THE JOINT RELATIONS AND THE SET D1 IN
CERTAIN CRYSTALLOGRAPHIC GROUPS
Shi Jian-yi
Department of Mathematics, East China Normal
University, Shanghai, 200062, P. R. China
and
School of Mathematics, Institute for
Advanced Study, Princeton, NJ 08540, USA
Abstract. We develop some results of Springer for the joint relations in certain
crystallographic groups W . We deduce further joint relations in W from some nonzero structural coefficients of the Hecke algebras H associated to W . In particular,
we make some detailed discussion for the set D1 . This includes the approach to the
proof of a conjecture of Lusztig concerning the characterization of the set D1 , the
connection between the finiteness of D1 and some joint relations in W , the explicit
(2)
description for some elements of D1 and the connectedness of the set D0 (Ω)∪D1 (Ω)
(see 3.2 and 5.2 for the notations).
For any Coxeter group W , D. Kazhdan and G. Lusztig introduced some joint
relations among the elements of W by which certain representations of W and its
associated Hecke algebra H are constructed [1]. Recently, in his letter to Lusztig [6],
T. A. Springer obtained some interesting results concerning the connection between
some non-zero structural coefficients of H and the joint relations in W when W is
certain crystallographic group. He investigated some properties of a subset D1 of
W (Springer denoted D1 by F in his letter) which plays an important role for the
joint relations in W .
In the present paper, we shall develop some results of Springer. We shall deduce
further joint relations in W from some non-zero structural coefficients of H. In
particular, we shall make some detailed discussion for the set D1 . This includes the
approach to the proof of a conjecture of Lusztig concerning the characterization of
the set D1 , the connection between the finiteness of D1 and some joint relations in
Key words and phrases. crystallographic groups, Hecke algebras, joint relations, distinguished
involutions.
I acknowledge NSF support (Grant DMS 8610730) at The Institute for Advanced Study
2
Jian-yi Shi
W , the explicit description for some elements of D1 and the connectedness of the
(2)
set D0 (Ω) ∪ D1 (Ω) (see 3.2 and 5.2 for the notations).
Complex Reflection Groups
3
§1. Preliminary.
1.1 Let W = (W, S) be a Coxeter group with S its Coxeter generator set. Let
≤ be the Bruhat order on W : y ≤ w in W means that there exist some reduced
forms w = s1 s2 · · · s` and y = si1 si2 · · · sit with si ∈ S such that i1 , i2 , . . . , it is a
subsequence of 1, 2, . . . , `. For w ∈ W , we denote by `(w) the length of w.
1.2 Let A = Z[u, u−1 ] be the ring of all laurent polynomials in an indeterminate u
with integer coefficients. The Hecke algebra H of W over A has two sets of A-bases
{Tx }x∈W and {Cw }w∈W which satisfy the relations:
½
(1.2.1)
Tw Tw0 = Tww0
(Ts − u−1 )(Ts + u) = 0
if `(ww0 ) = `(w) + `(w0 )
if s ∈ S
and
(1.2.2)
Cw =
X
u`(w)−`(y) Py,w (u−2 )Ty
y≤w
where Py,w (u) ∈ Z[u] satisfies that Pw,w = 1, Py,w = 0 if y 6≤ w and deg Py,w ≤
(1/2)(`(w) − `(y) − 1) if y < w. The Py,w ’s are called Kazhdan-Lusztig polynomials
[1].
1.3 We denote by µ(y, w) or µ(w, y) the coefficient of u(1/2)(`(w)−`(y)−1 in Py,w .
We say that y and w are joint, written y
w, if µ(y, w) 6= 0.
1.4 To any x ∈ W , we associate two subsets of S:


L(x) = {s ∈ S  sx < x} and R(x) = {s ∈ S  xs < x}.
We have the following relations: for any x ∈ W and s ∈ S,
(1.4.1)
 −1
 (u + u)Cx
P
Cs Cx =
µ(x, y)Cy
 y x
if s ∈ L(x)
if s ∈
/ L(x)
sy<y
and
(1.4.2)
 −1
 (u + u)Cx
P
Cx Cs =
µ(x, y)Cy
 y x
if s ∈ R(x)
if s ∈
/ R(x)
ys<y
where the numbers of the elements y occurring on the right hand sides of (1.4.1)
4
Jian-yi Shi
1.5 For any x, y, z ∈ W , we define hx,y,z ∈ A by
(1.5.1)
Cx Cy =
X
hx,y,z Cz
z
In the present paper, we assume once and forever that W is irreducible and crystallographic (i.e. for s, t ∈ S with s 6= t, the product st has order 2,3,4,6 or ∞)
satisfying the following conditions:
(a) There exists a positive integer N such that
(1.5.2)
uN hx,y,z ∈ Z[u] for all
x, y, z ∈ W
(b) Py,w has non-negative coefficients for any pair y, w ∈ W .
These include all the Weyl groups and all the affine Weyl groups [4].
We record some known results which are needed in the subsequent discussion.
The notations introduced here will be used subsequently without further comment.
1.6 By (1.4.1) and (1.4.2), we see that for x, y, z ∈ W , hx,y,z has non-negative
coefficients as a Laurent polynomial in u. By the assumption 1.5(a), for any z ∈ W ,
there exists an integer a(z) ≥ 0 such that
ua(z) hx,y,z ∈ Z[u]for all x, y ∈ W
ua(z)−1 hx,y,z ∈
/ Z[u]for some x, y ∈ W.
It is well known that
(1.6.1)
a(z) = 0 ⇐⇒ z = e
[3, Proposition 2.3]
where e is the identity of the group W . For x, y, z ∈ W with z 6= e, let γx,y,z and
δx,y,z be the coefficients of ua(z) and ua(z)−1 in hx,y,z , respectively.
1.7
For y, z ∈ W , we denote z ≤ y (resp. z ≤ y) if there exists a sequence of
L
R
elements z0 = z, z1 , . . . , zr = y in W with r ≥ 0 such that for every i, 1 ≤ i ≤ r,
zi−1
zi and L(zi−1 ) 6⊂ L(zi ) (resp. R(zi−1 ) 6⊂ R(zi )). We denote z ≤ y if there
LR
exists a sequence of elements z0 = z, z1 , . . . , zr = y in W with r ≥ 0 such that for
every i, 1 ≤ i ≤ r, either zi−1 ≤ zi or zi−1 ≤ zi . We write z ∼ y (resp. z ∼ y, z ∼ y)
L
R
L
R
LR
if z ≤ y ≤ z (resp. z ≤ y ≤ z, z ≤ y ≤ z). These are equivalence relations on W , the
Complex Reflection Groups
5
equivalence classes of W with respect to ∼ (resp. ∼, ∼ ) are called the left (resp.
L
R
LR
right, two-sided) cells of W [1].
1.8 Let δ(z) = deg Pe,z for z ∈ W . Lusztig showed that the inequality
(1.8.1)
`(z) − 2δ(z) − a(z) ≥ 0
holds for any z ∈ W [4, 1.3]. We define
(1.8.2)
i(z) = `(z) − 2δ(z) − a(z)
for
z∈W
Then i(z) ≥ 0. Define

Dm = {z ∈ W  i(z) = m}for m ≥ 0
(1.8.3)
Then W can be decomposed into a disjoint union of Dm ’s.
The following results are well known: For x, y, z ∈ W .
1.9 hx,y,z = hy−1 ,x−1 ,z−1 and hence a(z) = a(z −1 ). Since Pe,z = Pe,z−1 , this
implies that for any m ≥ 0 and z ∈ W ,
z ∈ Dm ⇐⇒ z −1 ∈ Dm
(1.9.1)
We also have
hx,y,z 6= 0 =⇒ `(z) ≤ `(x) + `(y), z ≤ x and z ≤ y.
R
L
Moreover, the coefficient of ui in hx,y,z is non-zero only if `(z) + i ≡ `(x) + `(y)
(mod 2).
1.10 γx,y,z = γy,z−1 ,x−1 = γz−1 ,x,y−1 and
γx,y,z 6= 0 =⇒ x ∼ y −1 , y ∼ zand x ∼ z.
L
L
R
1.11 x ≤ y =⇒ a(x) ≥ a(y). In particular,
LR
x ∼ y =⇒ a(x) = a(y).
LR
1.12 a(x) = a(y) and x ≤ y (resp. x ≤ y) =⇒ x ∼ y (resp. x ∼ y).
6
Jian-yi Shi
1.13 D0 is a finite set of involutions (Lusztig called them the distinguished involutions). Each left (resp. right) cell of W contains a unique element of D0 . (Thus
for x ∈ W , we can denote by d(x) the element of D0 with x ∼ d(x).)
L
1.14 γx,d(x),x = 1. Moreover, if γx,y,z 6= 0 and y ∈ D0 then x = z, y = d(x) and
γx,y,z = 1.
1.15 δx,y,z 6= 0 =⇒ either x ∼ z or y ∼ z. In particular, if a(x) = a(y), then
R
L
δx,y,z 6= 0 =⇒ x ∼ zand y ∼ z.
R
L
1.16 If either a(x) = a(y) or x 6∼ y −1 6∼ z −1 holds, then
L
R
X
δx,y,z =
γz −1 , x, vδv,y,d(z) .
ν
a(v)=a(z)
In particular, δd(z−1 ),y,z = δz−1 ,y,d(z) for any y, z ∈ W .
1.17 Let π(x) be the leading coefficient of Pe,x for x ∈ W . If x−1 6= y, then
µ(x, y) =
X
δx−1 ,y,d +
d∈D0
X
γx−1 ,y,f π(f ).
f ∈D1
1.18 Suppose that µ(x, y) 6= 0 and x 6∼ y. Then
R
a(y) ≤ a(x) =⇒ µ(x, y) = δx−1 ,y,d(x)
a(x) ≤ a(y) =⇒ µ(x, y) = δx−1 ,y,d(y)
a(x) = a(y) =⇒ x ∼ yand µ(x, y) =
L
X
γf,x,y π(f ).
f ∈D1
Similar results hold for the case when µ(x, y) 6= 0 and x 6∼ y.
L
1.19 µ(x, y) 6= 0 and a(y) < a(x) =⇒ x ≤ y and x ≤ y.
L
R
1.20 Assume that x ∈ W and d ∈ D0 satisfy that x ∼ d and x 6∼ d. Then
LR
R
µ(x, d) 6= 0 ⇐⇒ x ∼ dand x ∈ D1 .
L
When the equivalent conditions hold, we have µ(x, d) = π(x). Similar results hold
by replacing the assumption x 6∼ y by x 6∼ y.
R
L
1.21 Remark: Result 1.9 is obvious. 1.10 – 1.14 are due to Lusztig [3] [4]. The
Complex Reflection Groups
7
§2. The joint relations deduced from some non-vanishing coefficients of
hx,y,z .
From 1.17 – 1.20, we see that the knowledge of the γx,y,z ’s and the δx,y,z ’s
provide much information about the joint relations in W . So in this section, we
shall first study some properties of these coefficients. The following result can be
regarded as an analogue of 1.10.
2.1 Lemma. (a) If a(x) = a(z) > a(y), then δx,y,z = δy,z−1 ,x−1 = δz,y−1 ,x . So we
have x ≤ y −1 when δx,y,z 6= 0.
L
(b) If a(y) = a(z) > a(x), then δx,y,z = δz−1 ,x,y−1 = δx−1 ,z,y . So we have y ≤ x−1
R
when δx,y,z 6= 0.
Proof. It is enough to show (a) and then (b) follows by 1.9. From the associativity
formula (Cx Cy )Cz−1 = Cx (Cy Cz−1 ), we get
X
(2.1.1)
hx,y,v hv,z−1 ,d(z−1 ) =
v
X
hx,v,d(z−1 ) hy,z−1 ,v
v
Equating the coefficients of u2a(z)−1 on both sides, we find
X¡
¢
γx,y,v δv,z−1 ,d(z−1 ) + δx,y,v γv,z−1 ,d(z−1 ) =
v
(2.1.2)
=
X¡
¢
γx,v,d(z−1 ) δy,z−1 ,v + δx,v,d(z−1 ) γy,z−1 ,v
v
By 1.10, 1.14 and the condition a(x) = a(z) > a(y), we get
(2.1.3)
δx,y,z =
X
γx,v,d(z−1 ) δy,z−1 ,v = γx,x−1 ,d(z−1 ) δy,z−1 ,x−1
v
If x 6∼ z then δx,y,z = 0 = δy,z−1 ,x−1 by 1.12 and 1.15. On the other hand, if x ∼ z
R
R
then d(z −1 ) = d(x−1 ) and hence by 1.9, 1.10 and 1.14, (2.1.3) becomes
δx,y,z = δy,z−1 ,x−1 = δz,y−1 ,x
So the first assertion of (a) is true. Since
δx,y,z 6= 0 =⇒ δz,y−1 ,x 6= 0 =⇒ hz,y−1 ,x 6= 0,
this implies x ≤ y −1 and (a) is proved.
L
In contrast with 1.10, the conditions that a(x) = a(y) = a(z) and δx,y,z 6= 0 do
not imply x ∼ y −1 in general.
L
8
Jian-yi Shi
2.2 Corollary. If y, z ∈ W satisfy a(z) > a(y), then
(2.2.1)
δz,y,d(z−1 ) = δy,z,d(z)
Proof. By 1.16 and Lemma 2.1, we have
δz,y,d(z−1 ) = δd(z),y,z−1 = δy,z,d(z)
2.3 By 1.13, we may define an integer

N0 = max{`(d)  d ∈ D0 }
(2.3.1)
Corollary. For any x ∈ W , the set
(2.3.2)

J(x) = {z ∈ W  a(z) > a(x), z
x}
is finite.
Proof. Let z ∈ J(x). Then by 1.18 and Lemma 2.1, we have
0 6= µ(z, x) = δz−1 ,x,d(z) = δx,d(z),z .
This implies from 1.9 that
`(z) ≤ `(x) + d(z) ≤ `(x) + N0 .
But the set of all elements y of W with `(y) ≤ `(x) + N0 for a fixed x ∈ W is finite.
So J(x) is finite.
2.4 Corollary. If x
z and `(z) − `(x) > N0 then a(z) ≤ a(x). In particular, in
this case if a(x) = 1 then a(z) = 1.
Proof. Suppose a(z) > a(x). then by the proof of Corollary 2.3, we see that `(z) ≤
`(x) + N0 , i.e. `(z) − `(x) ≤ N0 which contradicts our condition. So a(z) ≤ a(x).
The second assertion follows by (1.6.1).
Next we shall deduce some joint relations of elements of W from some non-
Complex Reflection Groups
9
2.5 Proposition. Given x, y ∈ W with a(x) ≥ a(y). If δx,y,z 6= 0 for some z ∈ W ,
then there exists some element v ∈ W such that
x−1 ∼ v
y
R
(2.5.1)
In this case, if z ∈ D1 then v can be chosen such that x−1
v; if z ∈ D0 then we
can choose v = x−1 .
Proof. By 1.16, there exists some v with a(v) = a(x) such that δv−1 ,y,d(z) 6= 0 6=
γz−1 ,x,v−1 . Thus by 1.10 and 1.17, we see that v is a required element. Then the
remaining part of our proposition follows from 1.17 and 1.14.
For x, y, z ∈ W with a(z) ≥ 2, we denote by λx,y,z the coefficient of ua(z)−2 in
hx,y,z . Then we have
2.6 Corollary. Given x, y ∈ W . If λx,y,z 6= 0 for some z ∈ W with a(z) >
a(x), a(y). Then there exist two elements v, v 0 ∈ W such that
v ∼ v0
x−1
R
y
(2.6.1)
In particular, if z = d ∈ D0 then we can choose v, v 0 with v 0 = v.
Proof. Equating the coefficients of u2a(z)−2 on both sides of (2.1.1), we get
λx,y,z +
X
(δx,y,v δv,z−1 ,d(z−1 ) + γx,y,v λv,z−1 ,d(z−1 ) )
v
=
X
δx,v,d(z−1 ) δy,z−1 ,v .
v
Then by the positivity of the coefficients of the ha,b,c ’s and by the condition λx,y,z 6=
0, there exists some v ∈ W such that
(2.6.2)
δx,v,d(z−1 ) 6= 0 6= δy,z−1 ,v .
Clearly, we have a(v) = a(z) > a(y). So δv−1 ,y,z = δy,z−1 ,v 6= 0 by Lemma 2.1.
Hence this implies from Proposition 2.5 that there exists some v 0 ∈ W such that
(2.6.3)
v ∼ v0
R
On the other hand, we have
−1
y
10
Jian-yi Shi
by 2.15 and the fact δx,v,d(z−1 ) 6= 0. Thus (2.6.1) follows from (2.6.3) and (2.6.4).
If z = d ∈ D0 then (2.6.2) becomes
(2.6.5)
δx,v,d 6= 0 6= δy,d,v = δy−1 ,v,d
by Lemma 2.1. So our last assertion follows from 1.17 immediately.
Complex Reflection Groups
11
§3. The joint relations between D1 and D0 .
As we mentioned at the beginning of our paper, the set D1 plays a particularly
important role in the study of the joint relations in W . In this section, we shall
make some detailed discussion on this set. Let us start with a conjecture of Lusztig
which characterizes the set D1 .
3.1 Conjecture [5]. Let x ∈ W . Then there exists some d ∈ D0 with x ∼ d and
LR
µ(x, d) 6= 0 if and only if x ∈ D1 .
3.2 Let Ω be a two-sided cell of W . Kazhdan and Lusztig defined a W -graph Γ
with Ω as its vertex set [1]. Let Di (Ω) = Di ∩Ω, i = 0, 1. Then the above conjecture
is amount to say that D1 (Ω) is exactly the set of all vertices in Γ, each of which is
joined to some vertex in D0 (Ω).
Note that in the case of x 6∼ x−1 , Conjecture 3.1 is equivalent to 1.20 and so it
L
is true in this case.
The following result verifies one direction of the above conjecture.
3.3 Theorem. For any x ∈ W , define

D(x) = {d ∈ D0  d ∼ x, x
LR
d}.
Then D(x) ⊆ {d(x), d(x−1 )}. The equality holds if x ∈ D1 .
Proof. Let d ∈ D(x). Then by 1.17, we have
0 6= µ(x, d) =
X
d0 ∈D
δx−1 ,d,d0 +
0
X
γx−1 ,d,f π(f ).
f ∈D1
Hence one of the following cases must happen.
(a) δx−1 ,d,d0 6= 0 for some d0 ∈ D0 .
(b) γx−1 ,d,f 6= 0 for some f ∈ D1 .
In case (a), we have d ∼ d0 ∼ x−1 by 1.15. This implies from 1.13 that d = d0 = d(x).
L
R
In case (b), we have d = d(x−1 ) by 1.10. So our first assertion follows. Now assume
x ∈ D1 . By 1.14 and 1.10, we have
12
Jian-yi Shi
Hence by the positivity of the coefficients of the hx0 ,y0 ,z0 ’s and 1.17, we see that
µ(x, d(x)) = µ(x−1 , d(x)) ≥ γx,d(x),x π(x) = π(x) > 0
and
µ(d(x−1 ), x) ≥ γd(x−1 ),x,x π(x) = π(x) > 0.
This implies that {d(x), d(x−1 )} ⊆ D(x) for x ∈ D1 and hence D(x) = {d(x), d(x−1 )}
by the first assertion in this case.
3.4 Corollary. For any x ∈ W , we have
µ(x, d(x)) = µ(x, d(x−1 ))
(3.4.1)
Thus in the setup of Theorem 3.3 we have either d(x) = ∅ or D(x) = {d(x), d(x−1 )}.
Proof. By Theorem 2.3, it is enough to show (3.4.1). The result is trivial in the
case of x ∼ x−1 . Now assume x 6∼ x−1 . If x ∈
/ D1 , then by 1.20, we have
L
L
µ(x, d(x)) = 0 = µ(x, d(x−1 )).
If x ∈ D1 , then by 1.13, 1.18, 1.10 and 1.14, we have
µ(x, d(x−1 )) = µ(x−1 , d(x−1 )) =
X
γf,x−1 ,d(x−1 ) π(f ) = π(x)(3.4.2)
f ∈D1
and
µ(x, d(x)) =
X
γf,x,d(x) π(f ) = π(x−1 ).
(3.4.3)
f ∈D1
Since π(x) = π(x−1 ), our result follows.
3.5 For any d ∈ D0 , define two sets

∆(d) = {x ∈ W  δd,d,x 6= 0}
(3.5.1)
and

∆0 (d) = {x ∈ W  x ∼ d, x
LR
d}.
(3.5.2)
From 3.2 and Theorem 3.3, we see that the only part of Conjecture 3.1 remaining
Complex Reflection Groups
13
(A). For any d ∈ D0 , ∆0 (d) ⊆ D1 .
The following result gives an equivalent statement of (A).
Proposition. For any d ∈ D0 , we have
∆(d) ⊆ D1 ⇐⇒ ∆0 (d) ⊆ D1
(3.5.3)
Proof. First we claim ∆(d) ⊆ ∆0 (d). For, let x ∈ ∆(d). Then x 6= d by 1.9 and
1.14. We also have d = d(x) = d(x−1 ) by 1.15 and 1.13. Thus δx−1 ,d,d = δd,d,x 6= 0
by 1.16. This implies x ∈ ∆0 (d) by 1.17 and our claim is shown. So the implication
⇐ is obvious. Now assume ∆(d) ⊆ D1 . Let y ∈ ∆0 (d). Suppose y ∈
/ D1 . Then by
1.17 and 1.14, we have
0 6= µ(y, d) = δy−1 ,d,d0
for some
d0 ∈ D 0 .
Since y ∼ d, this implies a(y) = a(d) and hence by 1.15, we see d = d0 = d(y).
LR
Thus by 1.16,
δd(y−1 ),d,y = δy−1 ,d,d0 6= 0.
If d(y −1 ) 6= d, then y 6∼ d. This implies y ∈ D1 by 1.20 which contradicts our
R
assumption. If d(y −1 ) = d, then y ∈ ∆(d) ⊆ D1 which also contradicts our assumption. Therefore, we must have y ∈ D1 , i.e. ∆0 (d) ⊆ D1 .
The following result is an immediate consequence of Theorem 3.3 and Proposition
3.5.
3.6 Proposition. If ∆(d) ⊆ D1 for any d ∈ D1 , then Conjecture 3.1 is true.
3.7 Remark (a) From 1.13 and Proposition 3.5, we can at least say that the set
[
∆0 (d) − D1
d∈D0
is finite. Conjecture 3.1 says that this set is empty.
(b) From 1.9, 1.17 and Proposition 3.5, we see that
µ(x, d) =
X
½
γ
x−1 ,d,f
π(f ) =
π(x), if x ∈ D1 and d = d(x−1 )
14
Jian-yi Shi
for any x ∈ W and d ∈ D0 with `(x) > 2N0 (see (2.3.1) for the notation N0 ). So
Conjecture 3.1 is true for x ∈ W with `(x) > 2N0 .
(c) For any J ⊆ S, let WJ be the subgroup of W generated by J. Suppose that
the group WJ is finite. Then there exists a unique longest element wJ in WJ . It is
well known that CwJ CwJ = hwJ ,wJ ,wJ CwJ with γwJ ,wJ ,wJ = 1. This implies from
1.9 that δwJ ,wJ ,x = 0 for all x ∈ W . So ∆(wJ ) = ∅. By Proposition 3.5, we see
that the relation x
wJ implies the inclusion x ∈ D1 for any x ∈ W with x ∼ wJ .
LR
Let Ω1 be the two-sided cell of W of a-value 1. Then by [2, Proposition 3.8], Ω1 is
exactly the set of all non-identity elements of W whose reduced forms are unique.

We have D0 (Ω1 ) = {s  s ∈ S}. Then the above assertion shows that Conjecture
3.1 is true for any x ∈ Ω1 .
(d) In his letter to T. A. Springer [5], Lusztig showed that Conjecture 3.1 is true
in the case where W is a finite irreducible Weyl group and x is in the two-sided cell
Ω of W not containing the representation of degree 512 for the type E7 or 4096 for
the type E8 . His proof in this case is based on the fact that a(w) ≡ `(w) (mod 2)
for any w ∈ Ω with w ∼ w−1 . But this fact is no longer valid in general.
L
Complex Reflection Groups
15
§4 The finiteness of some subsets of D1
4.1 We know that for any two-sided cell Ω of W , each element of D1 (Ω) is joined
to some element of D0 (Ω) (see 3.2 for the notation Di (Ω)). But this fact does not
imply that the set D1 (Ω) is finite in general. In fact, when W is the affine Weyl
e2 and Ω is the two-sided cell of W of a-value 2, the set D1 (Ω) is
group of type B
infinite (here let me take the opportunity to thank Professor G. Lusztig for his
telling me this example). However, we have the following result.

(1)
(1)
4.2 Proposition. Let D1 = {x ∈ D1  x 6∼ x−1 }. Then D1 is a finite set.
L
Proof. By 1.13 and Theorem 3.3, it is enough to show that for any d ∈ D0 , the set

F (d) = {x ∈ D1  x 6∼ x−1 , d(x) = d}
(4.2.1)
L
is finite. Let x ∈ F (d). Then x 6∼ d. This implies from 1.16 and 1.18 that
R
δd(x−1 ),d(x),x = δx−1 ,d(x),d(x) = µ(x, d(x)) 6= 0.
But the set
(4.2.2)

M = {x ∈ W  δd,d0 ,x 6= 0
for some d, d0 ∈ D0 }
is finite since the set D0 is finite. So the set F (d) is finite.
4.3 For any two-sided cell Ω of W , let
(1)
(1)
D1 (Ω) = D1 ∩ Ω.
(4.3.1)
Then by the above proposition, we can define an integer
(4.3.2)

(1)
N1 (Ω) = max{`(x)  x ∈ D1 (Ω)} ≥ 0
Corollary. Let Ω be a two-sided cell of W . Suppose that x, y ∈ Ω satisfy the
relations x
y and x 6∼ y. Then x ∼ y and there exists a sequence of elements x0 =
R
L
x, x1 , . . . , xr = y in Ω with r ≤ N1 (Ω) such that xi−1
for every i, 1 ≤ i ≤ r.
Proof. By 1.18, we have
0 6= µ(x, y) =
X
γf,x,y π(f )
xi and L(xi−1 ) 6⊂ L(xi )
16
Jian-yi Shi
and then γf,x,y 6= 0 for some f ∈ D1 . So by 1.10, x ∼ y. Since x 6∼ y, we see
L
f∈
(1)
D1 (Ω)
R
and so `(f ) = r ≤ N1 (Ω).
Let f = sr sr−1 · · · s1 be a reduced form of f with si ∈ S. Then in the expression
Csr Csr−1 · · · Cs1 =
X
a w Cw ,
aw ∈ A,
w
we have af 6= 0. Hence by 6= 0 in the expression
X
Csr Csr−1 · · · Cs1 Cx =
bw Cw ,
bw ∈ A
w
This implies that there exists a sequence of elements x0 = x, x1 , . . . , xr = y such
that for every i, 1 ≤ i ≤ r, Cxi occurs in the expression
Csi Cxi−1 =
X
biw Cw ,
biw ∈ A
w
with bixi 6= 0. Let j1 , j2 , . . . , jt be the subsequence of 0, 1, 2, . . . , r such that j1 = 0,
xjk = xjk +1 = · · · = xjk+1 −1 6= xjk+1 for any k, 1 ≤ k < t, and xjk = xjt +1 = · · · =
xr . Then xj1 , xj2 , . . . , xjt is a required sequence.
The above corollary shows tht each pair x, y ∈ Ω with x
y and x 6∼ y (resp.
R
x 6∼ y) belong to a cycle of bounded length in the W -graph Γ(Ω) afforded by Ω,
L
where “a cycle in Γ(Ω)” means “a sequence of elements z0 , z1 , . . . , zr = z0 in Ω
with r > 1 and z1 , z2 , . . . , zr distinct such that for every i, 1 ≤ i ≤ r, zi−1
zi and
L(zi−1 ) q R(zi−1 ) 6⊆ L(zi ) q R(zi )”.

(0)
(0)
(0)
4.4 Let D1 = {x ∈ D1  x ∼ x−1 } and D1 (Ω) = D1 ∩ Ω for any two-sided cell
L
Ω of W . Then the following result provides some necessary and sufficient conditions
on the finiteness of the set D1 (Ω).
Theorem. Let Ω be a two-sided cell of W . Then the following statements are
equivalent.
(a) The set D1 (Ω) is finite.
(0)
(b) The set D1 (Ω) is finite.
(c) For any d ∈ D0 (Ω), the set

Complex Reflection Groups
17
is finite.
(d) There exists some integer m ≥ 0 such that µ(x, y) = 0 for any pair x, y ∈ Ω with
|`(x) − `(y)| > m.
Proof. The relation (a)⇔(b) follows from Proposition 4.2. Since each x ∈ D1 (Ω)
satisfies x
d(x) by Theorem 3.3, the implication (c)⇒(a) follows. Now we shall
show (c) by assuming (a). By 1.17 and 1.16, we see E(d) ⊆ M ∪ D1 (Ω) (see (4.2.2)
for the notation M ). This implies (c) by (a) and the finiteness of the set M . The
implication (d)⇒(a) follows from 1.13 and Theorem 3.3. Finally, we shall prove (d)
by assuming (a). Define

m = max{`(x)  x ∈ D1 (Ω) ∪ D0 (Ω)}.
Take any pair x, y ∈ Ω with |`(x) − `(y)| > m. Then we have
µ(x, y) =
X
δx−1 ,y,d +
d∈D0
X
γx−1 ,y,f π(f )
f ∈D1
by 1.17. If µ(x, y) 6= 0, then one of the following cases must occur.
(i) δx−1 ,y,d 6= 0 for some d ∈ D0 .
(ii) γx−1 ,y,f 6= 0 for some f ∈ D1 .
In case (i), we have d = d(x) = d(y) and hence by 1.16, we get
δd(x−1 ),y,x = δx−1 ,y,d(x) 6= 0
and
δd(y−1 ),x,y = δy−1 ,x,d(y) = δx−1 ,y,d(y) 6= 0.
Hence
hd(x−1 ),y,x 6= 0 6= hd(y−1 ),x,y .
So by 1.9, we have
`(d(x−1 )) ≥ `(x) − `(y) and
`(d(y −1 )) ≥ `(y) − `(x).
Thus one of `(d(x−1 )) and `(d(y −1 )) must be greater than m which is impossible.
In case (ii), we have
18
Jian-yi Shi
by 1.10. Then hx,f,y 6= 0 6= hy,f −1 ,x . By 1.9, we also get `(f ) ≥ |`(x) − `(y)| > m
which is impossible, too.
Therefore, it must be µ(x, y) = 0 and hence (d) follows.
4.5 Let Ω1 be the two-sided cell of W defined as in Remark 3.7(c). Then D0 (Ω1 ) =

{s  s ∈ S}. By the result of Lusztig in [3, 10.4], one can show that for x ∈ Ω1 and
s ∈ S with `(x) > 2, the relation x
s implies that x has the form x = y·v·r·t·v·y −1
with v, r, t ∈ S and s ∈ L(yv), where the notation z = z 0 · z 00 means that z = z 0 z 00
and `(z) = `(z 0 ) + `(z 00 ). So vrtv ∈ Ω1 with v, r, t ∈ S all distinct. But this could
e2 and {v, r, t} = S, in
happen if and only if W is the affine Weyl group of type A
this case, y = e. On the other hand, we see that the set

F1 = {st  s, t ∈ S, o(st) ≥ 3}
is contained in D1 (Ω1 ), where o(z) denotes the order of z. So by Remark 3.7(c)
and the proof of Theorem 4.4, we get the following result immediately.
Proposition. In the above setup, we have

½
F1 ∪ {srts  {s, r, t} = S},
D1 (Ω1 ) =
F1 ,
Thus the relation x
e2
if W has type A
otherwise.
y in Ω1 implies |`(x) − `(y)| ≤ 3. In particular, in the case
e2 , we have
where the type of W is not A
(4.5.1)
x
y ⇐⇒ x = sy or ys for some s ∈ S
for x, y ∈ Ω1 .
4.6 Corollary. For any x, z ∈ W with a(x) = 1 and `(z) − `(x) > N0 (N0 as
defined in (2.3.1)), we have µ(x, z) = 0.
e1 . The result in these two
Proof. Note that N0 ≥ 3 unless W has the type A1 or A
cases is obviously true. The result in the other cases follows from Corollary 2.4 and
Proposition 4.5.
Complex Reflection Groups
(1)
§5. The set D1
19
(2)
and D1 .
5.1 For x, y, ∈ W , we say that x, y are strongly joint (or call x, y a strongly joint
pair) if
(a) x
y
(b) either x 6∼ y or x 6∼ y.
R
L
We say that x, y are very strongly joint (or call x, y a very strongly joint pair) if
x, y are strongly joint and L(x) q R(x) 6= L(y) q R(y).

(2)
5.2 Let D1 = {x ∈ D1  L(x) 6= R(x)}. By [1, Proposition 2.4], the relation x ∼ y
(2)
L
(1)
(resp. x ∼ y) implies R(x) = R(y) (resp. L(x) = L(y)). So we have D1 ⊆ D1
R
(2)
general. By Propositioin 4.2, the set D1
As before, we define
(2)
D1 (Ω)
=
(2)
D1
in
is finite.
∩ Ω for any two-sided cell Ω of W .
We shall show that the strongly joint relation (resp. the very strongly joint
(1)
(2)
relation) in Ω is dominated by the set D1 (Ω) (resp. D1 (Ω)).
5.3 Lemma. Let Ω be a two-sided cell of W . Then for any strongly joint pair
(1)
(resp. for any very strongly joint pair) x, y ∈ Ω, there exists some f ∈ D1 (Ω)
(2)
(1)
(resp. f ∈ D1 (Ω)) such that γf,x,y 6= 0. Conversely, given any f ∈ D1 (Ω) (resp.
(2)
f ∈ D1 (Ω)), there eixsts some strongly joint pair (resp. some very strongly joint
pair) x, y, ∈ Ω with L(x) q L(y) = R(f ) q L(f ), and there also exists such a pair
with R(x) q R(y) = R(f ) q L(f ).
Proof. We may assume without loss of generality that x, y ∈ Ω are a strongly joint
pair with x 6∼ y (resp. a very strongly joint pair with L(x) 6= L(y)). Then by 1.18,
R
we have
0 6= µ(x, y) =
X
γf,x,y π(f ).
f ∈D1
This implies that γf,x,y 6= 0 for some f ∈ D1 . By 1.10, we have f ∼ y and f −1 ∼ x.
R
Hence f 6∼ f
−1
or equivalently f 6∼ f
R
(1)
f ∈ D1
−1
R
(resp. L(f ) = L(y) 6= L(x) = R(f )). So
L
(2)
(resp. f ∈ D1 ). The converse is obvious since we can take y = f and
x = d(f ) by Theorem 3.3.
(1)
(2)
From the above lemma, we see that the sets D1 (Ω) and D1 (Ω) play an important role for the joint relation in Ω. So subsequently, we shall investigate these
20
Jian-yi Shi
5.4 From Lemma 5.3 and [1, 2.3e, 2.3f], we see tht any x ∈ D1 must satisfy one of
the following conditions.
(a) L(x) = R(x);
(b) L(x)⊃R(x);
6=
(c) R(x)⊃L(x);
6=
(d) L(x) 6⊂ R(x) and R(x) 6⊂ L(x). There exists some s, t ∈ S with o(st) ≥ 3
such that sx = d(x) and xr = d(x−1 ) for some r ∈ {s, t}.
(2)
Clearly, D1
is exactly the set of all elements of D1 which satisfy (b), (c) or (d).
5.5 Lemma. Let Ω be a two-sided cell of W . Then for any d ∈ D0 (Ω) and s ∈ S,
(2)
(a) sd ∈ D1 (Ω) ⇔ sd ∈ Ω.
(2)
(b) ds ∈ D1 (Ω) ⇔ ds ∈ Ω.
Proof. The implication ⇒ of (a) is trivial. If x = sd ∈ Ω then d
x, x ∼ d and
L
x 6∼ d. So by 1.20, x ∈ D1 (Ω). Since L(x) 6= L(d) = R(d) = R(x), this implies
R
(2)
x ∈ D1 (Ω) and (a) follows. The proof of (b) is entirely similar.
5.6 For any d, d0 ∈ D0 with d 6= d0 , let
D(d, d0 ) = Ld ∩ Rd0 ∩ D1
(5.6.1)


where, Lx = {y ∈ W  y ∼ x} and Rx = {y ∈ W  y ∼ x}. In the remaining part
L
R
of this section, we shall give some explicit description for the elements d, d0 ∈ D0
and the set D(d, d0 ) in the case when D(d, d0 ) 6= ∅. To this end, we shall first study
some properties of an involution of W . We need a result which is an immediate
consequence of [3, Corollary 5.5].
Lemma. Suppose that x, y, z ∈ W satisfy that x = y · z, and R(x) 6⊂ R(z) (resp.
L(x) 6⊂ L(y)) (see 4.5 for the notation x = y · z). then a(x) > a(z) (resp. a(x) >
a(y)).
Complex Reflection Groups
21
5.7 Lemma. If x is an involution of W with |L(x) ∩ {s, t}| = 1 for some s, t ∈ S
with o(st) ≥ 3, then there exists a unique decomposition
(5.7.1)
x=α·y·β
with α, β ∈ hs, ti, s, t ∈
/ L(y) ∪ R(y) and 0 ≤ `(α) − `(β) ≤ 1, where hs, ti denotes
the subgroup of W generated by s, t. In this expression of x, we have y 2 = e and
β −1 α ∈ {e, s, t}. If β −1 α = r 6= e then ry = yr and a(y) < a(x).
Proof. The decomposition (5.7.1) obviously exists if we withdraw the requirement
0 ≤ `(α) − `(β) ≤ 1. The element y in (5.7.1) is uniquely determined by x as
the element of minimal length in the double coset hs, tixhs, ti. Since x = x−1 =
β −1 y −1 α−1 with α−1 , β −1 ∈ hs, ti and s, t ∈
/ L(y −1 ) ∪ R(y −1 ), this implies that
y = y −1 and so y is an involution. This also enables us to assume `(α) ≥ `(β).
By the conditions |L(x) ∩ {s, t}| = 1 and x2 = e, we have α = β −1 · γ for some
γ ∈ hs, ti and hence (γ · y)2 = e. γ has a unique reduced form γ = s1 s2 · · · s` with
si ∈ {s, t}. If ` > 0 then s1 ∈ L(γ · y) = R(γ · y) and hence s` y = ys1 by the
facts that s1 ∈ R(γ · y) − R(y) and L(γ · y) ∩ {s, t} = {s1 }. In this case, we have
a(y) < a(s` y) ≤ a(x) by Lemma 5.6. By applying induction on ` ≥ 0, we see tht
there exists some expression of x as in (5.7.1) with 0 ≤ `(α)−`(β) ≤ 1. In particular,
we have a(y) < a(x) in the case of `(α) − `(β) = 1. This shows the existence of a
decomposition (5.7.1) of x. Then the uniqueness of such a decomposition follows by
the facts that |L(x) ∩ {s, t}| = 1, 0 ≤ `(α) − `(β) ≤ 1 and `(α) + `(β) = `(x) − `(y).
5.8 Corollary. In the setup of Lemma 5.7, suppose that x0 = txs is also an
involution. Then
(a) ry = yr for r = s, t, if `(α) = `(β) + 1.
(b) sy = yt if `(α) = `(β).
In both cases, we have a(y) < a(x).
Proof. Since x is an involution, we may assume L(x) ∩ {s, t} = {s} without loss
of generality. Then α has the reduced form α = s1 s2 · · · s` with ` ≥ 1, s1 = s
and {si , si+1 } = {s, t} for 1 ≤ i < `. First assume `(α) = `(β) + 1. Then
22
Jian-yi Shi
x0 = ts1 s2 · · · s`−1 ys` s`−1 · · · s2 . This implies from x02 = e that (s`−1 y)2 = e and
hence s`−1 y = ys`−1 by the fact that s`−1 ∈ R(s`−1 y) − R(y), where we stipulate
that s`−1 = t if ` = 1. But we have shown that s` y = ys` in the proof of Lemma
5.7. This proves (a).
Next assume `(α) = `(β). Then β = s` s`−1 · · · s1 . Since x0 = txs =
= ts1 s2 · · · s` ys` s`−1 · · · s2 is an involution, this implies that s`−1 s` y is an involution, with the convention that s` s`−1 · · · s2 = e and s`−1 s` = ts if ` = 1. Then we
have s` y = ys`−1 by the facts that `(s`−1 s` y) = `(y) + 2, s`−1 ∈ R(s`−1 s` y) − R(y)
and s` ∈
/ L(s`−1 s` y). Since {s`−1 , s` } = {s, t} and y 2 = e, we get sy = yt and (b)
follows. Finally, the inequality a(y) < a(x) is an immediate consequence of (a) and
(b) by Lemma 5.7.
5.9 Corollary. If x and txs are two involutions of W with |L(x) ∩ {s, t}| = 1 for
some s, t ∈ S with o(st) ≥ 3, then there exists a unique decomposition x = z · y
satisfying that
(a) z ∈ hs, ti.
(b) s, t ∈
/ L(y) ∪ R(y), y 2 = e and a(y) < a(x).
(c) ry = yr for r = s, t if `(z) ≡ 1 (mod 2) or sy = yt if `(z) ≡ 0 (mod 2).
(d) L(x) − {s, t} = L(y).
Proof. The existence and uniqueness of such a decomposition with the conditions
(a), (b) and (c) follows immediately from Lemma 5.7 and Corollary 5.8. Then the
condition (d) follows from (b) and (c).
5.10 For any s, t ∈ S with s 6= t and o(st) = m < ∞, each coset hs, tiw can
be decomposed into four parts: one consists of the unique element x of minimal
length, one consists of the unique element y of maximal length, one consists of the
(m − 1) elements sx, tsx, stsx, · · · and one consists of the (m − 1) elements tx,
stx, tstx, · · · . The last two subsets are called (s, t)-strings. We shall regard them
as sequences (as above) rather than subsets.
5.11 If x
y,L(x) 6⊂ L(y) and L(y) 6⊂ L(x), then by [1, 2.3e], x, y must be two
neighboring terms in a (s, t)-string for some (and hence any) s ∈ L(x) − L(y)
Complex Reflection Groups
23
L(d0 ) 6⊂ L(d), and if z ∈ D(d, d0 ) (see (5.6.1) for the notation D(d, d0 )), then by
Theorem 3.3, z, d (resp. z −1 , d0 ) are two neighboring terms in a (s, t)-string for
some (and hence any) s ∈ L(d) − L(d0 ) and t ∈ L(d0 ) − L(d) with o(st) ≥ 3.
In this case, we have |`(d) − `(d0 )| = 0 or 2. If |`(d) − `(d0 )| = 2 then such an
element z ∈ D(d, d0 ), when it exists, must be unique. Now assume D(d, d0 ) 6= ∅
and |`(d) − `(d0 )| = 0, i.e. `(d) = `(d0 ). Then|D(d, d0 )| ≤ 2 and d0 = sdt = tds. In
particular, |D(d, d0 )| = 2 if and only if D(d, d0 ) = {sd, td} if and only if d (resp. d0 )
is a non-terminal term in the (s, t)-string containing it.
By Lemma 5.7 and Corollary 5.9, the above results can be summarized as below.
Theorem. For any d, d0 ∈ D0 with L(d) 6⊂ L(d0 ) and L(d0 ) 6⊂ L(d), we have
|D(d, d0 )| ≤ 2. Assume D(d, d0 ) 6= ∅. Then |`(d) − `(d0 )| = 0 or 2. Let s ∈
L(d) − L(d0 ) and t ∈ L(d0 ) − L(d). Then o(st) ≥ 3, and there exists some involution
y with s, t ∈
/ L(y) and some α, β, α0 , β 0 ∈ hs, ti with 0 ≤ `(α)−`(β), `(α0 )−`(β 0 ) ≤ 1
and 0 < `(α) + `(β), `(α0 ) + `(β 0 ) < 2 · o(st) − 1 such that
d=α·y·β
and
d0 = α 0 · y · β 0
(see 5.6).
The inequality a(y) < a(d) holds in the case when `(α) − `(β) = 1 (or equivalently,
`(α0 ) − `(β 0 ) = 1). If `(d) − `(d0 ) = 2 then d0 = sds and D(d, d0 ) = {sd}. If
`(d0 ) − `(d) = 2 then d0 = tdt and D(d, d0 ) = {td}. Now assume `(d) = `(d0 ).
Then d0 = tds. d and d0 can be rewritten as the forms d = z · y and d0 = z 0 · y
with z, z 0 ∈ hs, ti and the same y as above. y satisfies the further conditions that
a(y) < a(d), L(y) = L(d) − {s} and that sy = yt if `(z) ≡ 0 (mod 2) or ry = yr
for r = s, t if `(z) ≡ 1 (mod 2). We also


 {sd}
0
D(d, d ) =
{td}


{sd, td}
have
if `(z) = o(st) − 1
if `(z) = 1
if 1 < `(z) < 0(st) − 1
This theorem gives an explicit description for the structure of the elements d, d0 ∈
D0 and the set D(d, d0 ) in the case where L(d) 6⊂ L(d0 ), L(d0 ) 6⊂ L(d) and D(d, d0 ) 6=
∅.
5.12 If we know an element d ∈ D0 , then by using the above theorem, we can find
some other elements z ∈ D0 ∪ D1 with z ∼ d. The following result is served for this
LR
24
Jian-yi Shi
Proposition. Let y be an involution of W and let s, t ∈ S satisfy that o(st) ∈
{3, 4, 6} and s, t ∈
/ L(y). Define

G(y, s, t) = {z ∈ D0  z ∈ hs, tiyhs, ti, |L(z) ∩ {s, t}| = 1}
Suppose G(y, s, t) 6= ∅.
(a) If ry 6= yr for r = s, t, and sy 6= yt, then G(y, s, t) is the set of all elements of
the form zyz −1 , where z runs over one or two (s, t)-strings in hs, ti.
(b) Suppose that ry = yr for exactly one r ∈ {s, t}. Let

G1 = {zryz −1  z ∈ hs, ti, r ∈
/ R(z) and 0 ≤ `(z) < o(st) − 1}
and

G2 = {zyz −1  z ∈ hs, ti, r ∈
/ R(z) and 0 < `(z) ≤ o(st) − 1}.
Then G(y, s, t) = G1 , G2 or G1 ∪ G2 .
(c) If sy = yt then G(y, s, t) = {sty, tsy} or {ststy, tstsy}. The latter case could
occur only if o(st) = 6.
(d) Suppose ry = yr for r ∈ {s, t}.
(i) If o(st) = 3 then G(y, s, t) = {sy, ty}.
(ii) If o(st) = 4 then G(y, s, t) is one of the following sets: {sy, ty}, {sy, tsty},
{stsy, ty}, {stsy, tsty}.
(iii) If o(st) = 6 then
G(y, s, t) = {sy, ty} ⇔ {tsy, tstsy} ∩ D1 = {tsy},
G(y, s, t) = {stsy, tsty} ⇔ {tsy, tstsy} ⊆ D1 ,
G(y, s, t) = {ststsy, tststy} ⇔ {tsy, tstsy} ∩ D1 = {tstsy}.
Proof. According to 1.1.3 and Theorem 5.11, this can be shown by case-by-case
argument.
5.13 Next we shall consider a pair d, d0 ∈ D0 with L(d)⊃L(d0 ). Suppose z ∈
6=
Complex Reflection Groups
25
Thus L(d)⊃L(z) and R(z)⊃R(d0 ). This implies that either d = s·z (resp. z = d0 ·s)
6=
6=
0
for some s ∈ L(d) − L(d ) or d ≺ z (resp. z ≺ d0 ), where the notation x ≺ y means
that x
y and x < y. Thus we have the following result which describes the set
D(d, d0 ) in the case when L(d)⊃L(d0 ) and d ∼ d0 .
LR
6=
Proposition. Let d, d0 ∈ D0 with L(d)⊃L(d0 ) and d ∼ d0 . Define
LR
6=

e d0 ) = {z ∈ W  d ≺ z ≺ d0 , L(d) = R(z) and L(d0 ) = L(z)}.
D(d,
and

D(d, d0 ) = {z ∈ W  d
d0 , L(d) = R(z) and L(d0 ) = L(z)}
z
e d0 ) ⊆ D(d, d0 ) = D(d, d0 ) ⊆ D(2)
(a) D(d,
1
e d0 ) ⊆ {d0 · s, sd} for any x ∈ L(d) − L(d0 ). Thus D(d, d0 ) −
(b) D(d, d0 ) − D(d,
e d0 ) = ∅ if |L(d) − L(d0 )| ≥ 2.
D(d,
(c) For s ∈ L(d)−L(d0 ), d0 s ∈ D(d, d0 ) if and only if d0 s
sd ∈ D(d, d0 ) if and only if sd
d and L(d0 ) = L(d0 s);
d0 and R(sd) = R(d). When one of these equivalent
conditions is satisfied, we have L(d) = L(d0 ) ∪ {s}.
e d0 ) ⊆ D(d, d0 ) 6⊂ D(2) . From the above discussion, we
Proof. It is obvious that D(d,
1
see D(d, d0 ) ⊆ D(d, d0 ). Now let z ∈ D(d, d0 ). We have d ≤ z ≤ d0 by the facts that
L(d) 6⊆ L(z), R(z) 6⊆ R(d0 ) and d
L
R
d0 . Since a(d) ≥ a(z) ≥ a(d0 ) = a(d), this
z
implies from 1.12 that d ∼ z ∼ d0 . But we have d 6∼ z 6∼ d0 since L(d) =
6 L(z) and
L
R
R
L
R(z) 6= R(d0 ). So by 1.20, we get z ∈ D(d, d0 ) and (a) follows. the conclusion (b) is
obvious. Now let us show the first equivalent conditions of (c). The implication ⇒
follows from 1.20. Assume that d0 s
0
d and L(d0 ) = L(d0 s). Then d ≤ d0 s ≤ d0 by
0
0
0
L
R
the facts that L(d) 6⊆ L(d s) and R(d s) 6⊆ R(d ). Hence a(d) ≥ a(d s) ≥ a(d0 ) =
a(d). So d ∼ d0 s ∼ d0 by 1.12. Clearly, by L(d) 6= L(d0 s) and R(d0 s) 6= R(d0 ), we
L
R
get d 6∼ d0 s 6∼ d0 . So by 1.20, d0 s ∈ D(d, d0 ). This shows the first equivalence of (c).
R
L
Then the second equivalence of (c) follows by entirely the same argument. The last
assertion of (c) is obvious.
5.14 Finally, we shall consider the more general case where d 6= d0 in D0 with
0
26
Jian-yi Shi
Proposition. Let d 6= d0 in D0 with d ∼ d0 and let
LR

D[d, d0 ] = {z ∈ W  d
z
d0 , a(z) = a(d)}.
(1)
Then D(d, d0 )∪D(d0 , d) = D[d, d0 ] ⊆ D1 . There exists a bijective map from D[d, d0 ]
to itself and also from D(d, d0 ) to D(d0 , d) which sends z to z −1 .
(1)
Proof. The inclusions D(d, d0 ) ∪ D(d0 , d) ⊆ D[d, d0 ] and D(d, d0 ) ∪ D(d0 , d) ⊆ D1
follow by Theorem 3.3 and the fact that z 6∼ z −1 for any z ∈ D(d, d0 ) ∪ D(d0 , d).
L
Now let z ∈ D[d, d0 ]. Suppose z ∈
/ D(d0 , d). Then either z 6∼ d0 or z 6∼ d. We
L
R
may assume z 6∼ d0 . By 1.19 and the conditions that a(z) = a(d) = a(d0 ) and
L
d0 , we have d0 ∼ z and z ∈ D1 . We claim z 6∼ d. Since otherwise, it would be
z
R
R
d ∼ z ∼ d0 and thus d = d0 by 1.13 which gives rise to a contradiction. So z 6∼ d
R
R
R
and hence z ∼ d by the same argument as that implying z ∼ d0 from z 6∼ d0 . Thus
L
R
0
0
L
z ∈ D(d, d ). The implication to z ∈ D(d, d ) from z 6∼ d is by symmetry. This shows
R
that D(d, d0 ) ∪ D(d0 , d) = D[d, d0 ]. Since a(z) = a(z −1 ) and µ(z, d) = µ(z −1 , d) for
any z ∈ W and d ∈ D0 , we see that
(5.14.1)
z ∈ D[d, d0 ] ⇐⇒ z −1 ∈ D[d, d0 ].
On the other hand, we have
(5.14.2)
d ∼ z ∼ d0 ⇐⇒ d ∼ z −1 ∼ d0 .
L
R
R
L
Thus
(5.14.3)
z ∈ D(d, d0 ) ⇐⇒ z −1 ∈ D(d0 , d).
This verifies the last assertion of our proposition.
5.15 Corollary.
(5.15.1)
(1)
D1 (Ω) =
∪·
d,d0 ∈D0 (Ω)
d6=d0
D[d, d0 ]
for any two-sided cell Ω of W .
Proof. If z ∈ D[d, d0 ] then {d, d0 } = {d(z), d(z −1 )} with d(z) 6= d(z −1 ). So the union
(1)
on the right hand side of (5.15.1) is disjoint. By Theorem 3.3, any z ∈ D1 (Ω) is
in D[d(z), d(z −1 )] with d(z) 6= d(z −1 ). This implies the inclusion ⊆. Finally, the
opposite inclusion ⊇ follows by Proposition 5.14.
Complex Reflection Groups
27
(2)
§6. The connectedness of the set D0 (Ω) ∪ D1 (Ω)
6.1 We say a subset K of W to be connected if for any x, y ∈ K, there exists
a sequence of elements x0 = x, x1 , . . . , xr = y in K such that xi−1
xi and
L(xi−1 ) q R(xi−1 ) 6⊆ L(xi ) q R(xi ) for every i, 1 ≤ i ≤ r.
In this section, we shall show that for any two-sided cell Ω of W , the set D0 (Ω) ∪
(2)
D1 (Ω) is connected.
6.2 Lemma. Let P, Q be two left cells of W of the same a-value. Suppose that
there exists some x ∈ P , y ∈ Q and s ∈ S such that x
(a) For any x0 ∈ P , there exists some y 0 ∈ Q such that x0
y and s ∈ R(x) − R(y).
y 0 and x ∈ R(x0 )−R(y 0 ).
(b) For any y 00 ∈ Q, there exists some x00 ∈ P such that x00
y 00 and s ∈ R(x00 ) −
R(y 00 ).
Proof. To show (a), we may assume x
x0 and L(x0 ) 6⊂ L(x) without loss of
generality. Take t ∈ L(x0 ) − L(x). Then by (1.4.1), (1.4.2) and our assumption, we
have
hy,s,x 6= 0 6= ht,x,x0 .
Using the associativity formula
X
v
ht,y,v hv,s,x0 =
X
ht,v,x0 hy,s,v ,
v
we see that there exists some v ∈ W such that
ht,y,v 6= 0 6= hv,s,x0 .
By 1.9 and 1.11, we have v ∈ Q. Let y 0 = v. Then y 0 is a required element and so
(a) is shown. To show (b), we may assume y 00
y and L(y) 6⊂ L(y 00 ) without loss
of generality. The proof of (b) in this case is quite similar as above and so is left to
the reader.
6.3 Fix a two-sided cell Ω of W . Let L(Ω) be the set of all left cells of W in Ω.
It is well known that, in W , the intersection of a left cell with any right cell in the
same two-sided cell is non-empty. So for any P, Q ∈ L(Ω), there always exists a
28
Jian-yi Shi
every i, 1 ≤ i ≤ m, xi−1
xi and R(xi−1 ) 6⊂ R(xi ). Define δ(P, Q) = m if the
above sequence x0 , x1 , . . . , xm is the shortest possible with the described property.
By definition, we see that for any P, Q ∈ L(Ω), δ(P, Q) = 0 if and only if P = Q.
Suppose P, Q ∈ L(Ω) with δ(P, Q) = m > 0. Then by Lemma 6.2, we can show
by induction on m that for any x ∈ P , there exists a sequence of elements x0 = x,
x1 , . . . , xm in Ω with xm ∈ Q such that xi−1
xi and R(xi−1 ) 6⊂ R(xi ) for 1 ≤
i ≤ m. Moreover, we see from 1.12 and Lemma 6.2 that in such a sequence, xi ∼ xj
R
and xi 6∼ xj for any i 6= j, 0 ≤ i, j ≤ m. It is easily seen that δ(Lxi , Lxj ) = i − j for
L
0 ≤ i ≤ j ≤ m. Thus δ(P, Q) could be interpreted as the smallest integer m ≥ 0
with the property that there exists a sequence P0 = P , P1 , . . . , Pm = Q in L(Ω)
such that δ(Pi−1 , Pi ) = 1 for 1 ≤ i ≤ m. In particular, we have δ(P, Q) < |L(Ω)|
for any P, Q ∈ L(Ω).
(2)
6.4 Now we shall use the above concept to study the set D0 (Ω) ∪ D1 (Ω).
L. emma: Let d, d0 ∈ D0 (Ω). Then
δ(Ld , Ld0 ) = 1 ⇐⇒ D(d, d0 ) 6= ∅ and L(d) 6⊂ L(d0 ).
Proof. (⇒) By Lemma 6.2, there exists some x ∈ Ld with x
d0 and R(x) 6⊂ R(d0 ).
Thus x ∼ d0 and x ∈ D1 by 1.20. So x ∈ D(d, d0 ), i.e. D(d, d0 ) 6= ∅. On the other
R
hand, we have L(d) = R(d) = R(x) 6⊂ R(d0 ) = L(d0 ).
(⇐) We have d 6= d0 by L(d) 6⊂ L(d0 ). Let x ∈ D(d, d0 ). Then x
d0 by 1.20.
We also have R(x) = R(d) = L(d) 6⊂ L(d0 ) = R(d0 ). This implies δ(Ld , Ld0 ) = 1.
6.5 Let d, d0 ∈ D0 (Ω) with δ(Ld , Ld0 ) = m > 0. Then there exists a sequence
P0 = Ld , P1 , . . . , Pm = Ld0 in L(Ω) such that δ(Pi−1 , Pi ) = 1 for 1 ≤ i ≤ m. Let
di ∈ D0 ∩Pi . Then by Lemma 6.4, we see that D(di−1 , di ) 6= ∅ and L(di−1 ) 6⊂ L(di ).
Take xi ∈ D(di−1 , di ). Then by Theorem 3.3, di−1
xi
di with L(di−1 ) 6⊂ L(xi )
(2)
and R(xi ) 6⊂ R(di ). Since D(di−1 , di ) ⊆ D1 (Ω), this implies from Theorem 3.3
(2)
that for any x, y ∈ D0 (Ω) ∪ D1 (Ω), there exists a sequence of elements x0 = x,
(2)
x1 , . . . , xn = y in D0 (Ω) ∪ D1 (Ω) such that xi−1
xi and L(xi−1 ) q R(xi−1 ) 6⊂
Complex Reflection Groups
29
smallest possible. Then by Theorem 3.3 and Lemma 6.4, we see
(6.5.1)
n = 2 · min0 {δ(Ld , Ld0 )} + ε(x, y)
d,d
where d = x if x ∈ D0 (Ω) and d ranges over {d(x), d(x−1 )} with L(x) q R(x) 6⊂
(2)
L(d) q R(d) if x ∈ D1 (Ω); d0 = y if y ∈ D0 (Ω) and d0 ranges over {d(y), d(y −1 )}
(2)
(2)
with L(d0 ) q R(d0 ) 6⊂ L(y) q R(y) if y ∈ D1 (Ω); ε(x, y) = |D1 (Ω) ∩ {x, y}|.
The above results can be summarized as follows.
(2)
Theorem. Let Ω be any two-sided cell of W . Then the set D0 (Ω) ∪ D1 (Ω) is
(2)
connected (see 6.1). For any x, y ∈ D0 (Ω)∪D1 (Ω), there exists a shortest sequence
(2)
of elements x0 = x, x1 , . . . , xn = y in D0 (Ω) ∪ D1 (Ω) with the property that
xi−1
xi and L(xi−1 ) q R(xi−1 ) 6⊂ L(xi ) q R(xi ) for 1 ≤ i ≤ n, where n is as in
(6.5.1).
Acknowledgments
The author would like to thank Professor G. Lusztig and Professor T. A. Springer
for thier allowing me to use their unpublished results in this paper.
30
Jian-yi Shi
References
[1]
D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke
algebras, Invent. Math. 53 (1979), 165–184.
[2]
G. Lusztig, Some examples of square integrable representations of semisimple
p-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623–653.
[3]
G. Lusztig, Cells in affine Weyl groups, in “Algebraic groups and related topics”, Advanced Studies in Pure Math., Kinokunia and North-Holland, 1985,
pp. 255–287.
[4]
G. Lusztig, Cells in affine Weyl groups, II, J. Algebra, 109 (1987), 536–548.
[5]
G. Lusztig, A letter to T. A. Springer, 1987.
[6]
T. A. Springer, A letter to G. Lusztig, 1987.
email: jyshi@math.ecnu.edu.cn