2}^{∗}, …, d_{r}^{∗} called the codegrees of (V, W), defined by the following condition: the Poincaré polynomial of the graded module (V^{∗} ⊗ S_{W})^{W} is

q^{d_{1}^{∗}+1} + q^{d_{2}^{∗}+1} + ⋯ + q^{d_{r}^{∗}+1} .

We have

(q − d_{1}^{∗} − 1)(q − d_{2}^{∗} − 1) ⋯ (q − d_{r}^{∗} − 1) = ∑_{w∈W} det_{V} (w)q^{dimV^{⟨w⟩}} .

It follows that

∑_{j=1}^{j=r}(d_{j}^{∗} + 1) = ∑_{H∈A} 1 = ∑_{C∈A/W} N_{C} = N .

and so

N + N^{∗} = ∑_{j=1}^{j=r}(d_{j} + d_{j}^{∗}) = ∑_{C∈A/W} N_{C}e_{C} .

Remark. The “codegrees” have not been introduced as such in the quoted literature. Nevertheless, the sets of degrees and the codegrees are related to the sets of exponents {m_{1}, m_{2}, …, m_{r}} and coexponents {m_{1}^{∗}, m_{2}^{∗}, …, m_{r}^{∗}} (which are defined in [OrSo2]) by the formulae

m_{j} = d_{j} − 1 and m_{j}^{∗} = d_{j}^{∗} + 1 (j = 1, 2, …, r) .

• Algebra of invariants – More on degrees.

The algebra of invariants R is generated by r algebraically independent homogeneous elements of S respectively of degrees d_{1}, d_{2}, …, d_{r}.

The order of W is |W | = d_{1}d_{2} ⋯ d_{r}.

If W is irreducible, its center Z(W) has order |Z(W)| = d_{1} ∧ d_{2} ∧ ⋯ ∧ d_{r} (where we denote d_{1} ∧ d_{2} ∧ ⋯ ∧ d_{r} := gcd{d_{1}, d_{2}, …, d_{r}}).

• Cohomology of the hyperplane complements – More on codegrees.

We set M := V − ⋃_{H∈A} H. For H ∈ A, let us denote by α_{H} a linear form on V with kernel H, and let us define the holomorphic differential form ω_{H} on M by the formula

ω_{H} := \frac{1}{2iπ} \frac{dα_{H}}{α_{H}} ,

which we also write ω_{H} = \frac{1}{2iπ} dLog(α_{H}) . We denote by [ω_{H}] the corresponding de Rham cohomology class.

Brieskorn (cf. [Br2], Lemma 5) has proved the following result.

1.2. Let ℂ[(ω_{H})_{H∈A}] (resp. ℤ[(ω_{H})_{H∈A}]) be the ℂ-subalgebra (resp. the ℤ-subalgebra) of the ℂ-algebra of holomorphic differential forms on M which is generated by {ω_{H}}_{H∈A}. Then the map ω_{H} ↦ [ω_{H}] induces an isomorphism between ℂ[(ω_{H})_{H∈A}] and the cohomology algebra H^{∗}(M, ℂ) (resp. an isomorphism between ℤ[(ω_{H})_{H∈A}] and the singular cohomology algebra H^{∗}(M, ℤ)).

From now on, we write ω_{H} instead of [ω_{H}].

Orlik and Solomon (cf. [OrSo1]) have given a description of the algebra H^{∗}(M, ℂ). Before stating their result, we need to introduce more notation.

• Let ℂA := ⨁_{H∈A} e_{H} be the vector space with basis indexed by A, and let ΛA be its exterior algebra, endowed with the usual Koszul differential map δ: ΛA → ΛA of degree −1.

• For B = {H_{1}, H_{2}, …, H_{k}} ⊂ A, we denote by D_{B} the line generated by e_{H_{1}} ∧ e_{H_{2}} ∧ ⋯ ∧ e_{H_{k}}.

• We say that B is dependent if codim(⋂_{H∈B} H) < |B|.

• We denote by IΛA the (graded) ideal of ΛA generated by the δ(D_{B}) where B runs overs the set of all dependent subsets of A.

1.3. Theorem. (Orlik and Solomon) The map e_{H} ↦ ω_{H} induces an isomorphism of graded algebras between ΛA/IΛA and H^{∗}(M, ℂ).

Let Int(A) be the set of intersections of elements of A. For X ∈ Int(A), we set H^{(X)}(M, ℂ) := ∑ D_{B} where the summation is taken over all B ⊂ A, |B| = codim(X), ⋂_{H∈B} H = X, and where D_{B} is the complex line generated by ω_{H_{1}} ∧ ω_{H_{2}} ∧ ⋯ ∧ ω_{H_{k}} if B = (H_{1}, H_{2}, …, H_{k}).

Then it follows from Theorem 1.3 that

1.4. for any integer n, we have

H^{n}(M, ℂ) = ⨁_{(X∈Int(A)) (codim(X)=n)} H^{(X)}(M, ℂ) .

Moreover, we see that

1.5.

(1) the family (ω_{H})_{H∈A} is a basis of H^{1}(M, ℂ),

(2) for X an element of Int(A) with codimension 2, if H_{X} denotes a fixed element

of A which contains X,

• whenever H and H′ are two elements of A which contain X, we have ω_{H} ∧ ω_{H′} = ω_{H_{X}} ∧ ω_{H′} − ω_{H_{X}} ∧ ω_{H} ,

• the family (ω_{H_{X}} ∧ ω_{H})_{(H⊃X)(H≠H_{X})} is a basis of H^{(X)}(M, ℂ).

The codegrees are determined by the arrangement A, by the following consequence of Theorem 1.3.

1.6. The Poincaré polynomial P_{M}(q) := ∑_{n} q^{n}dim(H^{n}(M, ℂ)) of the cohomology algebra H^{∗}(M, ℂ) is given by the following formulae :

P_{M}(q) = (1 + (1 + d_{1}^{∗})q)(1 + (1 + d_{2}^{∗})q) ⋯ (1 + (1 + d_{r}^{∗})q)

= ∑_{w∈W} det_{V} (w)(−q)^{codim(V^{⟨w⟩})} .

B. Presentations.

The tables in Appendix 2 provide a complete list of the irreducible finite pseudo– reflection groups, as classified by Shephard and Todd, together with presentations of these groups symbolized by diagrams “à la Coxeter”, as well as some of the data attached to these groups. Many of these presentations were previously known. This is the case of the rank r groups which are generated by r reflections, studied by Coxeter [Cx]. Some others (the ones corresponding to the infinite series) occurred in [BrMa] or were inspired by [Ari].

The reader may refer to Appendix 2 to understand what follows.

Isomorphisms between diagrams.

We may notice that the only isomorphisms between the diagrams of our tables are between the diagrams of G(2, 1, 2) and G(4, 4, 2), between the diagrams of 𝔖_{4} and G(2, 2, 3), between the diagrams of 𝔖_{3} and G(3, 3, 2), and between the diagrams of 𝔖_{2} and G(2, 1, 1).

Coxeter diagrams.

Note (see tables) the following correspondence of notation :

• 𝔖_{r+1} (r ≥ 0) is the Coxeter group of type A_{r},

• G(2, 1, r) (r ≥ 2) is the Coxeter group of type B_{r},

• G(2, 2, r) (r ≥ 3) is the Coxeter group of type D_{r}.

Indeed (see table 2 for notation) since e = 2, t_{2} and t′_{2} commute, and it is enough to show that the “ double–link” braid relation t_{3}t′_{2}t_{2}t_{3}t′_{2}t_{2} = t′_{2}t_{2}t_{3}t′_{2}t_{2}t_{3} is a consequence of the other relations.

Applying successively the fact that t_{2} and t′_{2} commute, the braid relation between t_{3} and t′_{2}, and the braid relation between t_{3} and t_{2}, we get t_{3}t′_{2}t_{2}t_{3}t′_{2}t_{2} = t_{3}t_{2}t′_{2}t_{3}t′_{2}t_{2} = t_{3}t_{2}t_{3}t′_{2}t_{3}t_{2} = t_{2}t_{3}t_{2}t′_{2}t_{3}t_{2} = t_{2}t_{3}t′_{2}t_{2}t_{3}t_{2} = t_{2}t_{3}t′_{2}t_{3}t_{2}t_{3} = t_{2}t′_{2}t_{3}t′_{2}t_{2}t_{3} = t′_{2}t_{2}t_{3}t′_{2}t_{2}t_{3} .

• G(e, e, 2) (e ≥ 3) is the dihedral group of order 2e,

• G_{28} is the Coxeter group of type F_{4},

• G_{35} is the Coxeter group of type E_{6},

• G_{36} is the Coxeter group of type E_{7},

• G_{37} is the Coxeter group of type E_{8},

• G_{23} is the Coxeter group of type H_{3},

• G_{30} is the Coxeter group of type H_{4}.

Admissible subdiagrams and parabolic subgroups.

Let D be one of the diagrams. Let us define an equivalence relation between nodes by s ∼ s and, for s ≠ t,

s ∼ t ⟺ s and t are not in a homogeneous relation with support {s, t} .

Then we see that the equivalence classes have 1 or 3 elements, and that there is at most one class with 3 elements.

If there is no class with 3 elements, the rank r of the group is the number of nodes of the diagram, while it is this number minus 1 in case there is a class with 3 elements.

Thus s◯ 2_{5} ^{◯ 2 ^{t}}_{◯ 3 _{u}} has rank 2, as well as ◯_{s}_{4}  ◯_{t}_{3} .

Remark. One must point out that, in the first of the preceding two diagrams, s, t and u must be considered as linked by a line (so t and u do not commute).

An admissible subdiagram is a full subdiagram of the same type, namely a diagram with 1 or 3 elements per class.

Thus, the diagram s◯ 2_{5} ^{◯ 2 ^{t}}_{◯ 3 _{u}} has five admissible subdiagrams, namely the empty

diagram, the three diagrams consisting of one node, and the whole diagram.

1.7. Fact. Let D be the diagram of W as given in tables 1 to 4 in Appendix 2 below.

(1) If D′ is an admissible subdiagram of D, it gives a presentation of the corresponding subgroup W (D′) of W. This subgroup is a parabolic subgroup.

(2) Assume W is neither G_{27}, G_{29}, G_{33} nor G_{34}. If P_{1} ⊆ P_{2} ⊆ ⋯ P_{n} is a chain of parabolic subgroups of W, there exist g ∈ W and a chain D_{1} ⊆ D_{2} ⊆ ⋯ D_{n} of admissible subdiagrams of D such that

(P_{1}, P_{2}, …, P_{n}) = ^{g}(W (D_{1}), W (D_{2}), …, W (D_{n})) .

Remark.

For groups G_{27} and G_{29}, all isomorphism classes of parabolic subgroups are represented by admissible subdiagrams of our diagrams, but not all conjugacy classes of parabolics subgroups are represented by admissible subdiagrams, as noticed by Orlik.

For groups G_{33} and G_{34}, not all isomorphism classes of parabolic subgroups are represented by admissible subdiagrams of our diagrams. In these cases, it seems that a second diagram should be introduced, as suggested by [Hu]. Then all parabolic subgroups can be found somewhere inside one of the two diagrams given.

More precisely, for G_{33}, the second diagram is only needed for parabolic subgroups of type D_{4}, while for G_{34} it is needed for parabolic subgroups of type D_{4}, D_{5} and the second copy of A_{5}.

2. Braid groups and their diagrams

For X a topological space, we denote by P(X) its fundamental groupoid, where the composition of (classes of) paths is defined so that, if γ_{1} is a path going from x_{0} to x_{1} and γ_{2} is a path going from x_{1} to x_{2}, then the composite map going from x_{0} to x_{2} is denoted by γ_{2} · γ_{1}.

Given a point x_{0} ∈ X, we denote by π_{1}(X, x_{0}) (or π_{1}(X) if the choice of x_{0} is clear) the fundamental group with base point x_{0}. So we have π_{1}(X, x_{0}) = End_{P(X)}(x_{0}). If f: X → Y is a continuous map, we denote by P(f) the corresponding functor from P(X) to P(Y). We also denote by π_{1}(f, x_{0}) (or π_{1}(f)) the group homomorphism from π_{1}(X, x_{0}) to π_{1}(Y, f(x_{0})) induced by P(f).

We choose, once for all, a square root of (−1) in ℂ, which is denoted by i. Moreover, for every z ∈ ℂ^{×}, we identify π_{1}(ℂ^{×}, z) with ℤ by sending onto 1 the loop λ_{z}: [0, 1] → ℂ^{×} defined by λ_{z}(t) := z exp(2iπt).

A. Generalities about hyperplane complements.

What follows is probably well known to specialists of hyperplane complements and topologists. We include it for the convenience of the reader, and because of the lack of convenient references.

Let A be a finite set of affine hyperplanes (i.e., affine subspaces of codimension one) in a finite dimensional complex vector space V. We set M := V − ⋃_{H∈A} H .

Let x_{0} ∈ M. We shall give now some properties of the fundamental group π_{1}(M, x_{0}).

Generators of the monodromy around the hyperplanes.

In Appendix 1, we explain what we mean by the generator ρ_{[γ]} of the monodromy around H, associated to a path γ “from x_{0} to an affine hyperplane” H ∈ A.

For H ∈ A, let α_{H} be an affine map V → ℂ such that H = {x ∈ V | α_{H}(x) = 0} . Its restriction to M → ℂ^{×} induces a functor P(α_{H}): P(M) → P(ℂ^{×}) , and in particular a group homomorphism π_{1}(α_{H}, x_{0}): π_{1}(M, x_{0}) → ℤ

2.1. Lemma. For H, H′ ∈ A and γ a path from x_{0} to H (see Appendix 1), we have

π_{1}(α_{H′})(ρ_{[γ]}) = δ_{H,H′} .

Proof of 2.1.

Let us set M_{H} := H − ⋃_{{H′∈A \atop H′≠H}} H′ . Let x_{γ} := γ(1) and let B be an open ball

with center x_{γ} contained in M ∪ M_{H}. Let u ∈ [0, 1[ such that γ(t) ∈ B for t ≥ u. We set x_{1} := γ(u). Then, the restriction of α_{H} to B ∩ M induces an isomorphism π_{1}(α_{H}): π_{1}(B ∩ M, x_{1}) → ℤ . Let λ be a loop in B ∩ M, with origin x_{1}, whose image under π_{1}(α_{H}) is 1. Let γ_{u} be the “restriction” of γ to [0, u], defined by γ_{u}(t) := γ(ut) for all t ∈ [0, 1]. Define ρ_{γ,λ} := γ_{u}^{−1} · λ · γ_{u} . Then the loop ρ_{γ,λ} induces the generator of the monodromy ρ_{[γ]} (see Appendix 1), and

π_{1}(α_{H′})(ρ_{γ,λ}) = π_{1}(α_{H′})(λ) = δ_{H,H′} .

□

2.2. Proposition.

(1) The fundamental group π_{1}(M, x_{0}) is generated by all the generators of the monodromy around the affine hyperplanes H ∈ A.

(2) Let π_{1}(M, x_{0})^{ab} denote the largest abelian quotient of π_{1}(M, x_{0}). For H ∈ A, we denote by ρ_{H}^{ab} the image of ρ_{H,γ} in π_{1}(M, x_{0})^{ab}. Then

π_{1}(M, x_{0})^{ab} = ∏_{H∈A} ⟨ρ_{H}^{ab}⟩ ,

where each ⟨ρ_{H}^{ab}⟩ is infinite cyclic. Dually, we have

Hom(π_{1}(M, x_{0}), ℤ) = ∏_{H∈A} ⟨π_{1}(α_{H})⟩ .

Proof of 2.2.

(1) is a special case of Proposition A1 (see Appendix 1 below).

(2) is immediate and left to the reader. □

Remark. Let us recall that we have natural isomorphisms

π_{1}(M, x_{0})^{ab} ⟶^{∼} H_{1}(M, ℤ) and Hom(π_{1}(M, x_{0}), ℤ) ⟶^{∼} H^{1}(M, ℤ) .

Moreover, the duality between π_{1}(M, x_{0})^{ab} and H^{1}(M, ℤ) may be seen as follows. For γ a loop in M with origin x_{0} and for ω a holomorphic differential 1-form on M, we set ⟨γ, ω⟩ := ∫_{γ} ω . It is then clear that, under the isomorphism

Hom(π_{1}(M, x_{0}), ℤ) ⟶^{∼} H^{1}(M, ℤ) ,

the element π_{1}(α_{H}) is sent onto the class of the 1-form ω_{H} = \frac{1}{2iπ} \frac{dα_{H}}{α_{H}} (see 1.2 for more

details).

About the center of the fundamental group.

In this part, we assume the hyperplanes in A to be linear.

2.3. Notation. We denote by π the loop [0, 1] → M defined by

π: t ↦ x_{0} exp(2iπt) .

2.4. Lemma.

(1) π belongs to the center Z(π_{1}(M, x_{0})) of the fundamental group π_{1}(M, x_{0}).

(2) For all H ∈ A, we have π_{1}(α_{H})(π) = 1 .

Proof of 2.4.

(1) results from a more general lemma, for which we need to introduce more notation. Let z = |z|e^{iθ} be a complex number with argument θ, chosen so that −π < θ ≤ π.

For t ∈ [0, 1], we set z^{t} := |z|^{t}e^{tiθ}. For x ∈ M, we denote by γ_{z,x} the path in M, with initial point x and terminal point zx, defined by

γ_{z,x}: [0, 1] → M , t ↦ z^{t}x .

2.5. Lemma. Let γ be a path in M, with initial point x and terminal point y. Then the paths γ_{z,y} ·γ and zγ ·γ_{z,x} are homotopy equivalent (where zγ denotes the path defined by t ↦ zγ(t)).

The proof of Lemma 2.5 is easy and left to the reader. Note that Lemma 2.5 holds whenever M is a subset of V which is stable under multiplication by ℂ^{×}.

(2) is immediate since π_{1}(α_{H})(π) = \frac{1}{2πi} ∫_{0}^{1} \frac{d(α_{H}(π(t)))}{α_{H}(π(t))} = ∫_{0}^{1} dt . □

2.6. Proposition. Let \overline{M} be the image of M in (V − \{0\})/ℂ^{×}, and let \overline{x}_{0} denote the image of x_{0} in \overline{M}.

(1) The map π_{1}(M, x_{0}) → π_{1}(\overline{M}, \overline{x}_{0}) is surjective, and its kernel is ⟨π⟩.

(2) The center of π_{1}(M, x_{0}) is ⟨π⟩ if and only if the center of π_{1}(\overline{M}, \overline{x}_{0}) is trivial.

Proof of 2.6.

(1) Since the π(t) are scalar multiples of x_{0}, it is clear that π belongs to the kernel of the map π_{1}(M, x_{0}) → π_{1}(\overline{M}, \overline{x}_{0}).

The homotopy exact sequence ⋯ → π_{1}(ℂ^{×}) → π_{1}(M, x_{0}) → π_{1}(\overline{M}, \overline{x}_{0}) → 1 shows that the morphism π_{1}(M, x_{0}) → π_{1}(\overline{M}, \overline{x}_{0}) is surjective, and that its kernel is cyclic. Since π belongs to this kernel, it suffices to prove that π is a primitive element of π_{1}(M, x_{0}), i.e., that π has no proper root in π_{1}(M, x_{0}). But this results from Lemma 2.4, (2).

(2) Let us notice that, by Lemma 2.4, (2), the group ⟨π⟩ maps injectively into the largest abelian quotient of π_{1}(M, x_{0}). So it suffices to prove the following elementary lemma.

2.7. Lemma. Let G be a group, and let H a normal subgroup of G which maps injectively into the largest abelian quotient G/[G, G] of G. Then the natural morphism G → G/H sends the center of G onto the center of G/H.

Proof of 2.7. Indeed, let z be an element of G which becomes central in G/H. Then [z, G] ⊂ H. But by hypothesis we have H ∩ [G, G] = 1. Thus we have [z, G] = 1. □

□

Generating with one loop per hyperplane.

With a little more work, Proposition 2.2 can be made more precise ; one (wellchosen) generator of the monodromy around each affine hyperplane suffices to generate the fundamental group :

2.8. Proposition. There is a set R = {ρ_{H}}_{H∈A} of generators of π_{1}(M, x_{0}), where ρ_{H} is a generator of the monodromy around H.

Proof of 2.8.

We may assume that A is not empty. We prove the proposition by induction on the dimension of V.

• The linear case.

Let us first consider the case where the intersection of the affine hyperplanes of A is non trivial. Up to translation, we can assume that 0 is contained in this intersection, i.e., the hyperplanes of A are linear.

Let H_{2} be a hyperplane of A and H_{1} be the affine hyperplane of V parallel to H_{2} and containing x_{0}.

We consider the conic projection on H_{1} with center 0 :

f : (V − H_{2}) ⟶ H_{1} , x ⟼ ℂx ∩ H_{1}.

Both f and its restriction M → M_{1} = H_{1} ∩ M are locally trivial fibrations (see for example [Spa], chap. 2) with fiber F ≃ ℂ^{×}x.

The associated exact sequence of fundamental groups is

π_{1}(F, x_{0}) → π_{1}(M, x_{0}) ⟶^{π_{1}(f)}π_{1}(M_{1}, x_{0}) → 1 .

Let A′ = A − {H_{2}}. By induction, we can assume the proposition holds for the affine hyperplane arrangement A_{1} = {H ∩ H_{1}}_{H∈A′} in H_{1} : there is a set {ρ_{L}}_{L∈A_{1}} of generators of π_{1}(M_{1}, f(x_{0})) where ρ_{L} is a generator of the monodromy around L.

Let i be the inclusion M_{1} → M. Then, ρ_{H} = π_{1}(i)(ρ_{H∩H_{1}}) is a generator of the monodromy around H ∈ A′. Note that fi is the identity on M_{1}, hence π_{1}(f)π_{1}(i) = 1. In particular, the exact sequence shows that π_{1}(M, x_{0}) is generated by the set {ρ_{H}}_{H∈A′}, together with π, the image of the positive generator of π_{1}(F, x_{0}), which is central in π_{1}(M, x_{0}).

Let ρ_{H_{2}} be a generator of the monodromy around H_{2}. Then, there exists α in the subgroup generated by {ρ_{H}}_{H∈A′} such that π_{1}(f)(ρ_{H_{2}}α) = 1, that is, ρ_{H_{2}}α = π^{r} for some integer r. Since π_{1}(α_{H_{2}})(ρ_{H_{2}}) = 1, π_{1}(α_{H_{2}})(ρ_{H}) = 0 for H ∈ A′ and π_{1}(α_{H_{2}})(π) = 1 by Lemmas 2.1 and 2.4, we obtain r = 1. Hence, π is in the subgroup generated by {ρ_{H}}_{H∈A} and this proves that {ρ_{H}}_{H∈A} generates π_{1}(M, x_{0}).

• The affine case.

Let A′ be a finite set of affine hyperplanes of V disjoint from A and let M′ = V − ⋃_{H∈A∪A′} H. Assume x_{0} ∈ M′. Since one gets M′ by removing a sub–variety of (real) codimension 2 from M, the injection M′ ↪ M induces a surjection π_{1}(M′, x_{0}) ↠ π_{1}(M, x_{0}) (see for example [Go], chap. x, th. 2.3.). Under this morphism, a generator of the monodromy around a hyperplane H′ ∈ A′ becomes trivial. Hence, if the proposition holds for M′, then it holds for M. Note also that we can change the base point x_{0} in order to prove the result.

We choose an affine hyperplane H_{1} of V outside A, a new origin 0 ∈ V − H_{1} for V and a new base point x_{0} ∈ H_{1} ∩ M such that

• there is an open ball Ω with center 0, containing x_{0} and which doesn’t intersect any of the non-linear hyperplanes of M,

• the line ℂx_{0} never intersects two distinct affine hyperplanes of A at the same point, and

• no translate of the line ℂx_{0} lies in an affine hyperplane of A.

Then, adding to A

• the linear hyperplane H_{2} parallel to H_{1},

• the linear hyperplanes parallel to the affine hyperplanes of A,

• given two distinct non-linear hyperplanes H, H′ of A, the linear hyperplane containing H ∩ H′,

we may and will assume that A satisfies the following assumption :

Let A′ be the set of linear hyperplanes of A distinct from H_{2}, A′′ the set of non-linear hyperplanes of A, A_{1} = {H_{1} ∩ H}_{H∈A′} and M_{1} = H_{1} − ⋃_{L∈A_{1}} L. Then, the map

f : M ⟶ M_{1}
x ⟼ ℂx ∩ H_{1}

is a locally trivial fibration.

[figure: 0; +; x_{0}; ×; f(x_{0}); H_{2}; H_{1}; ℂx_{0}; Ω; •; •; •]

Note that the restriction of f to Ω ∩ M → M_{1} is also a locally trivial fibration. The associated exact sequences of fundamental groups give rise to the commutative

diagram :

[figure: π_{1}(Ω ∩ (ℂx_{0} − {0}), x_{0}); π_{1}(Ω ∩ M, x_{0}); π_{1}(M_{1}, f(x_{0})); 1; π_{1}(F, x_{0}); π_{1}(M, x_{0}); π_{1}(M_{1}, f(x_{0})); 1]

where F = f^{−1}(f(x_{0})) = ℂx_{0} − ({0} ∪ {ℂx_{0} ∩ H}_{H∈A′′}) .

The study of the linear case above shows that there is a set {ρ_{h}}_{H∈A′∪H_{2}} of generators of the monodromy around the linear hyperplanes in Ω∩M which generates π_{1}(Ω∩M, x_{0}).

There are generators of the monodromy ρ_{H} around the points ℂx_{0} ∩ H (H ∈ A′′) in F, such that, together with the image of π_{1}(Ω∩(ℂx_{0}−{0}), x_{0}), they generate π_{1}(F, x_{0}).

Now, the set {ρ_{H}}_{H∈A} generates π_{1}(M, x_{0}). □

B. Generalities about the braid groups.

More notation. We go back to notation introduced in §1. In particular, A is now the set of reflecting

hyperplanes of a finite subgroup W of GL(V) generated by pseudo–reflections. We denote by p: M → M/W the canonical surjection.

Notice first that we may assume that W is “essential”, i.e., that the subspace V^{W} of fixed points of V under W is trivial. Indeed, let us set

V_{0} := ⋂_{H∈A} H , \overline{V} := V/V_{0} , \overline{A} := \{H/V_{0} ; (H ∈ A)\} , \overline{M} := \overline{V} − ⋃_{\overline{H}∈\overline{A}} \overline{H} .

Then the natural epimorphism s: V ↦ \overline{V} induces a trivial fibration s: M ↦ \overline{M} with fibre V_{0}, hence a homotopy equivalence, and in particular an isomorphism

(2.9)  π_{1}(M, x_{0}) ⟶^{∼} π_{1}(\overline{M}, s(x_{0})) .

Let x_{0} ∈ M. We introduce the following notation for the fundamental groups:

P := π_{1}(M, x_{0}) and B := π_{1}(M/W, p(x_{0})) ,

and we call B and P respectively the braid group (at x_{0}) and the pure braid group (at x_{0}) associated to W. We shall often write π_{1}(M/W, x_{0}) for π_{1}(M/W, p(x_{0})).

The covering M → M/W is Galois by Steinberg’s theorem (see Theorem 1.1 above), hence the projection p induces a surjective map B ↠ W , σ ↦ \overline{σ}, as follows :

Let σ̃: [0, 1] → M be a path in M, such that σ̃(0) = x_{0}, which lifts σ. Then \overline{σ} is defined by the equality \overline{σ}(x_{0}) = σ̃(1) .

The map σ ↦ \overline{σ} is an anti–morphism. Indeed, if σ′ is another loop in M/W with origin \overline{x}_{0}, and if σ′ is lifted onto a path σ̃′ with origin x_{0} in M, we may lift the loop (σ′σ) onto the path \overline{σ}(σ̃′) · σ̃, whose image in W is clearly \overline{σ}\overline{σ}′ (here we set \overline{σ}(σ̃′)(t) := \overline{σ}(σ̃′(t))).

Denoting by W^{op} the group opposite to W, we have the following short exact sequence :

(2.10)  1 → P → B → W^{op} → 1 ,

where the map B → W^{op} is defined by σ ↦ \overline{σ}.

The spaces M and M/W are conjectured to be K(π, 1)-spaces.

The following result is due to Fox and Neuwirth [FoNe] for the type A_{n}, to Brieskorn [Br2] for Coxeter groups of type different from H_{3}, H_{4}, E_{6}, E_{7}, E_{8}, to Deligne [De1] for general Coxeter groups. The case of the infinite series of complex reflection groups G(de, e, r) has been solved by Nakamura [Na]. For the non-real Shephard groups (nonreal groups with Coxeter braid diagrams), this has been proven by Orlik and Solomon [OrSo3]. Note that the rank 2 case is trivial.

2.11. Theorem. Assume W has no irreducible component of type G_{24}, G_{27}, G_{29}, G_{31}, G_{33} or G_{34}. Then, M and M/W are K(π, 1)-spaces.

Generators of the monodromy around the hyperplanes.

For H ∈ A, we set ζ_{H} := exp(2iπ/e_{H}), and we denote by s_{H} the pseudo-reflection in W with reflecting hyperplane H and determinant ζ_{H}. We set

L_{H} := im(s_{H} − Id_{V}) .

For x ∈ V, we set x = pr_{H}(x) + pr_{H}^{⊥}(x) with pr_{H}(x) ∈ H and pr_{H}^{⊥}(x) ∈ L_{H}. Thus, we have s_{H}(x) = ζ_{H}pr_{H}^{⊥}(x) + pr_{H}(x) . If t ∈ ℝ, we set ζ_{H}^{t} := exp(2iπt/e_{H}), and we denote by s_{H}^{t} the element of GL(V) (a

pseudo–reflection if t ≠ 0) defined by :

(2.12)  s_{H}^{t}(x) = ζ_{H}^{t}pr_{H}^{⊥}(x) + pr_{H}(x) .

For x ∈ V, we denote by σ_{H,x} the path in V from x to s_{H}(x), defined by :

σ_{H,x}: [0, 1] → V , t ↦ s_{H}^{t}(x) .

For any path γ in M, with initial point x_{0} and terminal point x_{H}, the path defined by s_{H}(γ^{−1}): t ↦ s_{H}(γ^{−1}(t)) is a path in M going from s_{H}(x_{H}) to s_{H}(x_{0}).

Whenever γ is a path in M, with initial point x_{0} and terminal point x_{H}, we define the path σ_{H,γ} from x_{0} to s_{H}(x_{0}) as follows :

(2.13)  σ_{H,γ} := s_{H}(γ^{−1}) · σ_{H,x_{H}} · γ .

It is not difficult to see that, provided x_{H} is chosen “close to H”, the path σ_{H,γ} is in M, and its homotopy class does not depend on the choice of x_{H}, and the element it induces in the braid group B is actually a generator of the monodromy around the image of H in M/W (see Appendix 1 below).

The following properties are immediate.

2.14. Lemma.

(1) The image of s_{H,γ} in W is s_{H}.

(2) Whenever γ′ is a path in M, with initial point x_{0} and terminal point x_{H}, if τ denotes the loop in M defined by τ := γ′^{−1}γ, one has

σ_{H,γ′} = τ · σ_{H,γ} · τ^{−1}

and in particular s_{H,γ} and s_{H,γ′} are conjugate in P.

(3) The path ∏_{j=e_{H}−1}^{j=0} σ_{H,s_{H}^{j}(γ)} , a loop in M, induces the element s_{H,γ}^{e_{H}} in the braid group B, and belongs to the pure braid group P. It is homotopy equivalent, as a loop in M, to the generator ρ_{[γ]} of the monodromy around H in P (see Appendix 1).

2.15. Definition.

• A distinguished pseudo–reflection in W is a pseudo–reflection s with the following property : if H denotes its reflecting hyperplane, and if e_{H} is the order of the minimal parabolic subgroup W_{H}, then s is the element of W_{H} with determinant e^{2iπ/e_{H}}.

• Let s be a distinguished pseudo–reflection in W, with reflecting hyperplane H. An s–generator of the monodromy is a generator of the monodromy s around the image of H in M/W such that \overline{s} = s.

The discriminants.

Let C be an orbit of W on A. Recall that we denote by e_{C} the (common) order of the pointwise stabilizer W_{H} for H ∈ C. We call discriminant at C and we denote by δ_{C} the element of the symmetric algebra of V^{∗} defined (up to a non zero scalar multiplication) by

δ_{C} := ( ∏_{H∈C} α_{H})^{e_{C}} .

Since (see for example [Co], 1.8) δ_{C} is W -invariant, it induces a continuous function δ_{C}: M/W → ℂ^{×} , hence induces a functor P(δ_{C}): P(M/W) → P(ℂ^{×}) , and in particular it induces a group homomorphism π_{1}(δ_{C}): B → ℤ .

2.16. Proposition. For any H ∈ A, we have

π_{1}(δ_{C})(s_{H,γ}) = \{ \begin{matrix}1 if H ∈ C , \\ 0 if H ∉ C . \end{matrix}

Proof of 2.16.

Let us set C^{#} := C − {H} (so C^{#} = C if H ∉ C), and δ_{C^{#}} := (∏_{H′∈C^{#}} α_{H′})^{e_{C}} .

Recall that W_{H} denotes the (parabolic) subgroup of W generated by s_{H}. Then the maps

δ_{C^{#}} , α_{H}^{e_{H}}: M → ℂ^{×}

are both W_{H}-invariant, and so define maps

δ_{C^{#}} , α_{H}^{e_{H}}: M/W_{H} → ℂ^{×} .

The following diagram summarizes where the maps are defined :

\xymatrix{ &  & M/W \ar[ddr]^{δ_{C}} \\  & M/W_{H} \ar@{-->}[ur] \ar[drr]^{(α_{H})^{e_{H}} , δ_{C^{#}}} \\ M \ar[rrr]^{α_{H′}} \ar[ur] &  &  & ℂ^{×}}

The computation of π_{1}(δ_{C}(s_{H,γ})) may be performed at the level M/W_{H}, and so it suffices to check

(1) π_{1}(δ_{C^{#}})(s_{H,γ}) = 0 ,

(2) π_{1}(α_{H}^{e_{H}})(s_{H,γ}) = 1 . Let us check (1) and (2).

(1) It suffices to check that π_{1}(δ_{C^{#}})(s_{H,γ}^{e_{H}}) = 0 , and this follows from Lemmas 2.1 and 2.14, (3).

(2) We have

π_{1}(α_{H}^{e_{H}})(s_{H,γ}) = \frac{1}{2iπ} ∫_{0}^{1} \frac{d(α_{H}(s_{H}^{t}(x_{H}))^{e_{H}})}{α_{H}(s_{H}^{t}(x_{H}))^{e_{H}}} .

Since

α_{H}(s_{H}^{t}(x_{H}))^{e_{H}} = α_{H}(pr_{H}(x_{H}) + ζ_{H}^{t}pr_{H}^{⊥}(x_{H}))^{e_{H}}
= ζ_{H}^{te_{H}} α_{H}(pr_{H}^{⊥}(x_{H}))^{e_{H}}
= exp(2iπt)α_{H}(pr_{H}^{⊥}(x_{H}))^{e_{H}} ,

we see that π_{1}(α_{H}^{e_{H}})(s_{H,γ}) = ∫_{0}^{1} dt = 1 . □

Generators and abelianization of B.

2.17. Theorem.

(1) The group B is generated by the generators {s_{H,γ}} (for all hyperplanes H ∈ A and all paths γ from x_{0} to H in M) of the monodromy (in B) around the elements of A.

(2) We denote by B^{ab} the largest abelian quotient of B. For C ∈ A/W, we denote by s_{C}^{ab} the image of s_{H,γ} in B^{ab} for H ∈ C. Then

B^{ab} = ∏_{C∈A/W} ⟨s_{C}^{ab}⟩ ,

where each ⟨s_{C}^{ab}⟩ is infinite cyclic. Dually, we have

Hom(B, ℤ) = ∏_{C∈A/W} ⟨π_{1}(δ_{C})⟩ .

Remark. We have natural isomorphisms

B^{ab} ⟶^{∼} H_{1}(M/W, ℤ) and Hom(B, ℤ) ⟶^{∼} H^{1}(M/W, ℤ) ,

and, under the second isomorphism, we have

π_{1}(δ_{C}) ↦ e_{C} ∑_{H∈C} \frac{1}{2iπ} \frac{dα_{H}}{α_{H}} = \frac{1}{2iπ} dLog(δ_{C}) .

Proof of 2.17. The second assertion is immediate by the first one and by Proposition 2.16. Let us sketch a proof of (1).

Since W is generated by the set {s_{H}}_{(H∈A)} and since we have the exact sequence (2.10), it is enough to prove that the pure braid group P is generated by all the conjugates in P of the elements s_{H,γ}^{e_{H}}. This is a consequence of Proposition 2.2, (1). □

Let us denote by Gen(B) the set of all generators of the monodromy in B (see Definition 2.15 above). For s ∈ Gen(B), we denote by e_{s} the order of \overline{s}.

In other words, if s is a generator of the monodromy around the reflecting hyperplane H ∈ A, we set now (using the notation of Definition 2.15) : e_{s} := e_{H}.

The following property is a consequence of general results recalled in Appendix 1 below.

2.18. Proposition.

(1) The pure braid group P is generated by {s^{e_{s}}}_{s∈Gen(B)}.

(2) We have

W ≃ B/⟨s^{e_{s}}⟩_{s∈Gen(B)} .

Proof of 2.18. The two assertions are obviously equivalent. The first one results from Propositions A2 and A3, (2) (see Appendix 1 below). □

Length.

Let

δ := ∏_{C∈A/W} δ_{C}

be the discriminant, and let π_{1}(δ): B → ℤ be the corresponding group morphism.

Let b ∈ B. By Theorem 2.17 above, there exists an integer k and for 1 ≤ j ≤ k, H_{j} ∈ A, a path γ_{j} from x_{0} to H_{j} and an integer n_{j} such that

b = s_{H_{1},γ_{1}}^{n_{1}}s_{H_{2},γ_{2}}^{n_{2}} ⋯ s_{H_{k},γ_{k}}^{n_{k}} .

The following proposition results from Proposition 2.16 above.

2.19. Proposition. We have

π_{1}(δ)(b) = ∑_{j=1}^{j=k} n_{j} .

We call the length of b and we denote by ℓ(b) the integer π_{1}(δ)(b).

If {s} is a set of generators of the monodromy around hyperplanes which generates B, let us denote by B^{+} the sub–monoid of B generated by {s}. Then for b ∈ B^{+}, its length ℓ(b) coincide with its length on the distinguished set of generators {s} of the monoid B^{+}.

About the center of B.

2.20. Notation. We denote by β the path [0, 1] → M defined by

β: t ↦ x_{0} exp(2iπt/|Z(W)|) .

The following result is a consequence of Corollary 2.26. Notice that it generalizes a result of Deligne [De1], (4.21) (see also [BrSa]), from which it follows that if W is a Coxeter group, then ℓ(π) = 2N. It was noticed “experimentally” in [BrMi], (4.8).

2.21. Corollary. We have ℓ(β) = (N + N^{∗})/|Z(W)| and ℓ(π) = N + N^{∗} .

From now on, we assume that W acts irreducibly on V. Note that, since W is irreducible on V, it results from Schur’s lemma that

Z(W) = {exp(2iπk/|Z(W)|) | (k ∈ ℤ)} ,

and so in particular β defines an element of B, which we will still denote by β.

2.22. Lemma.

(1) The image \overline{β} of β in W is the scalar multiplication by exp(2iπ/|Z(W)|). It is a generator of the center Z(W) of W.

(2) We have β ∈ Z(B), π ∈ Z(P), and π = β^{|Z(W)|}.

Proof of 2.22. We only have to check that β ∈ Z(B). This results from Lemma 2.5. □

2.23. Proposition. Let \overline{M} be the image of M in (V − \{0\})/ℂ^{×}. Then, we have a commutative diagram, where all short sequences are exact :

\xymatrix{ & 1 \ar[d] & 1 \ar[d] & 1 \ar[d] \\ 1 \ar[r] & ⟨π⟩ \ar[d] \ar[r] & ⟨β⟩ \ar[d] \ar[r] & Z(W)^{op} \ar[d] \ar[r] & 1 \\ 1 \ar[r] & π_{1}(M, x_{0}) \ar[r] \ar[d] & π_{1}(M/W, x_{0}) \ar[r] \ar[d] & W^{op} \ar[r] \ar[d] & 1 \\ 1 \ar[r] & π_{1}(\overline{M}, \overline{x}_{0}) \ar[r] \ar[d] & π_{1}(\overline{M}/W, \overline{x}_{0}) \ar[r] \ar[d] & (W/Z(W))^{op} \ar[r] \ar[d] & 1 \\  & 1 & 1 & 1}

Proof of 2.23.

It is clear that β belongs to the kernel of the map π_{1}(M/W, x_{0}) → π_{1}(\overline{M}/W, \overline{x}_{0}). By Lemma 2.22, we know that the map ⟨β⟩ → Z(W) is onto. The three horizontal

sequences are exact, as well as the last vertical one. So it suffices to check that the first vertical sequence is exact, i.e., to show that ⟨π⟩ is equal to the kernel of the map π_{1}(M, x_{0}) → π_{1}(\overline{M}, \overline{x}_{0}). This is Proposition 2.6, (1). □

The following statement is known for Coxeter groups (see [De1] or [BrSa]). The result holds as well for G_{25}, G_{26}, G_{32}, since the corresponding braid groups are the same as braid groups of Coxeter groups. We shall prove it for all the infinite series in §3 below (see Propositions 3.4, 3.10, 3.33), and we give below a proof for the particular case of groups in dimension 2.

We conjecture it is still true in the case of G_{31}, as well as for G_{24}, G_{27}, G_{29}, G_{33}, G_{34}.

2.24. Theorem. Assume W different from G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, G_{34}.

The center Z(B) of B is infinite cyclic and generated by β, the center Z(P) of P is infinite cyclic and generated by π, and the short exact sequence (2.10) induces a short exact sequence

1 → Z(P) → Z(B) → Z(W) → 1 .

Note that, by Propositions 2.23 and 2.6, (2), Theorem 2.24 is equivalent to the following statement :

2.25. The center of the “projective braid group” π_{1}(\overline{M}, \overline{x}_{0}) is trivial.

Proof of 2.24 in dimension 2. Assume that V has dimension 2. The space \overline{M} is homeomorphic to ℙ^{1}(ℂ) minus N points, so π_{1}(\overline{M}, \overline{x}_{0}) is isomorphic to a free group F_{N−1} on N − 1 generators. Since W is irreducible, we have N > 2 and so 2.24 is proved. □

2.26. Corollary. Let β^{ab} be the image in B^{ab} of the central element β of B. Then we have

β^{ab} = ∏_{C∈A/W}(s_{C}^{ab})^{e_{C}N_{C}/|Z(W)|} .

Proof of 2.26. It suffices to prove that, for all C ∈ A/W, we have π_{1}(δ_{C})(β^{ab}) = e_{C}N_{C}/|Z(W)| . This is immediate :

π_{1}(δ_{C})(β^{ab}) = ∑_{H∈C} \frac{e_{C}}{2iπ} ∫_{0}^{1} \frac{dα_{H}(x_{0} exp(2iπt/|Z(W)|))}{α_{H}(x_{0} exp(2iπt/|Z(W)|))}

= \frac{e_{C}}{2iπ} ∑_{H∈C} \frac{2iπ}{|Z(W)|} ∫_{0}^{1} dt = e_{C}N_{C}/|Z(W)| .

□

C. The braid diagrams.

Let us first introduce some notation.

Let (V, W) be a finite irreducible complex reflection group. As previously, we set M := V − ⋃_{H∈A} H, B := π_{1}(M/W, x_{0}), and we denote by σ ↦ \overline{σ} the antimorphism B ↠ W defined by the Galois covering M ↠ M/W.

Let D be one of the diagrams given in tables 1, 2, 3 (Appendix 2 below) symbolizing a set of relations as described in Appendix 2.

• We denote by D_{br} and we call braid diagram associated to D the set of nodes of D subject to all relations of D but the orders of the nodes, and we represent the braid diagram D_{br} by the same picture as D where numbers insides the nodes are omitted.

Thus, if D is the diagram s◯ a e ^{◯ b t}_{◯ c u} , then D_{br} is the diagram s◯ e ^{◯ t}_{◯ u}

and represents the relations

\underbrace{stustu ⋯}_{e factors} = \underbrace{tustus ⋯}_{e factors} = \underbrace{ustust ⋯}_{e factors} .

Note that this braid diagram for e = 3 is the braid diagram associated to G(2d, 2, 2) (d ≥ 2), as well as G_{7}, G_{11}, G_{19}. Also, for e = 4, this is the braid diagram associated to G_{12} and for e = 5, the braid diagram associated to G_{22}. Similarly, the braid diagram

s◯_{5} ^{◯^{t}}_{◯_{u}} is associated to the diagrams of both G_{15} and G(4d, 4, 2).

• We denote by D^{op} and we call opposite diagram associated to D the set of nodes of D subject to all opposite relations (words in reverse order) of D. Thus, if D is the

diagram s◯ a e ^{◯ b t}_{◯ c u} , then D^{op} represents the relations

s^{a} = t^{b} = u^{c} = 1 and \underbrace{utsuts ⋯}_{e factors} = \underbrace{sutsut ⋯}_{e factors} = \underbrace{tsutsu ⋯}_{e factors} .

Note that D^{op} is the diagram u◯ c e ^{◯ b t}_{◯ a s} . Finally, we denote by D_{br}^{op} the braid dia-

gram associated with D^{op}. Thus, in the above case, D_{br}^{op} is the diagram u◯ ^{◯ t}_{◯ s} .

Note that if D_{br} is a Coxeter type diagram, then it is equal to D_{br}^{op}.

The following statement is well known for Coxeter groups (see for example [Br1] or [De1]). It has been noticed by Orlik and Solomon (see [OrSo3], 3.7) for the case of non real Shephard groups (i.e., non real complex reflection groups whose braid diagram – see above – is a Coxeter diagram). We shall prove it below for all the infinite series. We also checked it case by case for all the exceptional groups but G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, G_{34} – for the case of groups of rank 2, we made use of [Ba].

We conjecture it still holds for G_{31}. The question whether it is possible to find right diagrams for G_{24}, G_{27}, G_{29}, G_{33}, G_{34} is still open (see remark below).

2.27. Theorem. Let W be a finite irreducible complex reflection group, different from G_{24}, G_{27}, G_{29}, G_{33}, G_{34} – and also different from G_{31} for which the following assertions are still conjectural.

Let N (D) be the set of nodes of the diagram D for W given in tables 1–3 below, identified with a set of pseudo-reflections in W. For each s ∈ N (D), there exists an s–generator of the monodromy s in B (cf. Definition 2.15) such that the set {s}_{s∈N (D)}, together with the braid relations of D_{br}^{op}, is a presentation of B.

2.28. Questions. Let W be a finite irreducible complex reflection group, different from G_{24}, G_{27}, G_{29}, G_{33}, G_{34}. We denote by B^{+} the monoid defined by generators and relations as follows : a set of generators is {s}_{s∈N (D)}, subject to the braid relations represented by D_{br}^{op}.

(1) Is the natural morphism B^{+} → B injective ?

(2) Do we have

B = {π^{n}b | (n ∈ ℤ)(b ∈ B^{+})}?

Remark. This is true for Coxeter groups (see [De1]). But the answers to the above questions are negative for diagrams given above for G_{24}, G_{27}, G_{29}, G_{33}, G_{34}.

D. Parabolic subgroups.

From now on, we endow the complex vector space V with a W –invariant hermitian scalar product. The balls in V are defined with respect to this scalar product.

Let I be an intersection of reflecting hyperplanes : I = ⋂_{{H∈A \atop I⊆H}} H . Let us set

A_{I} := {H ∈ A | (I ⊆ H)} and M_{I} := V − ⋃_{H∈A_{I}} H .

We recall (see 1.1 above) that the parabolic subgroup W_{I}, the pointwise stabilizer of I, is generated by pseudo–reflections (more precisely, W_{I} is generated by all the cyclic groups W_{H} for H ∈ A_{I}).

For x_{0} ∈ M, we set

\begin{matrix}P := π_{1}(M, x_{0}) , P_{I} := π_{1}(M_{I}, x_{0}) \\ B := π_{1}(M/W, x_{0}) , B_{I} := π_{1}(M_{I}/W_{I}, x_{0}) . \end{matrix}

We shall define a morphism of short exact sequences (see 2.10 above)

\xymatrix{1 \ar[r] & P_{I} \ar@{^{(}->}[d] \ar[r] & B_{I} \ar@{^{(}->}[d] \ar[r] & W_{I}^{op} \ar@{^{(}->}[d] \ar[r] & 1 \\ 1 \ar[r] & P \ar[r] & B \ar[r] & W^{op} \ar[r] & 1}

where all vertical arrows are injective, and where the corresponding injection B_{I} ↪ B is well–defined up to P –conjugation.

Let ι: M ↪ M_{I} denote the inclusion, which induces a homomorphism

ι_{∗}: π_{1}(M, x_{0}) ⟶ π_{1}(M_{I}, x_{0}) .

Let us prove that ι_{∗} is a split surjection.

• We set M(I) := I − ⋃_{{H∈A \atop I⊈H}} (I ∩ H) .

• Let γ be a path from x_{0} to M(I) in M, i.e., a path in V such that γ(0) = x_{0}, γ(t) ∈ M for t ≠ 1, and γ(1) ∈ M(I).

Let ε be a positive real number such that the ball Ω with center γ(1) and radius ε does not intersect any subspace I ∩ H (for H ∈ A and I ⊈ H).

Let u ∈ [0, 1[ such that γ(t) ∈ Ω for t ≥ u. Set x_{1} := γ(u), and let γ_{u} be the “restriction” of γ to [0, u] (see Appendix 1).

• We define the homomorphism σ: π_{1}(Ω ∩ M, x_{1}) → π_{1}(M, x_{0}) by the condition

λ ↦ γ_{u}^{−1} · λ · γ_{u} .

Then it is easy to see that

ι_{∗}σ : π_{1}(Ω ∩ M, x_{1}) → π_{1}(M_{I}, x_{0})

is an isomorphism. In particular, ι_{∗} is a split surjection and σ a split injection, so we have an injection

σ(ι_{∗}σ)^{−1} : π_{1}(M_{I}, x_{0}) → π_{1}(M, x_{0}) ,

depending only on the M(I)–homotopy class of γ. It sends a generator of the monodromy around H ∈ A_{I} in M_{I} to a generator of the monodromy around H in M.

Since γ(1) ∈ I, the ball Ω is W_{I}-stable. In particular, the inclusion

κ : (Ω ∩ M)/W_{I} ↪ M/W_{I}

induces a homomorphism

κ_{∗}: π_{1}((Ω ∩ M)/W_{I}, x_{1}) → π_{1}(M/W_{I}, x_{0}) .

On the other hand, the injection λ: M/W_{I} ↪ M_{I}/W_{I} induces a homomorphism

λ_{∗}: π_{1}(M/W_{I}, x_{0}) → π_{1}(M_{I}/W_{I}, x_{0})

It is easily seen that

λ_{∗}κ_{∗}: π_{1}((Ω ∩ M)/W_{I}, x_{1}) → π_{1}(M_{I}/W_{I}, x_{0})

is an isomorphism.

2.29. Proposition.

(1) In the following diagram,

• all sequences but the vertical ones are exact,

• all triangles and squares but those involving both ι_{∗} and κ_{∗}(λ_{∗}κ_{∗})^{−1} are commutative.

\xymatrix{1 \ar[dr] \\ 1 \ar[r] & P \ar[r] \ar@{->>}[dd]_{ι_{∗}} \ar[dr] & B \ar[r] & W \ar[r] & 1 \\  &  & π_{1}(M/W_{I}, x_{0}) \ar@{->>}[d]_{λ_{∗}} \ar@{^{(}->}[u] \ar[dr] \\ 1 \ar[r] & P_{I} \ar@{^{(}->}[uu]_{σ(ι_{∗}σ)^{−1}} \ar[r] & B_{I} \ar[r] \ar@{^{(}->}[u]_{κ_{∗}(λ_{∗}κ_{∗})^{−1}} & W_{I} \ar@{^{(}->}[uu] \ar[r] \ar[dr] & 1 \\  &  &  &  & 1}

(2) The injective morphism κ_{∗}(λ_{∗}κ_{∗})^{−1} depends only on the M(I)–homotopy class of γ, hence is well–defined up to P –conjugation.

(3) If s is a distinguished pseudo–reflection of W_{I}, the injection B_{I} ↪ B defined by the preceding diagram sends an s–generator of the monodromy in B_{I} to an s–generator of the monodromy in B.

3. Proofs of the main theorems for the braid groups B(de, e, r)

In this paragraph, we shall prove Theorems 2.27 and 2.24 for the infinite series of irreducible complex reflection groups G(de, e, r).

A. Notation and prerequisites.

Notation.

Let d, e and r be positive integers. We denote by (z_{1}, z_{2}, …, z_{r}) a general element of ℂ^{r}. Let G(de, e, r) be the subgroup of GL_{r}(ℂ) whose elements are :

[\underline{a}, σ] : z_{j} ↦ a_{j}z_{σ(j)}

for σ ∈ 𝔖_{r} and \underline{a} = (a_{1}, …, a_{r}) where a_{j} ∈ ℂ, a_{j}^{de} = 1 and (a_{1} ⋯ a_{r})^{d} = 1.

The group G(de, e, r) is a subgroup of index e of G(de, 1, r) and G(de, 1, r) ≃ (ℤ/deℤ)≀ 𝔖_{r}.

For de ≠ 1 and (d, e, r) ≠ (1, 2, 2), the group G(de, e, r) acts irreducibly on ℂ^{r}, while G(1, 1, r) is isomorphic to 𝔖_{r} in its permutation action on ℂ^{r}.

Note that the center Z(G(de, e, r)) of G(de, e, r) is cyclic, of order d(e∧r). We denote by Δ(de, e, r) the abelian normal subgroup of G(de, e, r) given by

Δ(de, e, r) := \{[\underline{a}, 1]\} .

The group Δ(de, e, r) is of order d(de)^{r−1}.

For the following notation, we assume that de ≠ 1 and (d, e, r) ≠ (1, 2, 2) (i.e., that G(de, e, r) acts irreducibly on ℂ^{r}).

For m ∈ ℕ − {0}, let ζ_{m} := exp(2iπ/m). We set

(3.1)  \begin{aligned}s_{m} & := [(ζ_{m}, 1, …, 1), 1] \\ t′_{2}(m) & := [(ζ_{m}^{−1}, ζ_{m}, 1, …, 1), (1, 2)] \\ t_{j} & := [\underline{1}, (j − 1, j)] for 2 ≤ j ≤ r . \end{aligned}

Let S(de, e, r) denote the set of pseudo-reflections of G(de, e, r) given by

S(de, e, r) := \{ \begin{matrix}\{s_{d}, t′_{2}(de), t_{2}, …, t_{r}\} when e ≠ 1 , d ≠ 1 \\ \{s_{d}, t_{2}, …, t_{r}\} when e = 1 , \\ \{t′_{2}(e), t_{2}, …, t_{r}\} when d = 1 . \end{matrix}

The following result is proved, for example, in [Ari].

3.2. Proposition. The set S(de, e, r), together with the relations described in Appendix 2 and its tables 1 and 2, give a presentation by generators and relations of G(de, e, r).

Note that S(de, e, r) consists of distinguished pseudo-reflections (see Definition 2.15 above) for G(de, e, r).

Reflecting hyperplanes.

The following lemma is well known and easy to check.

3.3. Lemma. Let m be a positive integer.

(1) For e | m and e < m, the complement in ℂ^{r} of the union of the reflecting hyperplanes of G(m, e, r) is

M^{#}(m, r) := {(z_{1}, z_{2}, …, z_{r}) | (∀j, k, 1 ≤ j ≠ k ≤ r)(∀a ∈ ℤ)(z_{j} ≠ 0)(z_{j} ≠ ζ_{m}^{a}z_{k})} .

(2) For all e ∈ ℕ, the complement in ℂ^{r} of the union of the reflecting hyperplanes of G(e, e, r) is

M(e, r) := {(z_{1}, z_{2}, …, z_{r}) | (∀j, k, 1 ≤ j < k ≤ r)(∀a ∈ ℤ)(z_{j} ≠ ζ_{e}^{a}z_{k})} .

Choosing an appropriate base point, we denote by B(de, e, r) and P (de, e, r) respectively the corresponding braid group and pure braid group associated with G(de, e, r).

Remark. By Lemma 3.3 above, P (de, e, r) depends only on (de, r) :

P (de, e, r) = P (de, 1, r) (for all d ≠ 1) .

On the other hand, we shall prove (see Proposition 3.8 below) that B(de, e, r) depends only on (e, r) for d ≠ 1 (so that B(de, e, r) = B(2e, e, r)).

Preliminary : the case of the symmetric group.

Here we quote some well known results about the usual braid groups, mainly due to Artin [Ar] – see also [Bi], th. 1.8.

Let us introduce some specific notation.

We set M(r) := M(1, r) and M^{#}(r) := M^{#}(1, r) (see Lemma 3.3 above).

For all j ≤ r, we denote by H_{j}^{(r+1)} the hyperplane of ℂ^{r+1} defined by the equation z_{j} = z_{j+1}, and we denote by s_{j}^{(r+1)} (or simply s_{j}) the reflection in ℂ^{r+1} with respect to H_{j}^{(r+1)}. The set {s_{j}^{(r+1)}}_{(1≤j≤r)} generates a subgroup of GL_{r+1}(ℂ) which we identify with the symmetric group 𝔖_{r+1}, and the set M(r + 1) is the complement of the union of the reflecting hyperplanes of 𝔖_{r+1}.

We choose a base point x ∈ ℝ^{r+1} with coordinates x_{1}, x_{2}, …, x_{r+1} such that x_{1} < x_{2} < ⋯ < x_{r+1}. Note that x is in one of the alcôves of M(r + 1) ∩ ℝ^{r+1} delimited by (the real part of) the hyperplanes H_{j}^{(r+1)}.

We set

P (r + 1) := π_{1}(M(r + 1), x) and B(r + 1) := π_{1}(M(r + 1)/𝔖_{r+1}, x) .

For each j ≤ r, we denote by ξ_{j}^{(r+1)} (or simply ξ_{j}) the generator of the monodromy around H_{j}^{(r+1)} in M(r + 1)/𝔖_{r+1} associated to a path contained in ℝ^{r+1}/𝔖_{r+1}.

The following well known proposition establishes Theorem 2.27 for the case where G = 𝔖_{r+1}. The second assertion has been proved by Chow [Cho].

3.4. Proposition.

(1) The group B(r + 1) has a presentation described by the following diagram

◯_{ξ_{1}} ◯_{ξ_{2}} ⋯ ◯_{ξ_{r}}.

(2) Let π(r + 1) be the element of B(r + 1) defined by

π(r + 1) := (ξ_{1}ξ_{2} ⋯ ξ_{r})^{r+1} .

For r ≥ 2, we have

Z(B(r + 1)) = Z(P (r + 1)) = ⟨π(r + 1)⟩ .

We will often consider B(r) = π_{1}(M(r)/𝔖_{r}, (x_{2}, …, x_{r+1})) as a subgroup of B(r+1) through the injection ξ_{j}^{(r)} ↦ ξ_{j+1}^{(r+1)}. This induces an injection of the pure braid group P (r) = π_{1}(M(r), (x_{2}, …, x_{r+1})) into P (r + 1), as well as an injection of 𝔖_{r} in 𝔖_{r+1} as the subgroup fixing the first coordinate.

3.5. Proposition. The map

p_{r}: M(r + 1) → M(r) , (z_{1}, z_{2}, …, z_{r+1}) ↦ (z_{2}, …, z_{r+1})

is a locally trivial fibration, and it induces a short exact sequence

1 → F (r) → P (r + 1) → P (r) → 1 ,

where F (r) is a free subgroup, on the set of generators

{ξ_{1}^{2}, ξ_{2}ξ_{1}^{2}ξ_{2}^{−1}, …, ξ_{r} ⋯ ξ_{3}ξ_{2}ξ_{1}^{2}ξ_{2}^{−1}ξ_{3}^{−1} ⋯ ξ_{r}^{−1}} .

We have P (r + 1) = F (r) ⋊ P (r) .

B. Computation of B(de, e, r) and of its center for d ≠ 1.

B1. Proof of Theorem 2.27.

Let us use the notation introduced above about B(r + 1) and P (r + 1), as well as notation introduced in (3.1).

3.6. Theorem. Assume d ≠ 1.

(1) For s equal to respectively s_{d}, t_{2}, t_{3}, …, t_{r}, there exist s-generators of the monodromy denoted respectively by σ, τ_{2}, τ_{3}, …, τ_{r} in B(d, 1, r) and an injective group morphism

ϕ_{(d,1,r)} : B(d, 1, r) ↪ B(r + 1) , ϕ_{(d,1,r)} : \{ \begin{matrix}σ ↦ ξ_{1}^{2} \\ τ_{j} ↦ ξ_{j} for j ≥ 2 \end{matrix}

which induces an isomorphism of B(d, 1, r) onto the subgroup of B(r + 1) generated by {ξ_{1}^{2}, ξ_{2}, ξ_{3}, …, ξ_{r}} .

(2) This isomorphism, as well as the isomorphism between B(r) and the subgroup of B(r + 1) generated by {ξ_{2}, ξ_{3}, …, ξ_{r}}, induce the following commutative diagram :

[figure: 1; 1; 1; P (r); P (d,1,r); P (r+1); B(r); B(d,1,r); B(r+1); 𝔖_{r}; G(d,1,r); 𝔖_{r+1}; 1; 1; 1]

Proof of Theorem 3.6.

The map

(z_{1}, z_{2}, …, z_{r}) ↦ (z_{1}^{d}, z_{2}^{d}, …, z_{r}^{d})

identifies the quotient of M^{#}(d, r) by the action of the diagonal group Δ(d, 1, r) with the space M^{#}(r).

The map

f : M(r + 1) → M^{#}(r) , (z_{1}, z_{2}, …, z_{r+1}) ↦ (z_{1} − z_{2}, z_{1} − z_{3}, …, z_{1} − z_{r+1})

is a trivial fibration with fiber ℂ, which is 𝔖_{r}-equivariant with respect to the action of 𝔖_{r} on M(r + 1) defined by the embedding of 𝔖_{r} into 𝔖_{r+1} as the pointwise stabilizer of the first coordinate.

Since

G(d, 1, r) = Δ(d, 1, r) ⋊ 𝔖_{r} ,

we have the following commutative diagram :

\xymatrix{M^{#}(d, r) \ar[d] \\ M^{#}(d, r)/Δ(d, 1, r) \ar[d] \ar[r]^{∼} & M^{#}(r) \ar[d] & M(r + 1) \ar[l]_{ℂ−fibration} \ar[d] \\ M^{#}(d, r)/G(d, 1, r) \ar[r]^{∼} & M^{#}(r)/𝔖_{r} & M(r + 1)/𝔖_{r} \ar[l] \ar[d] \\  &  & M(r + 1)/𝔖_{r+1}}

The horizontal arrows induce isomorphisms between fundamental groups.

Let y ∈ M^{#}(d, r) with image f(x) in M^{#}(r). Let ψ be the isomorphism

π_{1}(M^{#}(d, r)/G(d, 1, r), y) → π_{1}(M(r + 1)/𝔖_{r}, x)

and ϕ(d, r) be the injection

π_{1}(M^{#}(d, r)/G(d, 1, r), y) → π_{1}(M(r + 1)/𝔖_{r}, x) → π_{1}(M(r + 1)/𝔖_{r+1}, x) .

Note that ξ_{1}^{2} is a generator of the monodromy around H_{1}^{(r+1)} in M(r+1)/𝔖_{r} (Proposition A3, Appendix 1). Since P (r + 1) is contained in the subgroup of B(r + 1) generated by ξ_{1}^{2}, ξ_{2}, …, ξ_{r} and since the image of this subgroup in 𝔖_{r+1} is 𝔖_{r}, it follows that π_{1}(M(r+1)/𝔖_{r}, x) is the subgroup of π_{1}(M(r+1)/𝔖_{r+1}, x) generated by ξ_{1}^{2}, ξ_{2}, …, ξ_{r}.

Let σ = ψ^{−1}(ξ_{1}^{2}) : this is a generator of the monodromy around the hyperplane z_{1} = 0. Let τ_{j} = ψ^{−1}(ξ_{j}) for j ≥ 2 : this is a generator of the monodromy around the hyperplane H_{j}^{(r)}. Then, π_{1}(M^{#}(d, r)/G(d, 1, r), y) is generated by σ, τ_{2}, …, τ_{r} and Theorem 3.6 follows. □

Let us now explain why Theorem 3.6 above implies Theorem 2.27 (for the case which is presently considered, namely the case of B(de, e, r) with d > 1).

1. The case e = 1.

By Theorem 3.6, the group B(d, 1, r) is isomorphic to the subgroup of B(r +1) generated by {ξ_{1}^{2}, ξ_{2}, ξ_{3}, …, ξ_{r}} , and from now on we identify B(d, 1, r) with this subgroup.

In particular, it follows from Theorem 3.6 above that P (r + 1) ⊂ B(d, 1, r) . Since the image of B(d, 1, r) in 𝔖_{r+1} (recall that 𝔖_{r+1} = B(r + 1)/P (r + 1)) is isomorphic to 𝔖_{r}, the index of B(d, 1, r) in B(r+1) is (r+1). Hence the set {1, ξ_{1}, ξ_{1}ξ_{2}, …, ξ_{1}ξ_{2} ⋯ ξ_{r}} is a set of right coset representatives of B(d, 1, r) in B(r+1). By construction, it is actually a Schreier set of right coset representatives, and it results from the Reidemeister–Schreier method (see [MKS], Theorem 2.8) that the braid relations defined by the diagram

◯_{ξ_{1}^{2}} ◯_{ξ_{2}} ◯_{ξ_{3}} ⋯ ◯_{ξ_{r}}

are indeed defining relations for B(d, 1, r).

Remark. The group G(2, 1, r) is actually a Coxeter group, since it is isomorphic to the Weyl group of type B_{r}. So we have reproved in this case a result which is known for all Coxeter groups by [De1] or [Br1].

Note also that 3.7. the injection B(d, 1, r) ↪ B(r + 1) induces an injection

P (d, 1, r) ↪ P (r + 1) .

2. The general case e > 1.

Let us set ξ′_{2} := ξ_{1}^{2}ξ_{2}ξ_{1}^{−2} . Note that (with notation introduced in Part A above) that ξ′_{2} is a t′_{2}(de)–generator of the monodromy in the braid group B(de, e, r).

Note also that, since we have the following coverings

M^{#}(de, r) → M^{#}(de, r)/G(de, e, r) → M^{#}(de, r)/G(de, 1, r) ,

it results from Proposition A3 (Appendix 1 below) that ξ_{1}^{2e} is an s_{d}–generator of the monodromy in the braid group B(de, e, r).

By Lemma 3.3 above we may identify P (de, 1, r) and P (de, e, r) for d ≠ 1. Further, G(de, e, r) acts as a subgroup of G(de, 1, r) on M^{#}(de, r), so we have the natural embeddings

P (de, 1, r) ↪ B(de, e, r) ↪ B(de, 1, r) ,

and the index of the latter embedding equals e. Let κ : B(de, 1, r) → G(de, 1, r) denote the canonical epimorphism. Set ξ := ξ_{1}^{2}. Now {1, κ(ξ), …, κ(ξ^{e−1})} is a set of right coset representatives of G(de, e, r) in G(de, 1, r), so, since

[κ(B(de, 1, r)) : κ(B(de, e, r))] = e = [B(de, 1, r) : B(de, e, r)] ,

the set {1, ξ, …, ξ^{e−1}} is a right (Schreier) transversal of B(de, e, r) in B(de, 1, r).

An application of the Reidemeister–Schreier method then proves, starting from the presentation for B(de, 1, r) on the set ξ_{1}^{2}, ξ_{2}, …, ξ_{r} (proved above) that the braid relations defined by the diagram

[figure: ξ_{1}^{2e}; ◯; e+1; ^{◯^{ξ_{2}}}; ⧹; _{◯_{ξ′_{2}}}; ⧸; ◯_{ξ_{3}}; ◯_{ξ_{4}}; ⋯; ◯_{ξ_{r}}]

are indeed defining relations for B(de, e, r). This proves Theorem 2.27 for B(de, e, r) and d > 1 assuming the corresponding statement for B(de, 1, r). □

Note that the above diagram is indeed the opposite diagram to the braid diagram describing the relations between the set S(de, e, r) of the corresponding family of distinguished generators of the finite group G(de, e, r), namely

s◯ d_{e+1} ^{◯ 2 ^{t′_{2}}} ⧹_{◯ 2 _{t_{2}}} ⧸◯_{t_{3}}_{2} ◯_{t_{4}}_{2}⋯◯_{t_{r}}_{2}.

It will be useful to note that we have proved for G(de, e, r) a statement similar to (and more general than) Theorem 3.6, (1), namely :

3.8. Proposition. For s equal to respectively s_{d}, t′_{2}(de), t_{2}, t_{3}, …, t_{r}, there exist sgenerators of the monodromy denoted respectively by σ^{e}, τ′_{2}, τ_{2}, τ_{3}, …, τ_{r} in B(de, e, r) and an injective group morphism

ϕ_{(de,e,r)} : B(de, e, r) ↪ B(r + 1) , ϕ_{(de,e,r)} : \{ \begin{matrix}σ^{e} ↦ ξ_{1}^{2e} \\ τ′_{2} ↦ ξ′_{2} (where ξ′_{2} = ξ_{1}^{2}ξ_{2}ξ_{1}^{−2}) \\ τ_{j} ↦ ξ_{j} for j ≥ 2 \end{matrix}

which induces an isomorphism of B(de, e, r) onto the subgroup of B(r + 1) generated by {ξ_{1}^{2e}, ξ′_{2}, ξ_{2}, ξ_{3}, …, ξ_{r}} .

On the pure braid group.

Let us note a result about the structure of P (d, 1, r) which is analogous to Proposition 3.5.

Let F (r) be the free subgroup of P (r + 1) introduced in Proposition 3.5. Let us set

φ_{1} := ξ_{1}^{2} and
φ_{j} := (ξ_{j} ⋯ ξ_{3}ξ_{2})ξ_{1}^{2}(ξ_{2}^{−1}ξ_{3}^{−1} ⋯ ξ_{j}^{−1}) for j = 2, …, r .

Then F (r) is the free group on {φ_{1}, φ_{2}, …, φ_{r}}.

The map π_{1}(M(r + 1), x) ≃ π_{1}(M^{#}(r), f(x)) → Δ(d, 1, r) provides by restriction a morphism F (r) → Δ(d, 1, r) . We denote by F (d, r) the kernel of this morphism. Thus we have a short exact sequence

1 → F (d, r) → F (r) → Δ(d, 1, r) → 1 .

3.9. Lemma.

(1) We have F (d, r) = F (r) ∩ P (d, 1, r), and F (d, r) is a free group on ((r − 1)d^{r} + 1) generators.

(2) We have P (d, 1, r) = F (d, r) ⋊ P (r).

Proof of 3.9.

The equality F (d, r) = F (r) ∩ P (d, 1, r) expresses the definition of F (d, r). Since F (d, r) is a subgroup of index d^{r} of the free group F (r), it is a free group on ((r−1)d^{r}+1) generators.

Since P (r + 1) = F (r) ⋊ P (r), the second assertion follows from the fact that P (r) ⊂ P (d, 1, r) ⊂ P (r + 1). □

B2. The center of B(de, e, r) for d ≠ 1.

Let us denote by β(de, e, r) the central element of B(de, e, r) defined as in 2.20.

The following proposition proves Theorem 2.24 for the braid groups B(de, e, r) with d > 1. We use the notation introduced in Proposition 3.8. In particular, B(de, e, r) is defined by generators and relations represented by the diagram

σ^{e}◯_{e+1} ^{◯^{τ_{2}}} ⧹_{◯_{τ′_{2}}} ⧸◯_{τ_{3}} ◯_{t_{4}}⋯◯_{τ_{r}}.

3.10. Proposition. We have :

(1) β(de, e, r) = σ^{er/(e∧r)}(τ′_{2}τ_{2}τ_{3} ⋯ τ_{r})^{e(r−1)/(e∧r)} ,

(2) Z(B(de, e, r)) = ⟨β(de, e, r)⟩,

(3) Z(P (de, e, r)) = Z(B(de, e, r)) ∩ P (de, e, r) = ⟨β(de, e, r)^{d(e∧r)}⟩ .

Proof of 3.10. Note that for e = 1 the result is already known by [De1], since by Theorem 3.6 we have B(d, 1, r) = B(2, 1, r), the braid group associated to a Weyl group.

In what follows, we identify B(de, e, r) with its image in B(r + 1) (see Proposition 3.8 above). Step 1. We prove that

(3.11)  Z(P (de, e, r)) ⊂ ⟨π(r + 1)⟩ .

Let z ∈ Z(P (de, e, r)).

Since P (r) ⊂ P (de, e, r) ⊂ P (r +1), the element z belongs to P (r +1) and centralizes P (r). Since (cf. Proposition 3.5) P (r+1) = F (r)⋊P (r), where F (r) is the normal closure in P (r + 1) of the subgroup generated by ξ_{1}^{2}, in order to prove that z ∈ Z(P (r + 1)) it suffices to prove that z centralizes ξ_{1}^{2}. But z centralizes ξ_{1}^{2de}. Thus the elements zξ_{1}^{2}z^{−1} and ξ_{1}^{2} both belong to the free group F (r), and their (de)-th powers are equal. This implies that they are equal (see for example [MKS], 1.4, ex. 2). This proves (3.11).

Thus we have

(3.12)  (Z(B(de, e, r)) ∩ P (de, e, r)) ⊂ Z(P (de, e, r)) ⊂ ⟨π(r + 1)⟩ ∩ B(de, e, r) .

Step 2. Let us now prove that

(3.13)  ⟨π(r + 1)⟩ ∩ B(de, e, r) = ⟨π(r + 1)^{e/(e∧r)}⟩ .

We have (see [Bi], 1.8.4) :

(3.14)  π(r + 1) = ξ_{1}^{2}(ξ′_{2}ξ_{1}^{2}ξ′_{2})(ξ_{3}ξ′_{2}ξ_{1}^{2}ξ′_{2}ξ_{3}) ⋯ (ξ_{r} ⋯ ξ_{3}ξ′_{2}ξ_{1}^{2}ξ′_{2}ξ_{3} ⋯ ξ_{r}) .

Since ξ_{2} = ξ_{1}^{−2}ξ′_{2}ξ_{1}^{2}, we have ξ′_{2}ξ_{1}^{2} = ξ_{1}^{2}ξ_{2}, and (3.14) becomes

(3.15)  π(r + 1) = ξ_{1}^{2}(ξ_{1}^{2}ξ_{2}ξ′_{2})(ξ_{3}(ξ_{1}^{2}ξ_{2}ξ′_{2})ξ_{3}) ⋯ (ξ_{r} ⋯ ξ_{3}(ξ_{1}^{2}ξ_{2}ξ′_{2})ξ_{3} ⋯ ξ_{r}) .

Since ξ_{1}^{2} commutes with (ξ_{2}ξ′_{2}), as well as with ξ_{j} for j ≥ 3, we deduce from (3.15) that

(3.16)  π(r + 1) = ξ_{1}^{2r}(ξ_{2}ξ′_{2})(ξ_{3}(ξ_{2}ξ′_{2})ξ_{3}) ⋯ (ξ_{r} ⋯ ξ_{3}(ξ_{2}ξ′_{2})ξ_{3} ⋯ ξ_{r}) .

and then for all n ∈ ℕ,

(3.17)  π(r + 1)^{n} = ξ_{1}^{2rn} ((ξ_{2}ξ′_{2})(ξ_{3}(ξ_{2}ξ′_{2})ξ_{3}) ⋯ (ξ_{r} ⋯ ξ_{3}(ξ_{2}ξ′_{2})ξ_{3} ⋯ ξ_{r}))^{n} .

Since, for e′ ∈ ℕ, we have ⟨B(de, e, r), ξ_{1}^{2e′}⟩ = B(de, (e ∧ e′), r) , it follows from (3.17) that π(r +1)^{n} ∈ B(de, e, r) if and only if e divides rn, i.e., if and only if e/(e∧r) divides n, which proves (3.13). Step 3. Let us now check that

(3.18)  (ξ_{2}ξ′_{2})(ξ_{3}(ξ_{2}ξ′_{2})ξ_{3}) ⋯ (ξ_{r} ⋯ ξ_{3}(ξ_{2}ξ′_{2})ξ_{3} ⋯ ξ_{r}) = (ξ_{2}ξ′_{2}ξ_{3} ⋯ ξ_{r})^{r−1} .

Let us introduce the group B(2, 1, r − 1) together with its distinguished set of generators {α, α_{2}, …, α_{r−1}}, which satisfy the relations described by the diagram :

◯_{α} ◯_{α_{2}} ◯_{α_{3}} ⋯ ◯_{α_{r−1}}

Then the map

α ↦ ξ_{2}ξ′_{2} , α_{j} ↦ ξ_{j+1} for j ≥ 2

defines a morphism B(2, 1, r −1) → B(de, e, r) . Thus, in order to prove (3.18), it suffices to prove that

α(α_{2}αα_{2})(α_{3}α_{2}αα_{2}α_{3}) ⋯ (α_{r−1} ⋯ α_{3}α_{2}αα_{2}α_{3} ⋯ α_{r−1}) = (αα_{2}α_{3} ⋯ α_{r−1})^{r−1} .

This last equality expresses a known property of reduced expressions of the longest element in the Weyl group G(2, 1, r − 1), i.e., the Weyl group of type B_{r−1} (see for example [Bou], chap. v, §6, ex. 2). The proof of (3.18) is complete.

Last step. Let us (temporarily) set β′ := π(r + 1)^{e/(e∧r)} . By (3.12) and (3.13), we see that

(3.19)  Z(B(de, e, r)) ∩ P (de, e, r) ⊂ ⟨β′⟩ ,

and by (3.18) and (3.16), we see that

β′ = ξ_{1}^{2re/(e∧r)}(ξ_{2}ξ′_{2}ξ_{3} ⋯ ξ_{r})^{e(r−1)/(e∧r)} ,

or, with the identification made in Proposition 3.8,

(3.20)  β′ = σ^{er/(e∧r)}(τ′_{2}τ_{2}τ_{3} ⋯ τ_{r})^{e(r−1)/(e∧r)} .

On the other hand, it is not difficult to check that 3.21. the canonical epimorphism κ: B(de, e, r) → G(de, e, r) sends β′ onto the scalar multiplication in V by exp(2iπ/|Z(G(de, e, r))|) .

Since the map κ: Z(B(de, e, r)) → Z(G(de, e, r)) is onto, and since (by (3.19)) its kernel is contained in ⟨β′⟩, it follows from 3.21 that

(3.22)  Z(B(de, e, r)) = ⟨β′⟩ .

Using (3.11), it is then easy to prove that Z(P (de, e, r)) = ⟨β′^{d(e∧r)}⟩ .

It remains to check now that β′ = β(de, e, r). This follows from the fact that,

• on one hand, we have (by (3.22) and Lemma 2.22, (2)) β(de, e, r) ∈ ⟨β′⟩,

• on the other hand, β(de, e, r) and β′ have both the same image under the discriminant map π_{1}(δ): B(de, e, r) → ℤ (cf. §2, B above). □

C. Computation of B(e, e, r) and of its center, e ≠ 1.

In this section we study the braid group of type B(e, e, r) and in particular prove Theorems 2.27 and 2.24 for type G(e, e, r).

Note that the construction in the proof of Theorem 3.6 above gives an identification of π_{1}(M^{#}(e, r)/G(e, e, r)) with the subgroup of the braid group B(r + 1) generated by ξ_{1}^{2e}, ξ′_{2}, ξ_{2}, ξ_{3}, …, ξ_{r} with presentation

(3.23)  [figure: ξ_{1}^{2e}; ◯; e+1; ^{◯^{ξ_{2}}}; ⧹; _{◯_{ξ′_{2}}}; ⧸; ◯_{ξ_{3}}; ◯_{ξ_{4}}; ⋯; ◯_{ξ_{r}}]

Since M^{#}(e, r) is obtained from M(e, r) by removing the hyperplanes {z_{j} = 0} for 1 ≤ j ≤ r we have a natural map

ψ : π_{1}(M^{#}(e, r)/G(e, e, r)) → π_{1}(M(e, r)/G(e, e, r)) ,

which is surjective since the complement M(e, r)/G(e, e, r) − M^{#}(e, r)/G(e, e, r) has complex codimension 1.

The following proposition follows from Proposition A1 in Appendix 1.

Proposition 3.24. The kernel of ψ is the normal closure of the subgroup generated by ξ_{1}^{2e} in the group π_{1}(M^{#}(e, r)/G(e, e, r)).

Note that Theorem 2.27 follows immediately from this. Indeed, by the above Proposition the presentation of B(e, e, r) is obtained from (3.23) by suppressing the node corresponding to ξ_{1}^{2e}.

Complements on B(e, e, r).

Theorem 3.25. Let e, r ≥ 2, and let B(e, e, r) be the braid group of type G(e, e, r), on standard generators τ_{2}, τ′_{2}, τ_{3}, …, τ_{r} ordered such that (τ_{2}τ′_{2}τ_{3})^{2} = (τ_{3}τ_{2}τ′_{2})^{2} :

e^{^{τ_{2}}◯ 2}⧹_{_{τ′_{2}}◯ 2}⧸◯_{τ_{3}}_{2} ◯_{τ_{4}}_{2}⋯◯_{τ_{r}}_{2}.

Let B̃(e, 1, r − 1) be the preimage of the subgroup G(e, 1, r − 1) of G(e, e, r) fixing the first coordinate. Then B̃(e, 1, r − 1) has index r in B(e, e, r) and has a presentation on generators

{α_{j}, β_{j,l}, | 2 ≤ j ≤ r, 0 ≤ l ≤ e − 1}

subject to

(3.26)  α_{i}^{−1}β_{j,l}α_{i} =\{ \begin{matrix}β_{j,l} & if i ≠ 2, j, j + 1 \\ β_{j+1,l} & if i = j + 1 \\ β_{j,l+1}^{−1}β_{j−1,l+1}β_{j,l} & if i = j ≠ 2 \\ β_{2,l+1}β_{j,l}β_{2,l}^{−1} & if i = 2 < j \\ β_{j,l+2} & if i = j = 2 \end{matrix}

(where the subscript l of β_{j,l} is taken modulo e),

(3.27)  β_{j,e−1}β_{j,e−2} ⋯ β_{j,0} = 1  for 2 ≤ j ≤ r,

and α_{2}, …, α_{r} satisfy the relations of the standard generators of B(e, 1, r − 1). In terms of the generators of B(e, e, r) we may take

α_{2} = τ_{2}τ′_{2},  α_{i} = τ_{i} for 3 ≤ i ≤ r,  β_{2,0} = τ′_{2}τ_{2}^{−1},  β_{2,1} = τ′_{2}^{−1}τ_{2} .

In particular, B̃(e, 1, r − 1) has a semidirect product decomposition

B̃(e, 1, r − 1) = F_{(e−1)(r−1)} ⋊ B(e, 1, r − 1) ,

where F_{(e−1)(r−1)} denotes the free group on the (e−1)(r −1) generators β_{2,0}, …, β_{r,e−1}.

Proof. The assertion can be proved by the Reidemeister–Schreier method (see for example [MKS], Theorem 2.8). Assume first that r = 2. Then B(e, e, 2) is generated by {τ_{2}, τ′_{2}} subject to the single relation τ_{2}τ′_{2}τ_{2} ⋯ = τ′_{2}τ_{2}τ′_{2} ⋯ with e factors on each side. A right transversal for B̃(e, 1, 1) in B(e, e, 2) is given by T := {1, τ_{2}}. Let ρ : B(e, e, 2) → T be the transversal map which to every element of B(e, e, 2) associates its coset representative in T. Then by the Reidemeister-Schreier theorem B̃(e, 1, 1) is generated by the elements (tg)^{−1}ρ(tg) where t ∈ T and g runs over the generators of B(e, e, 2). In our case this yields the generators

(3.28)  α_{2} := τ_{2}τ′_{2}, β_{2,0} := τ′_{2}τ_{2}^{−1}, γ_{2} := τ_{2}^{2}.

Furthermore, by the Reidemeister-Schreier algorithm the relations for B(e, e, 2) yield

(3.29)  α_{2}^{(e+1)/2} = β_{2,0}(γ_{2}β_{2,0})^{(e−1)/2}α_{2} = (γ_{2}β_{2,0})^{(e−1)/2}γ_{2} if e is odd,

(3.30)  α_{2}^{e/2} = (γ_{2}β_{2,0})^{e/2} = (β_{2,0}γ_{2})^{e/2} if e is even,

as defining relations for B̃(e, 1, 1). By introducing β_{2,1} := τ′_{2}^{−1}τ_{2} = α_{2}^{−1}γ_{2} we may eliminate γ_{2} and arrive at the statement of the theorem in this case.

Now assume that r = 3. Here a transversal is given by T := {1, τ_{2}, τ_{2}τ_{3}}. We obtain generators

α_{3} := τ_{3}, γ_{3} := τ_{2}τ_{3}^{2}τ_{2}^{−1}, δ := τ_{2}τ′_{2}^{−1}τ_{3}τ′_{2}τ_{2}^{−1} ,

and α_{2}, γ_{2}, β_{2,0} from (3.28) above, subject to the relations

(3.31)  \begin{matrix}(α_{2}α_{3})^{2} = (α_{3}α_{2})^{2}, α_{3}^{−1}γ_{2}α_{3} = γ_{3}, α_{3}γ_{3}γ_{2} = γ_{3}γ_{2}α_{3}, \\ β_{2,0}δ = α_{3}β_{2,0}, α_{2}α_{3}β_{2,0} = δγ_{3}, δγ_{3}α_{2} = γ_{3}α_{2}α_{3}, \\ γ_{2}β_{2,0}α_{3}δγ_{3} = α_{3}δγ_{3}γ_{2}β_{2,0} = γ_{3}γ_{2}β_{2,0}α_{3}δ, \end{matrix}

and the relations (3.29) respectively (3.30). We may eliminate γ_{3} = α_{3}^{−1}γ_{2}α_{3} and δ = β_{2,0}^{−1}α_{3}β_{2,0}. With β_{2,1} := α_{2}^{−1}γ_{2} as above we find the stated presentation.

Finally, for r ≥ 4 it follows easily from the presentation for B(e, e, r) given in Theorem 2.27 (proved by Proposition 3.24) that

T := {1, τ_{2}, τ_{2}τ_{3}, …, τ_{2}τ_{3} ⋯ τ_{r}}

is a right transversal for B̃(e, 1, r − 1) in B(e, e, r). This yields the generators

α_{i} := τ_{i} and γ_{i} := τ_{2}τ_{3} ⋯ τ_{i−1}τ_{i}^{2}τ_{i−1}^{−1} ⋯ τ_{3}^{−1}τ_{2}^{−1} (3 ≤ i ≤ r) ,

α_{2} := τ_{2}τ′_{2}, γ_{2} := τ_{2}^{2}, β_{2,0} := τ′_{2}τ_{2}^{−1}, δ := τ_{2}τ′_{2}^{−1}τ_{3}τ′_{2}τ_{2}^{−1} .

Furthermore, by the Reidemeister-Schreier algorithm the relations for B(e, e, r) yield as relations for B̃(e, 1, r − 1) the relations of the standard generators for B(e, 1, r − 1) on α_{2}, …, α_{r}, the commutator rules

[γ_{i}, α_{j}] = 1 for j ≠ i, i + 1, α_{i+1}^{−1}γ_{i}α_{i+1} = γ_{i+1} for 2 ≤ i ,

and

α_{i+1}γ_{i+1}γ_{i} = γ_{i+1}γ_{i}α_{i+1} for 2 ≤ i, ,

as well as

[β_{2,0}, α_{j}] = 1 for j ≥ 4, [δ, α_{j}] = 1 for j > 4, [δ, γ_{j}] = 1 for j ≥ 4 ,
δα_{4}δ = α_{4}δα_{4}, (α_{3}δα_{4})^{2} = (α_{4}α_{3}δ)^{2} .

and the relations (3.31), (3.29) respectively (3.30). Note that the generators γ_{3}, …, γ_{r} may be eliminated from this presentation using the relation γ_{i+1} = α_{i+1}^{−1}γ_{i}α_{i+1}. This reduces the assertion to the case r = 3. □

This result has some nice consequences, like the following analogue of Proposition 3.5 and Lemma 3.9.

Corollary 3.32. The pure braid group P (e, e, r) is a semidirect product

P (e, e, r) ≅ F_{(e−1)(r−1)} ⋊ P (e, 1, r − 1)

of the free group of rank (e − 1)(r − 1) with the pure braid group of type G(e, 1, r − 1).

This follows immediately by descent to the pure braid groups (see [Na], p. 6, for a related result). We also obtain Theorem 2.24 for type G(e, e, r):

Corollary 3.33. For e, r ≥ 2, (e, r) ≠ (2, 2), the center of the braid group B(e, e, r) is generated by (τ_{2} ⋯ τ_{r})^{e(r−1)/(e∧r)}.

Proof. Let κ : B(e, e, r) → G(e, e, r) be the canonical projection. If z is central in B(e, e, r) then so is κ(z) in G(e, e, r). Hence κ(z) = (τ_{2} ⋯ τ_{r})^{n} with n a multiple of e(r − 1)/(e ∧ r), and z = (τ_{2} ⋯ τ_{r})^{n}w for some w in ker(κ). But then we already have z ∈ B̃(e, 1, r − 1) (defined as above).

Let λ: B̃(e, 1, r − 1) → B(e, 1, r − 1) be the canonical projection with kernel F := F_{(e−1)(r−1)} the free group on (e − 1)(r − 1) generators emanating from Theorem 3.25. Since the center of λ(B̃(e, 1, r − 1)) = B(e, 1, r − 1) is generated by (α_{2} ⋯ α_{r})^{r} = (τ_{2} ⋯ τ_{r})^{r} we deduce that z = (τ_{2} ⋯ τ_{r})^{n}w for some w ∈ F. But (τ_{2} ⋯ τ_{r})^{n} is central in B̃(e, 1, r−1), while the center of the free group F is trivial (note that (e−1)(r−1) ≥ 2). Thus the center of B̃(e, 1, r − 1) is generated by (τ_{2} ⋯ τ_{r})^{e(r−1)/(e∧r)}. □

Remark 3.34. For the braid group B(2, 2, r) of Coxeter type D_{r} Theorem 3.25 specializes to the following: B̃(2, 1, r − 1) has a presentation on

{α_{i}, β_{j}, | 2 ≤ i, j ≤ r}

subject to

α_{i}^{−1}β_{j}α_{i} =\{ \begin{matrix}β_{j} & if i ≠ 2, j, j + 1 \\ β_{j+1} & if i = j + 1 \\ β_{j}β_{j−1}^{−1}β_{j} & if i = j ≠ 2 \\ β_{2}^{−1}β_{j}β_{2}^{−1} & if i = 2 < j \\ β_{j} & if i = j = 2 \end{matrix}

and α_{2}, …, α_{r} satisfy the relations of the standard generators of B(2, 1, r − 1).

Remark 3.35. The subgroup B̃(e, 1, 1) of index 2 of the braid group B(e, e, 2) of Coxeter type I_{2}(e) has a presentation on {α_{2}, β_{l}, | 0 ≤ l ≤ e − 1} subject to

α_{2}^{−1}β_{l}α_{2} = β_{l+2} for 0 ≤ l ≤ e − 1

(where the subscript of β_{l} has to be taken mod e), and

β_{e−1}β_{e−2} ⋯ β_{0} = 1 .

Remark 3.36. The action of B(e, 1, r−1) on F_{(e−1)(r−1)} in Theorem 3.25 can be extended to an action of the Artin braid group B(r). More precisely, let α̃_{2}, α_{3}, …, α_{r} be the standard generators of B(r). Then B(e, 1, r−1) is isomorphic to the subgroup generated by α̃_{2}^{2}, α_{3}, …, α_{r} by Theorem 3.6. We extend the action (3.26) of α_{2}, …, α_{r} to an action of B(r) on F_{(e−1)(r−1)} = ⟨β_{j,l} | ∏_{l=0}^{e−1} β_{j,l} = 1⟩ by

α̃_{2}^{−1}β_{j,l}α̃_{2} = \{ \begin{matrix}β_{2,l}β_{j,l}^{−1} & if j ≠ 2, \\ β_{j,l+1} & if j = 2. \end{matrix}

It is easy to verify that this does in fact extend the action of B(e, 1, r − 1).

On the other hand the action (3.26) can be viewed as an action on the free group F_{e(r−1)} on free generators ⟨β_{j,l} | 2 ≤ j ≤ r, 0 ≤ l ≤ e − 1⟩ by just omitting relations (3.27). The homomorphism defined by

ϕ : F_{e(r−1)} → ⟨t⟩ ≅ ℤ, β_{j,l} ↦ t for all j, l ,

is B(e, 1, r − 1)-equivariant (with trivial action on the right side), so it gives rise to a Magnus representation (see [Bi], Th. 3.9)

Φ : B(e, 1, r − 1) → GL_{e(r−1)}(ℤ(t))

with

Φ(α_{i})_{(j,l),(k,m)} =\{ \begin{matrix}δ_{jk}δ_{lm} & if i ≠ 2, j, j + 1 \\ δ_{j+1,k}δ_{lm} & if i = j + 1 \\ (δ_{j−1,k}δ_{l+1,m} − δ_{jk}δ_{l+1,m})t^{−1} + δ_{jk}δ_{lm} & if i = j ≠ 2 \\ δ_{2k}δ_{l+1,m} + (δ_{jk}δ_{lm} − δ_{2k}δ_{lm})t & if i = 2 < j \\ δ_{jk}δ_{l+2,m} & if i = j = 2 \end{matrix}

(where 2 ≤ i, j, k ≤ r, 0 ≤ l, m ≤ e − 1) of B(e, 1, r − 1).

A general statement for pure braid groups.

Let us first introduce some notation.

• We make the convention that G(1, 1, r) := 𝔖_{r+1}, and we denote by P (1, 1, r) the corresponding pure braid group.

• Let m_{r}^{∗}(de, e, r) be the co-exponent (see §1.A above) of G(de, e, r) such that the set of co-exponents of G(de, e, r) consists of the set of coexponents of G(de, 1, r − 1), together with m_{r}^{∗}(de, e, r). We have m_{r}^{∗}(de, e, r) = (r − 1)de + 1 for d ≠ 1 and m_{r}^{∗}(e, e, r) = (r − 1)(e − 1).

• For any natural integer m, let F_{m} be the free group on m generators.

3.37. Proposition. For all positive integers d, e, r, we have a split short exact sequence

0 → F_{m_{r}^{∗}(de,e,r)} → P (de, e, r) → P (de, 1, r − 1) → 0 .

In particular, P (de, e, r) is the semidirect product of a free group on m_{r}^{∗}(de, e, r) generators by the pure braid group associated with a complex reflection subgroup of G(de, e, r) of rank (r − 1).

Proof of 3.37. We assume d ≠ 1, since for d = 1, the result was proven in Corollary 3.32.

Consider the map f : M^{#}(de, r) → M^{#}(de, r − 1), (z_{1}, …, z_{r}) ↦ (z_{1}, …, z_{r−1}). This is a locally trivial fibration, with fiber isomorphic to ℂ minus (r − 1)de + 1 points. By Theorem 2.11, M^{#}(de, r − 1) is a K(π, 1)-space. Hence, we have a short exact sequence of fundamental groups associated to the fibration :

0 → F_{(r−1)de+1} → P (de, e, r) → P (de, 1, r − 1) → 0 .

The locally trivial fibration M^{#}(de, r) → M^{#}(r), (z_{1}, …, z_{r}) ↦ (z_{1}^{de}, …, z_{r}^{de}) induces a commutative diagram with exact rows and columns :

\xymatrix{ & 1 \ar[d] & 1 \ar[d] & 1 \ar[d] \\ 1 \ar[r] & F_{(r−1)de+1} \ar[r] \ar[d] & P (de, e, r) \ar[r] \ar[d] & P (de, e, r − 1) \ar[r] \ar[d] & 1 \\ 1 \ar[r] & F_{r} \ar[r] \ar[d] & P (r) \ar[r] \ar[d] & P (r − 1) \ar[r] \ar[d] & 1 \\ 1 \ar[r] & ℤ/deℤ \ar[r] \ar[d] & Δ(de, 1, r) \ar[r] \ar[d] & Δ(de, 1, r − 1) \ar[r] \ar[d] & 1 \\  & 1 & 1 & 1}

The splitting of the map P (r) → P (r − 1) together with the splitting of Δ(de, 1, r) → Δ(de, 1, r − 1) given by identifying Δ(de, 1, r − 1) with the subgroup of Δ(de, 1, r) of elements acting trivially on the last coordinate give then a splitting of P (de, e, r) → P (de, e, r − 1) and the proposition follows. □

4. Hecke algebras

We extend to the case of complex reflection groups the construction of generalized Knizhnik–Zamolodchikov connections for Weyl groups due to Cherednik ([Ch1], [Ch2], [Ch3]; see also the constructions of Dunkl [Du], Opdam [Op] and Kohno [Ko1]).^{2} This allows us to construct explicit isomorphisms between the group algebra of a complex reflection group and its Hecke algebra.

A. Background from differential equations and monodromy.

What follows is well known, and is introduced here at an elementary level for the convenience of the unexperienced reader, since we only need this elementary approach. For a more general approach, see for example [De2].

We go back to the setting of §1. Let A be a finite dimensional complex vector space. We denote by 1 a chosen non zero point of A — in the applications, A we will be an algebra. We set E := End(A). Let ω be a holomorphic differential form on M with values in E, i.e., a holomorphic map M → Hom(V, E) , where Hom(V, E) denotes the space of linear maps from V into E, such that (see 1.2 and 1.5, (1)) we have

ω = ∑_{H∈A} f_{H}ω_{H} ,

with ω_{H} = \frac{1}{2iπ} \frac{dα_{H}}{α_{H}}, and f_{H} ∈ E. For x ∈ M and v ∈ V, we have ω(x)(v) =

\frac{1}{2iπ} ∑_{H∈A} \frac{α_{H}(v)}{α_{H}(x)}f_{H} .

We consider the following linear differential equation

(Eq(ω))  dF = ω(F) ,

where F is a holomorphic function defined on an open subset of M with values in A. In other words, for x in this open subset, we have dF (x) ∈ Hom(V, A) , and we want F to satisfy, for all v ∈ V, dF (x)(v) = ω(x)(v)(F (x)) , or in other words dF (x)(v) =

\frac{1}{2iπ} ∑_{H∈A} \frac{α_{H}(v)}{α_{H}(x)}f_{H}(F (x)) .

For y ∈ M, let us denote by V(y) the largest open ball with center y contained in M. The existence and unicity theorem for linear differential equations shows that for each y ∈ M, there exists a unique function

F_{y}: V(y) → A , x ↦ F_{y}(x) ,

solution of Eq(ω) and such that F_{y}(y) = 1. From now on, we set

F (x, y) := F_{y}(x) .

Assume now that the finite group W acts linearly on A through a morphism φ: W → GL(A). Then it induces an action of W on the space of differential forms on M with values in E, and an easy computation shows that ω is W -stable if and only if, for all w ∈ W,

2This construction has also been noticed independently by Opdam, who is able to deduce from it some important consequences concerning the “generalized fake degrees” of a complex reflection group. We thank him for useful and friendly conversations.

(4.1)  ω(w(x)) = φ(w)(ω(x) · w^{−1})φ(w^{−1}) ,

which can also be written, for all x ∈ M and v ∈ V :

∑_{H∈A} ω_{H}(wx)(v)f_{H} = ∑_{H∈A} ω_{H}(x)(w^{−1}(v))φ(w)f_{H}φ(w^{−1}) .

An easy computation shows that this is equivalent to

(4.2)  ∑_{H∈A} f_{w(H)} \frac{dα_{w(H)}}{α_{H}} = ∑_{H∈A} φ(w)f_{H}φ(w^{−1})\frac{dα_{w(H)}}{α_{H}} .

In particular we see that

4.3. If f_{w(H)} = φ(w)f_{H}φ(w^{−1}) for all H ∈ A and w ∈ W, then the form ω is W -stable.

From (4.1) (and from the existence and unicity theorem), it follows that

4.4. If ω is W -stable, then for all y ∈ M, x ∈ V(y) and w ∈ W, the solution x ↦ F (x, y) satisfies

φ(w)(F (x, y)) = F (w(x), w(y)) .

The case of an interior W -algebra.

The following hypothesis and notation will be in force for the rest of this chapter. From now on, we assume that A is endowed with a structure of ℂ-algebra with unity,

and that ω takes its values in the subalgebra of E consisting of the multiplications by the elements of A – which, by abuse of notation, we still denote by A. With this abuse of notation, we may assume that

ω = ∑_{H∈A} a_{H}ω_{H} ,

where a_{H} ∈ A, and the equation Eq(ω) is written

dF = ωF or dF (x)(v) = \frac{1}{2iπ} ∑_{H∈A} \frac{α_{H}(v)}{α_{H}(x)}a_{H}F (x) .

Let γ be a path in M. From the existence and unicity of local solutions of Eq(ω), it results that the solution x ↦ F (x, γ(0)) has an analytic continuation t ↦ (γ^{∗}F)(t, γ(0)) along γ, which satisfies the following properties.

Let us say that a sequence of real numbers t_{0} = 0 < t_{1} < … < t_{n−1} < t_{n} = 1 is adapted to (γ, Eq(ω)) if for all 1 ≤ j ≤ n, we have γ([t_{j−1}, t_{j}]) ⊂ V(γ(t_{j})).

Then :

(1) there exists ε > 0 such that (γ^{∗}F)(t, γ(0)) = F (γ(t), γ(0)) for 0 ≤ t ≤ ε,

(2) whenever t_{0} = 0 < t_{1} < … < t_{n−1} < t_{n} = 1 is adapted to (γ, Eq(ω)), we have (γ^{∗}F)(t_{j}, γ(0)) = F (γ(t_{j}), γ(t_{j−1}))(γ^{∗}F)(t_{j−1}, γ(0)) for all j > 0 .

We see that

(4.5)  (γ^{∗}F)(1, γ(0)) = ∏_{j=n}^{j=1} F (γ(t_{j}), γ(t_{j−1})) .

Note that there is always an adapted sequence for (γ, Eq(ω)).

The case of an integrable form.

We recall that the form ω is said to be integrable if dω + ω ∧ ω = 0 .

The following fact was noticed, for example, by Kohno (see [Ko2], 1.2).

4.6. Lemma. The form ω = ∑_{H∈A} a_{H}ω_{H} is integrable if and only if, for all subspaces X of V with codimension 2, and for all H ∈ A such that X ⊂ H, a_{H} commutes with ∑_{{(H′∈A) \atop (H′⊃X)}} a_{H′}.

Indeed, this is an immediate consequence of 1.5, (2).

If ω is integrable, the value (γ^{∗}F)(1, γ(0)) depends only on the homotopy class of γ. By (4.5), we see that we get a covariant functor

S: \{ \begin{matrix}P(M) → A^{×} \\ γ ↦ (γ^{∗}F)(1, γ(0)) \end{matrix}

Action of W.

Assume now that A is an interior W -algebra, i.e., that there is a group morphism W → A^{×} (through which the image of w ∈ W is still denoted by w), which defines a linear operation φ of W on A by composition with the injection A^{×} ↪ GL(A) . So, with our convention, for w ∈ W and a ∈ A we have φ(w)(a) = wa.

The form ω is then W -stable if and only if, for all w ∈ W and x ∈ M,

ω(w(x)) = w(ω(x) · w^{−1})w^{−1} ,

which can also be written, for all x ∈ M and v ∈ V :

∑_{H∈A} ω_{H}(wx)(v)a_{H} = ∑_{H∈A} ω_{H}(x)(w^{−1}(v))wa_{H}w^{−1} .

By 4.3, we have the following criterion.

4.7. If, for all H ∈ A and w ∈ W, we have a_{w(H)} = wa_{H}w^{−1}, then the form ω is W -stable.

By 4.4, the solution F of Eq(ω) then satisfies

(4.8)  wF (x, y)w^{−1} = F (w(x), w(y)) .

4.9. Definition–Proposition. Assuming that ω is W -stable, we define a group morphism

T : π_{1}(M/W, x_{0}) → (A^{×})^{op}

(or, in other words, a group anti–morphism T : π_{1}(M/W, x_{0}) → A^{×} ), called the monodromy morphism associated with ω, as follows.

For σ ∈ B, with image \overline{σ} in W through the natural anti–morphism B → W (see 2.B above), we denote by σ̃ a path in M from x_{0} to \overline{σ}(x_{0}) which lifts σ. Then we set

T (σ) := S(σ̃^{−1})\overline{σ} .

Let us check that T is a group anti–morphism.

Notice first that, by (4.8) and by (4.5), for w ∈ W and γ a path in M, we have

wS(γ)w^{−1} = S(w(γ)) .

Thus we have

T (σ_{2})T (σ_{1}) = S(σ̃_{2}^{−1})\overline{σ}_{2}S(σ̃_{1}^{−1})\overline{σ}_{1}
= S(σ̃_{2}^{−1})S(\overline{σ}_{2}(σ̃_{1}^{−1}))\overline{σ}_{2}\overline{σ}_{1}
= S(σ̃_{2}^{−1}\overline{σ}_{2}(σ̃_{1}^{−1}))\overline{σ}_{2}\overline{σ}_{1}
= S((\overline{σ}_{2}(σ̃_{1})σ̃_{2})^{−1})\overline{σ_{1}σ_{2}}

which proves that T (σ_{2})T (σ_{1}) = T (σ_{1}σ_{2}) , since \overline{σ}_{2}(σ̃_{1})σ̃_{2} is indeed a path in M with origin x_{0} which lifts (σ_{1}σ_{2}). □

Dependence of parameters.

Suppose the form ω depends holomorphically on m parameters t_{1}, …, t_{m}. Denoting by O the ring of holomorphic functions of the variables t_{1}, …, t_{m}, we have ω = ∑_{H∈A} f_{H}ω_{H} where f_{H} ∈ O ⊗_{ℂ} E. Then, for y ∈ M, the function F_{y} is a holomorphic function of t_{1}, …, t_{m}, i.e., F_{y} has values in O ⊗_{ℂ} A.

Then, given a path γ in M, the analytic continuation t ↦ (γ^{∗}F)(t, γ(0)) depends holomorphically of t_{1}, …, t_{m}.

If ω is integrable and W -stable, then the monodromy morphism depends holomorphically on the parameters t_{1}, …, t_{m}. It follows that we have a monodromy morphism

T : π_{1}(M/W, x_{0})^{op} → (O ⊗_{ℂ} A)^{×}.

B. A family of monodromy representations of the braid group.

From now on, we assume that A = ℂW.

Notation and hypothesis.

We denote by O the ring of holomorphic functions of a set of ∑_{C∈A/W} e_{C} variables z = (z_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)} .

Let

t := (t_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)}

be a set of ∑_{C∈A/W} e_{C} complex numbers. For H ∈ C, we set t_{H,j} := t_{C,j}.

We put

q_{C,j} = exp(−t_{C,j}/e_{C}) for C ∈ A/W , 0 ≤ j ≤ e_{C} − 1 .

For H ∈ C, we set q_{H,j} := q_{C,j}.

Let C ∈ A/W and let H ∈ C. For 0 ≤ j ≤ e_{C} − 1, we denote by ε_{j}(H) the primitive idempotent of the group algebra ℂW_{H} associated with the character det_{V}^{j} of the group W_{H}. Thus we have

ε_{j}(H) = \frac{1}{e_{C}} ∑_{k=0}^{k=e_{C}−1} exp(\frac{−2iπjk}{e_{C}})s_{H}^{k} .

We set

a_{H} := ∑_{j=0}^{j=e_{H}−1} t_{H,j}ε_{j}(H) and ω := ∑_{H∈A} a_{H}ω_{H} .

In other words, we have

ω = ∑_{C∈A/W} ∑_{j=0}^{j=e_{C}−1} ∑_{H∈C} t_{C,j}ε_{j}(H)ω_{H} .

The following lemma is clear.

4.10. The map A → A, H ↦ a_{H} has the following properties :

(1) it is W -stable, i.e., for all w ∈ W and H ∈ A, we have a_{w(H)} = wa_{H}w^{−1},

(2) for all H ∈ A, a_{H} belongs to the image of ℂW_{H} in A.

The following property follows from 4.10.

4.11. Lemma. The form ω is W -stable and integrable.

Proof of 4.11. The form ω is W -stable by 4.7. It is integrable by 4.6. Indeed, let X be a codimension 2 subspace of V and let H be an element of A containing X. By 4.10 above, it is enough to check that, if w ∈ W_{H}, then w commutes with ∑_{(H′∈A)(H′⊃X)} a_{H′}. This is the case since w centralizes X, hence normalizes {H′ ∈ A | (H′ ⊃ X)}. □

The main theorem.

4.12. Theorem. We denote by T : B^{op} → (ℂW)^{×} the monodromy morphism associated with the differential form ω on M. For all H ∈ C, we have

∏_{j=0}^{j=e_{C}−1}(T (s_{H,γ}) − q_{H,j}det_{V} (s_{H})^{j}) = 0 .

Furthermore, T depends holomorphically on the parameters t_{C,j}, i.e., arises by specialization from a morphism T: B^{op} → (OW)^{×}.

Proof of 4.12. First step : case of rank 1

Here we assume dim(V) = 1. So we may assume that W is the cyclic group of order e generated by the multiplication s by exp(2iπ/e). We have M = ℂ^{×}.

For 0 ≤ j ≤ e − 1, let ε_{j} be the primitive idempotent of ℂW corresponding to the character of W which sends s onto exp(2iπj/e).

There are e complex numbers t_{0}, t_{1}, …, t_{e−1} such that

ω = ∑_{j=0}^{j=e−1} \frac{t_{j}}{2iπ} ε_{j} \frac{dz}{z} .

A function F: ℂ^{×} → ℂW may be written

F = ∑_{j=0}^{j=e−1} F_{j}ε_{j}

where F_{j}: ℂ^{×} → ℂ.

The equation Eq(ω) becomes

\frac{dF_{j}}{dz} = \frac{t_{j}}{2iπ} \frac{F_{j}(z)}{z} for 0 ≤ j ≤ e − 1 .

Hence the solution F (x, 1) is given by the formula

F (x, 1) = ∑_{j=0}^{j=e−1} x^{t_{j}/2iπ}ε_{j} .

The analytic continuation of F along the path σ : t ↦ exp(2iπt/e) gives

S(σ) = ∑_{j=0}^{j=e−1} exp(t_{j}/e)ε_{j}

hence

T (s) = S(σ)^{−1}s = ∑_{j=0}^{j=e−1} exp(−t_{j}/e) exp(2iπj/e)ε_{j} .

Thus we see that, with q_{j} := exp(−t_{j}/e), we have

(4.13)  ∏_{j=0}^{j=e−1}(T (s) − q_{j} exp(2iπj/e)) = 0 ,

as claimed. Second step : towards the reduction to the case of rank 1

We are back to the general case. Here we use notation introduced in §2.B.

1. First we prove that, to compute the relation satisfied by the monodromy of s_{H,γ}, we may assume that x_{0} is “close to H”, namely that x_{0} = x_{H}.

Let us denote by

T_{x_{H}}: π_{1}(M/W, x_{H})^{op} → A

the monodromy morphism associated with ω, and let us denote by s_{H} the element of π_{1}(M/W, x_{H}) defined by the path σ_{H,x_{H}}.

4.14. Lemma. For any path γ from x_{0} to s_{H}(x_{0}), we have

T (s_{H,γ}) = S(γ)^{−1}T_{x_{H}} (s_{H})S(γ) .

Proof of 4.14. By (2.13), we have σ_{H,γ} := s_{H}(γ^{−1})·σ_{H,x_{H}} ·γ , from which it follows that

T (s_{H,γ}) = S(σ_{H,γ}^{−1})s_{H}
= S(γ^{−1})S(σ_{H,x_{H}}^{−1} )S(s_{H}(γ))s_{H}
= S(γ^{−1})S(σ_{H,x_{H}}^{−1} )s_{H}S(γ)
= S(γ^{−1})T_{x_{H}} (s_{H})S(γ)

□

2. Now we prove that we may reduce to rank one.

Choose and fix H ∈ A. We still use notation introduced in §2.

The elements of the affine line (x_{H} + L_{H}) are the elements x_{H}(z) := pr_{H}(x_{H}) + zpr_{H}^{⊥}(x_{H}) with z ∈ ℂ. We may adjust the choice of x_{H} so that, if

D_{H}^{×} := {x_{H}(z) | 0 < |z| < 2} ,

we have D_{H}^{×} ⊂ M . Note that D_{H}^{×} is stable by the operation of the group W_{H}.

We have α_{H}(x_{H}(z)) = zα_{H}(pr_{H}^{⊥}(x_{H})). For H′ ≠ H, we set α_{H′}(pr_{H}^{⊥}(x_{H})) = u_{H′}, and α_{H′}(pr_{H}(x_{H})) = v_{H′}. Recall that x_{H} has been chosen so that

zu_{H′} + v_{H′} ≠ 0 on D_{H} if H′ ≠ H .

Then the function

R_{H}F: D_{H}^{×} → ℂW , x_{H}(z) ↦ F (x_{H}(z), x_{H}) ,

satisfies the following differential equation :

\frac{d(R_{H}F)}{dz} = (\frac{1}{2iπ} \frac{a_{H}}{z} + ∑_{H′≠H} \frac{1}{2iπ} \frac{u_{H′}}{u_{H′}z + v_{H′}} a_{H′} ) R_{H}F (x_{H}(z)) .

In other words, R_{H}F satisfies the differential equation associated with the differential form R_{H}ω defined on D_{H}^{×} by

(4.15)  R_{H}ω := \frac{1}{2iπ} (\frac{a_{H}}{z} + ∑_{H′≠H} \frac{u_{H′}}{u_{H′}z + v_{H′}} a_{H′} ) dz .

Note that R_{H}ω is W_{H}-stable.

3. Now we reduce to the case of the action of the cyclic group W_{H} on D_{H}^{×}.

Let R_{H}S: P(D_{H}^{×}) → ℂW be the monodromy functor associated with the form R_{H}ω. By the existence and unicity theorem for linear differential equations, since the loop σ_{H} takes its values in D_{H}^{×}, we see that

(4.16)  S(σ_{H}) = R_{H}S(σ_{H}) .

Let us still denote by s_{H} the image of the path σ_{H} in π_{1}(D_{H}^{×}/W_{H}). Let

R_{H}T_{x_{H}}: π_{1}(D_{H}^{×}/W_{H})^{op} → A

be the monodromy morphism associated with the differential form R_{H}ω. Then it results from (4.16) and from Lemma 4.14 that 4.17. T (s_{H,γ}) is conjugate (in (ℂW)^{×}) to R_{H}T_{x_{H}} (s_{H}).

Third step : reduction to the case of rank 1

Let T_{H}: π_{1}(D_{H}^{×}/W_{H}, x_{H})^{op} → ℂW_{H} be the monodromy morphism associated with the W_{H}-stable differential form defined on D_{H}^{×} by a_{H}ω_{H}. By (4.13), we know that the characteristic polynomial of T_{H}(s_{H}) (viewed as acting on ℂW_{H} by left multiplication) is

P_{H}(t) := ∏_{j=0}^{j=e_{C}−1}(t − q_{C,j} exp(2iπj/e_{C})) ,

where C denotes the W -orbit of H. We want to prove that P_{H}(R_{H}T_{x_{H}} (s_{H})) = 0. By the first two steps, it is clear that our problem may be reformulated as follows. We set D := {z ∈ ℂ | (|z| < 2)} and D^{×} := D − {0} , and we view D and D^{×} as

endowed with the action of W_{H} defined by (w, z) ↦ det_{V} (w)z for w ∈ W_{H} and z ∈ D.

In order to simplify the notation, we also set e := e_{C}(= e_{H}), and for 0 ≤ j ≤ e − 1, ε_{j} := ε_{j}(H) , t_{j} := t_{C,j} , q_{j} := q_{C,j} , and a := a_{H} = ∑_{j=0}^{e−1} t_{j}ε_{j} . We define a

holomorphic function b on D by the formula b(z) := \frac{1}{2iπ} ∑_{H′≠H} \frac{u_{H′}}{u_{H′}z + v_{H′}} a_{H′} (note

that u_{H′}z + v_{H′} ≠ 0 on D).

We have two differential equations for holomorphic functions (locally) defined on D^{×} with values in ℂW :

\frac{dF_{H}}{dz} (z) = \frac{a}{z}F_{H}(z) (Eq(a))

\frac{dF}{dz} (z) = (\frac{a}{z} + b(z))F (z) . (Eq(a, b))

Both these equations correspond to W_{H}-stable differential forms on D^{×} : a commutes with the action of W_{H}, and we have b(w.z) = det_{V} (w)^{−1}wb(z)w^{−1} for w ∈ W_{H} and z ∈ D.

We denote by S_{H} and S the functors from P(D^{×}) to (ℂW)^{×} associated respectively to the equations Eq(a) and Eq(a, b), and by T_{H} , T: π_{1}(D^{×}/W_{H}, 1) → (ℂW)^{×} the corresponding morphisms.

4.18. Proposition. Assume that, for all j, k , 0 ≤ j, k ≤ e − 1, q_{j}q_{k}^{−1} is not an e-th root of the unity. Then there exists an invertible element u of ℂW such that, for all σ ∈ π_{1}(D^{×}/W_{H}, 1), we have T (σ) = uT_{H}(σ)u^{−1} .

Proof of 4.18.

1. Equivalence of Eq(a) and Eq(a, b).

Here we follow [Ha], 1.4.

Let us consider the following differential equation

(Eq′(a, b))  \frac{dΦ}{dz}(z) = \frac{1}{z}(aΦ(z) − Φ(z)a) + b(z)Φ(z) ,

for Φ a function (locally) defined on D^{×} with values in ℂW. The following assertion is proved, for example, in [Ha], 1.4. (see in particular 1.4.2, and proof of 1.4.1). Here we

use the fact that the spectrum of the multiplication by a in ℂW is \{\frac{t_{0}}{2iπ} , \frac{t_{1}}{2iπ} , …, \frac{t_{e−1}}{2iπ} \}

(each \frac{t_{j}}{2iπ} with multiplicity |W : W_{H}|).

4.19. Lemma. Assume that, for all j, k , 0 ≤ j, k ≤ e−1, we have t_{j} −t_{k} ∉ 2iπℤ−{0}. Then there is a unique solution Φ of Eq′(a, b) satisfying the following two conditions :

(1) Φ is holomorphic on D,

(2) Φ(0) = 1.

Now it is immediate to check that, if z ↦ F_{H}(z, z_{0}) is the solution of Eq(a), defined in the neighbourhood of z_{0}, and such that F_{H}(z_{0}, z_{0}) = 1, then the function z ↦ F (z, z_{0}) := Φ(z)F_{H}(z, z_{0})Φ(z_{0})^{−1} is the solution of Eq(a, b), defined in the neighbourhood of z_{0}, and such that F (z_{0}, z_{0}) = 1.

By the formula (4.5), we see that, for all homotopy classes of paths γ in D^{×}, we have then

(4.20)  S(γ) = Φ(γ(1))S_{H}(γ)Φ(γ(0))^{−1} .

2. W_{H}-equivariance and proof of Proposition 4.18

By the unicity property of Φ (see Lemma 4.19), it follows from the W_{H} invariance of a and b that Φ(det_{V} (w)z) = wΦ(z)w^{−1} for all w ∈ W_{H} and z ∈ D. Then the formula (4.20), together with the definition 4.9 of the monodromy functors, imply Proposition 4.18 with u := Φ(1), i.e., for all σ ∈ π_{1}(D^{×}/W_{H}, 1), we have

T (σ) = Φ(1)T_{H}(σ)Φ(1)^{−1} .

□

Conclusion : end of proof of Theorem 4.12 From what precedes, we see that Theorem 4.12 is proved provided the family

q := (q_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)}

satisfies the condition that (for all C and all j, k) q_{C,j}q_{C,k}^{−1} is not an e_{C}-th root of the unity, i.e., if the family t has the property that t_{C,j} − t_{C,k} is not a non-zero integer, for all C and all j, k. Since the set of such families is a dense open subset in the space ℂ^{∑_{C∈A} e_{C}} of all families t, we see that Theorem 4.12 follows by continuity, since the solution x ↦ F (x, y) is a holomorphic function of t. □

C. Hecke algebras.

We define a set

u = (u_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)}

of ∑_{C∈A/W} e_{C} indeterminates. We denote by ℤ[u, u^{−1}] the ring of Laurent polynomials in the indeterminates u.

Let ℑ be the ideal of the group algebra ℤ[u, u^{−1}]B generated by the elements

(s_{H,γ} − u_{C,0})(s_{H,γ} − u_{C,1}) ⋯ (s_{H,γ} − u_{C,e_{C}−1})

where C ∈ A/W, H ∈ C, s_{H,γ} is a generator of the monodromy around H in B (cf. (2.13)) and s is the image of s_{H,γ} in W.

4.21. Definition. The Hecke algebra H_{u}(W) is the ℤ[u, u^{−1}]-algebra ℤ[u, u^{−1}]B/ℑ.

Now assume that W is a finite irreducible complex reflection group (see §2.C above for notation and references). Let D be the diagram of W, and let s ∈ N (D) be a node of D. We set u_{s,j} := u_{C,j} for j = 0, 1, …, e_{C} − 1, where C denotes the orbit under W of the reflecting hyperplane of s.

The following proposition is an immediate consequence of Theorem 2.27 :

4.22. Proposition. Assume W is different from G_{24}, G_{27}, G_{29}, G_{33}, G_{34} – and also different from G_{31} for which the following assertion is still conjectural. The Hecke algebra H_{u}(W) is isomorphic to the ℤ[u, u^{−1}]–algebra generated by elements (T_{s})_{s∈N (D)} such that

• the elements T_{s} satisfy the braid relations defined by D_{br}^{op},

• we have (T_{s} − u_{s,0})(T_{s} − u_{s,1}) ⋯ (T_{s} − u_{s,e_{s}−1}) = 0 .

Notice that through the specialization u_{s,j} ↦ det_{V} (s)^{j} (for s ∈ N (D) and 0 ≤ j ≤ e_{s} −1), the algebra H_{u}(W) becomes the group algebra of W^{op} over a suitable cyclotomic extension of ℤ.

Hecke algebras and parabolic subgroups.

Let I be an intersection of reflecting hyperplanes, and let W_{I} (the pointwise stabilizer of I) be the corresponding parabolic subgroup of W (see §2.D above). We recall that we denote by A_{I} the set of reflecting hyperplanes of W_{I}.

Let u = (u_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)} be a set of ∑_{C∈A/W} e_{C} indeterminates and v = (v_{C,j})_{(C∈A_{I}/W_{I})(0≤j≤e_{C}−1)} be a set of ∑_{C∈A_{I}/W_{I}} (e_{C}) indeterminates.

Let a : A_{I}/W_{I} → A/W be the map which sends a W_{I}-orbit of hyperplanes onto the corresponding W -orbit.

We have a morphism

ℤ[v, v^{−1}] → ℤ[u, u^{−1}] , v_{C,j} ↦ u_{a(C),j} .

The injection B_{W_{I}} ↪ B_{W}, defined up to conjugation (see 2.29 above) induces an inclusion (defined up to conjugation)

H_{v}(W_{I}) ⊗_{ℤ[v,v^{−1}]} ℤ[u, u^{−1}] → H_{u}(W) ,

whose image is called the parabolic Hecke sub–algebra of H_{u}(W) associated with I.

Specializing u_{C,j} to exp(−2iπj/e_{C}), this gives rise to the inclusion ℤW_{I} ↪ ℤW induced by the group inclusion W_{I} ⊆ W.

Hecke algebras and monodromy representations.

By Theorem 4.12, we see that the monodromy representation T factors through H_{u}(W). Indeed, let us set

t_{C,j} := det_{V} (s)^{j}u_{C,j}

for all (C, j), and

t := (t_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)} ,

and let us denote by O the ring of holomorphic functions of the set of variables t. Then we have the following commutative diagram :

\xymatrix{OB \ar[rr]^{T} \ar[dr] &  & OW^{op} \\  & O ⊗_{ℤ[u,u^{−1}]} H_{u}(W) \ar@{-->}[ur]}

Let K be the field of fractions of O.

The following lemma is a key point to understand the structure of H_{u}(W). It is wellknown to hold for Coxeter groups. For the infinite series of complex reflection groups, see [ArKo] for G(d, 1, r), [BrMa], (4.12) for G(2d, 2, r) and [Ari], Proposition 1.4 for the general case (it has been also checked for many of the remaining groups of small rank — see for example [BrMa], Satz 4.7). We conjecture it is true for all complex reflection groups.

4.23. Lemma. Assume W is Coxeter group or a complex reflection group in the infinite series.

The ℤ[u, u^{−1}]-module H_{u}(W) can be generated by |W | elements.

From this lemma, we can now deduce the following

4.24. Theorem. Assume W is Coxeter group or a complex reflection group in the infinite series.

The monodromy representation T induces an isomorphism of K-algebras

K ⊗_{ℤ[u,u^{−1}]} H_{u}(W) ⟶^{∼} KW^{op} .

Furthermore, H_{u}(W) is a free ℤ[u, u^{−1}]-module of rank |W |.

Proof. By Lemma 4.23, there is a surjective morphism of ℤ[u, u^{−1}]-modules

ϕ: ℤ[u, u^{−1}]^{|W |} → H_{u}(W) .

Let 𝔪 be the ideal of O of the functions vanishing at the point (t_{C,j} = 1). The morphism O_{𝔪} ⊗_{ℤ[u,u^{−1}]} H_{u} → O_{𝔪}W induced by the monodromy is surjective by Nakayama’s lemma, since it becomes an isomorphism after tensoring by (O_{𝔪})/𝔪. Composing with 1_{O_{𝔪}} ⊗ ϕ, we obtain an epimorphism O_{𝔪}^{|W |} → O_{𝔪}W : this must be an isomorphism. Hence, ker ϕ = 0, i.e., ϕ is an isomorphism and H_{u} is free of rank |W | over ℤ[u, u^{−1}].

Since the morphism K ⊗_{ℤ[u,u^{−1}]} H_{u} → KW is a surjective morphism between two K-modules with same dimensions, it is an isomorphism and Theorem 4.24 follows. □

5. Diagrams and tables

Information provided by the tables: invariants of braid diagrams.

Let us recall that a diagram where the orders of the nodes are “forgotten” and where only the braid relations are kept is called a braid diagram for the corresponding group.

The groups have been ordered by their diagrams, by collecting groups with the same braid diagram. Thus, for example,

• G_{15} has the same braid diagram as the groups G(4d, 4, 2) for all d ≥ 2,

• G_{4}, G_{8}, G_{16}, G_{25}, G_{32} all have the same braid diagrams as groups 𝔖_{3}, 𝔖_{4} and 𝔖_{5},

• G_{5}, G_{10}, G_{18} have the same braid diagram as the groups G(d, 1, 2) for all d ≥ 2, • G_{7}, G_{11}, G_{19} have the same braid diagram as the groups G(2d, 2, 2) for all d ≥ 2,

• G_{26} has the same braid diagram as G(d, 1, 3) for d ≥ 2.

The element β (generator of Z(W)) is given in the last column of our tables. Notice that the knowledge of degrees and codegrees allows then to find the order of Z(W), which is not explicitely provided in the tables.

The tables provide diagrams and data for all irreducible reflection groups.

• Tables 1 and 2 collect groups corresponding to infinite families of braid diagrams,

• Table 3 collects groups corresponding to exceptional braid diagrams (notice that the fact that the diagram for G_{31} provides a braid diagram is only conjectural), but G_{24}, G_{27}, G_{29}, G_{33}, G_{34},

• The last table (table 4) provides diagrams for the remaining cases (G_{24}, G_{27}, G_{29}, G_{33}, G_{34}). It is not known nor conjectural whether these diagrams provide braid diagrams for the corresponding braid groups.

Degrees and codegrees of a braid diagram.

The following property may be noticed on the tables. It generalizes a property already noticed by Orlik and Solomon for the case of Coxeter–Shephard groups (see [OrSo3], (3.7)).

5.1. Theorem. Let D be a braid diagram of rank r. There exist two families

(d_{1}, d_{2}, …, d_{r}) and (d_{1}^{∗}, d_{2}^{∗}, …, d_{r}^{∗})

of r integers, depending only on D, and called respectively the degrees and the codegrees of D, with the following property: whenever W is a complex reflection group with D as a braid diagram, its degrees and codegrees are given by the formulae

d_{j} = |Z(W)|d_{j} and d_{j}^{∗} = |Z(W)|d_{j}^{∗} (j = 1, 2, …, r) .

The zeta function of a braid diagram.

In [DeLo], Denef and Loeser compute the zeta function of local monodromy of the discriminant of a complex reflection group W, which is the element of ℚ[q] defined by the formula

Z(q, W) := ∏_{j} det(1 − qμ, H^{j}(F_{0}, ℂ))^{(−1)^{j+1}} ,

where F_{0} denotes the Milnor fiber of the discriminant at 0 and μ denotes the monodromy automorphism (see [DeLo]).

Putting together the tables of [DeLo] and our braid diagrams, one may notice the following fact.

5.2. Theorem. The zeta function of local monodromy of the discriminant of a complex reflection group W depends only on the braid diagram of W.

Remark. Two different braid diagrams may be associated to isomorphic braid groups. For example, this is the case for the following rank 2 diagrams (where the sign “∼” means that the corresponding groups are isomorphic) :

For e even, s◯_{e+1} ^{◯^{t}}_{◯_{u}} ∼ s◯ e ^{◯ t}_{◯ u} ,

for e odd, s◯_{e+1} ^{◯^{t}}_{◯_{u}} ∼ ◯_{s} ◯_{t},

and s ◯ ^{◯ t}_{◯ u}_{5} 4 ∼ ◯_{s} ◯_{t}.

It should be noticed, however, that the above pairs of diagrams do not have the same degrees and codegrees, nor do they have the same zeta function. Thus, degrees, codegrees and zeta functions are indeed attached to the braid diagrams, not to the braid groups.

Appendix 1 : Generators of the monodromy around an irreducible divisor

We define here what we mean by a “generator of the monodromy around an irreducible divisor” and recall some well known properties.

Let Y be a smooth connected complex algebraic variety, I a finite family of irreducible codimension 1 closed subvarieties (irreducible divisors) and Z := ∪_{D∈I}D. Let X := Y − Z and x_{0} ∈ X.

For D ∈ I, let D_{s} be the smooth part of D and D̃ := D_{s} − (D_{s} ∩ ⋃_{D′∈I,D′≠D} D′) . “A path from x_{0} to D in X” is by definition a path γ in Y such that γ(0) = x_{0},

γ(1) ∈ D̃ and γ(t) ∈ X for t ≠ 1.

Let γ′ be another path from x_{0} to D in X. We say that γ and γ′ are D-homotopic if there is a continuous map T : [0, 1] × [0, 1] → Y such that T (t, 0) = γ(t) and T (t, 1) = γ′(t) for t ∈ [0, 1] , T (0, u) = x_{0} and T (1, u) ∈ D̃ for all u ∈ [0, 1] and T (t, u) ∈ X for t ∈ [0, 1[ and u ∈ [0, 1]. We denote by [γ] the D-homotopy class of γ.

Given a path γ from x_{0} to D in X, let B be a connected open neighbourhood of γ(1) in X ∪ D̃ such that B ∩ X has a fundamental group free abelian of rank 1. Let u ∈ [0, 1[ such that γ(t) ∈ B for t ≥ u. Put x_{1} := γ(u). The orientation of B ∩ X coming from the orientation of X gives an isomorphism f: π_{1}(B ∩ X, x_{1}) ⟶^{∼} ℤ. Let λ be a loop in B ∩ X from x_{1} such that f([λ]) = 1.

Let γ_{u} be the “restriction” of γ to [0, u], defined by γ_{u}(t) := γ(ut) for all t ∈ [0, 1]. Define ρ_{γ,λ} := γ_{u}^{−1} · λ · γ_{u} . Then, the homotopy class of ρ_{γ,λ} in π_{1}(M, x_{0}) depends only on the D-homotopy class of γ and is denoted by ρ_{[γ]}. We call it the generator of the monodromy around D associated to [γ].

[figure: • ·; ·; γ]x_{0}

Given two paths γ and γ′ from x_{0} to D, the generators of monodromy ρ_{[γ]} and ρ_{[γ′]} are conjugate.

A1. Proposition. Let i be the injection of an irreducible divisor D in a smooth connected complex variety Y and x_{0} ∈ Y − D. Then, the kernel of the morphism π_{1}(i): π_{1}(Y − D, x_{0}) → π_{1}(Y, x_{0}) is generated by all the generators of the monodromy around D.

Sketch of proof of A1. Note that the singular points of D form a closed subvariety D_{sing} of D, distinct from D, hence of (complex) codimension at least 2 in Y. Therefore (see for example [Go], chap. x, 2.3) the natural morphism π_{1}(Y −D−D_{sing}, x_{0}) → π_{1}(Y −D, x_{0}) is an isomorphism, and in order to prove A1 we may assume D is smooth, which we do now.

The lemma then follows from the fact that given a locally constant sheaf F over Y − D, its extension i_{∗}F to Y is locally constant if and only if every generator of the monodromy around D acts trivially on F. □

A2. Proposition. Suppose that Y is simply connected. Then the fundamental group π_{1}(X, x_{0}) is generated by all the generators of the monodromy around the divisors D ∈ I.

Proof of A2.

This follows immediately from Proposition A1 by induction on |I|. □

Lifting generators of the monodromy.

Let p: Y → \overline{Y} be a finite covering between two smooth connected complex varieties. Let D be the branch locus of p and \overline{D} = p(D). We assume \overline{D} is an irreducible divisor. We set \overline{X} := \overline{Y} − \overline{D} and X := Y − D.

We shall see that a generator of the monodromy around \overline{D} (associated to a path \overline{γ} from \overline{x}_{0} to \overline{D} in \overline{Y}) may be naturally lifted to an element of P(X) (which depends only on the \overline{D}–homotopy class of \overline{γ}).

Indeed, let γ be the path from x_{0} to an irreducible component, say D_{γ}, of D, which lifts \overline{γ}. Let \overline{B} be an open neighbourhood of \overline{x}_{0} in \overline{Y} such that the fundamental group of \overline{B} ∩ \overline{X} is free abelian of rank 1 and B ∩ (X ∪ D̃_{γ}) → \overline{B} is unramified outside D_{γ}. Let u ∈ [0, 1[ such that \overline{γ}(t) ∈ \overline{B} for t ≥ u. Let \overline{λ} be a loop in \overline{B} ∩ \overline{X} with origin \overline{γ}(u) which is a positive generator of π_{1}(\overline{B} ∩ \overline{X}, \overline{γ}(u)).

Let λ be the path from γ(u) which lifts \overline{λ}. Let γ_{u} be the restriction of γ to [0, u]. Let γ_{u}^{∨} be the path from λ(1) which lifts (\overline{γ}_{u})^{−1}, where \overline{γ}_{u} is the “restriction” of \overline{γ} to [0, u].

The proof of the following proposition is left to the reader.

A3. Proposition. We define ρ_{γ} := γ_{u}^{∨} · λ · γ_{u} .

(1) The homotopy class of ρ_{γ} in P(X) depends only on the \overline{D}–homotopy class of \overline{γ}.

(2) Let e_{D} denote the ramification index of p on D̃. Then ρ_{γ}^{e_{D}} is the generator of the monodromy around D_{γ} associated to γ.

Appendix 2 : tables 1 to 5

Here are some definitions, notation, conventions, which will allow the reader to understand the diagrams.

The groups have presentations given by diagrams D such that

• the nodes correspond to pseudo-reflections in W, the order of which is given inside the circle representing the node,

• two distinct nodes which do not commute are related by “homogeneous” relations with the same “support” (of cardinality 2 or 3), which are represented by links beween two or three nodes, or circles between three nodes, weighted with a number representing the degree of the relation (as in Coxeter diagrams, 3 is omitted, 4 is represented by a double line, 6 is represented by a triple line). These homogeneous relations are called the braid relations of D.

More details are provided below.

Meaning of the diagrams.

This paragraph provides a list of examples which illustrate the way in which diagrams provide presentations for the attached groups.

• The diagram ◯_{s}_{d} ^{e} ◯_{t}_{d} corresponds to the presentation

s^{d} = t^{d} = 1 and \underbrace{ststs ⋯}_{e factors} = \underbrace{tstst ⋯}_{e factors}

• The diagram ◯_{s}_{5}  ◯_{t}_{3} corresponds to the presentation
s^{5} = t^{3} = 1 and stst = tsts .

• The diagram s◯ a e ^{◯ b t}_{◯ c u} corresponds to the presentation

s^{a} = t^{b} = u^{c} = 1 and \underbrace{stustu ⋯}_{e factors} = \underbrace{tustus ⋯}_{e factors} = \underbrace{ustust ⋯}_{e factors} .

• The diagram ◯_{v}_{2}⧸◯_{s}_{2} ◯_{t}_{2}  ◯_{w}_{2} ◯_{u}_{2}⧹ corresponds to the presentation
s^{2} = t^{2} = u^{2} = v^{2} = w^{2} = 1 ,
uv = vu , sw = ws , vw = wv ,
sut = uts = tsu ,
svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

• The diagram s◯ d_{e+1} ^{◯ 3 ^{t′_{2}}} ⧹_{◯ 2 _{t_{2}}} ⧸◯_{t_{3}}_{2} corresponds to the presentation

s^{d} = t′_{2}^{2} = t_{2}^{2} = t_{3}^{2} = 1 , st_{3} = t_{3}s ,

st′_{2}t_{2} = t′_{2}t_{2}s ,

t′_{2}t_{3}t′_{2} = t_{3}t′_{2}t_{3} , t_{2}t_{3}t_{2} = t_{3}t_{2}t_{3} , t_{3}t′_{2}t_{2}t_{3}t′_{2}t_{2} = t′_{2}t_{2}t_{3}t′_{2}t_{2}t_{3} ,

\underbrace{t_{2}st′_{2}t_{2}t′_{2}t_{2}t′_{2} ⋯}_{e+1 factors} = \underbrace{st′_{2}t_{2}t′_{2}t_{2}t′_{2}t_{2} ⋯}_{e+1 factors} .

• The diagram e^{^{t′_{2}}◯ 2}⧹_{_{t_{2}}◯ 2}⧸◯_{t_{3}}_{2} corresponds to the presentation

t′_{2}^{2} = t_{2}^{2} = t_{3}^{2} = 1 ,

t′_{2}t_{3}t′_{2} = t_{3}t′_{2}t_{3} , t_{2}t_{3}t_{2} = t_{3}t_{2}t_{3} , t_{3}t′_{2}t_{2}t_{3}t′_{2}t_{2} = t′_{2}t_{2}t_{3}t′_{2}t_{2}t_{3} ,

\underbrace{t_{2}t′_{2}t_{2}t′_{2}t_{2}t′_{2} ⋯}_{e factors} = \underbrace{t′_{2}t_{2}t′_{2}t_{2}t′_{2}t_{2} ⋯}_{e factors} .

• The diagram s◯ 2_{5} ^{◯ 2 ^{t}}_{◯ 3 _{u}} corresponds to the presentation

s^{2} = t^{2} = u^{3} = 1 , stu = tus , ustut = stutu .

• The diagram ◯_{s}_{2} △ ◯_{t}_{2} ⧸^{◯ 2 ^{u}} ⧹ ⧹ corresponds to the presentation

s^{2} = t^{2} = u^{2} = 1 , stst = tsts , tutu = utut , utusut = sutusu , sus = usu .

• The diagram ◯_{s}_{2} △ ◯_{t}_{2}^{5} ⧸^{◯ 2 ^{u}} ⧹ corresponds to the presentation

s^{2} = t^{2} = u^{2} = 1 , stst = tsts , tutut = ututu , utusut = sutusu , sus = usu .

• The diagram ◯_{s}_{2} ◯_{t}_{2} ◯_{u}_{2} ⧸^{◯ 2 ^{v}} ⧹ corresponds to the presentation

\begin{matrix}s^{2} = t^{2} = u^{2} = v^{2} = 1 , sv = vs , su = us , \\ sts = tst , vtv = tvt , uvu = vuv , tutu = utut , vtuvtu = tuvtuv . \end{matrix}

• The diagram s ◯_{2} ^{◯_{2} t}_{◯_{2} u}_{5} 4 corresponds to the presentation

s^{2} = t^{2} = u^{2} = 1 , ustus = stust , tust = ustu .

• The diagram ◯_{t}_{2}^{◯ 2 ^{s}}∗ ◯_{u}_{2}^{◯ 2 ^{v}} corresponds to the presentation

\begin{matrix}s^{2} = t^{2} = u^{2} = v^{2} = 1 , su = us , tv = vt , \\ sts = tst , tut = utu , uvu = vuv , vsv = svs , stuvstuvs = tuvstuvst . \end{matrix}

In the following tables, we denote by H ⋊ K a group which is a non-trivial split extension of K by H. We denote by H · K a group which is a non-split extension of K by H. We denote by p^{n} an elementary abelian group of order p^{n}.

name  diagram  degrees codegrees  β  field G/Z(G)

G(de, e, r)_{e>2,d,r≥2} [figure: s; ◯ d; e+1; ^{◯ 2 ^{t′_{2}}}; ⧹; _{◯ 2 _{t_{2}}}; ⧸; ◯_{t_{3}}_{2}; ◯_{t_{4}}_{2}; ⋯; ◯_{t_{r}}_{2}]{(ed,2ed,..., \atop (r−1)ed,rd)} {(0,ed,..., \atop (r−1)ed)} s^{\frac{r}{(e∧r)}} (t′_{2}t_{2}t_{3}···t_{r})^{\frac{e(r−1)}{(e∧r)}} ℚ(ζ_{de})

G_{15}  [figure: s; ◯ 2; 5; ^{◯ 2 ^{t}}; _{◯ 3 _{u}}]  12, 24 0, 24  ustut=s(tu)^{2}  ℚ(ζ_{24}) 𝔖_{4}

fS_{r+1}_{r≥1}  ◯_{t_{1}}_{2} ◯_{t_{2}}_{2}⋯◯_{t_{r}}_{2}  {(2,3,..., \atop ...,r+1)}  {(0,1,..., \atop ...,r−1)}  (t_{1}···t_{r})^{r+1}  ℚ
G_{4}  ◯_{s}_{3}  ◯_{t}_{3}  4, 6  0, 2  (st)^{3}  ℚ(ζ_{3})  𝔄_{4}
G_{8}  ◯_{s}_{4}  ◯_{t}_{4}  8, 12  0, 4  (st)^{3}  ℚ(i)  𝔖_{4}
G_{16}  ◯_{s}_{5}  ◯_{t}_{5}  20, 30 0, 10  (st)^{3}  ℚ(ζ_{5})  𝔄_{5}
G_{25}  ◯_{s}_{3}  ◯_{t}_{3}  ◯_{u}_{3}  6, 9, 12 0, 3, 6  (stu)^{4}  ℚ(ζ_{3}) 3^{2}⋊SL_{2}(3)
G_{32}  ◯_{s}_{3} ◯_{t}_{3} ◯_{u}_{3} ◯_{v}_{3} 12,18,24,30 0,6,12,18  (stuv)^{5}  ℚ(ζ_{3}) P Sp_{4}(3)
G(d, 1, r)_{d≥2} ◯_{s}_{d} ◯_{t_{2}}_{2} ◯_{t_{3}}_{2}⋯◯_{t_{r}}_{2}  {(d,2d,..., \atop ...,rd)} {(0,d,..., \atop ...,(r−1)d)}  (st_{2}t_{3}···t_{r})^{r}  ℚ(ζ_{d})
G_{5}  ◯_{s}_{3}  ◯_{t}_{3}  6, 12  0, 6  (st)^{2}  ℚ(ζ_{3})  𝔄_{4}
G_{10}  ◯_{s}_{4}  ◯_{t}_{3}  12, 24 0, 12  (st)^{2}  ℚ(ζ_{12}) 𝔖_{4}
G_{18}  ◯_{s}_{5}  ◯_{t}_{3}  30, 60 0, 30  (st)^{2}  ℚ(ζ_{15}) 𝔄_{5}
G_{26}  ◯_{s}_{2}  ◯_{t}_{3}  ◯_{u}_{3} 6, 12, 18 0, 6, 12  (stu)^{3}  ℚ(ζ_{3}) 3^{2}⋊SL_{2}(3)

name  diagram  degrees codegrees  β  field G/Z(G)

G(2d, 2, r)_{d,r≥2}◯_{s}_{d}  ^{◯ 2 ^{t′_{2}}} ⧹_{◯ 2 _{t_{2}}} ⧸◯_{t_{3}}_{2} ◯_{t_{4}}_{2}⋯◯_{t_{r}}_{2}{(2d,4d,... \atop 2(r−1)d,rd)} {(0,2d,... \atop 2(r−1)d)} s^{\frac{r}{(2∧r)}} (t′_{2}t_{2}t_{3}···t_{r})^{\frac{2(r−1)}{(2∧r)}} ℚ(ζ_{2d})

G_{7}  s◯ 2  ^{◯ 3 t}_{◯ 3 u}  12, 12  0, 12  stu  ℚ(ζ_{12}) 𝔄_{4}

G_{11}  s◯ 2  ^{◯ 3 t}_{◯ 4 u}  24, 24  0, 24  stu  ℚ(ζ_{24}) 𝔖_{4}

G_{19}  s◯ 2  ^{◯ 3 t}_{◯ 5 u}  60, 60  0, 60  stu  ℚ(ζ_{60}) 𝔄_{5}

G(e, e, r)_{e≥2,r>2} e^{^{t′_{2}}◯ 2}⧹_{_{t_{2}}◯ 2}⧸◯_{t_{3}}_{2} ◯_{t_{4}}_{2}⋯◯_{t_{r}}_{2}  {(e,2e,..., \atop (r−1)e,r)} {(0,e,...,(r−2)e, \atop (r−1)e−r)} (t′_{2}t_{2}t_{3}···t_{r})^{\frac{e(r−1)}{(e∧r)}}  ℚ(ζ_{e})
G(e, e, 2)_{e≥3}  ◯_{s}_{2} ^{e} ◯_{t}_{2}  2, e  0, e − 2  (st)^{e/(e∧2)}  ℚ(ζ_{e}+ζ_{e}^{−1})
G_{6}  ◯_{s}_{3}  ◯_{t}_{2}  4, 12  0, 8  (st)^{3}  ℚ(ζ_{12}) 𝔄_{4}
G_{9}  ◯_{s}_{4}  ◯_{t}_{2}  8, 24  0, 16  (st)^{3}  ℚ(ζ_{8}) 𝔖_{4}
G_{17}  ◯_{s}_{5}  ◯_{t}_{2}  20, 60  0, 40  (st)^{3}  ℚ(ζ_{20}) 𝔄_{5}
G_{14}  ◯_{s}_{3} ^{8} ◯_{t}_{2}  6, 24  0, 18  (st)^{4}  ℚ(ζ_{3},\sqrt{−2}) 𝔖_{4}
G_{20}  ◯_{s}_{3} ^{5} ◯_{t}_{3}  12, 30  0, 18  (st)^{5}  ℚ(ζ_{3},\sqrt{5}) 𝔄_{5}
G_{21}  ◯_{s}_{3} ^{10} ◯_{t}_{2}  12, 60  0, 48  (st)^{5}  ℚ(ζ_{12},\sqrt{5}) 𝔄_{5}

name  diagram  degrees codegrees  β  field  G/Z(G)

G_{12}  [figure: s; ◯ 2; ◯ 2 t; ◯ 2 u]  6,8  0,10  (stu)^{4} ℚ(√ ; \overline{−2})  𝔖_{4}

G_{13}  [figure: s; ◯_{2}; ◯_{2} t; ◯_{2} u; 5 4]  8,12  0,16  (stu)^{3}  ℚ(ζ_{8})  𝔖_{4}

G_{22}  s◯ 2 5 ^{◯ 2 t}_{◯ 2 u}  12,20  0,28  (stu)^{5} ℚ(i,\sqrt{5})  𝔄_{5}
G_{23}  ◯_{s}_{2} ^{5} ◯_{t}_{2}  ◯_{u}_{2}  2,6,10  0,4,8  (stu)^{5}  ℚ(\sqrt{5})  𝔄_{5}

G_{28}  ◯_{s}_{2} ◯_{t}_{2} ◯_{u}_{2} ◯_{v}_{2}  {2,6, \atop 8,12}  {0,4, \atop 6,10}  (stuv)^{6}  ℚ  2^{4}⋊(𝔖_{3}×𝔖_{3}) †

G_{30}  ◯_{s}_{2} ^{5} ◯_{t}_{2} ◯_{u}_{2} ◯_{v}_{2}  {2,12, \atop 20,30}  {0,10, \atop 18,28}  (stuv)^{15} ℚ(\sqrt{5})  𝔄_{5} ≀ 2

G_{35}  ◯_{s_{1}}_{2} ◯_{s_{3}}_{2} ◯_{s_{4}}_{2}^{◯ 2 ^{s_{2}}} ◯_{s_{5}}_{2} ◯_{s_{6}}_{2}  {2,5,6,8, \atop 9,12}  {0,3,4,6, \atop 7,10}  (s_{1}···s_{6})^{12}  ℚ  SO_{6}^{−}(2)′

G_{36}  ◯_{s_{1}}_{2} ◯_{s_{3}}_{2} ◯_{s_{4}}_{2}^{◯ 2 ^{s_{2}}} ◯_{s_{5}}_{2} ◯_{s_{6}}_{2} ◯_{s_{7}}_{2}  {2,6,8, \atop 10,12,} ; 14,18  {0,4,6, \atop 8,10,} ; 12,16  (s_{1}···s_{7})^{9}  ℚ  SO_{7}(2)

G_{37} ◯_{s_{1}}_{2} ◯_{s_{3}}_{2} ◯_{s_{4}}_{2}^{◯ 2 ^{s_{2}}} ◯_{s_{5}}_{2} ◯_{s_{6}}_{2} ◯_{s_{7}}_{2} ◯_{s_{8}}_{2} {2,8,12, \atop 14,18,20,} ; 24,30  {0,6,10, \atop 12,16,18,} ; 22,28 (s_{1}···s_{8})^{15}  ℚ  SO_{8}^{+}(2)

G_{31}  ◯_{v}_{2}⧸◯_{s}_{2} ◯_{t}_{2}  ◯_{w}_{2} ◯_{u}_{2}⧹  {8,12, \atop 20,24}  {0,12, \atop 16,28}  (stuvw)^{6} ℚ(i)  2^{4}⋊𝔖_{6} ⋆

Table 3

It is still conjectural whether the corresponding braid diagram for G_{31} provides a presentation for the associated braid group.

† The action of 𝔖_{3} × 𝔖_{3} on 2^{4} is irreducible.

⋆ The group G_{31}/Z(G_{31}) is not isomorphic to the quotient of the Weyl group D_{6} by its center.

name  diagram  degrees codegrees  β  field  G/Z(G)

G_{24}  ◯_{s}_{2} △ ◯_{t}_{2} ⧸^{◯ 2 ^{u}} ⧹ ⧹  4,6,14  0,8,10  (stu)^{7}  ℚ(\sqrt{−7})  GL_{3}(2)

G_{27}  ◯_{s}_{2} △ ◯_{t}_{2}^{5} ⧸^{◯ 2 ^{u}} ⧹  6,12,30  0,18,24  (stu)^{5} ℚ(ζ_{3},\sqrt{5})  𝔄_{6}

G_{29}  ◯_{s}_{2} ◯_{t}_{2} ◯_{u}_{2} ⧸^{◯ 2 ^{v}} ⧹  4,8,12,20  0,8,12,16  (stuv)^{5}  ℚ(i)  2^{4}⋊𝔖_{5} †

G_{33}  ◯_{s}_{2} ◯_{t}_{2} △ ◯_{u}_{2} ⧸^{◯ 2 ^{w}} ⧹ ◯_{v}_{2}  {4,6,10, \atop 12,18}  {0,6,8, \atop 12,14}  (ustvw)^{9}  ℚ(ζ_{3})  SO_{5}(3)′

◯_{t}_{2}^{◯ 2 ^{s}}∗ ◯_{u}_{2}^{◯ 2 ^{w′}}◯_{v}_{2}

G_{34} ◯_{s}_{2} ◯_{t}_{2} △ ◯_{u}_{2} ⧸^{◯ 2 ^{w}} ⧹ ◯_{v}_{2} ◯_{x}_{2} {6,12,18,24, \atop 30,42} {0,12,18,24, \atop 30,36} (stuvwx)^{7} ℚ(ζ_{3}) P SO_{6}^{−}(3)′·2

◯_{t}_{2}^{◯ 2 ^{s}}∗ ◯_{u}_{2}^{◯ 2 ^{w′}}◯_{v}_{2} ◯_{x}_{2}

Table 4

These diagrams provide presentations for the corresponding finite groups. It is not known nor conjectural whether they provide presentations for the corresponding braid groups.

† The group G_{29}/Z(G_{29}) is not isomorphic to the Weyl group D_{5}.

name  diagram  degrees  codegrees  β

B(de, e, r)_{e≥2,r≥2,d>1} [figure: σ; ◯; e+1; ^{◯^{τ_{2}}}; ⧹; _{◯_{τ′_{2}}}; ⧸; ◯_{τ_{3}}; ◯_{t_{4}}; ⋯; ◯_{τ_{r}}] {e, 2e, …, \atop (r−1)e, r}  {0, e, …, \atop (r−1)e} σ^{\frac{r}{(e∧r)}} (τ_{2}τ′_{2}τ_{3} ⋯ τ_{r})^{\frac{e(r−1)}{(e∧r)}}

B(1, 1, r)  ◯_{τ_{1}} ◯_{τ_{2}}⋯◯_{τ_{r}} 2, 3, …, r + 1 0, 1, …,r−1  (τ_{1} ⋯ τ_{r})^{r+1}
B(d, 1, r)_{d>1}  ◯_{σ} ◯_{τ_{2}} ◯_{τ_{3}}⋯◯_{τ_{r}} 1, 2, …, r 0, , …, (r−1)  (στ_{2}τ_{3} ⋯ τ_{r})^{r}

B(e, e, r)_{e≥2,r≥2} e^{^{τ_{2}}◯ 2}⧹_{_{τ′_{2}}◯ 2}⧸◯_{τ_{3}}_{2} ◯_{τ_{4}}_{2}⋯◯_{τ_{r}}_{2}  {e, 2e, …, \atop (r−1)e, r} {0, e, …, (r−2)e, \atop (r−1)e − r} (τ_{2}τ′_{2}τ_{3} ⋯ τ_{r})^{\frac{e(r−1)}{(e∧r)}}

Table 5 : braid diagrams

This table provides a complete list of the infinite families of braid diagrams and corresponding data. Note that the braid diagram B(de, e, r) for e = 2, d > 1 can also be described by a diagram as the one used for G(2d, 2, r) in Table 2. Similarly, the diagram for B(e, e, r), e = 2, can also be described by the Coxeter diagram of type D_{r}. The list of exceptional diagrams (but those associated with G_{24}, G_{27}, G_{29}, G_{33}, G_{34}) is identical with table 3.

References

[AlLu] 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.

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

[ArKo] 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.

[Ar] E. Artin, Theory of braids, Ann. of Math. 48 (1947), 101–126.

[Ba] E. Bannai, Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension 2, J. Math. Soc. Japan 28 (1976), 447–454.

[Bi] J. Birman, Braids, links and mapping class groups, Princeton University Press, Princeton, 1974.

[Bou] N. Bourbaki, Groupes et algèbres de Lie, chap. 4, 5 et 6, Hermann, Paris, 1968.

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

[Br1] E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 37–61.

[Br2] , Sur les groupes de tresses [d’après V.I. Arnold], Séminaire Bourbaki, 24ème année, 1971/72, Lecture Notes in Math., vol. 317, Springer-Verlag, Berlin, 1973.

[BrSa] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271.

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

[BMM] M. Broué, G. Malle, J. Michel, Generic blocks of finite reductive groups, Astérisque 212 (1993), 7–92.

[BMR] M. Broué, G. Malle, R. Rouquier, On complex reflection groups and their associated braid groups, Representations of Groups (B.N. Allison and G.H. Cliff, eds.), Canadian Mathematical Society, Conference Proceedings, vol. 16, Amer. Math. Soc., Providence, 1995, pp. 1–13.

[BrMi] 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.

[Ch1] I. Cherednik, On a calculation of monodromy of certain W -invariant local systems of type B, C and D, Funct. Anal. Appl. 23 (1989), 91–92.

[Ch2] , Generalized braid groups and local r-matrix systems, Soviet Math. Dokl. 40 (1990), 43–48.

[Ch3] , Monodromy representations for generalized Knizhnik-Zamolodchikov equations and Hecke algebras, Publ. RIMS Kyoto Univ. 27 (1991), 711–726.

[Ch] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782.

[Cho] W.-L. Chow, On the algebraic braid group, Ann. of Math. 49 (1948), 654–658.

[Co] A. M. Cohen, Finite complex reflection groups, Ann. scient. Éc. Norm. Sup. 9 (1976), 379– 436.

[Cx] H. S. M. Coxeter, Finite groups generated by unitary reflections, Abh. math. Sem. Univ. Hamburg 31 (1967), 125–135.

[De1] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273– 302.

[De2] , Équations différentielles à points singuliers réguliers, Lect. Notes in Math., vol. 163, Springer-Verlag, Berlin, 1970.

[DeMo] P. Deligne and G.D. Mostow, Commensurabilities among lattices in PU(1, n), Annals of mathematics studies, vol. 132, Princeton University Press, Princeton, 1993.

[DeLo] J. Denef and F. Loeser, Regular elements and monodromy of discriminants of finite reflection groups, Indag. Mathem. 6 (2) (1995), 129–143.

[Du] C.F. Dunkl, Differential-difference operators and monodromy representations of Hecke algebras, Pacific J. of Math. 159 (1993), 271–298.

[FoNe] R.H. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119–126.

[Go] C. Godbillon, Éléments de topologie algébrique, Hermann, Paris, 1971.

[Ha] A. Haefliger, Local theory of meromorphic connections in dimension one (Fuchs theory), A. Borel et al., Algebraic D-Modules, Perspectives in Math., vol. 2, Academic Press, Boston, 1987, pp. 129–149.

[Hu] M. C. Hughes, Complex reflection groups, Comm. Algebra 18 (1990), 3999–4029.

[Ko1] T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 (1987), 139–160.

[Ko2] , Integrable connections related to Manin and Schechtman’s higher braid groups, Illinois J. of Math. 34 (1990), 476–484.

[Lu] G. Lusztig, Exotic Fourier transform, Duke J. Math. 73 (1994), 243–248.

[MKS] W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory, Dover, New York, 1976.

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

[Ma2] , 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.

[Na] T. Nakamura, A note on the K(π, 1) property of the orbit space of the unitary reflection group G(m, ℓ, n), Sci. Papers College of Arts and Sciences Univ. Tokyo 33 (1983), 1–6.

[Op] E.M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Comp. Math. 85 (1993), 333–373.

[OrSo1] P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189.

[OrSo2] , Unitary reflection groups and cohomology, Invent. Math. 59 (1980), 77–94.

[OrSo3] , Braids and discriminants, Braids (J. S. Birman and A. Lebgober, eds.), Contemp. Math., vol. 78, AMS, Providence, 1988, pp. 605–613.

[OrTe] P. Orlik and H. Terao, Arrangements of hyperplanes, Springer-Verlag, Berlin - Heidelberg, 1992.

[ShTo] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.

[Spa] E. H. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966.

[Sp] T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198.

[St] R. Steinberg, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392–400.

Michel Broué : Université Paris 7 Denis–Diderot et Institut Universitaire de France, Case 7012, 2 Place Jussieu, F–75251 Paris Cedex 05, France

   E-mail address: broue@math.jussieu.fr

Gunter Malle : I.W.R., Im Neuenheimer Feld 368, D–69120 Heidelberg, Germany

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

Raphaël Rouquier : UMR 9994 du CNRS, Université Paris 7 Denis–Diderot, Case 7012, 2 Place Jussieu, F–75251 Paris Cedex 05, France

   E-mail address: rouquier@math.jussieu.fr