COHOMOLOGY ACTIONS AND CENTRALISERS IN UNITARY REFLECTION GROUPS

J. BLAIR AND G.I. LEHRER

Abstract. In an earlier work, the second author proved a general formula for the equivariant Poincaré polynomial of a linear transformation g which normalises a unitary reflection group G, acting on the cohomology of the corresponding hyperplane complement. This formula involves a certain function (called a Z-function below) on the centraliser C_{G}(g), which was proved to exist only in certain cases, e.g. when g is a reflection, or is G-regular, or when the centraliser is cyclic. In this work we prove the existence of Z-functions in full generality. Applications include reduction and product formulae for the equivariant Poincaré polynomials. The method is to study the poset L(C_{G}(g)) of subspaces which are fixed points of elements of C_{G}(g). We show that this poset has Euler characteristic one, which is the key property required for the definition of a Z-function. The fact about the Euler characteristic in turn follows from the “join-atom” property of L(C_{G}(g)), which asserts that if {X_{1}, ..., X_{k}} is any set of elements of L(C_{G}(g)) which are maximal (set theoretically) then their setwise intersection ∩_{i=1}^{k}X_{i} lies in L(C_{G}(g)).

1. Introduction

Let V be a finite dimensional vector space over ℂ. A reflection in V is a semi-simple linear automorphism with fixed-point subspace of codimension 1. A reflection group G in V is a finite group generated by reflections in V. Denote by M_{G} the smooth manifold obtained by removing from V the reflecting hyperplanes of G, i.e. those hyperplanes which are the fixed-point subspaces of reflections in G. If an automorphism g ∈ GL(V) normalises G, it permutes the reflecting hyperplanes of G and therefore acts on M_{G} and its de Rham cohomology. We are interested in this work in the (equivariant) Poincaré polynomial

2000 Mathematics Subject Classification. Primary 14R20, 55R80; Secondary 20C33, 20G40.

Research supported by grants from the Australian Research Council.

P_{G}(g, t) ∈ ℂ[t], defined by

P_{G}(g, t) = ∑_{i=0}^{ℓ} trace(g, H^{i}(M_{G}, ℂ)) · t^{i} (1)

The usual (non-equivariant) Poincaré polynomial of M_{G} was determined by Arn’old [1] in the case of the symmetric group, where the space M_{G} is the well known space of ordered configurations of n distinct points in ℂ. This was generalised to the case of an arbitrary finite Coxeter group by Brieskorn [3] and finally the general formula P_{G}(1, t) = ∏_{i}(1 + n_{i}t) was proved by Orlik and Solomon [17]. Here the n_{i} are the coexponents of the reflection group G (cf. [17]). The present work may be thought of as an equivariant generalisation of these results.

In [15], it was shown that, given the existence of a function f, called a Z-function below, on the centraliser C_{G}(g) := {x ∈ G; xg = gx} which satisfies two conditions (see (2.6) below), there is a simple closed formula for P_{G}(g, t) in terms of f. In this work we prove the existence of such a function f for any reflection group G and element g which normalises G and give a formula for one such function in terms of the partially ordered set L(C_{G}(g)) of fixed-point subspaces of the elements of the centraliser C_{G}(g). The key to the proof is the establishment of a certain property of this partially ordered set (the “Join-Atom” condition (see below)). This asserts that although it is not generally true that if X_{1}, X_{2}, ... is a collection of fixed-point subspaces of elements of C_{G}(g), their intersection is again a fixed-point subspace, it is true if the X_{i} are maximal with respect to inclusion (but not the whole of V).

A consequence of this property is that the poset L(C_{G}(g)) has Euler characteristic 1, which turns out to be sufficient for the definition of an explicit Z-function. The proof of the Join-Atom property for L(C_{G}(g)) involves a reduction to the case where G is irreducible, and then to the case where G has trivial centre. Using the Shephard-Todd classification of the irreducible groups, one is then reduced to just two cases, viz. the imprimitive groups G(r, r, ℓ) when r and ℓ are coprime, and the Weyl group of type E_{6}. These cases are explicitly checked.

The universal existence of Z-functions provides substantial information concerning the equivariant Poincaré polynomial, even when the function is not explicitly known (see 6.4 and 6.7 below). The property (called the JA property below) of the partially ordered set L(C_{G}(g)) of fixed points of elements of the centalisers C_{G}(g) appears to be rather special, and may to some extent characterise these as linear groups.

The authors wish to thank R.B. Howlett and D.E. Taylor for significant discussions concerning the verifications necessary for the proof of Theorem 6.1. In addition, the second author thanks the Université de Genève for hospitality while this work was in progress.

2. Preliminaries

Given a group G and an element x of G denote the centraliser of x in G by C_{G}(x). If G is a subgroup of the group H, write N_{H}(G) for the normaliser of G in H. For any endomorphism x of a vector space V, Fix(x) denotes the subspace of fixed points of x, i.e. Fix(x) = {v ∈ V : xv = v}. Definition 2.1. If G is a unitary reflection group (always assumed finite) on a vector space V, and if T is a subset of V then G_{T} denotes the subgroup {g ∈ G : Fix(g) ⊇ T }. These are known as the parabolic subgroups of G (generalising the terminology from real reflection groups).

We recall now several well known results we shall require later. 2.2 (Steinberg, [22] (1.5)). Let G ⊂ GL(V) be a finite unitary reflection group, and T be a subspace of V. Then the group G_{T} is a reflection group, generated by the reflections of G whose reflecting hyperplanes contain T. Definition 2.3. For any finite set S of subspaces of a vector space V, we denote by L(S) the lattice of intersections of subsets of S, ordered by the reverse of inclusion. By convention V ∈ L(S) corresponds to the empty subset of S.

Given a unitary reflection group G, we denote its set of reflecting hyperplanes by Arr(G), which we call the hyperplane arrangement of G. 2.4 ([18] Theorem 6.27). Let G ⊆ GL(V) be a finite unitary reflection group. Write L = L(Arr(G)), where Arr(G) denotes the hyperplane arrangement of G.

(1) If g ∈ G, then Fix(g) ∈ L.

(2) If X ∈ L, then there exists g ∈ G with Fix(g) = X.

2.5 (Lehrer, [15] Proposition 1.5). Let G(≠ 1) be a finite unitary reflection group on V = ℂ^{ℓ}. Suppose x ∈ N_{GL(V)}(G). Then C_{G}(x) is non-trivial.

If V = ℂ^{ℓ} and G is a finite subgroup of GL(V) then V has a positivedefinite G-invariant Hermitian form. Hence if U(V) is the (unitary) group of isometries of this form, then clearly G ≤ U(V). We shall assume such an invariant form to have been chosen and fixed throughout this work.

Let G be a finite unitary reflection group acting on V = ℂ^{ℓ}. Let M_{G} be the smooth manifold defined by

M_{G} = ℂ^{ℓ} \ ( ⋃_{H∈Arr(G)} H).

As was pointed out in §1, there is an action of N_{U(V)}(G) on H^{∗}(M_{G}, ℂ), the de Rham cohomology of M_{G}. We consider the Poincaré polynomial P_{G}(g, t) ∈ ℂ[t] defined in (1).

2.6 (Lehrer, [15] Corollary 2.2). Let G be a finite unitary reflection group acting on V = ℂ^{ℓ} and suppose g ∈ N_{U(V)}(G). Suppose f : C_{G}(g) → ℂ is a function which satisfies

(a) f(1) = 1 and

(b) for any subspace X ∈ L(Arr(G)) such that gX = X and G_{X} ≠ 1, we have

∑_{x∈C_{G_{X}} (g)} f(x) = 0.

Then

(∗)  P_{G}(g, t) = ∑_{x∈C_{G}(g)} f(x)(−t)^{c(x)}

where P_{G}(g, t) denotes the equivariant Poincaré polynomial of the reflection arrangement of G (see (1) above), and c(x) = codim_{V} Fix(x).

Definition 2.7. Let G be a finite unitary reflection group acting on V = ℂ^{ℓ} and suppose that g ∈ N_{U(V)}(G). We say that f : C_{G}(g) → ℂ is a Z-function if f satisfies (a) and (b) of (2.6) above.

The existence of easily computed Z-functions was proved generally in [15] for elements g which are regular or reflections. However the question of existence for Z-functions in general was left open. The main purpose of this paper is to prove that Z-functions always exist. We do this by determining an explicit Z-function η : C_{G}(g) → ℂ for any finite unitary reflection group G and any g ∈ N_{U(V)}(G). The function we produce in this work is not particularly easy to compute, but there are significant consequences of its existence (see §6 below).

3. The poset of fixed point spaces and the JA-condition

Definition 3.1. Let V be a vector space and let S be a subset of GL(V). Define the poset L(S) of subspaces of V by

L(S) = {Fix(s) : s ∈ S},

ordered by the reverse of inclusion.

Recall that if P is a locally finite poset and a, b ∈ P, we say a covers b if a > b and a ≥ x ≥ b implies that x = a or x = b. If P has a unique minimal element ˆ0, we say a is an atom of P if a covers ˆ0. We denote the set of atoms of P by At(P).

For any finite set Q of subspaces of the vector space V, recall (cf. (2.3)) that L(Q) is the lattice of intersections of all subsets of Q, ordered by the reverse of inclusion. This partially ordered set has minimal element ˆ0 = V ∈ L(Q), which corresponds to the intersection of the empty set. Definition 3.2. Let V be a vector space and H be a subgroup of GL(V). We say that H satisfies the Join-Atom condition (or simply the JA-condition) if

L( At (L(H))) ⊆ L(H),

where L(H) is the poset (cf. (3.1) of fixed-point subspaces of elements of H and L is defined in (2.3). This means simply that the set of (setwise) intersections of atoms of L(H) is contained in L(H). Remark 3.3. If G is a unitary reflection group then At(L(G)) = Arr(G). Hence (2.4) may be restated as L( At(L(G))) = L(G) for all finite unitary reflection groups G. In particular any reflection group G satisfies the JA-condition.

The key result to be proved below is that if G ≤ U(V) is a finite unitary reflection group and g is an element of the normaliser N_{U(V)}(G), then the centraliser C_{G}(g) satisfies the JA-condition. Remark 3.4. The JA-condition for H stipulates that any intersection of atoms of L(H) should again be in L(H). This property may hold even when arbitrary intersections of elements of L(H) are not in L(H). In particular, it is not generally the case that if G is a reflection group and g is an element of the normaliser N_{U(V)}(G) of G, then ⋂_{X∈L(C_{G}(g))} X ∈ L(C_{G}(g)). Example 3.5. An instance of the above remark is provided by the case G = W (D_{5}) and g ∈ G with cycle decomposition consisting of one positive one-cycle, one negative one-cycle and one negative three-cycle.

The intersection is then zero, but there is no element x of C_{G}(g) with Fix(x) = 0

4. Atoms of L(C_{G}(g)) for unitary reflection groups G

4.1. The posets L(G)^{g} and L(C_{G}(g)). Let V be a vector space over a field 𝕂. For any two elements g, h ∈ GL(V), we have h · Fix(g) = Fix(hgh^{−1}). Hence there is an obvious action of N_{GL(V)}(G) on the poset L(G). If L(G)^{g} is the set of points of L(G) fixed by g ∈ N_{GL(V)}(G), we discuss the relationship between the subposets L(C_{G}(g)) and L(G)^{g} of L(G). Definition 4.1. If P is any partially ordered set and γ is an automorphism of P, denote by P^{γ} the set of fixed points of γ on P. Lemma 4.2. Let G be a subgroup of GL(V) and let g ∈ N_{GL(V)}(G). Then, with notation as above, {V } ⊆ L(C_{G}(g)) ⊆ L(G)^{g},

Proof. Clearly {V } = Fix(1_{V}) ∈ L(C_{G}(g)), so it remains to show that L(C_{G}(g)) ⊆ L(G)^{g}. Let x ∈ C_{G}(g). Then g · Fix(x) = Fix(gxg^{−1}) = Fix(x), so that Fix(x) ∈ L(G)^{g} and L(C_{G}(g)) ⊆ L(G)^{g}. □

4.2. The atoms of the posets L(G)^{g} and L(C_{G}(g)). We shall prove that the posets L(C_{G}(g)) and L(G)^{g} have the same set of atoms, for G a unitary reflection group and g ∈ N_{GL(V)}(G). Denote by At(P) the set of atoms of P, as above. Lemma 4.3. Let P be a locally finite poset with a unique minimal element ˆ0. Let N be a subposet of P such that

{ˆ0} ∪ At(P) ⊆ N ⊆ P.

Then At(N) = At(P).

Proof. If an atom X of P lies in N, it is a fortiori an atom of N. Hence At(N) ⊇ At(P) ∩ N = At(P), since N ⊇ At(P). For the reverse inclusion, let X ∈ At(N). Then there exists Y ∈ At(P) such that Y ≤ X. But then Y ∈ N and Y ≠ ˆ0, so Y = X as X is an atom of N. Hence X ∈ At(P), i.e. At(N) ⊆ At(P). □

Proposition 4.4. Let G be a finite unitary reflection group in V and let g ∈ N_{GL(V)}(G). Then

At (L(C_{G}(g))) = At (L(G)^{g})

where L is as defined in (3.1).

Proof. By (4.2), we have

{V } ⊆ L(C_{G}(g)) ⊆ L(G)^{g}. (2)

We shall show that At (L(G)^{g}) ⊆ L(C_{G}(g)). Let X ∈ At (L(G)^{g}). Then gX = X, so (cf. Lehrer [15] Lemma 1.2) g ∈ N_{GL(V)}(G_{X}) where G_{X} = {h ∈ G : Fix(h) ⊇ X}. Now G_{X} is a non-trivial unitary reflection group by Steinberg (2.2) above, and hence C_{G_{X}}(g) is non-trivial by Lehrer (2.5) above. Hence there exists an element y ∈ C_{G_{X}}(g) such that y ≠ 1. Moreover V ⊃ Fix(y) ⊇ X, and Fix(y) ∈ L(G)^{g} since L(C_{G}(g)) ⊆ L(G)^{g} by Lemma 4.2. It follows that Fix(y) = X, because X ∈ At (L(G)^{g}). Thus

At (L(G)^{g}) ⊆ L(C_{G}(g)). (3)

The result is now immediate from (4.3). □

Remark 4.5. The above result does not hold in general if G is not assumed to be a unitary reflection group. For example, let V = ℂ^{5} with fixed standard basis, and let G ≤ GL(V) be the subgroup of permutation matrices with determinant one, so G is isomorphic to Alt(5) as an abstract group. This is a group generated by elements with fixed point spaces of codimension two (namely by three-cycles). Let g ∈ G be a permutation matrix corresponding to the cycle decomposition pattern of two 2-cycles and one 1-cycle. Then one may check that

At (L(C_{G}(g)) ≠ At (L(G)^{g}).

Remark 4.6. If g, G are as in 4.4 and H is a reflecting hyperplane of G, it is clear that ∩_{n≥0}g^{n}H ∈ L(G)^{g}. Moreover since any element of L(G)^{g} is contained in a hyperplane of G, it is clear that the atoms of L(G)^{g} are precisely the minimal (i.e. maximal as sets) elements among the intersections ∩_{n≥0}g^{n}H

5. Möbius functions and Z-functions

5.1. Möbius functions and joins of atoms. In the course of discussing various properties of the pair (G, g), where G is a unitary reflection group in V and g ∈ N_{U(V)}(g), we shall need to vary G and g. We shall therefore use the term unitary reflection pair, or simply pair for any such pair. From §4, it is clear that L(G)^{g} is a join semi-lattice which contains L(C_{G}(g)), since L(G) is a lattice, and if X, Y ∈ L(G)^{g}, their join X ∨Y in L(G) is g−stable. Moreover we have seen (4.4) that L(C_{G}(g)) and L(G)^{g} have the same set of atoms. The key fact which leads to a Z-function is that the Möbius function μ_{L(G)^{g}} of L(G)^{g} is supported on L(C_{G}(g)), and this is what we shall prove. We begin with a general result (5.2 below) concerning Möbius functions which is based on an idea of Phillip Hall ([8, §§2.3,2.4], cf. also [7, §5]). Definition 5.1. For any finite poset P with minimal element ˆ0, say that P has the JA-property if every set S of atoms of P has a (unique) least upper bound, which we call the join ∨S of S.

For such a poset P, let J(P) be the set of joins of sets of atoms of P. The minimal element ˆ0 ∈ J(P) is the join of the empty set of atoms. Proposition 5.2. Let P be a finite poset with the JA-property (5.1) and let J(P) ⊆ P be the set of joins of atoms of P. Let Q be any subposet of P such that J(P) ⊆ Q ⊆ P. Then for any element X ∈ P, we have

μ_{P} (X) = \begin{cases}μ_{Q}(X) & if X ∈ Q \\ 0 & otherwise, \end{cases}

where μ_{P} denotes the Möbius function of P, etc.

Proof. We proceed by induction on X ∈ P. Clearly the result holds for X = ˆ0. Note that by taking Q = J(P), it follows from 5.2 that

μ_{P} (X) = \begin{cases}μ_{Q}(X) = μ_{J(P)}(X) & if X ∈ J(P) \\ 0 & otherwise. \end{cases}

Now suppose 5.2 holds for all Y ∈ P with Y < X. Let Y_{0} be the join of all the atoms A ∈ P such that A ≤ X. Then Y_{0} ≤ X and Y_{0} ∈ J(P). If X ∉ Q, then X ∉ J(P), whence Y_{0} ≠ X. Thus μ_{P} (X) = − ∑_{Y <X} μ_{P} (Y) = − ∑_{Y ≤Y_{0}} μ_{P} (Y) by induction, and this latter sum is zero. If X ∈ Q, then μ_{P} (X) = − ∑_{Y <X} μ_{P} (Y) = − ∑_{Y <X,Y ∈Q} μ_{Q}(Y) = μ_{Q}(X), the second equality being an application of the induction hypothesis. □

Corollary 5.3. With notation as in 5.2, we have μ_{P} (X) = μ_{Q}(X) = μ_{J(P)}(X) if X ∈ J(P), μ_{P} (X) = μ_{Q}(X) for any X ∈ Q, and μ_{P} (X) = 0 otherwise.

The next lemma is implicit in the proof of 5.2, but we treat it separately for future reference. Lemma 5.4. Suppose P ⊇ Q ∋ {ˆ0} are locally finite posets such that μ_{P} (X) = 0 for X ∈ P \ Q. Then μ_{P} (X) = μ_{Q}(X) for X ∈ Q.

Proof. Let X ∈ Q. Then

μ_{P} (X) = − ∑_{Y <X} μ_{P} (Y) = − ∑_{Y <X,Y ∈Q} μ_{Q}(Y) = μ_{Q}(X).

□

Note that the condition on μ_{P} in Lemma (5.4) implies that Q contains all the atoms of P, since μ_{P} takes the value −1 on atoms. Our objective is to show that we have the situation of 5.4 for the posets P = L(G)^{g}, Q = L(C_{G}(g)).

5.2. Implications among properties of unitary reflection pairs. Consider the following assertions. For all reflection pairs (G, g):

Assertion Z: There is a Z-function f : C_{G}(g) → ℂ (see 2.6).

Assertion A: If X ∈ L(G)^{g} \ L(C_{G}(g)), then μ_{L(G)^{g}}(X) = 0.

Assertion B: The sum ∑_{X∈L(C_{G}(g))} μ_{L(C_{G}(g))}(X) equals zero, i.e., the Euler characteristic of the poset L(C_{G}(g)) is one.

Assertion C: The poset L(C_{G}(g)) enjoys the JA property (see 5.1), i.e., joins of atoms of L(C_{G}(g)) are again in L(C_{G}(g)).

Assertion D: The setwise intersection (or join in the poset L(G)^{g}) of all atoms of L(C_{G}(g)) lies in L(C_{G}(g)). Remark 5.5. Assertion D may be rephrased in terms of the poset L(C_{G}(g)). In analogy with Definition 3.2, say that a subgroup H of GL(V), satisfies the weak join-atom condition (or weak JA condition) if the (setwise) intersection of all the atoms of L(H) is in L(H). Then assertion D is that C_{G}(g) satisfies the weak JA condition.

The relationships among these assertions are described in

Theorem 5.6. We have the following implications among the assertions above:

D ⇒ C ⇒ B ⇒ A ⇒ Z.

Proof. We prove the four implications of the theorem separately.

A ⇒ Z : By lemma 5.4, if A holds, μ_{L(G)^{g}}(X) = μ_{L(C_{G}(g))}(X) for any X ∈ L(C_{G}(g)). Define f : C_{G}(g) → ℂ by

f(x) = \frac{μ_{L(G)^{g}}(Fix(x))}{ν(x)} for x ∈ C_{G}(g)

where ν(x) is the number of elements y ∈ C_{G}(g) such that Fix(y) = Fix(x).

Then f(1) = 1 and we need to check condition (b) of 2.6. Take X ∈ L(G)^{g} such that G_{X} ≠ 1. Then

∑_{x∈C_{G_{X}} (g)} f(x) = ∑_{x∈C_{G_{X}} (g)} \frac{μ_{L(G)^{g}}(Fix(x))}{ν(x)}

= ∑_{\begin{matrix}Y ≤X \\ Y ∈L(C_{G}(g)) \end{matrix}} μ_{L(C_{G}(g))}(Y)

= ∑_{\begin{matrix}Y ≤X \\ Y ∈L(G)^{g} \end{matrix}} μ_{L(G)^{g}}(Y) (by A, using Lemma 5.4)

= 0 (as X ∈ L(G)^{g} and X ≠ V = ˆ0).

B ⇒ A : Suppose (G, g) is a counterexample to A, with |G| minimal. Write T for the top element of L(G) (it is the intersection of all the reflecting hyperplanes of G). Then by minimality, for any element X ∈ L(G)^{g} with X ≠ T, the pair (G_{X}, g) satisfies A. In particular, we have, using A and 5.4, that if Y is any element of L(G)^{g}, Y ≠ T, then μ_{L(G)^{g}}(Y) = μ_{L(C_{G}(g))}(Y) if Y ∈ L(C_{G}(g)), while μ_{L(G)^{g}}(Y) = 0 if Y ∉ L(C_{G}(g)). Thus if (G, g) is a counterexample to A, we must have T ∈ L(G)^{g} \ L(C_{G}(g)) and μ_{L(G)^{g}}(T) ≠ 0. But ∑_{Y ∈L(G)^{g}} μ_{L(G)^{g}}(Y) = 0 = ∑_{Y ∈L(C_{G}(g))} μ_{L(G)^{g}}(Y) + μ_{L(G)^{g}}(T). Moreover T ∉ L(C_{G}(g)), together with B, yields that ∑_{Y ∈L(C_{G}(g))} μ_{L(G)^{g}}(Y) = 0. Thus μ_{L(G)^{g}}(T) = 0, whence if B holds there is no counterexample to A.

C ⇒ B : This follows directly from Proposition 5.2.

D ⇒ C : Suppose D holds and let {X_{1}, ..., X_{r}} be any set of atoms of L(C_{G}(g)). Write X = ∩_{i=1}^{r}X_{i}. Then (G_{X}, g) is a unitary reflection pair and X is the intersection of all the atoms of L(C_{G_{X}}(g)). Hence by D applied to (G_{X}, g), X ∈ L(C_{G_{X}}(g)) ⊆ L(C_{G}(g)). □

6. The JA property for L(C_{G}(g)) and its consequences

In this section we state our main theorem, reduce its proof to checking the irreducible groups, and give some of its consequences.

Theorem 6.1. Assertion D of §5.2 is true; i.e., for any unitary reflection pair (G, g), the join (setwise intersection) of all the atoms of the poset L(C_{G}(g)) of fixed point subspaces of elements of the centraliser C_{G}(g) is in L(C_{G}(g)), i.e. is of the form Fix(x) for some element x ∈ C_{G}(g).

Before giving the proof of Theorem 6.1, we point out some of its consequences. By Theorem 5.6, these include the assertions Z,A,B and C of 5.2, but we reformulate these below. Corollary 6.2. Given any reflection pair (G, g), the Euler characteristic of the poset L(C_{G}(g)) is one.

Proof. By 5.2 and 6.1, assertion B holds, which is the assertion 6.2, given that for any poset with bottom and top elements ˆ0, ˆ1, respectively, its Euler characteristic is 1 + μ(ˆ0, ˆ1) (see Rota [19]). □

Corollary 6.3. Given any reflection pair (G, g), there exists a Zfunction f : C_{G}(g) → ℂ.

Proof. In the proof that property A implies property Z of 5.2, it was shown that the function f(x) := μ_{L(G)^{g}}(Fix(x)(ν(x))^{−1} (x ∈ C_{G}(g)) is a Z-function, where ν(x) = |{y ∈ C_{G}(g) : Fix(y) = Fix(x)}|. □

Corollary 6.4. Maintain the above notation. Then

P_{G}(g, t) = ∑_{X∈L(C_{G}(g))} μ_{L(C_{G}(g))}(X)(−t)^{c(X)}

where c(X) = codim_{V} (X).

Proof. Combining the formula in 2.6(*) with the explicit formula for the Z-function given in the proof that A implies Z of 5.6, we see that

P_{G}(g, t) = ∑_{x∈C_{G}(g)} μ_{L(C_{G}(g))}(Fix(x))(ν(x))^{−1}(−t)^{c(x)}.

The formula 6.4 is now immediate. □

Remark 6.5. With the above notation, assertion A together with lemma 5.4 show that the formula above is equivalent to

P_{G}(g, t) = ∑_{X∈L(G)^{g}} μ_{L(G)^{g}}(X)(−t)^{c(X)},

which is the general formula of [16], 5.7. Corollary 6.6. Continuing in the notation of 6.1, suppose that p is the maximum value of c(x) among the elements x of C_{G}(g). Then trace(g, H^{j}(M_{G})) = 0 for j > p.

Corollary (6.6) follows from (e.g.) 6.4. Note that it provides group theoretic information about the character of G on the top cohomology of M_{G}.

Corollary 6.7. Still with the above notation, suppose that G has a reflection subgroup G′ which is normalised by g and is such that C_{G}(g) ≤ G′. Then P_{G}(g, t) = P_{G′}(g, t). In particular, if C_{G}(g) ≤ G_{1}×G_{2}, where the G_{i} are reflection subgroups of G, both normalised by g. Then

P_{G}(g, t) = P_{G_{1}}(g, t)P_{G_{2}}(g, t).

This is proved in a similar way to [15, (6.3),(6.5)]. Corollary 6.8. Maintain the notation of 6.1. If G_{1} is any reflection subgroup of G which is normalised by g and which is such that the sets of atoms of L(C_{G}(g)) and of L(C_{G_{1}}(g)) coincide, then P_{G}(g, t) = P_{G_{1}}(g, t).

The statement (6.8) is a straightforward consequence of the formula 6.4 and Proposition 5.2.

We turn now to the proof of Theorem 6.1. The first step is the reduction of the proof to the case where G is irreducible. Proposition 6.9. Suppose (6.1) holds for all pairs (G, g) with G irreducible. Then (6.1) holds for all pairs (G, g).

Proof. Let (G, g) be any pair. Write G̃ for the group < G, g > generated by G together with g. We show first that we may suppose that V is irreducible as G̃-space. For suppose that V_{1} is a non-trivial G̃ invariant subspace of V and let V_{2} = V_{1}^{⊥}. Then any reflection in G̃ acts trivially on either V_{1} or V_{2}. Thus G = G_{1} × G_{2}, where G_{i} is the pointwise stabiliser of V_{i}, and g normalises both of the G_{i}. It is then easily seen that Arr(G) = Arr(G_{1}) ⨿ Arr(G_{2}) and that L(G) ≅ L(G_{1}) × L(G_{2}). Clearly g acts on L(G) ≅ L(G_{1}) × L(G_{2}) componentwise, whence

L(C_{G}(g)) ≅ L(C_{G_{1}}(g)) × L(C_{G_{2}}(g)).

Further, (X_{1}, X_{2}) ∈ L(C_{G}(g)) ≅ L(C_{G_{1}}(g)) × L(C_{G_{2}}(g)) is an atom if and only if X_{i} is an atom of L(C_{G_{i}}(g)) for i = 1 or 2 and X_{j} = V_{j} for the other index j. It follows that if we know 6.1 for (G_{i}, g), i = 1, 2, then we know it for (G, g). Thus we may assume that V is an irreducible G̃-module.

Let V_{1} be a non-zero irreducible G-subspace of V. Then the distinct translates {g^{i−1}V_{1} := V_{i}, i = 1, 2, 3, ...} are all G-invariant and span a G̃-invariant subspace of V, whence they span V. Let r be minimal such that V = ∑_{i=1}^{r} V_{i}. For each i, V_{i} ∩ ∑_{j≠i} V_{j} is a G-submodule of V_{i}, hence is equal to either 0 or V_{i}. It follows (e.g. by induction) that V is the direct sum V = ⊕_{j=1}^{r}V_{j}. If we write G_{i} for the pointwise stabiliser in G of the subspace ∑_{j≠i} V_{j} (i = 1, 2, 3, ..., r) then G ≅ G_{1} × ... × G_{r}, where G_{i} acts as a reflection group on V_{i} and g permutes the pairs (G_{i}, V_{i}) cyclically, sending (G_{i}, V_{i}) to (G_{i+1}, V_{i+1}), the indices being taken modulo r. The latter fact follows because each of the reflections of G lies in one of the groups G_{i} (i = 1, ..., r), so that the irreducibility of (G_{i}, V_{i}) implies that g^{r} conjugates G_{r} into G_{1} and moves V_{r} into V_{1}.

It follows that Arr(G) = ⨿_{i=1}^{r} Arr(G_{i}) and that the arrangements Arr(G_{i}) are permuted cyclically by g, whence in particular they are isomorphic. Moreover L(G) ≅ (L(G_{1}))^{r} and g acts on L(G) by taking (X_{1}, ..., X_{r}) (X_{i} ∈ L(G_{1})) to (gX_{r}, gX_{1}, ..., gX_{r−1}). Hence (X_{1}, ..., X_{r}) ∈ L(G)^{g} precisely if it is of the form (X, gX, g^{2}X, ..., g^{r−1}X), where X ∈ L(G_{1}) and X = g^{r}X. This shows that

(α)  L(G)^{g} ≅ L(G_{1})^{g^{r}}.

Similarly, g acts on G ≅ G_{1}×...×G_{r} via (x_{1}, ..., x_{r}) ↦ (^{g}x_{r}, ^{g}x_{1}, ..., ^{g}x_{r−1}). It follows that (x_{1}, ..., x_{r}) ∈ C_{G}(g) exactly when

(x_{1}, ..., x_{r}) = (x, ^{g}x, ..., ^{g^{r−1}}x)

for some element x ∈ C_{G_{1}}(g^{r}). Hence

(β)  L(C_{G}(g)) ≅ L(C_{G_{1}}(g^{r}))

where the right hand side of (α) is interpreted as a subposet of L(G_{1}). Moreover under the identification β, the atoms of L(C_{G}(g)) correspond to those of L(C_{G_{1}}(g^{r})). It follows that the statement (6.1) for (G, g) is equivalent to (6.1) for (G_{1}, g^{r}). Since G_{1} is irreducible (on V_{1}), Theorem 6.9 follows. □

We next show that any minimal counterexample to 6.1 is strictly circumscribed. Corollary 6.10. Suppose that the pair (G, g) is a counterexample to the statement 6.1 which is minimal in the sense that the dimension in which G acts as a reflection group is minimal. Then

(1) G is irreducible

(2) The setwise intersection of the atoms of L(C_{G}(g)) is the zero subspace, and

(3) The centre Z(G) is trivial.

Proof. The arguments of the proof of 6.9 show that if (G, g) is a counterexample to 6.1 and G is not irreducible, then there is another counterexample (G_{1}, g_{1}) where G_{1} is a reflection group of lower dimension than G. Thus G is irreducible. Write T_{g} for the intersection of the atoms of L(C_{G}(g)) and G_{1} for the pointwise stabiliser in G of T_{g}. If T_{g} ≠ 0 then G_{1} acts in lower dimension (on T_{g}^{⊥}) than G and is normalised by g. Moreover since L(G_{1}) = {X ∈ L(G) | X ⊇ T_{g}} and the action of g on L(G_{1}) is compatible with its action on L(G), the atoms of L(C_{G_{1}}(g)) coincide with those of L(C_{G}(g)). Hence (G_{1}, g) is also a counterexample to 6.1, whence by minimality we must have T_{g} = 0. Finally, observe that if Z(G) ≠ 1, then C_{G}(g) always contains a nonidentity scalar, whose fixed point subspace is zero, which coincides with T_{g}. Thus Z(G) = 1. □

7. Completion of the proof of Theorem 6.1

Our task is to prove 6.1, which states that for any reflection pair (G, g), C_{G}(g) satisfies the weak JA-condition (i.e. has property D). If this assertion were false, then there is a pair (G, g) which is a minimal counterexample in the sense of 6.10. We shall prove that for any pair which is of the form given in the statement 6.10, the assertion of 6.1 holds. This will prove the Theorem.

7.1. Further reductions. If (G, g) is a minimal counterexample then by 6.10 (i) and (iii), we know that G is irreducible and that the centre Z(G) is trivial. Inspection of the list of irreducible groups confirms that Lemma 7.1. Suppose that G is an irreducible reflection group with trivial centre. Then either

(1) G = G(r, r, ℓ), with g.c.d.(r, ℓ) = 1, or

(2) G = G_{35} in the Shephard-Todd notation, i.e. G is the Weyl group of type E_{6}.

It therefore remains only to verify Theorem 6.1 when G is one of the two types in Lemma (7.1).

We next make some further reductions. Lemma 7.2. Let G ≤ GL(V), where V is a finite dimensional vector space. Let H ≤ N_{GL(V)}(G) and suppose h_{1}, h_{2} ∈ H are such that h_{1} is conjugate to h_{2} in H. Then C_{G}(h_{1}) satisfies the weak JA condition if and only if C_{G}(h_{2}) satisfies the weak JA condition. Lemma 7.3. Let G be a finite unitary reflection group acting on V. Let x ∈ N_{U(V)}(G). Then x = λ · y for some λ ∈ ℂ^{×} and y ∈ N_{U(V)}(G) of finite order.

Proof. The element x acts on the set of reflecting hyperplanes of G as a permutation k, say. Thus x^{k} stabilises each reflecting hyperplane H of G, and x^{k} also stabilises H^{⊥}, as x^{k} ∈ U(V). Hence x^{k} ∈ C_{U(V)}(r) for every reflection r of G. Hence x^{k} centralises G, and so x^{k} = β · ι_{V}, for some β ∈ ℂ^{×}, as G is irreducible. Let λ ∈ ℂ^{×} be such that β = λ^{k}. Then y = λ^{−1} · x satisfies y^{k} = id_{V}. □

Remark 7.4. The last two observations show that to check that C_{G}(h) satisfies the weak JA condition for all elements h of N_{U(V)}(G), it is enough to check it on a set of representatives of those conjugacy classes of N_{U(V)}(G) whose elements have finite order. Lemma 7.5. Property D is enjoyed by all reflection pairs (G, g) where (i) G is two dimensional or (ii) G = G(1, 1, ℓ)

Proof. (i) Let K be any finite group of rank 2 (not necessarily a reflection group). Suppose that there exist H_{1}, H_{2} ∈ L(K) such that codim_{V} (H_{i}) = 1, for i = 1, 2, and such that H_{1} ≠ H_{2}. Let r_{i} ∈ K denote a reflection with fixed point space H_{i} (i = 1, 2). It is then easy to see that Fix(r_{1}r_{2}) = 0, whence K has the weak JA property, proving (i).

(ii) Here G is the symmetric group Sym_{ℓ}, and we may assume g ∈ G. If g has cycle decomposition (i^{m_{i}}) (i.e. m_{i} cycles of length i, ∑_{i} im_{i} = ℓ), then C_{G}(g) ≅ C_{1}×C_{2}×..., where C_{i} is the centraliser in Sym_{im_{i}} = G_{i} of the homogeneous element i^{m_{i}}. If V is the ℓ-dimensional reflection representation of Sym_{ℓ}, then correspondingly V = V_{1} ⊕ V_{2} ⊕ ... where dim(V_{i}) = im_{i}, and G_{i} acts on V_{j} trivially if i ≠ j.

Now for x = (x_{1}, x_{2}, ...) ∈ G_{1} × G_{2} × ..., Fix_{V} (x) =Fix_{V_{1}}(x_{1})⊕ Fix_{V_{2}}(x_{2}) ⊕ .... Hence

L(C_{G}(g)) = ∏_{i} L(C_{G_{i}}(g_{i}).

It follows that the atoms of L(C_{G}(g)) are of the form V_{1}⊕V_{2}⊕...⊕A_{i}⊕..., where A_{i} is an atom of L(C_{G_{i}}(g_{i}) acting on V_{i}, and hence that the intersection I of all these atoms is I = I_{1} ⊕ I_{2} ⊕ ..., where I_{j} is the intersection of all the atoms of L(C_{G_{j}}(g_{j})).

Hence it is sufficient to consider the case where g = g_{i}, i.e., we may assume that all cycles of g have the same length i. However in this case g is centralised by an n-cycle, whose fixed point subspace is the top element of L(G), which is the intersection of the atoms of L(C_{G}(g)). □

By 7.5, we may assume that r ≥ 2 and that ℓ ≥ 3. Hence by (7.3) and Cohen [5] (2.13) we may take g ∈ G(r, 1, ℓ). Moreover the assumption g.c.d.(r, ℓ) = 1 implies that G(r, 1, ℓ) ≅ Z ×G(r, r, ℓ) where Z is the centre of G(r, 1, ℓ). Hence we may take g ∈ G. If G = G_{35}, then by Cohen [5], (5.14) we may assume g ∈ G. The next statement summarises the above reductions. Lemma 7.6. Any minimal counterexample (G, g) to Theorem 6.1 satisfies

(a) The intersection of the atoms of L(C_{G}(g)) is zero, and either (b) G = G(r, r, ℓ) with g.c.d.(r, ℓ) = 1, r ≥ 2, ℓ ≥ 3 and g runs over

the G̃-conjugacy class representatives of G, where G̃ = G(r, 1, ℓ), or

(c) G = G_{35} and g runs over a set of representatives of the conjugacy classes of G.

7.2. The groups G(r, m, ℓ). For any r ∈ ℕ, let μ_{r} denote {θ ∈ ℂ^{×} : θ^{r} = 1}, the set of complex r-th roots of unity. Then for ℓ ∈ ℕ write G(r, 1, ℓ) for the group of ℓ × ℓ monomial matrices whose nonzero entries are in μ_{r}. As in 7.1, Sym(ℓ) denotes the group of ℓ × ℓ permutation matrices; A(r, ℓ) will denote the group of ℓ × ℓ diagonal matrices with diagonal entries in μ_{r}. Then it is standard that G(r, 1, ℓ) is the semidirect product of Sym_{ℓ} and A(r, ℓ). Let σ : G(r, 1, ℓ) → Sym_{ℓ} denote the homomorphism taking an element g ∈ G(r, 1, ℓ) to its underlying permutation matrix σ(g).

Let τ : G(r, 1, ℓ) → μ_{r} be the homomorphism given by τ(g) = det(g)· det(σ(g)). Then τ takes g to the product of the non-zero entries of g. We call τ(g) the type of g, and also speak of the type of a cycle of g. For any triple r, m, ℓ ∈ ℕ with m|r, define the group G(r, m, ℓ), by G(r, m, ℓ) = ker(τ^{\frac{r}{m}}). We shall consistently use the notation G = G(r, r, ℓ) and G̃ = G(r, 1, ℓ), with appropriate modification when the parameters vary.

In this section we shall prove

Theorem 7.7. Let G = G(r, r, ℓ) and G̃ = G(r, 1, ℓ), for any integers r, ℓ and let g ∈ G̃. If the atoms of L(C_{G}(g)) have join (i.e. setwise intersection) zero, then there is an element x ∈ G with Fix(x) = 0.

Since the case ℓ = 2 has been covered in 7.5, we may assume ℓ ≥ 3. Also, in view of 7.5(ii) we may (and do) assume r ≥ 2. We begin by collecting some facts concerning the groups.

The conjugacy classes of G(r, 1, ℓ) are described by the decomposition pattern into cycles, where each cycle has a type in μ_{r} (see [12], Theorem 4.2.8).

Definition 7.8. We say that an element g ∈ G(r, 1, ℓ) is homogeneous if all of its cycles are of the same length and type.

Proposition 7.9. Let r, m, ℓ ∈ ℕ, satisfy m|r. Let V = ℂ^{ℓ}. Then

(1) G(r, m, ℓ) ⊴ G(r, 1, ℓ), so the group G(r, 1, ℓ) acts on the group G(r, m, ℓ) by conjugation.

(2) Let Ω = [ℓ] × μ_{r}, where [ℓ] = {1, ..., ℓ}. The set of conjugacy classes of G(r, 1, ℓ) is in bijection with the set of functions

{c : Ω → ℤ_{≥0} | ∑_{(n,θ)∈Ω} n · c(n, θ) = ℓ}.

(3) Let g ∈ G(r, 1, ℓ) and write g = g_{1} ⋯ g_{k} for the unique decomposition of g as a product of homogeneous elements g_{i}. Then

C_{G(r,1,ℓ)}(g) ≅ C_{G(r,1,ℓ_{1})}(g_{1}) × … × C_{G(r,1,ℓ_{k})}(g_{k})

where ℓ_{i} ∈ ℕ is such that g_{i} ∈ G(r, 1, ℓ_{i}) for 1 ≤ i ≤ k. There is a corresponding decomposition

V ≅ V_{1} ⊕ … ⊕ V_{k},

where dim(V_{i}) = ℓ_{i} and G_{i} ≅ G(r, 1, ℓ_{i}) acts trivially on each V_{j} such that j ≠ i.

(4) If m ≠ r then

Arr(G(r, m, ℓ)) = {H_{ij}^{α} : 1 ≤ i < j ≤ ℓ, α ∈ μ_{r}} ∪ {H_{k} : 1 ≤ k ≤ ℓ}

while if m = r then

Arr(G(r, r, ℓ)) = {H_{ij}^{α} : 1 ≤ i < j ≤ ℓ, α ∈ μ_{r}},

where, for 1 ≤ i < j ≤ ℓ, α ∈ μ_{r} = {θ ∈ ℂ^{×} : θ^{r} = 1}, and 1 ≤ k ≤ ℓ, H_{ij}^{α} and H_{k} are the subspaces of ℂ^{ℓ} defined by

H_{ij}^{α} = {(x_{1}, …, x_{ℓ}) ∈ ℂ^{ℓ} : x_{i} = αx_{j}},
H_{k} = {(x_{1}, …, x_{ℓ}) ∈ ℂ^{ℓ} : x_{k} = 0}.

The atoms of L(G)^{g} for G = G(r, 1, ℓ), ℓ ≥ 2 and g ∈ G have been described explicitly by Hanlon (see [10] Theorem 2.1). We shall make liberal use of those results below. Definition 7.10. Let θ ∈ μ_{r}. Define (cf. Hanlon [10] p435) the standard ℓ-cycle of type θ in G(r, 1, ℓ)

σ(ℓ, θ) = ( \begin{matrix}0 & 0 & 0 & … & 0 & θ \\ 1 & 0 & 0 & … & 0 & 0 \\ 0 & 1 & 0 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & 0 & 0 \\ 0 & 0 & 0 & … & 1 & 0 \end{matrix} ) .

If d, e ∈ ℤ_{≥0}, define the standard homogeneous element σ(d, θ)^{(e)} of G(r, 1, de) to be the product of e “consecutive” standard d-cycles of type θ. In matrix terms, this corresponds to a diagonal block matrix with e diagonal blocks, each equal to σ(d, θ). Note that if d divides ℓ, σ(ℓ, θ)^{d} is conjugate to the standard homogeneous element σ(\frac{ℓ}{d}, θ)^{(d)} of G̃.

For the remainder of this section, we assume that r ≥ 2. With notation as in 7.7, write g = g_{1}...g_{k} for the homogeneous decomposition of g. Since g may be replaced by a conjugate, we may assume that g_{i} is a standard homogeneous element of G̃_{i} ≅ G(r, 1, ℓ_{i}). Thus we write g_{i} = σ(d_{i}, θ_{i})^{(e_{i})}. Then as in 7.9(iii),

CG̃(g) = CG̃_{1}(g_{1}) × ... × CG̃_{k}(g_{k}). (4)

As indicated above, G_{i} will denote G(r, r, ℓ_{i}) ≤ G̃_{i}. There is a decomposition of V corresponding to the decomposition 4, and for any element x = (x_{1}, ..., x_{k}) ∈ CG̃(g) (x_{i} ∈ CG̃_{i}(g_{i})) we have

(5) Fix_{V} (x) = Fix_{V_{1}}(x_{1}) ⊕ ... ⊕ Fix_{V_{k}}(x_{k}).

Lemma 7.11. Let h = σ(d, θ)^{(e)} be a homogeneous element of G̃ = G(r, 1, de) (any d, e) and recall r > 1. Write G = G(r, r, de). If e ≠ 1, then C_{G}(h) contains an element x with Fix(x) = 0.

Proof. Think of the element g in block matrix terms as the matrix with each diagonal block equal to σ(d, θ). If I_{d} is the identity matrix of size d and α is a non trivial element of μ_{r}, let x be the block matrix with αI_{d} in the (1, 1) block, \overline{α}I_{d} in the (2, e) block and I_{d} in blocks (i, i − 1) for i = 3, 4, ..., e. Then x ∈ G and clearly centralises g. It is easily verified that Fix(x) = 0 □

Lemma 7.12. If g ∈ G̃ has no 1-cycles, then At(L(C_{G}(g))) = At(L(CG̃(g))).

Proof. Recall that by 4.6, the atoms of L(C_{G}(g)) are the maximal (as sets) elements among the intersections ∩_{n≥0}g^{n}H, where H is a reflecting hyperplane of G. Since by 7.9(iv), Arr(G̃) = Arr(G) ⨿ {H_{1}, ..., H_{ℓ}}, it suffices to show that ∩_{n≥0}g^{n}H_{i} ∉ At(L(CG̃(g))) for each i ∈ [ℓ]. But by hypothesis, any i ∈ [ℓ] lies in a g orbit of size at least 2. Let j ≠ i be in the g orbit of i. Then ∩_{n≥0}g^{n}H_{ij}^{α} clearly contains ∩_{n≥0}g^{n}H_{i}, where α is any element of μ_{r}. □

We now require a brief discussion of the atoms in L(C_{G}(g)) in the cases we are considering. These have been described explicitly by Hanlon [10] (see [10] Theorem 2.1). For the next two results, we drop the restrictions on r, ℓ but otherwise use the notation above. In particular G = G(r, r, ℓ) and G̃ = G(r, 1, ℓ). We take g to be the standard ℓ-cycle of type θ ∈ μ_{r}. Let Ω = [ℓ] × μ_{r}, as in 7.9.

Definition 7.13. Define the map ψ_{g} : Ω → L(G(r, 1, ℓ))^{g} (cf. [10], p.437) by

ψ_{g}(k, α) = \begin{cases}⋂_{n∈ℕ} g^{n}H_{1,1+k}^{α} & if k ≠ ℓ \\ V & if k = ℓ \end{cases}

Lemma 7.14. (cf. Hanlon [10] Lemma 2.1) Let g = σ(ℓ, θ) and take (k, α) ∈ Ω.

(i) ψ_{g}(k, α) ∉ At(L(C_{G}(g))) unless k divides ℓ.

(ii) Suppose k|ℓ, k ≠ ℓ, and α ∈ μ_{r}. Then ψ_{g}(k, α) ≠ {0} if and only if α^{\frac{ℓ}{k}} = θ.

The next result is now easily verified, given 4.6 and 7.14. Lemma 7.15. Let G = G(r, m, ℓ), with no restrictions on the parameters. Suppose g = σ(ℓ, θ) is a standard ℓ-cycle of type θ. Then

At(L(C_{G}(g))) = { ψ_{g}(p, α) | p is prime, p|ℓ and α ∈ μ_{r}, α^{p} = θ }

The next result is easily deduced from 7.15 Lemma 7.16. Let G = G(r, m, ℓ), with r ≥ 2 and ℓ ≥ 2. For any integer n, denote by ρ(n) the square-free part of n, i.e. the product of the distinct prime divisors of n. Then

⋂_{X∈At(L(C_{G}(g)))} X = \begin{cases}ψ_{g}(\frac{ℓ}{ρ(ℓ)}, β) ≠ 0 & if gcd(r, ℓ) = 1 \\ 0 & otherwise \end{cases}

where β is some element of μ_{r}.

We shall also require the following description of the atoms of L(C_{G}(g)) when g is a product of 1-cycles. Lemma 7.17. Suppose g ∈ G̃ = G(r, 1, ℓ) is a product of one-cycles. Then

At(L(C_{G}(g))) = {H_{i,j}^{α} : α ∈ μ_{r}, 1 ≤ i < j ≤ ℓ, with i and j

being one-orbits of g of the same type} ⋃ {H_{i} ∩ H_{j} : 1 ≤ i < j ≤ ℓ, with i and j being one-orbits of g of different type,}

while

At(L(CG̃(g))) = {H_{i,j}^{α} : α ∈ μ_{r}, 1 ≤ i < j ≤ ℓ, with i and j

being one-orbits of g of the same type} ⋃ {H_{j} : 1 ≤ j ≤ ℓ},

where H_{i} and H_{i,j}^{α} are as in (7.9).

The proof of Lemma 7.17 is a straightforward application of the characterisation of atoms in 4.6 and the description of the hyperplanes in 7.9.

Lemma 7.18. Let σ(d, θ) be a standard cycle in G̃ = G(r, 1, d) (any d ≠ 1). Write G = G(r, r, d). If the join of the atoms of L(C_{G}(g)) is trivial, then C_{G}(g) contains an element x with Fix(x) = 0

Proof. From Lemma 7.16 we see that the join of the atoms of L(C_{G}(g)) is trivial if and only if gcd(r, d) ≠ 1. Take a prime p which divides both r and d and let θ be a primitive p^{th} root of unity. Then θ · Id_{V} ∈ C_{G}(g) and Fix(θ · Id_{V}) = 0. □

Proposition 7.19. Theorem 7.7 holds if g has no cycle of length 1.

Proof. We use the notation of 7.9, 4 and 5 without further explanation. In view of 7.5 we may take r ≥ 2. Let g = g_{1}...g_{k} be the homogeneous decomposition of g. Then any x ∈ C_{G}(g) is of the form x = x_{1}...x_{k}, with x_{i} ∈ CG̃_{i}(g_{i}). Thus if we write L_{i} = L(C_{G_{i}}(g_{i})) and L̃_{i} = L(CG̃_{i}(g_{i})) (i = 1, ..., k), it follows from Lemma 7.12 that

At(L_{i}) = At(L̃_{i}) for i = 1, ..., k.

It is therefore evident that the sets of atoms of L_{1}×...×L_{k} and L̃_{1}×...× L̃_{k} coincide, both sets consisting of the subspaces V_{1} ⊕...⊕A_{i} ⊕...⊕V_{k}, 1 ≤ i ≤ k, where A_{i} is an atom of L_{i} and the V_{i} are as in 7.9. Since

L_{1} × ... × L_{k} ⊆ L(C_{G}(g)) ⊆ L̃_{1} × ... × L̃_{k},

it follows that the atoms of L(C_{G}(g)) coincide with those of L_{1}×...×L_{k}. Hence if the join of the atoms of L(C_{G}(g)) is 0, the same is true of those of L_{i}, for each i. Now if g_{i} is not a single cycle, there is an element x_{i} ∈ C_{G_{i}}(g_{i}) with Fix_{V_{i}}(x_{i}) = 0 by Lemma 7.11. If g_{i} is a single cycle, then since the intersection of the atoms of L_{i} is 0, it follows from Lemma 7.18 that such an element x_{i} again exists. The element x = x_{1}x_{2}...x_{k} ∈ C_{G}(g) now satisfies Fix(x) = 0, as required. □

Lemma 7.20. Suppose all cycles of g ∈ G̃ have length 1. If (i) g has more than one cycle type, or (ii) g consists of a single cycle of multiplicity > 1, then there is an element x ∈ C_{G}(g) with Fix(x) = 0. If g is not as described in (i) or (ii), then the intersection of the atoms of L(C_{G}(g)) is not 0.

Proof. As usual, let g = g_{1}...g_{k} be the homogeneous decomposition of g, and note that since all cycle lenths are 1, the type of a homogeneous component is defined by an element of μ_{r}, whence k ≤ r. Let γ be a generator of μ_{r}. Now σ(ℓ_{i}, θ) ∈ G̃_{i} has no non-zero fixed points on V_{i} if θ ≠ 1. In case (i), let x_{i} = σ(ℓ_{i}, γ) for i = 1, ..., k − 1 and let x_{k} = σ(ℓ_{i}, γ^{−(k−1)}). Since k ≤ r, γ^{−(k−1)} ≠ 1. Hence x = x_{1}...x_{k} satisfies the requirements. In case (ii), the element x = σ(1, γ)σ(ℓ − 1, γ^{−1}) has the required properties.

If g is not of type (i) or (ii), ℓ = 1 and G = 1. Hence At(L(C_{G}(g))) is empty, and the Lemma is proved. □

We are now able to give the

Proof of Theorem 7.7. Let g = h_{1}h_{2}, where h_{1} is the product of the homogeneous components of g which have cycle length one, and h_{2} is the product of the other homogeneous components. Then CG̃(g) ≅ CG̃_{1}(h_{1}) × CG̃_{2}(h_{2}), where G̃_{i} ≅ G(r, 1, ℓ_{i}), i = 1, 2, ℓ_{1} + ℓ_{2} = ℓ in notation analogous to that of 7.9, 4 and 5. Write L_{i} (resp. L̃_{i}) for L(C_{G_{i}}(h_{i})) (resp. L(CG̃_{i}(h_{i}))), i = 1, 2. Then

L̃_{1} × L̃_{2} ⊇ L(C_{G}(g)) ⊇ L_{1} × L_{2}.

Moreover by 7.12, the atoms of L̃_{2} and L_{2} coincide. It follows that At(L(C_{G}(g))) includes all the spaces of the form V_{1} ⊕ A_{2}, where A_{2} is an atom of L_{2}, since these spaces are atoms of L̃_{1} × L̃_{2} and of L_{1} × L_{2}. Now suppose Y_{1} ⊕ Y_{2} ∈ At(L(C_{G}(g))), where Y_{i} ⊆ V_{i}. If Y_{2} ≠ V_{2}, then there is an atom X_{2} of L̃_{2} such that X_{2} ⊇ Y_{2}. Since At(L̃_{2}) = At(L_{2}), it follows that V_{1}⊕X_{2} ∈ L(C_{G}(g)), whence Y_{1} = V_{1} and Y_{2} = X_{2}. Thus all atoms of L(C_{G}(g)) are of the form V_{1}⊕Y_{2} with Y_{2} ∈ At(L_{2}) or Y_{1}⊕V_{2} with Y_{1} ∈ L̃_{1}. But if Y_{1} ⊕ V_{2} = Fix(x) for x = x_{1}x_{2} ∈ G ⊆ G̃_{1} × G̃_{2}, then x_{2} = 1, whence x_{1} ∈ G_{1}, and so Y_{1} ∈ L_{1}. Hence Y_{1} ∈ At(L_{1}).

The intersection condition on the atoms of L(C_{G}(g)) therefore implies that the intersections of the atoms of both L_{1} and of L_{2} is zero. By Proposition 7.19 applied to the pair G_{2}, k_{2}, there is an element x_{2} ∈ C_{G_{2}}(h_{2}) with Fix(x_{2}) = 0. Similarly, by Lemma 7.20, there is an element x_{1} ∈ C_{G_{1}}(h_{1}) with Fix(x_{1}) = 0. The element x = x_{1}x_{2} ∈ G_{1} × G_{2} ≤ G centralises g and has zero fixed point set. This completes the proof of Theorem 7.7. □

Proposition 7.21. Let G = G(r, m, ℓ) and let g ∈ G̃ = G(r, 1, ℓ) ⊇ G. For any linear group Γ, let L^{0}(Γ) be the poset of setwise intersections of the elements of At(Γ).

(i) If g has no cycles of length 1, then L^{0}(C_{G}(g)) = L^{0}(CG̃(g))

(ii) If g = h_{1}h_{2}, where h_{1} is the product of the homogeneous components of g which have cycle length one, and h_{2} is the product of the other homogeneous components, then L^{0}(C_{G}(g)) ≅ L^{0}(C_{G_{1}}(h_{1})) × L^{0}(C_{G_{2}}(h_{2})), where the notation is analogous to that in the proof of 7.7.

The proof of 7.21 is essentially contained in the proof of 7.7.

We now turn to the final case.

7.3. The group G_{35}. We shall prove Theorem 7.22. Let G = G_{35} be the Weyl group of type E_{6}. Then for each element g ∈ G, we have either

(i) C_{G}(g) is contained in a proper parabolic subgroup, or

(ii) C_{G}(g) contains an element x such that Fix(x) = 0. Corollary 7.23. Let G = G_{35} be the Weyl group of type E_{6}. If g ∈ G and the join of the atoms of L(C_{G}(g)) is 0, then there exists an element x ∈ C_{G}(g) such that Fix(x) = 0.

Proof. If the join of the atoms of L(C_{G}(g)) is 0, then C_{G}(g) is not contained in any proper parabolic subgroup of G, because any proper parabolic subgroup is the isotropy group of a non-zero vector v, which is in Fix(y), for each y ∈ C_{G}(g), and hence in the join of the atoms of L(C_{G}(g)). Hence by Theorem 7.22, there is an element x ∈ C_{G}(g) with Fix(x) = 0. □

Combining Lemma 7.6, Theorem 7.7 and Corollary 7.23 completes the proof of Theorem 6.1. It therefore remains only to prove Theorem 7.22.

The proof of 7.22 is by explicit computation, using the table of conjugacy classes of G given in Carter [5], p.50. This reveals that there are 25 conjugacy classes. Our notation for the discussion below is taken from that source. For g ∈ G, we say the reflection length c(g) of g is the minimal length of g as a product of reflections. Since G is a real group, c(g) is equal to the codimension of Fix(g). In the notation of [5], loc. cit., we note that W_{1} is a reflection subgroup of G which contains g as an element of maximal length, equal to the rank of W_{1}. The parabolic subgroup \overline{W_{1}} is the “parabolic closure” of g, i.e. the isotropy group of Fix(g). The reflection subgroup W_{2} is generated by all reflections in roots orthogonal to those of W_{1}. Clearly W_{2} ⊆ C_{G}(g). Observe also that if x centralises g, the x fixes Fix(g), whence x normalises \overline{W_{1}}. Thus C_{G}(g) ≤ N_{G}(\overline{W_{1}}). Now the structure of the normaliser of a parabolic subgroup has been discussed by Howlett in [11]. We shall adopt the perspective of that work, which starts with the elementary statement

7.24. N_{G}(\overline{W_{1}}) is the semidirect product of \overline{W_{1}} with the subgroup N_{1} of G which preserves the set of positive roots of \overline{W_{1}}. Lemma 7.25. (a) If the rank of W_{1}×W_{2} = 6, then there is an element x ∈ C_{G}(g) with Fix(x) = 0. This accounts for the classes ∅, A_{1}, A_{2}, A_{2}^{2}, A_{2}^{3}, A_{5}, A_{5} × A_{1}, E_{6}, E_{6}(a_{1}) and E_{6}(a_{2}).

(b) If N_{G}(\overline{W_{1}}) is contained in a proper parabolic subgroup of G, the same is true of C_{G}(g).

(c) If \overline{W_{1}} = D_{5} or A_{4} × A_{1} , then N_{G}(\overline{W_{1}}) = \overline{W_{1}} . This accounts for the classes A_{3} × A_{1}^{2}, A_{4} × A_{1}, D_{5} and D_{5}(a_{1}).

Proof. In case (a), we may take x to be g multiplied by a Coxeter element of W_{2}. The statement (b) is clear in view of 7.24, while in case (c), one checks easily using the ideas of [11], that \overline{W_{1}} is selfnormalising. □

Lemma 7.26. The classes A_{1}^{4} and D_{4}(a_{1}) are respectively contain the 6^{th} and 3^{rd} power of a regular element (in the sense of Springer [21]) of order 12, which has no fixed points. Thus these classes satisfy 7.22(ii).

This is easily verified using the properties of regular elements proved in [21]. Lemma 7.27. Suppose g is of type

(a) A_{2}^{2} × A_{1}, or

(b) D_{4}, or

(c) A_{1}^{3}.

Then C_{G}(g) contains an element x with no non-zero fixed points.

Proof. In case (a), g is a Coxeter element of order 6 in W_{1} ≅ A_{2}^{2} × A_{1} Here the group N_{1} of 7.24 has order 2, and if x′ is the non-trivial element, g may be chosen so that x′ centralises g. Hence so does x = gx′ and one checks that Fix(x) = 0.

In case (b), g is a Coxeter element in W_{1} ≅ D_{4} and N_{1} induces the group Sym_{3} of triality automorphisms of W_{1}. If g is appropriately chosen, N_{1} centralises g. If x′ is an element of order 3 in N_{1} a short matrix calculation shows that if x = gx′, Fix(x) = 0.

In case (c), N_{1} has order 12, and acts on Π_{1}^{⊥} as a reflection group of type A_{2} ×A_{1}. The element g is the product of 3 commuting reflections, which are permuted by N_{1}. Thus N_{1} centralises g. There is an element y (of order 3) in N_{1} such that Fix_{Π_{1}^{⊥}}(y) = 0. This y permutes the three reflections above, whence Fix(gy) = 0. □

Lemma 7.28. Suppose that g is of type (a) A_{4}, or

(b) A_{3} × A_{1}, or

(c) A_{2} × A_{1}, or

(d) A_{2} × A_{1}^{2}, or

(e) A_{3}, or

(f) A_{1}^{2}.

Then \overline{W_{1}} is of the same type as g and N_{G}(\overline{W_{1}}) is contained in a proper parabolic subgroup of G.

Proof. The statement concerning \overline{W_{1}} may be verified by inspection. In case (a), one verifies that N_{1} = W_{2} ≅ A_{1} and that N_{G}(\overline{W_{1}}) = W_{1} ×W_{2}, which is proper parabolic in G. In case (b), since Aut_{W} (Π_{1}) = 1, N_{1} centralises \overline{W_{1}}, whence again N_{1} = W_{2}, so that N_{G}(\overline{W_{1}}) is the proper parabolic subgroup W_{1} × W_{2}. The argument for case (c) is the same as for (b).

In case (d), \overline{W_{1}} has rank 4, and N_{G}(\overline{W_{1}}) is the semidirect product of \overline{W_{1}} with N_{1}, which has order 2, and is generated by a single element v(a, J) (see [11]). Hence N_{G}(\overline{W_{1}}) is contained in a parabolic subgroup of rank 5 (and type D_{5}). In case (e), the argument is similar to that for (d), except that \overline{W_{1}} has rank 3 and N_{1} is generated by two commuting elements v(a_{i}, J) (i = 1, 2). So N_{G}(\overline{W_{1}}) is again contained in a parabolic subgroup of rank 5. The argument in case (f) is again similar, with \overline{W_{1}} here having rank 2, but N_{1} having 3 generators. The conclusion again is that N_{G}(\overline{W_{1}}) is contained in a parabolic subgroup of type D_{5}. This completes the proof of 7.28. □

Lemmas 7.25, 7.26, 7.27 and 7.28 exhaust all possibilities for g up to conjugacy. Thus Theorem 7.22 is proved. This completes the proof of Theorem 6.1.

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School of Mathematics and Statistics,, The University of Sydney, Sydney, Australia 2006.

  E-mail address: gusl@maths.usyd.edu.au