Vogan classes and cells in the unequal parameter case

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Cédric Bonnafé

To cite this version:

Cédric Bonnafé. Vogan classes and cells in the unequal parameter case. 2014. �hal-00995159v4�

PARAMETER CASE

by

C ÉDRIC B ONNAFÉ

Abstract. — Kazhdan and Lusztig proved that Vogan classes are unions of cells in the equal parameter case. We extend this result to the unequal parameter case.

Let (W,S) be a Coxeter system and let ϕ : S → Z

be a weight function. To this da- tum is associated a partition of W into left, right and two-sided cells. Determining these partitions is a difficult problem, with deep connections (whenever W is a fi- nite or an affine Weyl group) with representations of reductive groups, singularities of Schubert cells, geometry of unipotent classes.

In their original paper, Kazhdan and Lusztig described completely this partition whenever W is the symmetric group in terms of the Robinson-Schensted correspon- dence. Their main tool is the so-called ∗ -operation. It is defined in any Coxeter group (whenever there exists s , t ∈ S such that s t has order 3 ): they proved that it provides some extra-properties of cells, whenever ϕ is constant. Our aim in this paper is to prove that Kazhdan-Lusztig result relating cells and the ∗ -operation holds in full generality (this result has also been proved independently and simultaneously by M. Geck [Ge2]).

Commentary. In [Bon], the author used improperly the ^∗ -operation in the unequal parameter context. The present paper justifies a posteriori what was, at that time, a big mistake!

Acknowledgement. I would like to thank M. Geck for pointing out some references and for his useful remarks.

Notation. We fix in this paper a Coxeter system ⁽ W, S ) and a totally ordered abelian group ^Γ . We use an exponential notation for the group algebra A = Z[Γ] :

A = ⊕

γ∈Γ

Z v

,

The author is partly supported by the ANR (Project No ANR-12-JS01-0003-01 ACORT).

with v

v

= v

for all γ , γ

∈ Γ . If γ

∈ Γ , we set Γ

= {γ ∈ Γ | γ ¾ γ

},

Γ

= { γ ∈ Γ | γ > γ

}, A

= ⊕

γ¾γ0

Zv

, A

= ⊕

γ>γ0

Z v

and similarly for ^Γ

, ^Γ

, A

and A

. We denote by : A → A the involutive automorphism such that v

= v

.

1. Preliminaries

Hypothesis and notation. In this section, and only in this section, we fix an A -module M and we assume that:

(P1) M admits an A -basis (m

)

, where X is a poset. We set M

= ⊕

A

m

.

(P2) M admits a semilinear involution : M → M . We set M

= {m ∈ M | m + m = 0}.

(P3) If x ∈ X , then m

≡ m

mod

y

⊕

Am

(P4) If x ∈ X , then the set ^{{y ∈} X | y ¶ x } is finite.

Proposition 1.1. — The ^Z -linear map

M

−→ M

m 7−→ m − m is an isomorphism.

Proof. — First, note that the corresponding result for the A -module A itself holds.

In other words,

(1.2) The map ^A

→ A

, ^{a 7→ a − a} is an isomorphism.

Indeed, if a ∈ A

, write a = P

γ∈Γ

r

v

, with r

∈ R . Now, if we set a

= P

γ>0

r

v

, then a = a

− a

. This shows the surjectivity, while the injectivity is trivial.

Now, let ^Λ : M

→ M

, m 7→ m − m . For X ⊂ X , we set M

= ⊕

A m

and

M

= ⊕

A

m

. Assume that, for all x ∈ X and all y ∈ X such that y ¶ x , then

y ∈ X . By hypothesis, ^M

is stabilized by the involution . Since X is the union

of such finite ^X (by hypothesis), it shows that we may, and we will, assume that

X is finite. Let us write X = {x

, x

, . . . , x

} in such a way that, if x

¶ x

, then i ¶ j

(this is always possible). For simplifying notation, we set m

= m

. Note that, by hypothesis,

(∗) m

∈ m

+

⊕

0¶j<i

A m

. In particular, m

= m

.

Now, let m ∈ M

be such that m = m and assume that m 6= 0 . Write m = P

i=0

a

m

, with r ¶ n , a

∈ A

and a

6= 0 . Then, by hypothesis, m ≡ a

m

mod

⊕

0¶j<i

Am

.

Since m = m , this forces a

= a

, which is impossible (because a

∈ A

and a

6= 0 ).

So ^Λ is injective.

Let us now show that ^Λ is surjective. So, let m ∈ M

, and assume that m 6= 0 (for otherwise there is nothing to prove). Write m = P

i=0

a

m

, with r ¶ n , a

∈ A and a

6= 0 . We shall prove by induction on r that there exists µ ∈ M such that m = µ− µ . If r = 0 , then the result follows from (1.2) and the fact that m

= m

. So assume that r > 0 . By hypothesis,

m + m ≡ (a

+ a

)m

mod M

,

where ^X

= {x

, x

, . . . , x

} . Since m + m = 0 , this forces a

∈ A

. So, by (1.2), there exists a ∈ A

such that a − a = a

. Now, let m

= m − a m

+ a m

. Then m

+ m

= 0

and m

∈ ⊕

A m

. So, by the induction hypothesis, there exists µ

∈ M

such

that m

= µ

− µ

. Now, set µ = a m

+ µ

. Then µ ∈ M

and m = µ − µ = Λ( µ ) , as desired.

Corollary 1.3. — Let m ∈ M . Then there exists a unique M ∈ M such that (

M = M ,

M ≡ m mod M

.

Proof. — Setting M = m + µ , the problem is equivalent to find µ ∈ M

such that m + µ = m + µ . This is equivalent to find µ ∈ M

such that µ − µ = m − m : since m − m ∈ M

, this problem admits a unique solution, thanks to Theorem 1.1.

The Corollary 1.3 can be applied to the A -module A itself. However, in this case, its proof becomes obvious: if a

= P

γ∈Γ

a

v

, then a = P

γ¶0

a

v

+ P

γ>0

a

v

is the unique element of A such that a = a and a ≡ a

mod A

.

Corollary 1.4. — Let X be a subset of X such that, if x ¶ y and y ∈ X , then x ∈ X . Let

M ∈ M be such that M = M and M ∈ M

+ M

. Then M ∈ M

.

Proof. — Let M

∈ M

be such that M ≡ M

mod M

. From the existence state- ment of Corollary 1.3 applied to M

, there exists M

∈ M

such that M

= M

and

M

≡ M

mod M

. The fact that M = M

∈ M

now follows from the uniquenes

statement of Corollary 1.3.

Corollary 1.5. — Let x ∈ X . Then there exists a unique element M

∈ M such that (

M

= M

,

M

≡ m

mod M

.

Moreover, M

∈ m

+ ⊕

A

m

and (M

)

is an A -basis of M .

Proof. — The existence and uniqueness of M

follow from Corollary 1.3. The state- ment about the base change follows by applying this existence and uniqueness to M

, where X

= {y ∈ X | y ¶ x } .

Finally, the fact that (M

)

is an A -basis of M follows from the fact that the base change from (m

)

to (M

)

is unitriangular.

Corollary 1.5 gathers in a single general statement the argument given by Lusztig [Lus1] for the construction of the Kazhdan-Lusztig basis of a Hecke algebra (which is different from the argument contained in the original paper by Kazhdan and Lusztig [KaLu]) and the construction, still due to Lusztig [Lus2, Theorem 3.2], of the canonical basis associated with quantum groups.

2. The main result

2.A. Kazhdan-Lusztig basis. — We fix in this paper a weight function ϕ : S → Γ

(i.e. ϕ ( s ) = ϕ ( t ) whenever s and t are conjugate in W ). We denote by ^H the Hecke algebra associated with ⁽ W,S, ϕ ) : as an A -module, ^H admits an A -basis ⁽ T

)

and the multiplication is completely determined by the following rules:

(

T

T

= T

if ℓ (w w

) = ℓ (w ) + ℓ (w

) , (T

− v

)(T

+ v

) = 0, if s ∈ S .

Here, ℓ : W → Z

denote the length function on W .

We denote by : H → H the involutive antilinear automorphism of H such that T

= T

.

The triple (H , (T

)

, ) satisfies the properties (P1), (P2), (P3) and (P4) of the pre- vious section. Therefore, if w ∈ W , there exists a unique element C

∈ H such that

C

≡ T

mod H

(see Corollary 1.5) and (C

)

is an A -basis of H (see Corollary 1.4), called the Kazhdan-Lusztig basis.

2.B. Cells. — In this context, we define the preorders ^¶

, ^¶

, ^¶

and the equiv- alence relations ∼

, ∼

and ∼

as in [KaLu] or [Lus3]. If C is a left cell (i.e. an equivalence class for ∼

) of W , we set

H

= ⊕

w¶LC

A C

, H

= ⊕

w<LC

A C

and M (C ) = H

/H

.

By the very definition of the preorder ^¶

, H

and H

are left ideals of H , so M (C ) inherits a structure of H -module. If w ∈ C , we denote by c

the image of C

in M (C ) : then (c

)

is an A -basis of M (C ) .

2.C. Parabolic subgroups. — If I ⊂ S , we set W

= 〈I 〉 : it is a standard parabolic subgroup of W and (W

, I ) is a Coxeter system. We also set

H

= ⊕

w∈WI

A T

.

It is a subalgebra of H , naturally isomorphic to the Hecke algera associated with (W

, I , ϕ

) , where ϕ

: I → Z

denotes the restriction of ϕ .

We denote by X

the set of elements x ∈ W which have minimal length in x W

: it is well-known that the map X

→ W /W

, x 7→ x W

is bijective and that

X

= {x ∈ W

| ∀ s ∈ I , ℓ (x s ) > ℓ (x )}

= {x ∈ W

| ∀ w ∈ W

, ℓ (x w ) = ℓ (x ) + ℓ (w )}.

As a consequence, the right H

-module H is free (hence flat) with basis (T

)

. This remark has the following consequence (in the next lemma, if E is a subset of H , then H E denotes the left ideal generated by E ):

Lemma 2.1. — If I and I

are left ideals of ^H

such that I _⊂ I

, then:

( a ) H I _{= ⊕}

I

T

I .

(b) The natural map H ⊗

I _{→ H} I is an isomorphism of H -modules.

(c) The natural map H ⊗

( I

_/ I _{) → H} I

_/H I is an isomorphism.

We will now recall results from Geck [Ge1] about the parabolic induction of cells.

First, it is clear that (C

)

is the Kazhdan-Lusztig basis of ^H

. We can then define

a preorder ^¶

and its associated equivalence class ^∼

on W

in the same way as ^¶

and ∼

are defined for W . If w ∈ W , then there exists a unique x ∈ X

and a unique w

∈ W

such that w = x w

: we then set

G

= T

C

.

Finally, if C is a left cell in W

, then we define the left H

-module M

(C ) similarly as M (C

) was defined for left cells C

of W .

Theorem 2.2 (Geck). — Let E be a subset of W

such that, if x ∈ E and if y ∈ W

is such that y ¶

x , then y ∈ E . Let I _{= ⊕}

_{A C}

. Then

H I _{= ⊕}

w∈XI·E

A G

= ⊕

w∈XI·E

A C

.

Moreover, the transition matrix between the A -basis (C

)

and the A -basis (G

)

is unitriangular (for the Bruhat order) and its non-diagonal entries belong to A

.

Corollary 2.3 (Geck). — We have:

(a) ¶

and ∼

are just the restriction of ^¶

and ∼

to W

(and so we will use only the notation ^¶

and ∼

).

(b) If C is a left cell in W

, then X

· C is a union of left cells of W .

3. Generalized ^∗ -operation

Hypothesis and notation. We fix in this section, and only in this section, a subset I of S , two left cells C

and C

of W

, and we assume that:

(V1) There exists a bijection σ : C

→ C

such that the ^A -linear map M

(C

) → M

(C

) , c

7→ c

is in fact H

-linear.

(V2) If {i , j } = {1, 2} , then {w ∈ W

| w ∈ C

and w <

C

} = ∅ . We set E

= {w ∈ W

| w <

C

or w <

C

} , E

= X

∪ ˙ C

for i ∈ {1, 2} and

H

= ⊕

w∈Ei

AC

for i ∈ {0, 1, 2} .

R

- If W is finite and if we assume that Lusztig’s Conjectures [Lus3, Conjec-

tures P1 to P15] hold for ⁽ W

, I , ϕ

) , then (V2) is a consequence of (V1). ^„

Note that H

is a left H

-module for i ∈ {0, 1, 2} . By (V2), H

/H

is a left H

- module isomorphic to M

(C

) (for i ∈ {1, 2} ) so it admits an A -basis (c

)

. By (V1), the map σ induces an isomorphism of H

-modules

(♣) H

/H

−→ H

/H

. Let σ

: X

· C

−→

X

· C

denote the bijection induced by σ (i.e. σ

( x w ) = xσ ( w ) if x ∈ X

and w ∈ C

). By Lemma 2.1 and Theorem 2.2,

(♦) H H

= ⊕

w∈XI·Ei

A G

= ⊕

w∈XI·Ei

A C

.

Now, if i ∈ {1, 2} and w ∈ X

· C

, we denote by g

(respectively ^c

) the image of G

(respectively C

) in the quotient ^{H H}

/H H

. By Lemma 2.1, the isomorphism (♣) induces an isomorphism of H -modules

σ

: H H

/H H

−→ H H

/H H

which is defined by

σ

(g

) = g

for all w ∈ X

· C

. The key result of this section if the following one:

Theorem 3.1. — If w ∈ X

· C

, then σ

( c

) = c

.

Proof. — Let i ∈ {1, 2} . For simplification, we set M [i ] = H H

/H H

. By (♦) , (g

)

and ^{( c}

)

are both A -bases of M [i ] . We set

M [i ]

= ⊕

w∈XI·Ci

A

g

.

By Geck’s Theorem,

(♥) c

≡ g

mod M [i ]

.

Moreover, the antilinear involution on H stabilizes H H

and H H

so it in- duces an antilinear involution, still denoted by , on M [i ] . It is also clear that the isomorphism σ

: M [1] → M [2] , g

7→ g

satisfies σ

(m ) = σ

(m ) and σ

(M [1]

) =

M [2]

. Therefore, it follows from ^(♥) that, if w ∈ X

· C

, then

( σ

( c

) = σ

( c

),

σ

( c

) ≡ g

mod M [2]

.

But it follows again from Geck’s Theorem that the datum (M [i ], ( g

)

, ) sat-

isfies the properties (P1), (P2), (P3) and (P4) of §1. So σ

( c

) = c

by Corol-

lary 1.5.

We can now state the main consequence of Theorem 3.1. We first need a notation:

if C

6= C

(which is the interesting case...), we extend σ

to an involution of the set W , by setting

σ

(w ) =







w if w 6∈ X

· C

∪ ˙ X

· C

, σ

(w ) if w ∈ X

· C

,

( σ

)

(w ) if w ∈ X

· C

. Note that σ

: W → W is an involution.

Theorem 3.2. — Let w , w

∈ W . Then w ∼

w

if and only if σ

(w ) ∼ σ

(w

) .

Proof. — First, let us write

C

C

= X

z∈W

h

C

, where h

∈ A .

Now, assume that w ∼

w

. According to Corollary 2.3(b), there exists a unique cell C in W

such that w , w

∈ X

· C . If C 6∈ {C

, C

} , then σ

(w ) = w and σ

(w

) = w

, so σ

(w ) ∼

σ

(w

) . So we may assume that C ∈ {C

,C

} . Since σ

is involutive, we may assume that C = C

. Therefore, w , w

∈ X

· C

.

By the definition of ^¶

and ∼

, there exists four sequences x

,. . . , x

, y

,. . . , y

, w

,. . . , w

, w

,. . . , w

such that:



 



 



w

= w, w

= w

, w

= w

, w

= w ,

∀ i ∈ {1, 2, . . . , m − 1}, h

6= 0,

∀ j ∈ {1, 2, . . . ,n − 1}, h

j,w_j+1^′

6= 0.

Therefore, w = w

¶

w

¶

· · · ¶

w

= w

= w

¶

w

¶

· · · ¶

w

= w and so w = w

∼

w

∼

· · · ∼

w

= w

= w

∼

w

∼

· · · ∼

w

= w . Again by Corollary 2.3(b), w

, w

∈ X

· C

. So it follows from Theorem 3.1 that h

= h

and h

= h

for all x ∈ W . Therefore,

(

∀ i ∈ {1, 2, . . . ,m − 1}, h

6= 0,

∀ j ∈ {1, 2, . . . , n − 1}, h

j),σ^L(w^′_j+1)

6= 0.

It then follows that

σ

( w ) = σ

( w

) ¶

σ

( w

) ¶

· · · ¶

σ

( w

) = σ

( w

) = σ

( w

)

¶

σ

( w

) ¶

· · · ¶

σ

( w

) = σ

( w ) ,

and so σ

(w ) ∼

σ

(w

) , as expected.

Corollary 3.3. — Let C be a left cell of W . Then σ

(C ) is a left cell of W and the A -linear map M (C ) → M ( σ

(C )) , c

7→ c

is an isomorphism of H -modules.

Proof. — This follows immediately from Theorems 3.1 and 3.2.

4. Generalized Vogan classes

4.A. Dihedral parabolic subgroups. — Let E

denote the set of pairs (s , t ) of ele- ments of S such that one of the following holds:

(O) s t has odd order ^¾ 3 ; or

(E) ϕ (s ) < ϕ (t ) and s t has even order ^¾ 4 .

We fix in this subsection a pair (s , t ) ∈ E

. Let w

denote the longest element of W

. We then set:

R

= {w ∈ W

| ℓ (w s ) < ℓ (w ) and ℓ (w t ) > ℓ (w )} = (W

∩ X

) \ X

and R

= {w ∈ W

| ℓ (w s ) > ℓ (w ) and ℓ (w t ) < ℓ (w )} = (W

∩ X

) \ X

. If we are in the case (O), we then set

Γ

= R

and ^Γ

= R

while, if we are in the case (E), we set

Γ

= R

\ {s } Γ

= R

\ {w

s }.

Finally, if we are in the case (O), we ^Æ

: Γ

→ Γ

, w 7→ w

w while, in the case (E), we set ^Æ

: Γ

→ Γ

, w 7→ w s . Then, by [Lus3, §7], we have:

Lemma 4.1. — If (s , t ) ∈ E

, then ^Γ

and ^Γ

are two left cells of W

and ^Æ

: Γ

→ Γ

is a bijection which satisfies the properties (V1) and (V2) of §2.

So ^Æ

induces a bijection ^Æ

: W → W and, according to Theorem 3.2, the fol- lowing holds:

Corollary 4.2. — If (s , t ) ∈ E

and if C is a left cell of W , then C

= Æ

(C ) is also a left cell and the map M (C ) → M (C

) , c

7→ c

s,t(w)

is an isomorphism of H -modules.

R

- The bijection ^Æ

is called the ∗ -operation and is usually denoted by

w 7→ m

. ^„

4.B. Generalized Vogan classes. — Let V

be the group of bijections of W gener- ated by all the ^Æ

, where (s , t ) runs over E

. We will call it the left Vogan group (associated with ϕ ). Let P (S) denotes the set of subsets of S and, if w ∈ W , we set

R(w ) = {s ∈ S | ℓ (w s ) < ℓ (w )}.

It is called the right descent set of w . It is well-known that the map ^R : W → P ( S ) is constant on left cells [Lus3, Lemma 8.6].

E

- It can be checked by using computer computations in GAP that

|V

| = 2

· 3

· 5

· 7

· 11

whenever (W, S) is of type H

. ^„

Now, let Maps(V

, P (S)) denote the set of maps V

→ P (S) . Then, to each w ∈ W , we associate the map τ

∈ Maps(V

, P (S)) which is defined by

τ

( σ ) = R ( σ ( w ))

for all σ ∈ V

. The fiber of the map τ

: W → Maps(V

, P (S)) are called the general- ized Vogan left classes. In other words, two elements x and y of W lie in the same generalized Vogan left class if and only if

∀ σ ∈ V

, R ( σ (x )) = R ( σ (y )).

It follows from Corollary 4.2 that:

Theorem 4.3. — Generalized Vogan left classes are unions of left cells.

4.C. Knuth classes. — Let s ∈ S . We now define a permutation κ

of W as follows:

κ

(w ) =







s w if there exists t ∈ S such that t w < w < s w < t s w and ϕ (s ) ¶ ϕ (t ) , s w if there exists t ∈ S such that t s w < s w < w < t w and ϕ (s ) ¶ ϕ (t ) , w otherwise.

Then κ

is an involution of W . We denote by K

the group of permutations of W generated by the κ

, for s ∈ S . A Knuth left class is an orbit for the group K

. The following result is well-known [Lus1]:

Proposition 4.4. — Every left cell is a union of Knuth left classes.

4.D. Knuth classes and Vogan classes. — If w ∈ W , we set L (w ) = {s ∈ S | ℓ (s w ) < ℓ (w )}.

It is called the left descent set of w . If σ ∈ V

, then

(4.5) ^{L (} σ ( w )) = L ( w ) .

Proof. — We only need to prove the result whenever σ = Æ

for some (s , t ) ∈ E

. Now, write w = x w

with w ∈ X

and w

∈ W

and let u ∈ S . Then σ (w ) = x σ (w

) . By Deodhar’s Lemma, two cases may occur:

• If u x ∈ X

, then u ∈ L (w ) (or L ( σ (w )) ) if and only if u ∈ L (x ) . So u ∈ L (w ) if and only if u ∈ L ( σ (w )) , as desired.

• If u x 6∈ X

, then u x = x v , for some v ∈ {s , t } . Therefore, u ∈ L (w ) (respectively u ∈ L (w ) ) if and only if v ∈ L (w

) (respectively v ∈ L ( σ (w

)) ). But it is easy to check directly in the dihedral group W

that ^L (w

) = L ( σ (w

)) . So again u ∈ L (w ) if and only if u ∈ L ( σ (w )) , as desired.

Proposition 4.6. — If C is a Knuth left class and if σ ∈ V

, then σ (C ) is also a Knuth left class.

Proof. — It is sufficient to show that, if s ∈ S , if (t , u ) ∈ E

and if w ∈ W , then ^Æ

(w ) and ^Æ

⁽ κ

(w )) are in the same Knuth left class. If κ

(w ) = w , then this is obvious.

So we may (and we will) assume that κ

(w ) 6= w . Therefore, there exists s

∈ S such that s

w < w < s w < s

s w or s

s w < s w < w < s

w , and ϕ (s ) ¶ ϕ (s

) . So κ

(w ) = s w and, by replacing if necessary w by s w , we may assume that s

w < w < s w < s

s w . We write σ = Æ

and w = x w

, with w

∈ W

. Two cases may occur:

First case: assume that s x ∈ X

. Then σ (s w ) = s σ (w ) . By (4.5), we have s

σ (w ) < σ (w ) < s σ (w ) = σ (s w ) < s

σ (s w ) = s

s σ (w ).

So κ

( σ ( w )) = σ ( s w ) , so σ ( w ) and σ ( κ

( w )) are in the same Knuth left class.

Second case: assume that s x 6∈ X

. Then s x = x t

with t

∈ {t , u } by Deodhar’s Lemma.

Therefore,

s

x w

< x w

< x t

w

< s

x t

w

.

This shows that w

< t

w

and s

x 6∈ X

. Therefore, again by Deodhar’s Lemma, we have s

x = x u

for some u

∈ {t , u } . Hence

u

w

< w

< t

w

< u

t

w

and t

6= u

. So ^{t , u } = {t

, u

} . Moreover, ϕ (s ) = ϕ (t

) and ϕ (s

) = ϕ (u

) . In this

situation, two cases may occur:

• Assume that t u has odd order. In this case, ϕ (t

) = ϕ (u

) and so ϕ (s ) = ϕ (s

) . Moreover,

σ (w ) = x w

w

and σ (s w ) = x w

t

w

= x u

w

w

= s

σ (w ).

Therefore,

s σ ( w ) < σ ( w ) < s

σ ( w ) = σ ( s w ) < s s

σ ( w ) ,

and so σ ( κ

( w )) = κ

( w ) since ϕ ( s ) = ϕ ( s

) . This shows again that σ ( w ) and σ ( κ

( w )) are in the same Knuth left cell.

• Assume that t u has even order. Since {t

, u

} = {t , u } , ϕ (t ) < ϕ (u ) and ϕ (t

) = ϕ (s ) ¶ ϕ (s

) = ϕ (u

) , we have t

= t and u

= u . In particular,

u w

< w

< t w

< u t w

.

This shows that w

, t w

∈ Γ

∪ ˙ Γ

, so σ (w

) = w

t and σ (t w

) = t w

t = t σ (w

) . Again, by (4.5),

u σ (w

) < σ (w

) < t σ (w

) = σ (t w

) < u t σ (w

) and so

s

σ (w ) < σ (w ) < s σ (w ) = σ (s w ) < s

s σ (w ) and so σ (s w ) = κ

( σ (w )) , as desired.

5. Commentaries

5.A. Since the map W → W , w 7→ w

exchanges left cells and right cells, and ex- changes left descent sets and right descent sets, all the results of this paper can be transposed to results about right cells.

5.B. As it has been seen in Type H

, the group ^V

can become enormous, even in small rank, so it is not reasonable to compute generalized Vogan left classes by computing completely the map τ

. Computation can be performed by imitating the inductive definition of classical Vogan left classes. With our point-of-view, this amounts to start with the partition given by the fibers of the map R : W → P (S) , and to refine it successively using the action of the generators of V

, and to stop whenever the partition does not refine any more.

More precisely, let V

(k ) denote the set of elements of V

which can be expressed

as the product of at most k involutions of the form ^Æ

, for (s , t ) ∈ E

and let τ

(k ) :

W → Maps(V

(k ), P (S)) be the map obtained in a similar way as τ

. This map can be

easily computed inductively for small values of k , and gives rise to a partition ^C (k )

of W which is a priori coarser than the partition into generalized Vogan left classes.

However, when C (k ) = C (k + 1) , this means that C (k ) coincides with the partition into generalized Vogan left classes.

For instance, in type H

, this algorithm stops at k = 5 . Computing the generators of ^V

takes less than 4 minutes on a very basic computer, while the deduction of Vogan classes is then almost immediate.

References

[Alv] D. A

, The left cells of the Coxeter group of type

, J. Algebra 107 (1987), 160- 168.

[Bon] C. B

, On Kazhdan-Lusztig cells in type

, Journal of Alg. Comb. 31 (2010), 53-82.

[Ge1] M. G

, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), 608-614.

[Ge2] M. G

, A generalised

-invariant for the unequal parameter case, preprint (2014), arXiv:1405.5736 .

[KaLu] D. K

& G. L

, Representations of Coxeter groups and Hecke alge- bras, Invent. Math. 53 (1979), 165-184.

[Lus1] G. L

, Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983), 99-111, Lecture Notes in Math. 1024, Springer, Berlin, 1983.

[Lus2] G. L

, Canonical bases arising from quantized enveloping algebras, J. Amer.

Math. Soc. 3 (1990), 447-498.

[Lus3] G. L

, Hecke algebras with unequal parameters, CRM Monograph Series 18, American Mathematical Society, Providence, RI (2003), 136 pp.

[Vog] D. V

, A generalized

-invariant for the primitive spectrum of a semisimple Lie algebra, Math. Ann. 242 (1979), 209-224.

July 11, 2014

CÉDRICBONNAFÉ, Institut de Mathématiques et de Modélisation de Montpellier (CNRS: UMR 5149), Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34095 MONTPEL- LIER Cedex, FRANCE ^• E-mail :cedric.bonnafe@univ-montp2.fr