Semicontinuity properties of Kazhdan-Lusztig cells

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Cédric Bonnafé

To cite this version:

Cédric Bonnafé. Semicontinuity properties of Kazhdan-Lusztig cells. 2010. �hal-00312825v3�

SEMICONTINUITY PROPERTIES OF KAZHDAN-LUSZTIG CELLS

C´EDRIC BONNAF´E

Abstract. Computations in small Coxeter groups or dihedral groups suggest that the partition into Kazhdan-Lusztig cells with unequal parameters should obey to some semicontinuity phenomenon (as the parameters vary). The aim of this paper is to provide a rigorous theoretical background for supporting this intuition that will allow to state several precise conjectures.

Let (W, S) be a Coxeter group and assume, for simplification in this introduction, that S is finite and that S = S

∪ ˙ S

, where S

and S

are two non-empty subsets of S such that no element of S

is conjugate to an element of S

. Let ℓ : W → N = {0, 1, 2, . . . } be the length function and ℓ

: W → N the S

-length function (if w ∈ W , ℓ

(w) is the number of occurrences of elements of S

in a reduced decomposition of w). Let us fix two non-zero natural numbers a and b and let L

: W → Z, w 7→ aℓ

(w) + bℓ

(w). Then L

is a weight function (in the sense of Lusztig) so it is possible to define a partition of W into left Kazhdan-Lusztig cells.

It is clear that this partition depends only on b/a: we shall denote it by L

(W ).

Explicit computations suggest the following conjecture:

Conjecture 0. There exist rational numbers 0 < r

< · · · < r

(depending only on (W, S)) such that (setting r

= 0 and r

= +∞), if θ and θ

are two positive rational numbers, then:

(a) If 0 6 i 6 m and r

< θ, θ

< r

, then L

(W ) = L

(W ).

(b) If 1 6 i 6 m and r

< θ < r

< θ

< r

, then L

i

(W ) is the finest partition of W which is less fine than L

(W ) and less fine than L

(W ).

Remarks - (1) One can obviously state similar conjectures for the partitions into right or two-sided cells.

(2) In the case where W is finite, the existence of rational numbers 0 < r

<

· · · < r

satisfying (a) is easy (see Proposition 5.7). However, even in this case, the statement (b) is still a conjecture.

Date: March 24, 2010.

1991Mathematics Subject Classification. According to the 2000 classification: Primary 20C08;

Secondary 20C15.

The author is partly supported by the ANR (Project No JC07-192339).

1

The aim of this paper is to provide a theoretical background that allows us to generalize this conjecture in the following two directions:

• It is of course possible to work with a partition of S into more than two subsets (this does not occur if W is finite and irreducible).

• One can also be interested in weight functions with values in any totally ordered abelian group, and whose values on simple reflections are not necessarily positive.

These two generalisations lead us to define equivalence classes of weight functions (in the easy case detailed in this introduction, this has been done by working with the ratio b/a instead of the pair (a, b)) and to use a topology on the set of such equivalence classes. This is done by using the notion of positive subsets of an abelian group (as defined in [4, §1]): we shall review some of the results of [4] in an Appendix.

This paper is organized as follows. After two sections devoted to recollections of well-known facts about Hecke algebras and Kazhdan-Lusztig theory with unequal parameters, we shall state our main conjecture in the third one: it is only concerned with the partition into cells and will be given into different forms (see Conjectures A, A’ and A”). In the fourth section, we shall also state a conjecture saying, roughly speaking, that Conjecture A is compatible with the construction of cell representa- tions (see Conjecture B). In the last section, we illustrate these conjectures by giving a detailed account of the following examples:

Examples - (1) Dihedral groups: If |S| = 2 and if |S

| = |S

| = 1, then the Conjecture 0 holds by taking m = 1 and r

= 1 (see [18, §8.8]).

(2) Type F

: If (W, S) is of type F

and if |S

| = |S

| = 2, then the Conjecture 0 holds by taking m = 3 and r

= 1/2, r

= 1 et r

= 2 (see [11, Corollaire 4.8]).

(3) Type B

: Assume that (W, S) is of type B

(with n > 2) and that |S

| = n − 1 and |S

| = 1. In [5, Conjectures A and B], the Conjecture 0 is made more precise: it should be sufficient to take m = n − 1 and r

= i. This has been checked for n 6 6.

(4) Type G ˜

: Assume that (W, S) is the affine Weyl group of type ˜ G

and that S

and S

are chosen in such a way that |S

| = 2 and |S

| = 1. Then Guilhot [15] has shown that the Conjecture 0 holds by taking m = 3 and r

/1, r

= 3/2 and r

= 2.

(5) Type B ˜

: Guilhot [15] has also shown that Conjecture 0 holds if (W, S) is an affine Weyl group of type ˜ B

. In this case, there are several possibilities for the choice of the partition S = S

∪ ˙ S

, but Guilhot has proved that Conjecture 0 holds for all possible choices.

Acknowledgements. Part of this work was done while the author stayed at the

MSRI during the winter 2008. The author wishes to thank the Institute for its

hospitality and the organizers of the two programs for their invitation. The author

also thanks M. Geck, L. Iancu and J. Guilhot for many fruitful discussions.

Contents

1. Hecke algebras 3

1.A. Preliminaries 3

1.B. Hecke algebras with unequal parameters 4

1.C. Generic Hecke algebra 5

2. Kazhdan-Lusztig cells 5

2.A. Kazhdan-Lusztig basis 5

2.B. Cell representations 6

2.C. Strictly increasing morphisms 7

2.D. Changing signs 7

2.E. Zero values of ϕ 8

3. Conjectures about cells 11

3.A. Positive subsets of Z[ ¯ S ] 11

3.B. Conjectures 13

3.C. Another form of Conjecture A 14

3.D. Essential hyperplanes 15

3.E. Example: the case where | S ¯ | = 2 16

4. Conjectures about cell representations 19

5. Examples 20

5.A. Finite dihedral groups 20

5.B. Infinite dihedral group 22

5.C. Type F

22

5.D. Type B 23

5.E. Type G e

23

5.F. Type B e

24

5.G. General results about finite Coxeter groups 24 Appendix A. Positive subsets of an abelian group 25

References 30

1. Hecke algebras

1.A. Preliminaries. Let (W, S) be a Coxeter system (S and W can be infinite). If s, t ∈ S, we shall write s ∼ t if they are conjugate in W . Let ¯ S = S/∼. If s ∈ S, we denote by ¯ s its class in ¯ S. Let Z [ ¯ S] the free Z -module with basis ¯ S.

Let ℓ : W → N be the length function associated with S. If ω ∈ S ¯ and if

w ∈ W , we denote by ℓ

(w) the number of occurrences of elements of ω in a

reduced expression of w (it is well-known that it does not depend on the choice of

the reduced expression). We set

ℓ : W −→ Z[ ¯ S]

w 7−→ X

ω∈S¯

ℓ

(w)ω.

If Γ is an abelian group, a map ϕ : W → Γ is called a weight function if ϕ(ww

) = ϕ(w) + ϕ(w

) for all w, w

∈ W such that ℓ(ww

) = ℓ(w) + ℓ(w

) (see [18, §3.1]).

The maps ℓ

(viewed as functions with values in Z) and ℓ are weight functions. In fact, the map ℓ is universal in the following sense:

Lemma 1.1. Let ϕ : W → Γ be a weight function. Then there exists a unique morphism of groups ϕ ¯ : Z[ ¯ S] → Γ such that ϕ = ¯ ϕ ◦ ℓ.

Proof. Clear.

Let Maps( ¯ S, Γ) be the set of maps ¯ S → Γ and let Weights(W, Γ) be the set of weight functions W → Γ. The lemma 1.1 shows that there are canonical bijections

Weights(W, Γ) ←→

Maps( ¯ S, Γ) ←→

Hom(Z[ ¯ S], Γ).

We shall identify these three sets all along this paper. More precisely, we shall work with maps ϕ : ¯ S → Γ that we shall see, according to our needs, as morphisms of groups Z [ ¯ S] → Γ or as weight functions W → Γ. In particular, we can talk as well about ϕ(w) (for w ∈ W ) as about ϕ(λ) (for λ ∈ Z[ ¯ S]): we hope that it will not lead to some confusion. For instance, Ker ϕ is a subgroup of Z[ ¯ S] (and not of W !).

1.B. Hecke algebras with unequal parameters. Let us first fix the notation that will be used throughout this paper.

Notation. Let Γ be an abelian group and let ϕ : ¯ S → Γ be a map.

We shall use an exponential notation for the group algebra of Γ: Z [Γ] = ⊕

γ∈Γ

Z e

, where e

· e

= e

for all γ, γ

∈ Γ. We then denote by H (W, S, ϕ) the Hecke algebra with parameter ϕ that is, the free Z[Γ]-module with basis (T

)

endowed with the unique Z[Γ]-bilinear associative multiplication completely determined by the following rules:

( T

T

= T

if ℓ(ww

) = ℓ(w) + ℓ(w

), (T

− e

)(T

+ e

) = 0 if s ∈ S.

If necessary, we denote by T

the element T

to emphasize that it lives in the Hecke algebra with parameter ϕ.

This algebra is endowed with several involutions. We shall use only the following:

if γ ∈ Γ and w ∈ W , we set e

= e

and T

= T

(note that T

is invertible).

This extends by Z-linearity to a Z[Γ]-antilinear map H (W, S, ϕ) → H (W, S, ϕ), h 7→ h which is an involutive antilinear automorphism of ring.

The previous construction is functorial. If ρ : Γ → Γ

is a morphism of abelian groups, then ρ induces a map

ρ

: H (W, S, ϕ) −→ H (W, S, ρ ◦ ϕ)

defined as the unique Z[Γ]-linear map sending T

on T

(here, H (W, S, ρ ◦ ϕ) is viewed as a Z[Γ]-algebra through the morphism of rings Z[Γ] → Z[Γ

] induced by ρ). It is then easily checked that

(1.2) ρ

is a morphism of Z[Γ]-algebras.

If h ∈ H (W, S, ϕ), then

(1.3) ρ

(h) = ρ

(h).

Moreover, if σ : Γ

→ Γ

is another morphism of abelian groups, then (1.4) (σ ◦ ρ)

= σ

◦ ρ

.

The next lemma is an obvious consequence:

Lemma 1.5. The morphism of groups ρ is injective (respectively surjective, respec- tively bijective) if and only if the morphism of algebras ρ

is.

1.C. Generic Hecke algebra. Let R be the group algebra Z[Z[ ¯ S]]. Let i : ¯ S → Z[ ¯ S] be the canonical map. The Hecke algebra H (W, S, i) will then be denoted by H (W, S): it is called the generic Hecke algebra.

It is universal in the following sense: if we identify the map ϕ : ¯ S → Γ with the morphism of groups ϕ : Z [ ¯ S] → Γ, then the Hecke algebra H (W, S, ϕ) is now equipped with a canonical morphism of R-algebras ϕ

: H (W, S) → H (W, S, ϕ), so that H (W, S, ϕ) is a specialization of H (W, S), i.e.

H (W, S, ϕ) = Z[Γ] ⊗

H (W, S).

2. Kazhdan-Lusztig cells

2.A. Kazhdan-Lusztig basis. To define the Kazhdan-Lusztig basis, we shall need to work all along this paper under the following hypothesis:

Hypothesis and notation. Until the end of this paper, we assume that Γ is endowed with a total order 6 compatible with the group law.

We denote respectively by Γ

, Γ

, Γ

and Γ

the set of positive,

non-negative, negative, non-positive elements of Γ.

If E is any subset of Γ, we set Z[E ] = ⊕

γ∈E

Ze

. For instance, Z[Γ

] is a subring of Z[Γ] and Z[Γ

] is an ideal of Z[Γ

]. Let

H

(W, S, ϕ) = ⊕

w∈W

Z[Γ

] T

.

Then, if w ∈ W , there exists [18, Theorem 5.2] a unique element C

∈ H (W, S, ϕ)

such that (

C

= C

,

C

≡ T

mod H

(W, S, ϕ).

Again, if necessary, the element C

will be denoted by C

.

The family (C

)

is a Z[Γ]-basis of H (W, S, ϕ) (called the Kazhdan-Lusztig basis [18, Theorem 5.2]). If x, y ∈ W

, then we shall write x ←−

y (respectively x ←−

y, respectively x

←− y) if there exists h ∈ H

such that the coefficient of C

in the decomposition of hC

(respectively C

h, respectively hC

or C

h) is non-zero.

We denote by 6

(respectively 6

, respectively 6

) the transitive closure of ←−

(respectively ←−, respectively

←−). Then 6

, 6

and 6

are preorders on W and we denote respectively by ∼

, ∼

and ∼

the associated equivalence relations [18, Chapter 8]. An equivalence class for ∼

(respectively ∼

, respectively ∼

) is called a left (respectively right, respectively two-sided) cell (for (W, S, ϕ)). We recall the following result [18, §8.1]: if x, y ∈ W

, then

(2.1) x ∼

y ⇐⇒ x

∼

y

. If w ∈ W , and if ? ∈ {L, R, LR}, we set

Cell

(w) = {x ∈ W | x ∼

w}.

2.B. Cell representations. Let ? ∈ {L, R, LR}. For simplification, we define a

?-ideal to be a left ideal if ? = L, a right ideal if ? = R and a two-sided ideal if

? = LR. Similarly, if A is a ring, a ?-module (or an A-?-module) is a left A-module if ? = L, a right A-module if ? = R and an (A, A)-bimodule if ? = LR.

If C is a cell in W for ∼

, we set, following [18, §8.3], H (W, S, ϕ)

= ⊕

w6^ϕ_?C

Z[Γ] C

, H (W, S, ϕ)

?C

= ⊕

w<^ϕ_?C

Z[Γ] C

and M

= H (W, S, ϕ)

/ H (W, S, ϕ)

?C

. Then, by definition, H (W, S, ϕ)

and H (W, S, ϕ)

?C

are ?-ideals and M

is an H (W, S, ϕ)-?-module. Note that M

is a free Z [Γ]-module with basis the image of (C

)

.

Let aug : Z[Γ] → Z be the augmentation morphism and view Z (or Q) as a Z[Γ]-

algebra through aug. We have an isomorphism of rings Z ⊗

H (W, S, ϕ) ≃ ZW .

Let ZM

(respectively QM

) be the ZW -?-module (respectively QW -?-module) Z ⊗

M

(respectively Q ⊗

M

).

If W is finite, we denote by χ

the character of QM

(if ? = LR, it is a character of W × W ).

2.C. Strictly increasing morphisms. The main goal of this paper is to study how the partition into cells behaves whenever the datum (Γ, 6 , ϕ) varies. The first easy remark is that this partition does not change by composition with a strictly increasing morphism of groups:

Proposition 2.2. Let Γ

be a totally ordered abelian group, let ρ : Γ → Γ

be a strictly increasing morphism of groups and let ? ∈ {L, R, LR}. Then:

(a) ρ

is injective.

(b) If w ∈ W , then ρ

(C

) = C

.

(c) The relations 6

et 6

coincide (as well as the relations ∼

et ∼

).

(d) If w ∈ W , then Cell

(w) = Cell

(w).

Proof. The injectivity of ρ

follows from the fact that ρ itself is injective (see Lemma 1.5). Hence (a) holds. Let us show (b). Let C = ρ

(C

). Then, by 1.3, we get C = C. The fact that ρ is strictly increasing implies that ρ

(Z[Γ

]) ⊆ Z[Γ

], so that

(2.3) ρ

( H

(W, S, ϕ)) ⊆ H

(W, S, ρ ◦ ϕ).

Therefore, C − T

∈ H

(W, S, ρ ◦ ϕ). So C = C

by the characterization of the elements of the Kazhdan-Lusztig basis.

Now (c) follows immediately from (a) and (b), while (d) is an immediate conse-

quence of (c).

2.D. Changing signs. We shall show in this subsection that changing the sign of some values of the function ϕ has no effect on the partition into cells (and that it has little effect on the cell representations). For this, we shall work under the following hypothesis:

Hypothesis and notation. In this subsection, and only in this sub- section, we fix a partition S = S

∪ ˙ S

of S such that no element of S

is conjugate to an element of S

. Let ϕ

: ¯ S → Γ be the (well- defined) map such that

ϕ

(¯ s) =

( ϕ(¯ s) if s ∈ S

,

−ϕ(¯ s) if s ∈ S

.

If w ∈ W , we set

ℓ

(w) = X

ω∈S¯±

ℓ

(w),

so that ℓ(w) = ℓ

(w) + ℓ

(w). Let θ : H (W, S, ϕ) −→ H (W, S, ϕ

) be the unique Z[Γ]-linear map such that

θ(T

) = (−1)

T

. An elementary computation shows that:

Proposition 2.4. The map θ : H (W, S, ϕ) −→ H (W, S, ϕ

) is an isomorphism of Z[Γ]-algebras. It sends H

(W, S, ϕ) onto H

(W, S, ϕ

). If h ∈ H (W, S, ϕ), then θ(h) = θ(h). Consequently, if w ∈ W , then

θ(C

) = (−1)

C

.

Corollary 2.5. If ? ∈ {L, R, LR}, then the relations 6

and 6

coincide. Simi- larly, the relations ∼

and ∼

coincide.

Let γ : W → {1, −1} denote the unique linear character of W such that γ(s) =

( 1 if s ∈ S

,

−1 if s ∈ S

.

In other words, γ(w) = (−1)

. Let Z

(respectively Z

) denote the (left or right) ZW -module (respectively the (ZW, ZW )-bimodule) of Z-rank 1 on which W (respectively W × W ) acts through the linear character γ (respectively the linear character γ ⊠ γ : W × W → {1, −1}, (w, w

) 7→ γ(w)γ(w

)). If C is a cell for ∼

, then an easy computation using the isomorphism θ shows that

(2.6) ZM

≃

( Z

⊗

ZM

if ? ∈ {L, R}, Z

⊗

ZM

if ? = LR.

2.E. Zero values of ϕ . It follows from Corollary 2.5 that the computation of Kazhdan-Lusztig cells can be reduced to the case where ϕ has non-negative values.

We shall study here what happens when ϕ vanishes on some elements of S. We shall work under the following hypothesis:

Hypothesis and notation. In this subsection, and only in this sub- section, we fix a partition S = I ∪ ˙ J of S such that no element of I is conjugate to an element of J . Let W

denote the subgroup of W generated by I and we set

J e = {wtw

| w ∈ W

and t ∈ J }.

Let W f be the subgroup of W generated by J. We shall also assume e that

(∗) if s ∈ I, then ϕ(s) = 0.

If w ∈ W , let ℓ

(w) and ℓ

(w) denote respectively the number of occurrences of elements of I and J in a reduced expression of w.

By a result of Gal [17, Proposition 2.1] (see also [3, Theorem 1.1]), ( W , f J e ) is a Coxeter system and

(2.7) W = W

⋉ f W

We shall denote by ˜ ℓ : W f → N the length function on f W with respect to J. In fact e [3, Corollary 1.3], if w ∈ W f , then

(2.8) ℓ(w) = ˜ ℓ

(w).

If ˜ t ∈ J, we denote by e ν(˜ t) the unique element of J such that ˜ t is W

-conjugate to ν(˜ t) (see [3, (1.3)]). We set

˜

ϕ(˜ t) = ϕ(ν(˜ t)).

By [3, Corollary 1.4] that, if ˜ t and ˜ t

are two elements of J e which are conjugate in f W , then

(2.9) ϕ(˜ ˜ t) = ˜ ϕ(˜ t

).

This shows that we can define a Hecke Z[Γ]-algebra H (f W , J, e ϕ). The group ˜ W

acts on W f and stabilizes J e and the map ˜ ϕ, so it acts naturally on the Hecke algebra H (f W , J, e ϕ). We can then define the semidirect product of algebras ˜

W

⋉ H ( f W , J, e ϕ). ˜ It is a Z[Γ]-algebra with Z[Γ]-basis (x · T

)

I,w∈fW

. Finally, let f

H = ⊕

w∈Wf

Z[Γ] T

⊆ H (W, S, ϕ).

Proposition 2.10. Recall that ϕ(s) = 0 if s ∈ I . Then:

(a) H f is a sub-algebra of H (W, S, ϕ).

(b) The unique Z[Γ]-linear map θ : W

⋉ H ( W , f J, e ϕ) ˜ −→ H (W, S, ϕ) that sends x · T

on T

(x ∈ W

, w ∈ f W ) is an isomorphism of Z[Γ]-algebras. It sends H ( W , f J, e ϕ) ˜ isomorphically onto H f .

(c) If x ∈ W

and w ∈ W , then C

= T

, C

= T

C

, C

= C

T

. If moreover w ∈ W f , then θ(C

) = C

.

Proof. Since ϕ(I) = {0}, we have, for all s ∈ I,

(T

)

= 1 and T

= T

.

So it follows by an easy induction argument that:

(2.11) If ℓ

(xy ) = ℓ

(x) + ℓ

(y), then T

T

= T

.

In particular, if x ∈ W

and w ∈ W , then T

T

= T

et T

T

= T

. We deduce that C

= T

, C

= T

C

, C

= C

T

. This proves the first assertion of (c).

Consequently, if x ∈ W

and t ∈ J , then

T

= T

T

(T

)

and so, if we set ˜ t = xtx

, we have

(T

− e

)(T

+ e

) = 0.

To show (a) and (b), it only remains to show the following: if w and w

are two elements of W f such that ˜ ℓ(ww

) = ˜ ℓ(w) + ˜ ℓ(w

) (here, ˜ ℓ denotes the length function on W f associated to J e ), then

T

T

= T

. But this follows from 2.8 and 2.11.

In order to show the last assertion of (c), it is sufficient to notice that, if h ∈ H (f W , J, e ϕ), then ˜ θ(h) = θ(h), and to use the characterization of the elements of

the Kazhdan-Lusztig basis.

Corollary 2.12. Let a, b ∈ W

, x, y ∈ f W . Then:

(a) ax 6

by if and only if x 6

y.

(b) xa 6

yb if and only if x 6

y.

(c) ax 6

yb if and only if there exists c ∈ W

such that x 6

cyc

.

Proof. (b) and (c) follow easily from (a). So let us prove (a). It is sufficient to show that ax ←−

by if and only if x ←−

y.

First assume that ax ←−

by. Then there exists w ∈ W such that C

occurs in the decomposition of T

C

. Let us write w = cz, with c ∈ W

and z ∈ f W . Then

T

C

= T

T

C

.

This shows that cb = a and that C

occurs in the decomposition of T

C

. By applying the Proposition 2.10 (and the isomorphism θ), we get that x ←−

y.

Conversely, assume that x ←−

y. Then there exists w ∈ f W such that C

occurs in the decomposition of T

C

. By applying the Proposition 2.10 (and the isomorphism θ), we get that C

occurs in the decomposition of T

C

. So ax ←−

by.

We can immediately deduce the following corollary:

Corollary 2.13. Assume that ϕ(I) = {0}. Then the left (respectively right, re-

spectively two-sided) cells for (W, S, ϕ) are of the form W

· C (respectively C · W

,

respectively W

· C · W

), where C is a left (respectively right, respectively two-sided)

cell for ( W , f J, e ϕ). ˜

At the level of cell representations, we get:

Corollary 2.14. If C if a left cell for ( f W , J, e ϕ), then ˜ ZM

≃ Ind

ZM

.

Proof. This follows easily from the Proposition 2.10 (and its proof) and the Corol-

laries 2.12 and 2.13.

3. Conjectures about cells

From now on, and until the end of this paper, we assume that S is finite. We shall use the notion, notation and results of the Appendix:

positive subsets of Z[ ¯ S], topology on P os(Z[ ¯ S]), hyperplane arrange- ments, Pos(ϕ), U (λ), H

...

As explained before, the main aim of this paper is to study the behaviour of the relations ∼

as the datum (Γ, 6 , ϕ) varies. The first step is to show that the relations 6

(and ∼

) depend only on the positive subset Pos(ϕ) ∈ P os(Z[ ¯ S]) (see Corollary 3.2). This allows us to define relations 6

and ∼

for X ∈ P os(Z[ ¯ S]).

Our conjectures are then about the behaviour of the relations ∼

whenever X runs over P os(Z[ ¯ S]): this involves the topology of this set together with the notion of hyperplane arrangements (see the Appendix).

3.A. Positive subsets of Z[ ¯ S ]. Let X be a positive subset of Z[ ¯ S]. Let Γ

= Z[ ¯ S]/(X ∩ (−X)) and let 6

be the total order on Γ

defined in A.4. Let ϕ

: S ¯ → Γ

be the canonical map (it is the composition of the natural map ¯ S → Z[ ¯ S]

with the canonical map can

: Z [ ¯ S ] → Γ

of the Appendix). For simplification, the relations 6

et ∼

will be denoted by 6

and ∼

. Similarly, if w ∈ W , we will denote by Cell

(w) the subset Cell

(w) and, if C is a ?-cell for (W, S, ϕ

), we denote by ZM

the ZW -module ZM

(and, if W is finite, we denote by χ

the character χ

). The next proposition and its corollary show that the family

(Γ

, 6

, ϕ

)

X∈Pos(Z[ ¯S])

is essentially exhaustive.

Proposition 3.1. Let X = Pos(ϕ). Then there exists a unique morphism of groups

¯

ϕ : Γ

→ Γ such that ϕ = ¯ ϕ ◦ ϕ

. This morphism ϕ ¯ is strictly increasing.

Proof. Indeed, Ker ϕ = X ∩ (−X) so the existence, the unicity and the injectivity of ¯ ϕ are clear. On the other hand, if γ, γ

∈ Γ

are such that γ 6 γ

and if λ ∈ Z [ ¯ S]

is a representative of γ

− γ, then λ ∈ X by A.3. So ϕ(λ) > 0. In other words

¯

ϕ(γ

− γ) > 0 that is, ¯ ϕ(γ) 6 ϕ(γ ¯

). So ¯ ϕ is increasing: the fact that it is strictly

increasing follows from the injectivity.

Corollary 3.2. Let X = Pos(ϕ) and let ? ∈ {L, R, LR}. Then:

(a) The relations 6

and 6

are equal (as well as the relations ∼

and ∼

).

(b) If w ∈ W , then Cell

(w) = Cell

(w).

(c) If C = Cell

(w), then Z M

≃ Z M

.

Thanks to Corollary 3.2, studying the behaviour of the relations ∼

as the weight function ϕ varies is equivalent to studying the behaviour of relations ∼

as X runs over P os(Z[ ¯ S]). The following remark can be useful for switching from one point of view to the other:

Remark 3.3 - Let λ ∈ Z [ ¯ S] and X ∈ P os( Z [ ¯ S]). Then the following hold:

(a) X ∈ U (λ) if and only if ϕ

(λ) <

0.

(b) X ∈ H

if and only if ϕ

(λ) = 0.

(c) X ∈ U (λ) if and only if ϕ

(λ) 6

0.

In the same spirit, the Corollary 2.5 can be translated into the Proposition 3.4 below. We first need some notation. If ω ∈ S, we denote by ¯ τ

the Z-linear symmetry on Z [ ¯ S] such that τ

(ω) = −ω and τ

(ω

) = ω

if ω

6= ω. It is an automorphism of Z[ ¯ S]: it induces an homeomorphism τ

of P os(Z[ ¯ S]).

Proposition 3.4. If ω ∈ S, if ¯ X ∈ P os( Z [ ¯ S]) and if ? ∈ {L, R, LR}, then the relations ∼

and ∼

coincide.

Proof. The map τ

induces a strictly increasing isomorphism τ

: Γ

−→

Γ

.

Therefore, the relations ∼

and ∼

coincide (see the Proposition 2.2).

Now, let ϕ

= τ

◦ ϕ

: ¯ S → Γ

. But ϕ

(ω

) =

( ϕ

(ω

) if ω 6= ω

,

−ϕ

(ω) if ω = ω

.

So the result follows from Corollary 2.5.

3.B. Conjectures. A rational hyperplane arrangement A in P os(Z[ ¯ S]) is called complete if H

∈ A for all ω ∈ S. Recall that ¯

H

= {X ∈ P os(Z[ ¯ S]) | ω ∈ X ∩ −X}

= {X ∈ P os(Z[ ¯ S]) | ϕ

(ω) = 0}.

If A is a complete arrangement and if F is an A -facet, we denote by W

the standard parabolic subgroup generated by S

ω∈S,¯F⊆H_ω

ω. As in the Appendix, we denote by Fac( A ) and Cham( A ) the set of A -facets and A -chambers respectively.

We denote by R el(W ) the set of equivalence relations on W . If R and R

are two equivalence relations on W , we write R 4 R

if R is finer than R

(i.e. if x R y implies x R

y). The poset ( R el(W ), 4 ) is a lattice (i.e. every pair of two elements has a supremum and an infimum): sup( R , R

) is the equivalence relation generated by R and R

while inf( R , R

) is the equivalence relation R

defined by x R

y if and only if x R y and x R

y (for x, y ∈ W ). Finally, if ? ∈ {L, R, LR} and if H is a subgroup of W , we denote by trans

the equivalence relation on W defined by the action of H (or H × H) by translation on W :

x trans

y ⇔

 



 

∃ w ∈ H, y = wx, if ? = L,

∃ w ∈ H, y = xw, if ? = R,

∃ w, w

∈ H, y = wxw

, if ? = LR.

Conjecture A. Assume that S is finite. Then there exists a finite complete rational hyperplane arrangement A in P os( Z [ ¯ S]) satisfying the following properties (for all

? ∈ {L, R, LR}):

(a) If X and Y are two positive subsets of Z[ ¯ S ] belonging to the same A -facet F , then the relations ∼

and ∼

coincide (we will denote it by ∼

).

(b) Let F be an A -facet. Then the cells for the relation ∼

are the minimal subsets C of W satisfying the following conditions:

(b1) For every chamber C such that F ⊆ C , C is a union of cells for ∼

; (b2) C is stable by translation by W

(on the left if ? = L, on the right if

? = R, on the left and on the right if ? = LR).

Remark 3.5 - With the notation of the above Conjecture, the statement (b) is equivalent to the following one:

(b

) Let F be an A -facet. Then

∼

= sup

trans

, sup

C∈Cham(A) F⊆C

∼

.

Here, the suprema are computed in the set R el(W ).

Remark 3.6 - The statement (b2) in the Conjecture A comes from Corollary 2.13.

It is necessary, as it can already be seen in the case where |S| = 1.

The previous remark (together with the Corollary A.16 of the Appendix) implies immediately the following proposition (which “justifies” the title of the paper):

Proposition 3.7. If the Conjecture A holds for (W, S), then the map P os( Z [ ¯ S]) −→ R el(W )

X 7−→ ∼

is upper semicontinuous (for all ? ∈ {L, R, LR}).

Corollary 3.8. Let ? ∈ {L, R, LR}, let w ∈ W and let P (W ) denote the power set of W . If the Conjecture A holds for (W, S), then the map

P os( Z [ ¯ S]) −→ P (W ) X 7−→ Cell

(w) is upper semicontinuous.

3.C. Another form of Conjecture A. Let us now give a translation of this con- jecture in terms of maps ϕ : ¯ S → Γ. For this, let SGN = {+, −, 0} and let E be a finite set of elements in Z[ ¯ S] \ {0}. Let SGN

denote the set of maps E → SGN. If X ∈ P os(Z[ ¯ S]), we set:

sgn

(X) : E −→ SGN

λ 7−→

 



 

+ if λ 6∈ −X, 0 if λ ∈ X ∩ (−X),

− if λ 6∈ X.

This defines a map

sgn

: P os(Z[ ¯ S]) −→ SGN . Similarly, we set

sgn

(ϕ) : E −→ SGN

λ 7−→

 



 

+ if ϕ(λ) > 0, 0 if ϕ(λ) = 0,

− if ϕ(λ) < 0.

In other words,

(3 . 9) sgn

(X) = sgn

(ϕ

) and sgn

(ϕ) = sgn

(Pos(ϕ)).

Let A

be the finite rational hyperplane arrangement { H

| λ ∈ E }. By the very definition of facets, the A

-facets are exactly the non-empty fibers of the map sgn

: P os( Z [ ¯ S]) −→ SGN

. If F is an A

-facet, then we denote by sgn

( F ) the element sgn

(X) ∈ SGN

, where X is some (or any) element of F .

We now endow SGN with the unique partial order 4 such that 0 4 +, 0 4 − and

+ and − are not comparable. This defines a partial order on SGN

(componentwise)

which is still denoted by 4 . Then, by Proposition A.14, we have, for all facets F and F

,

(3.10) F ⊆ F

if and only if sgn

( F ) 4 sgn

( F

).

(Note that this implies, by Corollary A.16, that the map sgn

is lower semicontinu- ous.) Finally, we say that the map ϕ is E -open if Pos(ϕ) belongs to an A

-chamber:

equivalently, ϕ is E -open if sgn

(ϕ)(λ) 6= 0 for all λ ∈ E \ {0}.

By the above discussion, we get that the next Conjecture is clearly equivalent to Conjecture A (here, we denote by W

the parabolic subgroup generated by {s ∈ S | ϕ(s) = 0}).

Conjecture A’. Assume that S is finite. Then there exists a finite set E in Z[ ¯ S]\{0}

containing S ¯ and such that, for all ? ∈ {L, R, LR}, we have:

(a) If Γ and Γ

are two abelian ordered groups and if ϕ : ¯ S → Γ and ϕ

: ¯ S → Γ

are two maps such that sgn

(ϕ) = sgn

(ϕ

), then the relations ∼

and ∼

coincide.

(b) If Γ is a totally ordered abelian group and if ϕ : ¯ S → Γ is a map, then the cells for ∼

are the minimal subsets C of W such that:

(b1) For all totally ordered abelian group Γ

and for all E -open maps ϕ

: S ¯ → Γ

such that sgn

(ϕ) 4 sgn

(ϕ

), C is a union of cells for ∼

. (b2) C is stable by translation by W

(on the left if ? = L, on the right if

? = R, on the left and on the right if ? = LR).

An element λ ∈ Z[ ¯ S] is called reduced if λ 6= 0 and Z[ ¯ S]/Zλ is torsion-free. If H is a rational hyperplane, then there exist only two reduced elements λ ∈ Z[ ¯ S]

such that H = H

(one is the opposite of the other). A subset E of Z[ ¯ S] is called reduced if all its elements are reduced. It is called complete if ¯ S ⊆ E . It is called symmetric if E = − E . If A is a rational hyperplane arrangement, then there exists a unique reduced symmetric subset E of Z[ ¯ S] such that A = A

. In this case, A is complete if and only if E is.

3.D. Essential hyperplanes. The Remark 3.5 implies easily the following

Proposition 3.11. If the Conjecture A holds for (W, S) and two finite complete rational hyperplane arrangements A and A

of P os(Z[ ¯ S]), then it holds for (W, S) and for the finite complete arrangement A ∩ A

.

Proof. Clear.

The Proposition 3.11 shows that, if the Conjecture A holds for (W, S), there

exists a unique minimal finite complete rational hyperplane arrangement A such

that the statements (a) and (b) of Conjecture A hold. We call the elements of this

minimal arrangement the essential hyperplanes of (W, S): indeed, if (W, S) is finite,

they should be the same as the essential hyperplanes defined by M. Chlouveraki [9,

§4.3.1], which appear when she studied the Rouquier blocks of cyclotomic Hecke algebras associated to complex reflection groups.

Similarly, the Proposition 3.11 shows that, if the Conjecture A’ holds for (W, S), there exists a unique minimal finite complete symmetric reduced subset E of Z[ ¯ S]

such that the statements (a) and (b) of Conjecture A’ holds: they will be called the essential elements of (W, S).

Remark 3.12 - The Proposition 3.4 shows that, if the Conjecture A holds for (W, S), then the set of essential hyperplanes (respectively the set of essential ele- ments) of (W, S) is stable by the action of all the symmetries τ

(respectively τ

), ω ∈ S. ¯

Remark 3.13 - Let I be a subset of S. If s, t ∈ I, we shall write s ∼

t if s and t are conjugate in W

= hIi. We set ¯ I = I/∼

. Note that ¯ I is not necessarily the image of I in ¯ S. Nevertheless, the inclusion I ֒ → S induces a map ¯ I → S, ¯ which extends by linearity to a map γ

: Z[ ¯ I] → Z[ ¯ S]. By functoriality (see the Appendix), this induces a map γ

: P os(Z[ ¯ S]) → P os(Z[ ¯ S]). Assume here that Conjecture A holds for (W, S) and (W

, I): let A (respectively A

) denote the set of essential hyperplanes for (W, S) (respectively (W

, I)). Then, since any left cell of W

is the intersection of a left cell of W with W

(see [10]), we get that (γ

)

( A

) is contained in A ∪ { P os(Z[ ¯ S])}.

Equivalently, if E and E

denote the sets of essential elements of (W, S) and (W

, I) respectively, then γ

( E

) ⊆ E ∪ {0}.

Example 3.14 - If | S| ¯ = 1, then Conjecture A is obviously true: the set of essential hyperplanes is { H

} (note that ¯ S = {S}) and the set of essential elements is {S, −S} ⊆ Z[ ¯ S]. Indeed, if ϕ : ¯ S → Γ and ϕ

: ¯ S → Γ

are two maps such that ϕ(S) > 0 and ϕ

(S) > 0, then the relations ∼

and ∼

coincide. On the other hand, if ϕ(S) = 0, then W contains only one cell for ∼

, namely W itself: it is clearly the smallest subset of W which is stable by translation by W

= W .

3.E. Example: the case where | S| ¯ = 2 . Since the Conjecture A is expressed in terms of the topology of the set P os(Z[ ¯ S]), it might be difficult to understand it concretely. The purpose of this example is to give a concrete version of this statement whenever | S| ¯ = 2: this will show that Conjecture A contains the Conjecture 0 of the introduction, and gives some precision for the case where ϕ vanishes at some simple reflections.

So assume here that | S| ¯ = 2 and write ¯ S = {ω

, ω

}. We shall identify Z [ ¯ S] with Z

(through (λ, µ) 7→ λω

+ µω

). If r is a rational number, we shall denote by H

the hyperplane H

, where d is a non-zero natural number such that dr ∈ Z.

Then H

= {Z[ ¯ S], X

, X

}, where

X

= {(λ, µ) ∈ Z

| λ + rµ > 0} and X

= {(λ, µ) ∈ Z

| λ + rµ 6 0}.

We set H

= H

. Then H

= {Z[ ¯ S], X

, X

}, where

X

= {(λ, µ) ∈ Z

| µ > 0} and X

= {(λ, µ) ∈ Z

| µ 6 0}.

Now, since Γ is torsion-free, the natural map Γ → Q ⊗

Γ is injective, so we shall view Γ as embedded in the Q-vector space Q ⊗

Γ: in particular, if r ∈ Q and γ ∈ Γ, then rγ is well-defined. Moreover, the order on Γ extends uniquely to a total order on Q ⊗

Γ that we still denote by 6 . Now, let

ϕ(ω

) = a and ϕ(ω

) = b.

Then, one can immediately translate in concrete terms the fact that Pos(ϕ) belongs or not to one of these hyperplanes:

Lemma 3.15. With the above notation, we have:

(a) Pos(ϕ) ∈ H

(respectively Pos(ϕ) ∈ H

) if and only if b = ra (respectively a = 0).

(b) Pos(ϕ) = X

(respectively X

) if and only if b = ra and a > 0 (respectively a < 0).

(c) Pos(ϕ) = X

(respectively X

) if and only if a = 0 and b > 0 (respectively b < 0).

For simplification, we set τ

= τ

. Now, let A be a finite complete rational hyperplane arrangement in P os( Z [ ¯ S]) which is stable under the actions of τ

and τ

. Since τ

◦ τ

= − Id

, this is equivalent to say that it is stable under τ

. Since all hyperplanes of P os(Z[ ¯ S]) are of the form H

for r ∈ Q ∪ {∞}, and since τ

( H

) = H

if r ∈ Q (and τ

( H

) = H

), there exist positive rational numbers 0 < r

< r

< · · · < r

such that

A = { H

, H

1

, H

1

, H

2

, H

2

, . . . , H

m

, H

m

, H

}.

Let us draw Pos

( A ) in R[ ¯ S]

= Rω

⊕ Rω

, where (ω

, ω

) is the dual basis of

(ω

, ω

):