Cohomology of Coxeter arrangements and Solomon's descent algebra

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_L$ is a parabolic subgroup of $W$ all of whose irreducible factors are of type $A$.


Introduction
Suppose V is a finite-dimensional, complex vector space.A linear transformation t in GL(V ) is called a reflection if it has finite order and the fixed point set of t is a hyperplane in V , or equivalently, if t is diagonalizable with eigenvalues 1, with multiplicity dim V − 1, and ζ, where ζ is a root of unity, with multiplicity 1. Suppose that W is a finite subgroup of GL(V ) generated by a set of reflections T .For each t in T , let H t = Fix(t) denote the fixed point set of t in V and set A = { H t | t ∈ T } and M W = V \ ∪ t∈T H t .Then (V, A) is a hyperplane arrangement and the complement M W is an open, W -stable subset of V .
The action of W on M W determines a representation of W on the singular cohomology of M W .For p ≥ 0 let H p (M W ) denote the p th singular cohomology space of M W with complex coefficients and let H * (M W ) = p≥0 H p (M W ) denote the total cohomology of M W , a graded C-vector space.It follows from a result of Brieskorn [10] that dim H * (M W ) = |W | and so a naive guess would be that the representation of W on H * (M W ) is the regular representation of W .A simple computation for the symmetric group S 3 shows that this is not the case.
In 1987, Lehrer [24] determined the character of the representation of W on H * (M W ) when W = S n is a symmetric group by explicitly computing the "equivariant Poincaré polynomials" P Sn (g, t) = p≥0 trace(g, H p (M W ))t p , for g in S n (here t is an indeterminate).Subsequently, equivariant Poincaré polynomials were computed case-by-case for other groups by various authors.In 2001, Blair and Lehrer [8] showed that for any complex reflection group, the equivariant Poincaré polynomials have the form P W (g, t) = x∈Z W (g) f g (x)(−t) c(x) where f g : Z G (g) → C is explicitly given and c(x) is the codimension of the fixed point space of x in V .Felder and Veselov [13] have found an elegant description of the character of H * (M W ) when W is a Coxeter group that precisely describes how the character of H * (M W ) differs from the regular character ρ of W . Specifically Felder and Veselov show that the character of H * (M W ) is given as where the sum runs over a set of "special" involutions t in W .
In contrast, while the representation of W on H * (M W ) is well-understood, much less is known about the representations of W on the individual graded pieces H p (M W ) for p ≥ 0. When W is a symmetric group Lehrer and Solomon [25] have described these representations as sums of representations induced from linear characters of centralizers of elements in W .They conjecture that a similar decomposition exists in general.
For the symmetric group S n , Barcelo and Bergeron [1] construct an explicit S n -stable subspace of the exterior algebra of the free Lie algebra on n letters that affords the character of H * (M W ) tensored with the sign character.Their construction could be used to study the characters of the individual cohomology spaces H p (M W ).
For the hyperoctahedral group W (B n ), the first author [11] extended Lehrer and Solomon's construction and expressed each H p (M W ) as a sum of representations induced from linear characters of subgroups.However, the subgroups appearing are not always centralizers of elements of W (B n ).At the same time, Bergeron used the free Lie algebra on 2n letters to construct a representation of W (B n ) analogous to the representation of S n constructed in [1].The character of this representation of W (B n ) is again the character of H * (M W ) tensored with the sign character.He then uses this construction to study the characters of the individual cohomology spaces H p (M W ).
In this paper we state a conjecture for a finite Coxeter group W (Conjecture 2.1) that both refines the conjecture of Lehrer and Solomon [25,Conjecture 1.6] and directly relates the representation of W on H p (M W ) to a subrepresentation of the right regular representation of W .It is straightforward to see that Conjecture 2.1 holds for W if and only if it holds for every irreducible factor of W . Thus, to prove the conjecture we may assume that W is irreducible.Conjecture 2.1 is proved for symmetric groups in this paper (Theorem 6.3).The conjecture has been proved for all rank two Coxeter groups [12] and has been checked using the computer algebra system GAP [31] and the package CHEVIE [15] for all Coxeter groups with rank six or less [6], [7].More generally, in §7 we extend the constructions used in the proof of Theorem 6.3 and prove a "relative" version of Conjecture 2.1 for pairs (W, W L ), where now W is any finite Coxeter group and W L is a parabolic subgroup of W , all of whose irreducible factors are of type A. If the conclusion of Theorem 7.1 were to hold for every parabolic subgroup W , not just those that are products of symmetric groups, then Conjecture 2.1 would hold for W .The statement of the conjecture, along with an expository review of the background material from the theories of Coxeter groups and hyperplane arrangements we use later in the paper, is given in §2.
The subrepresentations of the right regular representation of W that we consider arise from a decomposition of a subalgebra of the group algebra of W , known as "Solomon's descent algebra," into projective, indecomposable modules.Projective, indecomposable modules in an artinian C-algebra are generated by idempotents.The idempotents in the descent algebra we use in this paper were discovered by Bergeron, Bergeron, Howlett, and Taylor [4]; we call them BBHT idempotents.In §3 we study the relationships between BBHT idempotents for W and BBHT idempotents for parabolic subgroups.We also compare the BBHT idempotents for W with those of its irreducible factors when W is reducible.In §4 we show that the right ideals in the group algebra of W generated by BBHT idempotents afford induced representations.It then follows that the BBHT idempotents give rise to a decomposition of the group algebra that is the direct analog of the decomposition of H * (M W ) given by Brieskorn's Lemma [10, Lemme 3].
The proof of Conjecture 2.1 for symmetric groups is given in §5 and §6.As a consequence, we obtain a decomposition of the group algebra CS n as a direct sum of representations induced from one-dimensional representations of centralizers, one for each conjugacy class.Similar decompositions of the group algebra of the symmetric group have been proved independently by Bergeron, Bergeron, and Garsia [3], Hanlon [18], and more recently, Schocker [32], all using different methods.
For readers familiar with the literature on the free Lie algebra and its connections with the descent algebras and group algebras of symmetric groups, Theorems 5.1(a) and 6.3(a) will seem familiar.Indeed, Theorem 5.1(a) was likely known in some form to Brandt [9], Wever [38], and Klyachko [21], and a proof of Theorem 6.3(a) can be extracted from results in [3, §4], [30,Theorem 8.24], and [14, §4].In contrast with these references, where the methods are combinatorial and the emphasis is on the connections between the group algebra CS n and the free Lie algebra on n letters, our methods are mainly group-and representation-theoretic, using the theory of Coxeter groups in conjunction with special features of symmetric groups, and our focus is on the connections between the group algebra CS n and the representation of S n on the cohomology spaces H p (M W ).Moreover, our line of reasoning, which is motivated by the arguments in Lehrer and Solomon [25], demonstrates a striking parallelism between the Orlik-Solomon algebras and group algebras of symmetric groups that to our knowledge has not been observed before.We hope that the approach taken in this paper will lead to a deeper understanding of the topological and geometric properties of general Coxeter arrangements.For example, as we show in §7, the constructions in §5 and §6 have natural extensions when the focus is shifted from considering not just a single symmetric group to considering a product of symmetric groups embedded as a parabolic subgroup in a larger finite Coxeter group W , where the type of W is arbitrary.
Bergeron and Bergeron conjecture in [2] that there might be a decomposition of the group algebra CW (B n ) analogous to the decomposition of CS n studied by Bergeron-Bergeron-Garsia [3] and Hanlon [18].In [5] Bergeron gives a decomposition of the group algebra of a hyperoctahedral group as a direct sum of induced representations induced from linear characters of subgroups.Unfortunately, this decomposition is not the decomposition proposed in Conjecture 2.1, it is the analog for group algebras of hyperoctahedral groups of the decomposition of H * (M W ) found in [11].
2. The Orlik-Solomon algebra and Solomon's descent algebra In the rest of this paper we assume that V is a finite-dimensional, complex vector space and W ⊆ GL(V ) is a finite Coxeter group with Coxeter generating set S. Then each s in S acts on V as a reflection with order two and W is generated by S subject to the relations (st) ms,t = 1, where m s,s = 1 and m s,t = m t,s > 1 for s = t in S. Let T denote the set of all reflections in W .
We assume also that a positive, definite, Hermitian form • , • on V is given and that W is a subgroup of the unitary group of V with respect to this form.It is known that Fix(W ) ⊥ , the orthogonal complement of the space of fixed points of W on V , has a basis ∆ = { α s | s ∈ S } so that α s , α t = − cos(π/m s,t ) for s and t in S. Then s acts on V as the reflection through the hyperplane orthogonal to α s and Φ = { w(α s ) | w ∈ W, s ∈ S } is a root system in Fix(W ) ⊥ with base ∆.
2.1.Shapes and conjugacy classes.We begin by recalling a parameterization of the conjugacy classes in W due to Geck and Pfeiffer (see [17, §3.2]) in a form compatible with the arrangement (V, A) of W .
The lattice of A, denoted by L(A), is the set of subspaces of V that arise as intersections of hyperplanes in A: to be the pointwise stabilizer of X in W .It follows from Steinberg's Theorem [36] that W X is generated by {t ∈ T | X ⊆ H t }.It then follows that X = Fix(W X ), and so the assignment X → W X defines an injection from L(A) to the set of subgroups of W . Notice that W X is again a Coxeter group.
The action of W on A induces an action of W on L(A).Obviously wW X w −1 = W w(X) and so for X and Y in L(A), the subgroups W X and W Y are conjugate if and only if X and Y lie in the same W -orbit.Thus, the assignment X → W X induces a bijection between the set of orbits of W on L(A) and the set of conjugacy classes of subgroups W X .
By a shape of W we mean a W -orbit in L(A).We denote the set of shapes of W by Λ.For example, if W is the symmetric group S n , then Λ is in bijection with the set of partitions of n, and with the set of Young diagrams with n boxes.When λ is a shape and X is a subspace in λ we say λ is the shape of W X .
It is shown in [29, §6.2] that the assignment w → Fix(w) defines a surjection from W to L(A).Composing with the map that sends an element X in L(A) to its W -orbit, we get a map sh : W → Λ.
We say sh(w) is the shape of w.Thus, sh(w) is the W -orbit of Fix(w) in L(A).Clearly, sh is constant on conjugacy classes and so we can define the shape of a conjugacy class to be the shape of any of its elements.
An element w in W , or its conjugacy class, is called cuspidal if Fix(w) = Fix(W ).For example, if W is the symmetric group S n , then the conjugacy class consisting of n-cycles is the only cuspidal class.In general, there is more than one cuspidal conjugacy class.Cuspidal elements and conjugacy classes are called elliptic by some authors.
Suppose that λ is a shape, X in L(A) has shape λ, and C is a conjugacy class in W with shape λ.If w is in C, then Fix(w) is in the W -orbit of X and so C ∩ W X is a non-empty union of cuspidal W X -conjugacy classes.).In the following we use the presentation of this algebra given by Orlik and Solomon [27].
Recall that the set T of reflections in W parametrizes the hyperplanes in A. The Orlik-Solomon algebra of W is the C-algebra, A = A(A), with generators { a t | t ∈ T } and relations • a t 1 a t 2 = −a t 2 a t 1 for t 1 and t 2 in T and The algebra A is a skew-commutative, graded, connected C-algebra that is isomorphic as a graded algebra to H * (M W ). Let A p denote the degree p subspace of A. Then See [29, §3.1] for details.
The action of W on A extends to an action of W on A as algebra automorphisms.An element w in W acts on a generator a t of A by wa t = a wtw −1 .With this W -action A is isomorphic to H * (M W ) as graded W -algebras.
Orlik and Solomon [28] have shown that the normalizer of W X in W is the setwise stabilizer of X in W , that is For X in L(A), define A X to be the span of Proofs of the following statements may be found in [29, Corollary 3.27 and Theorem 6.27].
• If codim X = p, then A X ⊆ A p .
• There are vector space decompositions A p ∼ = codim X=p A X and A ∼ = X∈L(A) A X .
• For w in W , wA X = A w(X) .Thus, A X is an N W (W X )-stable subspace of A.
For a shape λ in Λ, set A λ = X∈λ A X .Suppose X is a fixed subspace in λ and that codim X = p.Then A λ is a W -stable subspace of A p and we have isomorphisms of CWmodules (see [25]).
2.3.Solomon's descent algebra.In contrast with the Orlik-Solomon algebra A, which is defined for every complex reflection group, Solomon's descent algebra is defined using the Coxeter generating set S of W and so has no immediate analog for complex reflection groups that are not Coxeter groups.
Suppose that I is a subset of S. Define Then X I is in L(A) and codim X I = |I|.It follows from Steinberg's Theorem [36] that Then X I is the orthogonal complement of the span of ∆ I .
Orlik and Solomon (see [29, §6.2]) have shown that each orbit of W on L(A) contains a subspace X I for some subset I of S. For subsets I and J of S define I ∼ J if there is a w in W with w(∆ I ) = ∆ J .Then ∼ is an equivalence relation.It is well-known that W I and W J are conjugate if and only if I ∼ J (see [34]).It follows that the assignment I → X I induces a bijection between S/∼, the set of ∼-equivalence classes, and Λ, the set of shapes of W .
Next, let ℓ denote the length function of W determined by the generating set S and define Then W I is a set of left coset representatives of W I in W . Also, define x I = w∈W I w in the group algebra CW .Solomon [33] has shown that the span of { x I | I ⊆ S } is in fact a subalgebra of CW .This subalgebra is denoted by Σ(W ) and called the descent algebra of W .It is not hard to see that { x I | I ⊆ S } is linearly independent and so dim Σ(W ) = 2 |S| .Notice that x S = 1 is the identity in both CW and its subalgebra Σ(W ).
Bergeron, Bergeron, Howlett, and Taylor [4] have defined a basis { e I | I ⊆ S } of Σ(W ) that consists of quasi-idempotents and is compatible with the set of shapes of W .For λ in Λ define Then each e λ is idempotent and { e λ | λ ∈ Λ } is a complete set of primitive, orthogonal idempotents in Σ(W ).(See §3 for more details.)In particular, λ∈Λ e λ = 1 in CW .
Define E λ = e λ CW .In §3 we show that e I CW I affords an action of N W (W I ) and that if I is in S λ , then E λ is induced from e I CW I .Thus, in analogy with the decomposition in §2.2 of the Orlik-Solomon algebra A, we have isomorphisms of CW -modules 2.4.Centralizers and complements.The last ingredient we need in order to state the conjecture is a certain set of characters of centralizers of elements of W .These characters, together with the sign character, should quantify the difference between the representation of W on H p (M W ) and a subrepresentation of the regular representation.They naturally arise in work of Howlett and Lehrer [20] and in recent results of the second author [22] that describe the structure of the centralizer of an element in W .
Suppose that I is a subset of S and C is a conjugacy class in W such that C ∩ W I is a cuspidal conjugacy class in W I .Howlett [19] has shown that W I has a complement, N I , in for z in Z W (c).

2.5.
Relating the Orlik-Solomon algebra and the descent algebra.We now have all the concepts we need in order to state the conjecture.
Let ǫ denote the sign character of W .For c in W , define X c = Fix(c), rk(c) = codim X c , and Associated with each λ in Λ we have the W -stable subspace A λ of the Orlik-Solomon algebra A, the right ideal E λ in CW , and the set of conjugacy classes with shape λ.We conjecture that A λ and E λ are related to the set of conjugacy classes with shape λ as follows.
where ǫ c denotes the restriction of ǫ to Z W (c).
In particular, As stated in the introduction, we prove Conjecture 2.1 for symmetric groups in §5 and §6.
The conjecture is known to be true for all Coxeter groups with rank up to six [12], [6], [7].
We in fact prove more than is stated in the conjecture.First, we show that the character ϕ c of Z W (c) may be chosen to be a trivial extension of a character of Z Wc (c).Second, we construct explicit CW -module isomorphisms In §7 we extend the constructions in §5 and §6 and show if W is any finite Coxeter group, λ is in Λ, c is in W with sh(c) = λ, and the irreducible components of W c are all of type A, then the character ϕ c of Z Wc (c) constructed in §6 extends to a character ϕ c of Z W (c).Moreover, we construct explicit CW -module isomorphisms Note that with the given assumptions we have |C λ | = 1, and so the sums in Conjecture 2.1 (a) and (b) reduce to a single summand.Also, in contrast with the case when the ambient group W is a symmetric group and ϕ c is the trivial extension, in the general case, ϕ c may not be the trivial extension of ϕ c .
A direct proof of the conjecture involves finding suitable linear characters ϕ c of Z W (c).The computations in [6] and [7], as well as calculations in type B, show that some natural guesses about the characters ϕ c are not true.For example, Z W (c) acts on the eigenspaces of c in V and so the powers of the determinant character of Z W (c) acting on an eigenspace of c are linear characters of Z W (c).An example in [6, §4] shows that it is not always possible to choose ϕ c to be one of these characters for any eigenspace.
On the other hand, suppose that c is an involution and Then in all the cases that have been computed so far, it turns out that ϕ c may be chosen to be the "trivial" extension of ǫ Wc to Z W (c) in the sense that ϕ c (wn) = ǫ Xc (w) for w in W Xc and n in N Xc .The representations Ind W Z W (c) ϕ c play a role in understanding the characters of finite reductive groups [23], [16], and the corresponding representations of the Iwahori-Hecke algebra of W play a role in the representation theory of complex reductive groups [26].The fact that these representations seem to be closely related to the representation of W on H * (M W ) is quite mysterious.

BBHT idempotents
In this section we collect several preliminary results about the descent algebra of W and the BBHT idempotents.
For subsets I, J, and K of S define Then W IJ is the set of minimal length (W I , W J )-double coset representatives in W . Solomon [33] has shown that Let 2 S denote the power set of S and fix a function σ : where Φ + is the positive system determined by ∆.Thus, for J ⊆ K and w in W K we have w(∆ J ) ⊆ Φ + .For subsets J and K of S define Because σ(I) > 0 for all subsets I of S we have m σ JJ = 0 for all J and so the system of equations Then e σ K is in Σ(W ), n σ KK = (m σ KK ) −1 for all subsets K of S, and n σ JK = 0 when J ⊆ K. Bergeron, Bergeron, Howlett, and Taylor have shown [4, §7] that e σ I is a quasi-idempotent in Σ(W ).Precisely, for λ in Λ define σ(λ) = I∈S λ σ(I).Then In the special case when the function σ(I) = 1 for all subsets I of S we do not include it in the notation.Thus, (3.4) Notice that the quantities m σ JK , e σ J , m JK , e J , . . .are defined relative to an ambient Coxeter system (W, S).Below we also consider the analogous quantities defined relative to a parabolic subsystem (W L , L).To help keep things straight, in this section and the next we use the following conventions.
• σ always denotes a function from 2 S to R >0 .
• When σ(I) = 1 for all I ⊆ S, the BBHT quasi-idempotents in CW defined with respect to σ are denoted by e J .
• τ always denotes a function from 2 L to R >0 where L is a subset of S.
• When τ (I) = 1 for I ⊆ L, the BBHT quasi-idempotents in CW L defined with respect to τ are denoted by e L J .Thus, e S J = e J .For example, {S} is a shape of W and {L} is a shape of W L .Then e σ {S} = σ(S)e σ S , e {S} = e S , and e L L = e τ L = e τ {L} when τ (I) = 1 for all subsets I of L. The following lemmas give some translation properties for the quantities defined above.Lemma 3.5.Suppose that K ⊆ S and d is in ) for all L ⊆ K and so This proves (b).
Using (a) and (b) we see that for J ⊆ K, Recall that we have fixed a positive, definite, Hermitian form on V such that W is a subgroup of U(V ), the unitary group of V .Define Notice that if n is in N(W ), then nSn −1 = S. Thus, N(W ) acts on S and on 2 S , and Lemma 3.6.Suppose that n is in N(W ) and that σ(I n ) = σ(I) for all I ⊆ S. Then e σ I n = n −1 e σ I n for I ⊆ S. In particular, n centralizes e σ S in CW .
and so W J ∩ P = ∐ J⊆K (Y K ∩ P ) for every subset P of W .For P ⊆ W define Then p P J = J⊆K q P K and f P J = J⊆K g P K , and so by Möbius inversion, (3.8) Taking J = ∅ and P = W in (3.8), we have Y ∅ = {w 0 } and so (3.9) , where w J is the longest element in W J , and so g P J = 1 and f P K = m JK .Therefore, Using (3.9) and (3.10), we have Now suppose that J ⊆ S and µ ∈ Λ are such that J ∈ S µ .By (3.3), we have e µ e J = e J and so w 0 e J = λ∈Λ (−1) d λ e λ e µ e J = (−1) dµ e µ e J = (−1) |J| e J .
For subsets J and K of S we have m JK = m J w 0 K w 0 .Thus, n JK = n J w 0 K w 0 and it follows from (3.1) that w 0 e J w 0 = e J w 0 .Finally, e J w 0 = w 0 (w 0 e J w 0 ) = (−1) |J w 0 | e J w 0 = (−1) |J| e J w 0 .
This completes the proof of the lemma.It is straightforward to check that for a in CW L and wn in W L N L , the assignment (a, wn) → a • wn = n −1 awn defines an action of the group and so left multiplication by x L defines an N W (W L )-equivariant embedding of CW L into CW .For later reference we record this fact in the following lemma.
and so (b) holds.
Suppose n is in N L .Then using (b) and Lemma 3.5 we have J n and e σ L J n are both in CW L and x L (n −1 e σ L J n) = x L e σ L J n , so we conclude from Lemma 3.11 that n −1 e σ L J n = e σ L J n .This proves (c).
We conclude this section with a description of the quasi-idempotents e σ K in the case when W is reducible.
Suppose that W is reducible, say W ∼ = W 1 × W 2 and S = S 1 ∐ S 2 is the disjoint union of S 1 and S 2 where W 1 = S 1 and Every element in W has a unique expression as a product w 1 w 2 with w 1 in W 1 and w 2 in Now suppose that σ : 2 S → R >0 has the property that σ(J 1 ∐ J 2 ) = σ(J 1 )σ(J 2 ) for J 1 ⊆ S 1 and J 2 ⊆ S 2 .Then for J ⊆ K ⊆ S we have where σ i is the restriction of σ to 2 S i for i = 1, 2. Conversely, if we are given functions σ i : 2 S i → R >0 for i = 1, 2 and define σ : With σ as above, set and so f σ J = e σ J .This proves the following proposition.

E λ is an induced representation
Suppose λ is in Λ and σ : 2 S → R >0 .Define E σ λ = e σ λ CW to be the right ideal in CW generated by e σ λ .Similarly, using the notation in (3.4), define We have seen in §2.
In this section we show that E σ λ has a similar description as an induced representation and we analyze how E σ λ depends on the choice of σ.In particular, we show in Corollary 4.8 that for L in S λ , The next lemma follows immediately from (3.3).
Lemma 4.1.Suppose that λ is in Λ and I is in S λ .Then E σ λ = e σ I CW .
The next proposition shows that up to isomorphism, E σ λ does not depend on σ.Proposition 4.2.Suppose that λ is in Λ and that σ and σ 1 are functions from 2 S to R >0 .Then there is a unit u in Σ(W ) such that left multiplication by u defines an isomorphism of right CW -modules Proof.Let rad(Σ(W )) denote the Jacobson radical of Σ(W ) and let θ denote the natural projection from Σ(W ) to Σ(W )/rad(Σ(W )).Bergeron, Bergeron, Howlett, and Taylor [4, §7] have shown that θ(e σ λ ) = θ(e σ 1 λ ) is a primitive idempotent in Σ(W )/rad(Σ(W )).Thus, it follows from [37, Theorem 3.1] that there is a unit u in 1 + rad(Σ(W )) such that ue σ λ = e σ 1 λ u.Then left multiplication by u defines an isomorphism of right CW -modules e σ λ CW ∼ = e σ 1 λ CW .
Suppose that N is a subgroup of N(W ), so W N is a subgroup of N U(V ) (W ).Then as in Lemma 3.11, W N acts on CW by a • wn = n −1 awn for a in CW , w in W , and n in N. If N centralizes e σ λ , then clearly E σ λ is a W N-submodule of CW .It follows from Lemma 3.6 that if σ is constant on N-orbits in 2 S , then N centralizes e σ {S} = e σ S .More generally, if σ is constant on N-orbits in 2 S and S λ is N-stable, then N centralizes e σ λ .Let Σ(W ) N denote the algebra of N-invariants in Σ(W ).Proposition 4.3.Suppose that λ is in Λ and that σ and σ 1 are functions from 2 S to R >0 such that S λ is N-stable, N centralizes e σ λ , and σ 1 is constant on N-orbits in 2 S .Then there is a unit v in Σ(W ) N such that left multiplication by v defines an isomorphism of right CW N-modules E σ λ and E σ 1 λ .
Proof.We saw in the proof of Proposition 4.2 that there is a unit u in 1 + rad(Σ(W )) such that ue σ λ = e σ 1 λ u and we observed in the proof of Lemma 3.6 that n −1 x I n = x I n for I ⊆ S and n in N. It follows that Σ(W ) and rad(Σ(W )) are stable under conjugation by N and so ) and hence is a unit.By assumption, N centralizes e σ λ and e σ 1 λ and it follows that ve σ λ = e σ 1 λ v. Therefore, left multiplication by v defines a N U(V ) (W )-module isomorphism e σ λ CW ∼ = e σ 1 λ CW .
In the next proposition, ℓ x denotes left multiplication by x.
Proposition 4.4.Suppose L is a subset of S, σ : 2 S → R >0 , τ : 2 L → R >0 , and τ is constant on N L -orbits.Then there is a unit Proof.It follows from Lemma 3.12(c) and Lemma 3.6 that e σ L L CW L and e τ L CW L are N W (W L )stable right ideals of CW L , where N W (W L ) acts on CW L as in Lemma 3.11.By Lemma 3.12(b) we have e σ L = x L e σ L L and so it follows from Lemma 3.5(a) that e σ L CW L and x L e τ L CW L are stable under right multiplication by N W (W L ).It now follows from Lemma 3.11 that the vertical maps are isomorphisms of N W (W L )-modules.
The hypotheses of Proposition 4.3 are satisfied with L, {L}, and σ L in place of S, S λ , and σ.Thus, there is a unit v in Σ(W L ) N L such that ℓ v : e σ L L CW L → e τ L CW L is an isomorphism.The conclusion of the proposition is now clear as The decomposition 1 = λ∈Λ e σ λ gives a decomposition |, the number of elements in W with shape λ.In the next lemma we compute |sh −1 (λ)| in terms of cuspidal elements in a parabolic subgroup of W with shape λ.Lemma 4.5.Suppose that λ is a shape in Λ, X is a subspace in λ, and C is a conjugacy class in W with shape λ.Then Proof.Notice that C ∩ W X is a cuspidal conjugacy class in W X .Thus, it follows from (1) and ( 2 This proves (a).Statement (b) follows from (a) and the observation that sh −1 (λ) is the union of those conjugacy classes in W whose intersection with W X is a cuspidal conjugacy class in W X .
Corollary 4.6.Suppose λ is in Λ, J is in S λ , σ : 2 S → R >0 , and τ : This proves (b).It remains to show that dim e σ J CW J = dim e τ J CW J = |sh −1 (λ) ∩ W J |. Using Lemma 3.12(b), Lemma 3.11, Proposition 4.2, and (b) applied to the shape {J} of W J , we have The next proposition and its corollary are the main results in this section.Proposition 4.7.Suppose that σ : 2 S → R >0 , λ is in Λ, X is in λ, and L is in S λ .
(a) N W (W L ) acts on e σ L CW L by right multiplication and We prove (a).
It was shown in Proposition 4.4 that e σ L CW L is stable under right multiplication by N W (W L ).By Lemma 4. 1, E σ λ = e σ L CW .Therefore, to prove that This map is obviously a surjection.Using Lemma 4.5 and Corollary 4.6, we have CW and so the multiplication map is indeed a bijection.
We saw in Proposition 4.4 that e σ L CW L , e σ L L CW L , e τ L CW L , and x L e τ L CW L all afford equivalent representations of N W (W L ) when τ : 2 L → R >0 is chosen only subject to the restriction that it is constant on N L -orbits.Thus, Proposition 4.7(a) implies the following corollary.
In particular, if σ(I) = 1 and τ (J) = 1 for all I ⊆ S and J ⊆ L, then 5. Symmetric groups: λ = (n) In this section and the next we prove Conjecture 2.1 for symmetric groups.In these two sections we take W to be the symmetric group on n letters with n ≥ 2 and we identify W with the subgroup of GL n (C) that acts on the basis {v 1 , v 2 , . . ., v n } as permutations.Here, v i is the column vector whose j th entry is 0 for j = i and 1 for j = i.For 1 ≤ i ≤ n − 1 let s i denote the matrix in W that interchanges v i and v i+1 and fixes v j for j = i, i + 1.Then S = {s 1 , s 2 , . . ., s n−1 } is a Coxeter generating set for W .
By a partition of n we mean a non-increasing finite sequence of positive integers whose sum is n.
It is well-known that for W = S n we may identify Λ with the set of partitions of n.We make this identification precise as follows.Suppose that λ is a partition of n with p parts.Define partial sums τ i for i = 0, 1, . . ., p by τ 0 = 0 and Then W λ is isomorphic to the product of symmetric groups S λ 1 × • • • × S λp , where the factor S λ i acts on the subset Then X λ is in L(A) and W X λ = W λ .We have seen in Proposition 4.7 that It is well-known and straightforward to check that { X λ | λ is a partition of n } is a complete set of orbit representatives for the action of W on L(A) and that { I λ | λ is a partition of n } is a complete set of representatives for S/ ∼.
Notice that in the extreme case when all parts of λ are equal 1 we have I λ = ∅ and W λ = W ∅ = {1}.At the other extreme, when λ = (n), we have I λ = S and W λ = W S = W .We first prove Conjecture 2.1 when λ = (n).
For the rest of this section we take λ = (n).Then W λ = N W (W λ ) = W and so is the top, non-zero graded piece of A. To simplify the notation, we denote A (n) , E (n) , and e I (n) by A n , E n , and e n respectively.
Define c 1 = 1 in W and for 1 < i ≤ n define c i = s i−1 • • • s 2 s 1 , so c i acts on the basis {v 1 , v 2 , . . ., v n } as an i-cycle.Also, set c = c n .Then, • c is a cuspidal element in W , • the set of cuspidal elements in W is precisely the conjugacy class of c, and  [25].However, the character ϕ of Z W (c) is the same as in [25].
Theorem 5.1.With the preceding notation we have that Statement (b) has been proved by Stanley [35,Theorem 7.2] and by Lehrer and Solomon [25,Theorem 3.9].As mentioned in the introduction, a proof of (a) may be extracted from classical results about the representation of S n on the free Lie algebra on n letters.In contrast, our proof below that the character of W on E n is Ind W Z W (c) (ϕ) follows the Lehrer-Solomon argument and demonstrates a parallelism between the group algebra and the Orlik-Solomon algebra that we expect will apply in some form to all finite Coxeter groups.In addition, our argument is valid not only for E n and e n , but more generally for E σ (n) and e σ (n) for any function σ : 2 S → R >0 .
To emphasize and differentiate the parallel arguments, we use the convention that the superscript + denotes quantities associated with E n and the superscript − denotes quantities associated with A n .Notice that with the notation of §2, we have X c = {0} and so α c = det | Xc is the trivial character.
Suppose t is an indeterminate.For 0 respectively (the k th factor in the product on the left-hand side of the last equation is Lehrer and Solomon [25, §3] prove the following statements. (iii) Consider the homomorphism of left CW -modules from CW to A n given by right multiplication by a n .The kernel of this mapping is the left Next we show that the analogous statements hold with A n replaced by E n and b − (n, k) replaced by b + (n, k).
For this, we consider elements in W as acting on {1, . . ., n}.That is, we identify the vector v j with j for 1 ≤ j ≤ n.Then . If j ≥ i k , then w(j) ≤ j < j + 1 = w(j + 1).Suppose that k < j < i k .Choose r minimal such that Proof.Using the definition and Lemma 5.2 we have Proposition 5.4.The following analogs of (i)-(iv) above hold.
(c) Consider the endomorphism of CW considered as a right CW -module given by left multiplication by e n .The kernel of this mapping is the free, right Proof.The first statement follows immediately from the definitions.
We prove (b) by recursion.It is clear that . It follows from [4,Theorem 7.8] that e n x J = 0 unless J = S. Thus, it follows from Corollary 5.3 that On the other hand, it follows from the definition that Therefore, Next, consider the endomorphism of CW given by x → e n x.Let K denote the kernel of this mapping and let Therefore K 1 = K.This proves (c).
Finally, define idempotents f + and f − in CZ W (c) by Obviously, the lines Cf + and Cf − in CW are stable under left and right multiplication by Z W (c) and afford the characters ϕ and ǫϕ of Z W (c) respectively.Moreover, Ind W Z W (c) (ϕ) is afforded by the right CW -module f + CW and ǫInd W Z W (c) (ϕ) = Ind W Z W (c) (ǫϕ) is afforded by the left CW -module CW f − .Thus, to prove Theorem 5.1 it is enough to find CW -isomorphisms Lemma 5.5.The idempotent f + acts invertibly by right multiplication on e n and the idempotent f − acts invertibly by left multiplication on a n .
Proof.Lehrer and Solomon [25, §3] show that f − acts invertibly on a n .Their argument is easily modified to show that f + acts invertibly by right multiplication on e n as follows.
We have (1

Multiply both sides on the left by
1 n e n and use Proposition 5.4(b) to get Proof of Theorem 5.1.(See [25, §3].)Consider the mapping from f + CW to E n given by x → e n x.It follows from Lemma 5.5 and the discussion preceding it that e n f + = 0, that Z W (c) acts on the line Ce n f + in E n as the character ϕ, and that the mapping is a surjection.Since dim f + CW = |W : Z W (c)| = (n − 1)! = dim E n , the mapping is also an injection.Thus, we have an isomorphism of right CW -modules, E n ∼ = f + CW .
As in [25, §3], similar reasoning applies to the mapping from CW f − to A n given by x → xa n and shows that A n ∼ = CW f − .

Symmetric groups: arbitrary λ
In this section we consider the case of an arbitrary partition of n and complete the proof of Conjecture 2.1 for symmetric groups.
Suppose λ = (λ 1 , λ 2 , . . ., λ p ) is a partition of n.Recall that I λ = S \ {s τ 1 , s τ 2 , . . ., s τ p−1 } and that W λ = I λ is isomorphic to the product of symmetric groups S λ 1 × • • • × S λp , where the factor S λ i acts on • the set of cuspidal elements in W λ is precisely the conjugacy class of c λ , and With λ as above, for 1 ≤ i ≤ p, define ϕ λ i to be the character of g λ i with ϕ λ i (g −1 λ i ) = e 2πi/λ i .Then ϕ λ i is the analog of the character ϕ in §5 for the factor S λ i of W λ .Next, define the character Note that this notation is not consistent with that of Lehrer and Solomon; our character ϕ λ corresponds to the character ϕ λ ǫ in [25].Applying the special case λ = (n) considered in §5 to each factor S λ i of W λ , for 1 ≤ i ≤ p define Finally, define idempotents f + λ and f − λ in CZ λ by Obviously the lines Cf + λ and Cf − λ in CW are stable under left and right multiplication by Z λ and afford the characters ϕ λ and ǫϕ λ of Z λ respectively.Now consider the canonical complement N X λ of W λ in N W (W λ ).Set N λ = N X λ .If λ has m j parts equal j, then N λ is isomorphic to the product of symmetric groups j S m j (see [19] or [25]).In particular, N λ has one Coxeter generator, say r i , for each i such that λ i = λ i+1 .The generator r i acts on the set {v 1 , v 2 , . . ., v n } by interchanging v τ i−1 +j and v τ i +j for 1 ≤ j ≤ λ i , and fixing v k for k ≤ τ i−1 and k > τ i+1 .
Proof.Suppose that i is such that λ i = λ i+1 and consider the generator r i of N λ .Then r i is an involution and it follows from the description of the action of r i on the basis {v 1 , . . ., v n } of V that Since ϕ λ (g λ i ) = ϕ λ (g λ i+1 ), it follows that r i stabilizes ϕ λ and ǫϕ λ .
The group N λ is generated by { r i | λ i = λ i+1 , } and so N λ stabilizes the characters ϕ λ and ǫϕ λ of Z λ .Moreover, N λ acts on {g λ 1 , . . ., g λp } by conjugation as a group of permutations.Thus, it follows from the definition of f + λ i and f − λ i that conjugation by N λ permutes {f + λ 1 , . . ., f + λp } and {f − λ 1 , . . ., f − λp }.Since the f + λ i 's pairwise commute and the f − λ i 's pairwise commute, we see that Set α λ = α X λ .Then α λ is a character of N W (W λ ) and α λ (r i ) = −1.Note that this notation is not consistent with that of Lehrer and Solomon; our character α λ corresponds to the character α λ ǫ in [25] as ǫ(r i ) = (−1) λ i .Theorem 6.2.Suppose that λ is a partition of n.Then the N W (W λ )-modules e I λ CW λ and A X λ , and the character ϕ λ of Z W (c λ ), are related by (a) the character of the right N W (W λ )-module e I λ CW λ is Ind Proof.Statement (b) has been proved by Lehrer and Solomon [25,Theorem 4.4].Their argument may be rephrased as follows.Extending the definition of the element a n in A n when λ = (n), Lehrer and Solomon define an element a λ in A X λ on which f − λ acts invertibly.Then: To prove (a) we first note that by Proposition 4.4, e I λ CW λ and e I λ I λ CW λ are isomorphic right N W (W λ )-modules and so it suffices to prove that e I λ I λ CW λ affords the character Ind Z W (c λ ) (ϕ λ ).We argue as for A X λ with e I λ I λ in place of a λ .For the rest of this proof we fix a partition λ = (λ 1 , . . ., λ p ) of n.To simplify the notation, set I = I λ and e = e I I .It suffices to show that the line Cef + λ in the right N W (W λ )-module eCW λ satisfies properties analogous to (i), (ii), and (iii) above.
. For 1 ≤ j ≤ p, the idempotent e τ I j in CW λ j is defined using the partition (λ j ) of λ j as in §5 and so the idempotent f + λ j acts as a unit on e τ I j by Lemma 5.5.Therefore, f + λ acts invertibly by right multiplication on e and so ef + ) that the mapping is surjective.Moreover, using Corollary 4.6 we have and so the mapping is an isomorphism.This completes the proof of the theorem.
The proof of Conjecture 2.1 for symmetric groups now follows from Proposition 4.7, Theorem 6.2, and transitivity of induction.Theorem 6.3.For each partition λ of n there is a linear character , where ǫ λ denotes the restriction of ǫ to Z W (c λ ).
In particular,

Parabolic subgroups of type A
In this section we return to the case when W is an arbitrary finite Coxeter group and prove a relative version of Theorem 6.3.
Suppose that λ is in Λ, c is in W with sh(c) = λ, and that all the irreducible components of W c are of type A. Without loss of generality we may assume that W c = W L is a standard parabolic subgroup.Suppose that L = p i=1 L i where We assume that l 1 ≥ • • • ≥ l p , L i = {s i,1 , . . ., s i,l i }, and ∆ L i = {α i,1 , . . ., α i,l i }, where the labeling is such that s i,j and s The rest of this section is devoted to the proof of the following theorem.
Theorem 7.1.The character ϕ c of Z W L (c) extends to a character ϕ c of Z W (c) such that The proof follows the same outline as in §6: We find lines in e L L CW L and A X L such that the analogs of statements (i), (ii), (iii) and (i ′ ), (ii ′ ), and (iii ′ ) in the proof of Theorem 6.2 hold, and so Z W (c) (ǫα c ϕ c ).We then apply Ind W N W (W L ) to both sides of both equations and the theorem follows from Proposition 4.7 by transitivity of induction.The argument in this section is complicated by the fact that the subgroup N L is not necessarily contained in Z W (c).
In case W is a symmetric group, it was shown in 6.1 that ϕ c is the trivial extension of ϕ c .In the general case, this is no longer so.Formulas for ϕ c are given in the proof of Lemma 7.3 below.Notice that c is an involution if and only if l 1 = 1 for 1 ≤ i ≤ p, and that in this case ϕ c is the sign character of Z W L (c) and ϕ c is the trivial extension of ϕ c to Z W (c).
Although N L is not necessarily contained in Z W (c), Konvalinka, Pfeiffer, and Röver [22] have shown that Z W L (c) does have a complement, N c , in Z W (c), and N c is also a complement to By [19], the group N L is generated by { r i , g j | 1 ≤ i ≤ p − 1, 1 ≤ j ≤ p }, where r i and g i act on W L as follows.
In particular, r i is an involution, and r i is in Z W (c).
• Either g i = 1 or g i acts on L by In particular, g i is an involution and if g i = 1, then Notice that if W is a symmetric group, then g i = 1 for 1 ≤ i ≤ p.
For 1 ≤ i ≤ p, let w i denote the longest element in W L i and define Then h i is in Z W (c j ) for 1 ≤ i, j ≤ p.It is shown in [22] that As in §6 define Then the lines Cf + L and Cf − L in CW are stable under left and right multiplication by Z W L (c) and afford the characters ϕ c and ǫϕ c of Z W L (c), respectively.Because h i centralizes c j for 1 ≤ i, j ≤ p, the proof of the second statement in Lemma 6.1 applies word-for-word to N c and proves the next lemma.Proof.The argument in statement (i ′ ) in §6 applies verbatim to f + L and shows that f + L acts invertibly by right multiplication on e L L and so e L L f + L = 0. Similarly, the argument in Lehrer and Solomon [25, §3] shows that f − i acts as a unit on a i for 1 ≤ i ≤ p.It follows that f − L acts invertibly by left multiplication on a L and so f − L a L = 0. Therefore, Z W L (c) acts on Ce L L f + L and Cf − L a L by ϕ c and ǫϕ c , respectively, and ϕ c extends ϕ c .If n i = n i+1 , then as in §6 we have (7.4) L g i w i )f + L = (e L L w i )f + L = (−1) l i e L L f + L , (the last equality follows from Lemma 3.7), and (7.7) since h i centralizes L. It follows that Z W (c) acts on the lines Ce L L f + L and Cf − L a L .Moreover, from (7.4) and (7.6) we see that ϕ c (r i ) = 1 and ϕ c (h i ) = (−1) l i .
To complete the proof we need to show that Z W (c) acts on the line Cf − L a L as ǫα c ϕ c .For w in Z W L (c) we have α c (w) = 1 and ϕ c (w) = ϕ c (w). Hence For n in N L it is shown in [12, Lemma 2.1] that ǫ(n)α c (n) is the sign of the permutation of L induced by conjugation by n.
Therefore, using (7.7) we see that (3.2) e σ I e σ J = σ(λ) −1 e σ J when I and J are in S λ .Thus, if we set e σ λ = I∈S λ σ(I)e σ I , it follows from (3.2) that e σ λ is an idempotent in Σ(W ) and hence an idempotent in CW .We call the quasi-idempotents e σ I BBHT quasi-idempotents and the idempotents e σ λ BBHT idempotents.By definition we have 1 = x S = J⊆S n σ JS e σ J and so 1 = λ∈Λ e σ λ in Σ(W ) and CW .It follows that { e σ λ | λ ∈ Λ } is a set of pairwise orthogonal idempotents in CW and that (3.3) e σ λ e σ I = e σ I and e σ I e σ λ = σ(λ) −1 e σ λ for I ∈ S λ .
Thus, I m σ IJ (e σ I d) = I m σ IJ e σ I d .Now fix a subset L of K, multiply both sides by n σ JL , and sum over J, to get e σ L d = e σ L d .(Note that n σ JL = 0 unless J ⊆ L.) This proves (c).

For a subset L
of S, the pair (W L , L) is a Coxeter system.Because W L is a complete set of left coset representatives of W L in W , left multiplication by x L defines an embedding of CW L into CW .For I ⊆ L define W I L = W L ∩W I and x L I = w∈W I L w. Then { x L I | I ⊆ L } is a basis of Σ(W L ).It is well-known and easy to prove that W L W I L = W I , and so x L x L I = x I .If n is in N L , then n(∆ L ) = ∆ L and so by Lemma 3.5(a) we have x L n = x L .It follows that x L CW L is stable under right multiplication by elements of N W (W L ).

Lemma 3 . 11 .
Suppose that L is a subset of S. Then N W (W L ) acts on CW L by a • wn = n −1 awn, for a ∈ CW L , w in W L , and n ∈ N L , and left multiplication by x L defines an N W (W L )-equivariant embedding of CW L into CW .Recall that the quasi-idempotents e σ I are defined relative to the ambient set S and the function σ.Define σ L : 2 L → R >0 by σ L (I) = m σ IL .Then for J ⊆ L we have the quasi-idempotent e σ L J = I⊆L n σ L IJ x L I in CW L defined relative to the set L and the function σ L .Lemma 3.12.Suppose that I, J, and L are subsets of S with I, J ⊆ L. Then (a) m σ L IJ = m σ IJ and n σ L IJ = n σ IJ ; (b) x L e σ L J = e σ J ; and (c) n −1 e σ L J n = e σ L J n for n in N L .Proof.It is shown in [4, Theorem 7.5] that m σ L IJ = m σ IJ .It then follows from the definitions that n σ L IJ = n σ IJ .This proves (a).Now is the cyclic group of order n generated by c. Set ζ = e 2πi/n in C and define ϕ : Z W (c) → C by ϕ(c −1 ) = ζ.The elements we have denoted by c i are denoted by c −1 i by Lehrer and Solomon (i ′ ) Z W (c λ ) acts on the line Cef + λ via the character ϕ λ : We have seen in Lemma 3.6 that N λ centralizes e and in Lemma 6.1 that N λ centralizes f + λ .Thus, N λ centralizes ef + λ and Z W (c λ ) = N λ Z λ acts on the line Cef + λ via the character ϕ λ if ef + λ = 0. Let τ : 2 I → R >0 be the function that takes the constant value 1.We have I = p j=1 I j where W λ j = I j and so e = e τ I = e τ I 1 • • • e τ Ip by Proposition 3.13.Therefore, ef

Lemma 7 . 2 . 2 . 7 . 3 .
The subgroup N c of Z W (c) centralizes the idempotents f + L and f − L in CZ W L (c).For 1 ≤ i ≤ p, define a i = a s i,1 • • • a s i,l i .Set a L = a 1 • • • a p .Then a L is in A X L .The next lemma is the analog of statements (i) and (i ′ ) in the proof of Theorem 6.LemmaThe lines Ce L L f + L in e L L CW L and Cf − L a L in A X L are non-zero and Z W (c)-stable. Let ϕ c be the character of Z W (c) acting by right multiplication on the line Ce L L f + L .Then ϕ c is an extension of ϕ c and the character of Z W (c) acting by left multiplication on the line Cf − L a L is ǫα c ϕ c .
This completes the proof of the lemma.Because f + λ acts invertibly on e L L , N c acts on the line Ce L L f + L by scalars, andN W (W L ) = N c W L , we see that (7.8) e L L CW L = e L L f + L CW L = e L L f + L CN c W L = e L L f + L CN W (W L ).Lehrer and Solomon [25, Proposition 4.4(ii)] have shown that A X L = CW L a L .Thus, because f − λ acts invertibly on a L , N c acts on the line Cf − L a L by scalars, andN W (W L ) = W L N c , we see that (7.9)A X L = CW L a L = CW L f − L a L = CW L N c f − L a L = CN W (W L )f − L a L .Equations (7.8) and (7.9) are the analogs of statements (ii) and (ii ′ ) in the proof of Theorem 6.2.The dimension computation in the proof of statement (iii ′ ) now applies to show that the analogs of statements (iii) and (iii ′ ) both hold in the present situation: The multiplication maps(7.10)Ce L L f + L ⊗ CZ W (c) CN W (W L ) → e L L CW L and CN W (W L ) ⊗ CZ W (c) Cf − L a L → A X L are isomorphisms of CN W (W L )-modules.It follows from (7.10) and Lemma 7.3 thate L L CW L ∼ = Ind N W (W L ) Z W (c) ( ϕ c ) and A X L ∼ = Ind N W (W L ) Z W (c) (ǫα c ϕ c ). Apply Ind W N W (W L )to both sides of these last two equations and use transitivity of induction to getInd W N W (W L ) e L L CW L ∼ = Ind W Z W (c) ( ϕ c ) and Ind W N W (W L ) (A X L ) ∼ = Ind W Z W (c) (ǫα c ϕ c).By Proposition 4.7 and Corollary 4.8 we haveE λ ∼ = Ind W Z W (c) ( ϕ c) and A λ ∼ = Ind W Z W (c) (ǫα c ϕ c ).This completes the proof of the theorem.
Geck and Pfeiffer [17,  §3.2]have shown that in fact C ∩ W X is a single W X -conjugacy class.It follows that C → C ∩ W X defines a bijection between the set of conjugacy classes in W with shape λ and the set of cuspidal conjugacy classes in W X .Fix a set {X λ | λ ∈ Λ } of W -orbit representatives in L(A).Summarizing the preceding discussion we see that conjugacy classes in W are parametrized by pairs (λ, C λ ), where λ is a shape and C λ is a cuspidal conjugacy class in W X λ .