A SURVEY ON THE CELL THEORY OF AFFINE WEYL GROUPS
Shi Jianyi
Department of Mathematics, East China Normal
University, Shanghai, 200062, P.R. China
Abstract. This paper makes a survey concerning the contributions of Chinese mathematicians on the
cell theory of affine Weyl groups. The main points for their works are as follows. The establishment of
the coordinate forms and the sign types for the elements of an irreducible affine Weyl group Wa ; the
exploration of some relations among the cells of Wa ; explicit descriptions for the lowest two-sided cell
W(ν) of Wa and all the left cells of Wa in W(ν) , for all the two-sided cells of the affine Weyl group of
∼
∼
∼
type D4 and for all the cells of the affine Weyl groups of types Al , l ≥ 1, and B 3 .
§1. Introduction.
1.1 Cells of a Coxeter group W , as defined by Kazhdan and Lusztig [1], are certain equivalence
classes of W . They are important not only in the representation theory of W and its corresponding
Hecke algebra, but also in the representation theory of algebraic groups, finite groups, finite groups
of Lie type, Lie algebras and enveloping algebras.
One of the main themes in the cell theory of W is to give an explicit description of the cells of
W and to investigate the structural properties of these cells.
This task is far from being completed except for a few cases.
1.2 In this paper, we mainly concentrate the case when W is an affine Weyl group, which is a
special kind of infinite Coxeter group. The cell theory of affine Weyl groups is of interest for the
representation theory of affine Hecke algebras.
The cell theory of affine Weyl groups was first studied by G. Lusztig. He described explicitly all
the cells of the affine Weyl groups of ranks ≤ 2 in l979 [2]. By the knowledge of his results, Lusztig
raised a number of conjectures on this subject in 1980 and in 1983 [3]. After that, he published a
series of papers to study the cells in affine Weyl groups and the representations of the corresponding
Hecke algebras afforded by them [2] [3] [4] [5] [6].
1.3 The Chinese mathematicians have been studying the cell theory of affine Weyl groups since
1981 and have made a remarkable contribution to this theory. They are Du Jie, Xi Nanhua and the
∼
∼
author. In 1983, the author described all the cells of affine Weyl groups Wa (An−1 ) of type Al (l ≥ 1)
explicitly [7]. In his book [7], the author introduced the coordinate forms and the sign types for the
Key words and phrases. affine Weyl group, left cells, two-sided cells, Hecke algebras.
Supported by the Science Fund of the Chinese Academy of Sciences.
1980 Mathematics subject classifications: 20H15.
Typeset by AMS-TEX
1
2
Jian-yi Shi
∼
elements of Wa (An−1 ). These concepts were generalized to the case of a general affine Weyl group
by the author in 1985 [8] [9]. These concepts are very useful in the description of cells of affine Weyl
groups. Then in 1986, the author found out the lowest two-sided cell, written W(ν) , of an affine
Weyl group of arbitrary type and described all the left cells contained in W(ν) [10] [11].
On the other hand, Du gave an explicit description for all the cells of the affine Weyl group of
∼
∼
type B 3 in 1986 [12] and also for the two-sided cells of the affine Weyl group of type D4 in 1988
[13].
In early 1988, Xi got some nice results on the relations among the cells of affine Weyl groups in
his joint works with Lusztig [14].
1.4 Besides the above people, T. Springer, R. Bédard and G. M. Lawton also got some results in
this field [5] [15] [16].
1.5 In the present paper, we shall mainly introduce the achievements of Chinese mathematicians
in the cell theory of affine Weyl groups.
The content of our paper is organized as follows. In §§2-3, we collect some definitions which are
applicable for any Coxeter group. Then in the subsequent sections, we shall only consider the case
of (irreducible) affine Weyl groups. In §4, we define an affine Weyl group Wa in a geometric way
and then associate each element of Wa with its coordinate form, some properties of elements of Wa
in terms of their coordinate forms are stated. The concepts of an (admissible) sign type of type Φ
and an ST -class of Wa are defined in §5, where Φ is the root system determined by Wa . These two
concepts are essentially the same. A result on the number of ST -classes of Wa is stated. Then in
∼
§§6-7, we give a detailed description for all the cells of affine Weyl groups of types Al , l ≥ 1, and
∼
B 3 . In §8, we state some partial results on the cells of the affine Weyl groups of arbitrary types or
∼
type D4 . These results include the description for the lowest two-sided cell W(ν) of an affine Weyl
group Wa and all the left cells of Wa in W(ν) , the description for all the two-sided cells of the affine
∼
Weyl group of type D4 , and some relations among the cells of an affine Weyl group.
§2 The definition of cells.
2.1 Let u be an indeterminate and let A = Z[u, u−1 ] be the ring of Laurent polynomials in u.
Let W = (W, S) be a Coxeter group with S the set of Coxeter generators. Let ≤ be the Bruhat
order on W and let l(x) be the length function of W . We denote by H the Hecke algebra associated
to W : H is a free A-module with basis Tw , w ∈ W , and the multiplication is defined by
Tw Tw0 = Tww0 , if l(ww0 ) = l(w) + l(w0 ),
(Ts + u)(Ts − u−1 ) = 0, if s ∈ S.
As an A-module, H also has the basis (Cw )w∈W :
Cw =
X
ul(w)−l(y) Py,w (u−2 )Ty ,
y≤w
where the Py,w (u) are known as the Kazhdan-Lusztig polynomials in u, which satisfy that deg Py,w ≤
1
2 (l(w) − l(y) − 1) if y < w and Pw,w = 1.
Cell theory of affine Weyl groups
3
2.2 Given y, w ∈ W, write y ≺ w if y < w and deg Py,w = 12 (l(w) − l(y) − 1). Write y—w if either
y ≺ w or w ≺ y
For w ∈ W, define L(w) = {s ∈ S|sw < w} and R(w) = {s ∈ S|ws < w}. We define a preorder
≤L on W as follows. Write y ≤L w if there is a sequence of elements y0 = y, y1 , ..., yr = w in W
such that for every i, 1 ≤ i ≤ r, yi−1 —yi and L(yi−1 ) 6⊂ L(yi ). Write y ∼L w, if y ≤L w ≤L y. The
equivalence classes with respect to ≤L are called left cells of W . Write y ∼R w, if y −1 ∼L w−1 . The
corresponding equivalence classes are called right cells of W . Write y ≤LR w, if there is a sequence
of elements y0 = y, y1 , ..., yr = w in W such that for every i, 1 ≤ i ≤ r, either yi−1 ≤L yi or
−1
yi−1
≤L yi−1 . Write y ∼LR w, if y ≤LR w ≤LR y. The corresponding equivalence classes are called
two-sided cells of W .
2.3 It is known that each cell of W (i.e. left, right or two-sided cell) affords a representation of
W (resp. H) [1].
For example, let Γ be a left cell of W and let EΓ be the free A-module with basis {ew }w∈Γ . Define,
for each s ∈ S, an endomorphism Cs : EΓ −→ EΓ by

(u + u−1 )ew , if s ∈ L(w),


 X

µ(y, w)ey , if s ∈
/ L(w) .
Cs (ew ) =

y−−w


 sy<y
y∈Γ
1
where µ(y, w) = µ(w, y) is the coefficient of u 2 (|l(w)−l(y)|−1) in Py,w if y ≺ w, or in Pw,y if w ≺ y.
Conversely, any irreducible representation of W (resp. H) occurs as a component of a representation afforded by some cell of W .
§3 SL-classes, P L-classes and generalized τ -invariant.
We still assume that (W, S) is a Coxeter group in this section.
3.1 Fix a pair of elements s, t ∈ S with o(st) > 2, where the notation o(x) stands for the order of
x. We define
DL (s, t) = {w ∈ W |L(w) ∩ {s, t} has exactly one element}.
DR (s, t) = {w ∈ W |R(w) ∩ {s, t} has exactly one element}.
If w ∈ DL (s, t) then at least one of sw, tw is in DL (s, t). We define ∗ w to be the subset {sw, tw}
if both of sw, tw are in DL (s, t), or to be {w0 , w0 } if exactly one of sw, tw, say w0 , is in DL (s, t).
In particular, the latter case always happens in the case when o(s, t) = 3, in that case, we identify
{w0 , w0 } with the element w0 .
On the other hand, when w ∈ DR (s, t), we can defined w∗ in the similar way as the above by
replacing sw, tw by ws, wt.
Clearly, if ∗ w = {w0 , w00 } then w0 ∼L w ∼L w00 . The map w 7→ ∗ w, when it makes sense, is called
a left star operation on w, Note that the resulting subset ∗ w under a left star operation is dependent
of the choice of the pair s, t ∈ S. If w0 ∈ ∗ w then we say that w0 can be obtained from w by a left
star operation. Clearly, in that case, w can be obtained from w0 by a left star operation, also.
3.2 We define an equivalence relation ∼SL on W generated by x ∼SL tx with t ∈ S, x ∈ W ,
L(x) 6⊂ L(tw) and L(x) 6⊃ L(tw). We call the corresponding equivalence classes by SL-classes.
4
Jian-yi Shi
Clearly, two elements of W are in the same SL-class if and only if one can be obtained from the
another by a succession of left star operations. We also see that every SL-class of W is contained
in some left cell of W .
3.3 If y, w ∈ DL (s, t) and y—w for some s, t ∈ S with o(s, t) > 2, and if the left star operations
w 7→ ∗ w and y 7→ ∗ y are both applied in DL (s, t), then by a result of Lusztig [2], there exist
some elements w0 ∈ ∗ w and y 0 ∈ ∗ y such that y 0 —w0 . In this case, we say that w0 and y 0 can be
obtained from w and y respectively by the same left star operation. More generally, if there exist
two sequences of elements y0 = y, y1 , ..., yr = y 0 and w0 = w, w1 , ..., wr = w0 in W for some
r ≥ 0 such that for every i, 1 ≤ i ≤ r, yi and wi can be obtained from yi−1 and wi−1 respectively
by the same left star operation, then we say that y 0 can be obtained from y by the same succession
of left star operations as w0 from w. By definition, we have y—w and y 0 —w0 . If they satisfy the
additional condition that either L(y) 6⊂ L(w), L(w0 ) 6⊂ L(y 0 ) or L(w) 6⊂ L(y), L(y 0 ) 6⊂ L(w0 ) hold,
then this implies that y ∼L w, in that case, we call {yi , wi } a left primitive pair for any i, 0 ≤ i ≤ r.
Clearly, if {y, w} forms a left primitive pair, then y ∼L w.
3.4 For x, y ∈ W , write x ∼P L y, if there exists a sequence of elements x0 = x, x1 , ..., xr = y in
W for some r ≥ 0 such that for every i, 1 ≤ i ≤ r, {xi−1 , xi } forms a left primitive pair. This is an
equivalence relation on W . The corresponding equivalence classes of W are called the P L-classes of
W.
By definition, we see that every P L-class of W is a union of SL-classes of W and is contained in
some left cell of W .
3.5 For any subset K ⊆ W , we call K −1 = {x | x−1 ∈ K} the inverse set of K. Thus we call a
subset of W a right primitive pair, an SR-class or a P R-class of W if its inverse set is a left primitive
pair, an SL-class or a P L-class of W , respectively.
3.6 We say that w, y ∈ W are equivalent to order zero, written w ≈0 y, if R(w) = R(y).
Inductively, we define equivalence to order n for n ≥ 1. We say that w, y are equivalent to order
n. written w ≈n y, if w ≈n−1 y and for every s, t ∈ S with o(st) = 3 and y, w ∈ DR (s, t), we have
y 0 ≈n−1 w0 , where y 0 = y ∗ and w0 = w∗ with respect to s, t. We say that w, y have the same
generalized τ -invariant if w ≈n y for any n ≥ 0.
It is well known that if w ∼L y in W then w, y have the same generalized τ -invariant [1].
All the elements of W having the same τ -invariant form an equivalence class of W , called a left
V -cell of W .
There are the following inclusion relations.
{SL-classes of W } ≤ {P L-classes of W } ≤ {left cells of W } ≤ {left V -cells of W },
where for two sets X, Y of subsets of a set, the expression X ≤ Y means that every element of X
is contained in some element of Y and that every element of Y contains some element of X also.
§4 Affine Weyl groups.
From now on, we concentrate our attention to the affine Weyl group (Wa , S). We assume that
Wa is irreducible once and forever.
Theorem 4.1 [7]. The Coxeter graph of an affine Weyl group (Wa , S) has one of the following
forms.
Cell theory of affine Weyl groups
∼
A1
∼
Al (l ≥ 2)
∼
B l (l ≥ 3)
∼
C l (l ≥ 2)
∼
Dl (l ≥ 4)
∼
E6
∼
E7
∼
E8
0
◦
∞
2
◦
|
|
|
◦
1
1
◦
...
◦
0
1
◦
5
l−1
◦
|
|
|
◦
l
2
◦
|
|
|
|
◦
0
3
◦
...
l−1
◦
4
l
◦
1
◦
2
◦
...
l−1
◦
4
l
◦
1
◦
2
◦
|
|
|
|
◦
0
3
◦
...
1
◦
3
◦
4
◦
|
|
|
|
◦2
|
|
|
|
◦0
5
◦
6
◦
0
◦
1
◦
3
◦
4
◦
|
|
|
|
◦
2
1
◦
3
◦
4
◦
|
|
|
|
◦
2
5
◦
0
◦
4
l−3
◦
l−2
◦
|
|
|
|
◦
l
l−1
◦
5
◦
6
◦
7
◦
6
◦
7
◦
8
◦
0
◦
6
Jian-yi Shi
∼
F4
∼
G2
0
◦
1
◦
0
◦
1
◦
2
◦
6
4
3
◦
4
◦
2
◦
∼
where X denotes the type of the corresponding affine Weyl group Wa (Sometimes we use the notation
∼
Wa (X) instead of Wa to indicate the type of Wa ). These groups are pairwise non-isomorphic.
∼
4.2 An affine Weyl group Wa of type X can be defined geometrically as follows [8].
Let Φ be the root system of type X and let E be the euclidean space spanned by Φ with inner
product h , i such that |α|2 = hα, αi = 1 for any short root α ∈ Φ. Then the affine Weyl group Wa
can be regarded as a group of right isometric transformations on E. More precisely, let W be the
Weyl group of Φ generated by the reflections sα on E for α ∈ Φ: sα sends x ∈ E to x − hx, α∨ i α,
where α∨ = 2α/ hα, αi . Let Q denote the root lattice ZΦ. Let N denote the group consisting of all
translations Tλ , λ ∈ Q on E: Tλ sends x to x + λ. Then Wa can be regarded as the semi-direct
product W 7× N .
Let Π = {α1 , ..., αl } be a choice of simple root system of Φ and let Φ+ be the corresponding
positive root system. Let −α0 be the highest short root in Φ. We define s0 = sα0 · T−α0 , and
si = sαi , 1 ≤ i ≤ l. Then S = {s0 , s1 , ..., sl } forms a Coxeter generator set of Wa . By properly
numbering these simple roots, we may assume that the subscripts of s0 , s1 , ..., sl are compatible
with the Coxeter graph of Wa (see Theorem 4.1).
4.3 For any α ∈ Φ+ and k ∈ Z, we define a hyperplane
Hα;k = {v ∈ E | hv, α∨ i = k}
and a stripe
1
1
Hα;k
= H−α;−k
= {v ∈ E | k < hv, α∨ i < k + 1}
We call any non-empty simplex of
E−
[
Hα;k
α∈Φ
k∈Z
T
1
an alcove of E. Each alcove of E has the form α∈Φ Hα;k
for a Φ-tuple (kα )α∈Φ over Z. In
α
T
1
particular, A1 = α∈Φ Hα;0 is an alcove of E.
It is well known that the right action of Wa on E induces a simply transitive permutation on the
set Ξ of alcoves of E. Hence we have a bijection w 7→ (A1 )w = Aw from Wa to Ξ. We shall identify
w with Aw .
4.4 Since a Φ-tuple over Z does not always determine an alcove as in the above,
to
T it is desirable
1
give a necessary and sufficient condition on a Φ-tuple (kα )α∈Φ over Z such that α∈Φ Hα,kα ∈ Ξ.
For this purpose, the author defined a set E(Φ) consisting of all Φ-tuples k = (kα )α∈Φ which
satisfy the following two conditions.
(a) kα = −kα ∈ Z, for α ∈ Φ.
(b) For any α, β ∈ Φ+ with α + β ∈ Φ+ , the inequality
(4.5)
|α|2 kα + |β|2 kβ < |α + β|2 (kα+β + 1) < |α|2 kα + |β|2 kβ + |α|2 + |β|2 + |α + β|2
holds. Then the author showed
Cell theory of affine Weyl groups
7
∼
Theorem 4.6 [8]. Let Wa be an affine Weyl group of type X and let Φ be the root system of type
X. Then there exists a bijective map
w=
\
1
Hα;k(w,α)
7→ kw = (k(w, α))α∈Φ
α∈Φ
from Wa to E(Φ).
The above result characterizes a Φ-tuple (kα )α∈Φ over Z which corresponds to an alcove
Ξ.
T
α∈Φ
1
Hα;k
∈
α
As in the above theorem, we call kw the coordinate form of w. We shall identify kw with w. We
can see from the following results that the coordinate forms are very useful in the study of the affine
Weyl groups.
Proposition 4.7 [8]. Let w = (k(w, α))α∈Φ ∈ Wa . Write w = wTλ with w ∈ W and λ ∈ Q. Then
(a) for any α ∈ Φ, we have
(4.8)
k(w, α) = hλ, α∨ i + k(w, α)
where the integer k(w, α) is given by
½
k(w, α) =
(b)
(c)
(d)
(e)
0,
−1,
if (α)w −1 ∈ Φ+ ,
if (α)w −1 ∈ Φ− .
L(w) = {sj ∈ S | k(w, (αj )w) > 0}.
R(w) =P
{sj ∈ S | k(w, αj ) < 0}.
l(w) = α∈Φ+ |k(w, α)|.
Let w0 = sj w with w ∈ Wa and 0 ≤ j ≤ l. Then for any α ∈ Φ, we have
k(w0 , α) = k(w, α) + k(sj , (α)w −1 ).
(f ) Let w0 = wsj with w ∈ Wa and 0 ≤ j ≤ l. Then for any α ∈ Φ, we have
k(w0 , α) = k(w, (α)sj ) + k(sj , α).
(g) k(w−1 , α) = k(w, −(α)w) for any α ∈ Φ.
§5 Sign types.
5.1 Let Φ be an irreducible root system of type 6= G2 and let Φ+ be a choice of positive root
system of Φ. In his paper [9], the author introduced a sign type of type Φ.
A Φ-tuple X = (Xα )α∈Φ is called a sign type of type Φ (or briefly, a sign type), if the set
{Xα , X−α } is either {o,o} or {+, −} for any α ∈ Φ. Since X is determined uniquely by the Φ+ tuple (Xα )α∈Φ+ . We shall identify (Xα )α∈Φ+ with X. Let Θ = Θ(Φ) be the set of all sign types of
type Φ. We now define the following sets.
∆1 =
+
++,
+
+o,
+
+−,
o
+−,
−
+−,
+
o+,
+
o o,
o
o o,
o
o−,
−
o−,
+
−+,
o
−+,
−
−+,
o
−o,
−
−o,
−
−−
.
8
Jian-yi Shi

 o
∆2 = o o

o,
o
+o
o,
+
−−
−,
o
+o
+,
+
+−
−,
−
oo
o,
+
+−
+,
−
−o
o,
−
o+
o,
−
−o
−,
−
o+
+,
+
+o
+,
−
−+
o,
o
o−
o,
o
++
+,
o
o−
−,
−
++
+,
o
+−
o,
−
−+
+,
o
−−
−,
−
−+
−,
+
o−
−,
+
+−
o,
−
−−
−

+ 
++ .

+
For any subsystem Φ0 of Φ, Φ0+ = Φ+ ∩ Φ0 is a positive subsystem of Φ.
Given an irreducible positive subsystem Φ0+ of Φ of rank 2, we say that a sign type (Xα )α∈Φ0+ is
admissible, if one of the following conditions is satisfied:
(a) Φ0+ has type A2 , say Φ0+ = {α, β, α + β}. Then
Xα+β
∈ ∆1
Xα
Xβ
(b) Φ0+ has type B2 , say Φ0+ = {α, β, α + β, 2α + β}. Then
Xβ
Xα+β
Xα ∈ ∆2 .
X2α+β
We say that a sign type (Xα )α∈Φ is admissible if for any irreducible positive subsystem Φ0+ of
Φ of rank 2, the sign type (Xα )α∈Φ0+ is admissible. Let Θ = Θ(Φ) be the set of all admissible sign
types in Θ(Φ).
Remark 5.2. In the case when Φ is of type G2 , we can also define a sign type (resp. an admissible
sign type) of type Φ in the same way as above [9]. So in the subsequent discussion involving the
sign types, the restriction on the type Φ can be removed.
The author showed
∼
Theorem 5.3 [9]. Let Wa be an affine Weyl group of type X and let Φ be the root system of type
X. Then there exists a surjective map ζ : Wa −→ Θ which sends x = (k(x, α))α∈Φ to X = (Xα )α∈Φ
such that for any α ∈ Φ,
k(x, α) > 0 =⇒ Xα = +,
k(x, α) = 0 =⇒ Xα = o,
k(x, α) < 0 =⇒ Xα = −.
5.4 We call X the sign type of x in the above theorem. We usually denote the sign types of
elements x, y, ... of Wa by the corresponding capital latter X, Y, ....
Thus we have a decomposition of Wa :
[
Wa =
ζ −1 (X)
X∈Θ
Cell theory of affine Weyl groups
9
We call these ζ −1 (X)’s the ST -classes of Wa . By Theorem 5.3, we can identify Θ with the set of all
ST -classes of Wa .
5.5 There is a geometrical interpretation of admissible
sign types. Let Υ = {Hα;k | α ∈ Φ+ , k =
S
0, 1}. Then the connected components of P = E− H∈Υ H are all open simplices of E. By definition,
we see that any alcove of E is contained in some connected component of P and that two alcoves
correspond to the same sign type if and only if they are in the same connected component of P . So
∼
by Theorem 5.3, the map ζ induces a bijection between the set P of connected components of P
∼
and the set Θ. Thus Θ can also be identified with P under this correspondence.
5.6 Given a subset K of Wa , we say K is left (resp. right) connected, if for any x, y ∈ K, there
∈S
exists a sequence of elements x0 = x, x1 , ..., xr = y in K for some r ≥ 0 such that xi−1 x−1
i
(resp. x−1
x
∈
S)
for
every
i,
1
≤
i
≤
r.
i−1 i
Clearly, any SL-class (resp. SR-class) of Wa is left (resp. right) connected.
The following properties of the ST -classes of Wa were showed by the author.
Theorem 5.7 [9]. (a) Any ST -class of Wa is left connected and contains a unique shortest element.
(b) The number of ST -classes of Wa is equal to (h + 1)l , where h and l are the Coxeter number and
the rank of Φ, respectively.
Remark 5.8 The assertion in Theorem 5.7 (b) was conjectured by R. W. Carter.
∼
§6 The cells of Wa (An−1 ), n ≥ 2.
6.1 In the cell theory of affine Weyl groups Wa , the most important thing is to describe the cells
of Wa as explicit as possible. But up to now, this has been done for all the cells of Wa only in the
∼
∼
∼
∼
∼
cases when Wa is of type Al , l ≥ 1, B 2 , B 3 , C 3 and G2 [7] [4] [2] [12] [15]. For other cases, there
are some partial results. In the subsequent sections, we shall mainly introduce the results of Du, Xi
and the author on this aspect.
∼
6.2 It is well known that Wa (An−1 ) can be defined as a permutation group on Z as follows.
(
∼
Wa (An−1 ) =
σ : Z → Z | σ(i + n) = σ(i) + n, for any i, and
n
X
i=1
σ(i) =
n
X
)
i .
i=1
∼
Let si ∈ Wa (An−1 ), 0 ≤ i < n, corresponds to the permutation

if t 6= i, i + 1,
 t
t 7→ t − 1, if
t = i + 1,

t + 1, if
t = i.
where j 7→ j is the natural map from Z to Z/(n). Then S = {s0 , s1 , ..., sn−1 } forms a Coxeter
∼
generator set of Wa (An−1 ) such that for 0 ≤ i, j

 1,
o(si sj ) = 2,

3,
≤ n,
if
if
if
i = j,
i 6= j, j ± 1,
i=j±1
10
Jian-yi Shi
6.3 The relation between the above definition and the previous geometrical definition of the
∼
∼
group Wa (An−1 ) given before (see §4) is as follows. For any w ∈ Wa (An−1 ), let (k(w, α))α∈Φ be the
coordinate form of w. Then we have
(6.4)
w(j) − w(i) = k(w, αij ) · n + r(w, αij ), if 1 ≤ i < j ≤ n,
where αij = αi + αi+1 + ... + αj−1 and the integer r(w, αij ) satisfies 1 ≤ r(w, αij ) < n [7]. The
following result can be easily checked
∼
Proposition 6.5 [7]. For any w ∈ Wa (An−1 ), we have
(a) L(w) = {st ∈ S | w−1 (t + 1) < w−1 (t)}.
(b) R(w) = {st ∈ S | w(t + 1) < w(t)}.
P ¯¯h y(j)−y(i) i¯¯
(c) l(w) =
¯
¯,
n
1≤i<j≤n
where [h] is the integer part of h for any rational number h.
∼
6.6 With each σ ∈ Wa (An−1 ), Lusztig associated a set {dk | 1 ≤ k ≤ n} of integers as follows:
dk is the maximal cardinal of a subset of Z whose elements are non-congruent to each other mod n
and which is a disjoint union of k subsets each of which has its natural order reversed by σ. By a
result of C. Greene, we have
d1 ≥ d2 − d1 ≥ d3 − d2 ≥ ... ≥ dn − dn−1 .
∼
Thus each element σ of Wa (An−1 ) determines a partition φ(σ) = {d1 ≥ d2 − d1 ≥ ... ≥ dn − dn−1 }
∼
of n. Let Λn be the set of partitions of n. Then we have a well defined map φ : Wa (An−1 ) −→ Λn .
Clearly, φ is surjective. Lusztig had the following
∼
Conjecture 6.7 [3]. (a) Let σ, σ 0 ∈ Wa (An−1 ). Then
σ ∼LR σ 0 ⇔ φ(σ) = φ(σ 0 ).
(b) Given λ ∈ Λn , let µ = {µ1 ≥ ... ≥ µn } be the dual of λ. Then the number of left cells in φ−1 (λ)
is equal to
n!
Qm
j=1 µj !
This conjecture has been proved by Lusztig and the author. The author proved part (b) and the
direction ” ⇐ ” of part (a) [7]. Then Lusztig proved the remaining part [4].
6.8 The proof of the author is purely combinatorial. The main tool of his proof is an affine matrix
of period n. Each affine matrix is a generalized permutation matrix, say M = (aij )i,j∈Z such that
½
aij =
1,
0,
if j = σ(i),
otherwise.
Cell theory of affine Weyl groups
∼
11
∼
for some σ ∈ Wa (An−1 ). Then Wa (An−1 ) can be identified with the set of all affine matrices of
period n.
6.9 Let Cn be the set of all Young tableaux X of rank n such that the numbers in each row of X
increase from left to right. Then Cn contains all standard Young tableaux of rank n as its subset.
Example 6.10
1
2
5
are two elements of C7 , where the first
3 7
2 5 7
4
and 1 4
6
3 6
one is standard but the second one is not.
6.11 Each element X of Cn determines a unique partition ψ(X) = {λ1 ≥ ... ≥ λt } of n, where
λi is the cardinal of the set of numbers in the i-th column of X. Thus X 7→ ψ(X) gives a surjective
map ψ : Cn 7→ Λn . It is easily seen that for any λ ∈ Λn , the cardinal of the set ψ −1 (λ) is
n!
Qm
j=1
µj !
,
where {µ1 ≥ ... ≥ µm } ∈ Λn is the dual partition of λ.
∼
The author defined a surjective map T : Wa (An−1 ) 7→ Cn in terms of affine matrices of period n
as follows.
6.12 For any λ = {λ1 ≥ ... ≥ λr } ∈ Λn , let µ = {µ1 ≥ ... ≥ µm } be the dual partition of λ.
∼
We call an element w ∈ Wa (An−1 ) a normalized element of type λ if there exists a set of non-zero
entries {e(i + t, j(t)) | 1 ≤ t ≤ n} of w for some i ∈ Z, where e(i, j) denotes the entry of w in the
position (i, j), satisfying the following conditions: Let
αu = i +
u
X
λh , 0 ≤ u ≤ r.
h=1
Then (i) j(αu − 1) < j(αu − 2) < ... < j(αu−1 ),
(ii) For any t with 1 ≤ t ≤ λ1 , we have
j(λ1 − t) < j(λ2 − t) < ... < j(λµt − t) < j(λ1 − t) + n.
In that case, we can check that φ(w) = λ.
Example 6.13 Let n = 9, λ = {4 ≥ 3 ≥ 2}. Then µ = {3 ≥ 3 ≥ 2 ≥ 1} is the dual partition of
λ. The following element y is a normalized element of type λ:

text1strow




1







1st column

1
1
1
1
1
1
1
1












12
Jian-yi Shi
+where, y has the non-zero entry set { e(1, 8), e(2, 6), e(3, 3), e(4, 0), e(5, 7), e(6, 5), e(7, 2), e(8, 10),
e(9, 4) }, which has the required properties.
∼
Let N be the set of all normalized elements of Wa (An−1 ).
6.14 We defined a map T : N 7→ Cn as follows. Let w be an element of N of type λ with a
set of non-zero entries {e(i + t, j(t)) | 1 ≤ t ≤ n} satisfying the conditions as above. We define a
Young tableau of Cn such that the set Xt of numbers in the t-th row of this tableau satisfying that
X t = {j(αh − t) | 1 ≤ h ≤ µt } for 1 ≤ t ≤ m, where i 7→ i is the natural map from Z to Z/nZ and
for any subset X ⊂ Z, we denote X = {x | x ∈ X} ⊆ Z/nZ. We see that such a Young tableau
is only dependent on w and is independent on the choice of this entry set of w. Let us denote this
tableau by T (w). Then the map T is well defined. Clearly, T is surjective.
In the above example, we have
2 4 9
1 3 5
T (y) =
6 7
8
The set N has the following properties.
∼
Proposition 6.15 [7]. (a) The intersection of the set N with each SL-class of Wa (An−1 ) is nonempty.
∼
(b) The intersection of the set N with each P L-class of Wa (An−1 ) is non-empty which has the form
T −1 (X) for some X ∈ Cn . Conversely, any T −1 (X) with X ∈ Cn is the intersection of N with
∼
∼
some P L-class of Wa (An−1 ). In particular, T −1 (X) is contained in some left cell of Wa (An−1 ).
∼
(c) If X 6= X 0 in Cn , then T −1 (X) and T −1 (X 0 ) belong to two different left cells of Wa (An−1 ).
6.16 By Proposition 6.15(a), we can extend the map T : N 7→ Cn to the whole group T :
∼
∼
Wa (An−1 ) 7→ Cn in the following way. For any w ∈ Wa (An−1 ), let y be any element of N with
y ∼SL w (There is an algorithm to find such an element y from w). We define T (w) by T (y).
6.17 Robinson defined a map from the symmetric group Sn to the set of standard Young tableaux
of the same shape and size n: σ 7→ (P (σ), P (σ −1 )). Then Schensted proved that this map is bijective.
This map is called the Robinson-Schensted map which has the following properties: A left cell of Sn
is the set of elements of Sn corresponding to the set of such pairs (P, Q) with P fixed; A two-sided
cell of Sn is the set of elements of Sn corresponding to the set of such pairs (P, Q) with P, Q of
fixed shape.
Now we are ready to state
Theorem 6.18 [7]. (a) The diagram
∼
Wa (An−1 )
φ
T
−→
Cn
ψ
Λn
is commutative. The maps in this diagram are all surjective.
∼
(b) For x, y ∈ Wa (An−1 ), x ∼L y ⇔ T (x) = T (y).
Cell theory of affine Weyl groups
∼
(c) For any λ ∈ Λn , φ−1 (λ) is in some two-sided cell of Wa (An−1 ) which is a union of
13
Qmn!
∼
j=1
µj !
left
(resp. right) cells of Wa (An−1 ), where {µ1 ≥ ... ≥ µm } is the dual partition of λ
∼
(d) Let W be the subgroup of Wa (An−1 ) generated by s1 , ..., sn−1 , which is isomorphic to Sn . Then
T (w) = P (w) for any w ∈ W . Thus the map T generalizes the Robinson-Schensted map for Sn .
6.19 By using Theorem 6.18, Lusztig proved that for any λ ∈ Λn , the fibre φ−1 (λ) is itself a
∼
two-sided cell of Wa (An−1 ) [4]. His proof relies on a deep result of intersection cohomology. He
introduced a function a on any affine Weyl group Wa which has the following properties.
Theorem 6.20 [2] [5]. (a) For any z ∈ Wa , a(z) is an integer satisfying 0 ≤ a(z) ≤ 21 |Φ|, where
Φ is the root system determined by Wa (see §4).
(b) The function a is constant on any two-sided cell of Wa .
(c) If x, y ∈ Wa satisfy x ≤L y and a(x) = a(y) then x ∼L y.
By Theorem 6.20 (c), we see that each two-sided cell Ω of Wa is a smallest non-empty set with
the property that Ω is both a union of left cells of Wa and a union of right cells of Wa . This, together
∼
with Theorem 6.18 (c), implies that for any λ ∈ Λn , φ−1 (λ) is a single two-sided cell of Wa (An−1 ).
6.21 Recall that in any Coxeter group W , we have
{P L-classes of W } ≤ {left cells of W } ≤ {left V -cells of W } (see §3)
The author showed that these three families of equivalence classes coincide with each other in the
∼
case of Wa (An−1 ). That is,
∼
Theorem 6.22 [7]. Let K ⊂ Wa (An−1 ). Then the following statements are equivalent.
∼
(a) K is a P L-class of Wa (An−1 ).
∼
(b) K is a left cell of Wa (An−1 ).
∼
(c) K is a left V -cell of Wa (An−1 ).
(d) There exists some X ∈ Cn such that K = T −1 (X).
∼
(e) There exists a two-sided cell Ω of Wa (An−1 ) such that K occurs as a maximal left connected
subset of Ω.
∼
Remark 6.23. The above theorem gives five different ways to describe a left cell of Wa (An−1 ).
However, the statements (a), (c) and (d) will be no larger equivalent to (b) for a general affine Weyl
∼
group Wa instead of Wa (An−1 ). But one may expect that the statements (b) and (e) would be still
equivalent to each other for a general Wa .
∼
6.24 The image of an element w ∈ Wa (An−1 ) under the map φ can be interpreted in terms of
its coordinate form as follows. Let φ(w) = {λ1 ≥ ... ≥ λr } ∈ Λn . Then for 1 ≤ t ≤ r, the integer
Pt
S
i=1 λi is the maximal cardinal of subsets X of n = {1, 2, ..., n}, where X =
1≤h≤t Xh is such
that if i, j ∈ Xh , i < j and 1 ≤ h ≤ t then k(w, αij ) 6= 0 with αij = αi + αi+1 + ... + αj−1 . This
∼
interpretation of φ acturally gives an alternative description of two-sided cells of Wa (An−1 ). By
Theorem 6.22 (b), (c), the following result is immediate.
14
Jian-yi Shi
∼
∼
Corollary 6.25 [7]. Any left cell of Wa (An−1 ) is a union of some ST -classes of Wa (An−1 ).
∼
Note that the conclusion of Corollary 6.25 will be no longer valid when Wa (An−1 ) is replaced by
∼
a general affine Weyl group Wa (e.g. a counterexample can be found in the case of Wa = Wa (B 2 )).
Nevertheless, we may still expect
∼
∼
∼
Conjecture 6.26. If the affine Weyl group Wa is of simple laced type A, D or E, then any left cell
of Wa is a union of some ST -classes of Wa .
∼
Finally, there is a formula for computing the a-value of an element w ∈ Wa (An−1 ).
Pm
Pm
(6.27)
a(w) = i=1 (i − 1)µi = i=1 iµi − n,
where {µ1 ≥ ... ≥ µm } ∈ Λn is the dual partition of φ(w).
∼
§7 The cells of Wa (B 3 ).
7.1 As Lusztig has made explicit description of all cells for the affine Weyl groups Wa of rank
≤ 2, it is natural to ask if one can do the same with Wa of higher rank. Thus the groups Wa of rank
∼
∼
∼
∼
3 are taken into consideration. They are Wa (A3 ), Wa (B 3 ) and Wa (C 3 ). The case Wa (A3 ) has been
∼
∼
included in §6. The cases Wa (B 3 ) and Wa (C 3 ) were treated by Du Jie and R. Bédard, respectively.
∼
Now we shall state the results of Du on the cells of the group Wa (B 3 ).
∼
7.2 In his paper [12], Du first found out all SR-classes of Wa (B 3 ), each of which is represented
∼
by an element contained in it. Then he merged these SR-classes into P R-classes of Wa (B 3 ) and
∼
wrote down a set of representatives for the P R-classes of Wa (B 3 ) explicitly. Then he showed that,
∼
in Wa (B 3 ), the set of P R-classes coincides with the set of right cells. Finally, he merged all the right
∼
∼
cells of Wa (B 3 ) into eight two-sided cells of Wa (B 3 ).
∼
7.3 Suppose that the set S = {s0 , s1 , s2 , s3 } of Coxeter generators of Wa (B 3 ) satisfies the
conditions that o(s0 s2 ) = o(s1 s2 ) = 3 and o(s3 s4 ) = 4. That is, the subscript i of si is compatible
with the Coxeter graph
∼
For J ⊆ {0, 1, 2, 3}, let wJ be the longest element of WJ which is the subgroup of Wa (B 3 )
generated by {si | i ∈ J}. Following Du [12], we define some elements as below.
a123 = w{1,2,3} , â023 = w{0,2,3} , b012 = w{0,1,2} , c23 = w{2,3} ,
d013 = w{0,1,3} , e13 = w{1,3} , f01 = w{0,1} , g0 = s0 , h∅ = 1.
Then we define some more elements which occur as vertices in the following diagrams
i
where two vertices x, y are jointed by a labelled segment −−− if {x, y} forms a left primitive pair
with xy −1 = si . The lengths of the elements in each diagram increase from the top to bottom. The
subscripts of these elements w indicate the subsets L(w) of S, for example, L(a013 ) = {s0 , s1 , s3 }.
Du showed the following results.
Cell theory of affine Weyl groups
15
∼
Theorem 7.4 [12]. (a) In Wa (B 3 ), the right cells coincide with the P R-classes.
(b) All the elements occurring as the vertices in the above diagrams form a set of representatives for
∼
∼
the right cells of Wa (B 3 ). Thus the number of the right cells of Wa (B 3 ) is 112 .
(c) Let X̂ be the set of all elements occurring as the vertices in the diagram (X), let  = Â1 ∪ Â2
and let
Σ = {Â, B̂, Ĉ, D̂, Ê, F̂ , Ĝ, Ĥ}
∼
Then any X̂ ∈ Σ is contained in some two-sided cell of Wa (B 3 ). If X̂ 6= Ŷ in Σ then X̂ and Ŷ
∼
belong to two different two-sided cells of Wa (B 3 ).
∼
7.5 Denote the two-sided cell of Wa (B 3 ) containing X̂ by the corresponding bold face X. Then
by Theorem 7.4, we see that
[
∼
X
Wa (B 3 ) =
X̂∈Σ
∼
is the decomposition of Wa (B 3 ) into two-sided cells and that any X̂ ∈ Σ is a set of representatives
∼
of right cells of Wa (B 3 ) in X.
∼
To describe the two-sided cells of Wa (B 3 ) more precisely, Du defined the following subsets:
S(i) = {I ⊂ S | l(wI ) = i}, for i ≥ 0
S(2)1 = {{s0 , s1 }}, S(2)2 = S(2) − S(2)1 .
∼
For any i with i 6= 2 and S(i) 6= ∅, define by C(i)0 the set of all elements w of Wa (B 3 ) which
satisfy the following conditions:
∼
(a) w = x · wI · z for some x, z ∈ Wa (B 3 ) and I ∈ S(i).
∼
(b) w 6= x · wJ · z for any x, z ∈ Wa (B 3 ) and J ⊆ S with J 6⊂
means that x = yz and l(x) = l(y) + l(z).
Si
j=0
S(j), where the notation x = y · z
∼
Define C(2)j , j = 1, 2, to be the set of all elements w ∈ Wa (B 3 ) such that
∼
(a0 ) w = x · wI · z for some x, z ∈ Wa (B 3 ) and I ∈ S(2)j .
∼
(b0 ) w 6= x · wJ · z for any x, z ∈ Wa (B 3 ) and J ∈
/ S(0) ∪ S(1) ∪ S(2)j .
Then we define the following sets.
C(0) = {1}, C(i) = C(i)0 , i = 1, 6, 9.
C(3) = C(3)0 − C(3)00 and C(4) = C(4)0 ∪ C(3)00 ,
∼
where C(3)00 = {x · y · z | y ∈ {s1 s2 s1 s3 s2 s1 , s0 s2 s0 s3 s2 s0 }, x, z ∈ Wa (B 3 )}. Therefore Du proved
16
Jian-yi Shi
Theorem 7.6 [12]. (a) A = C(9), B = C(6), C = C(4), D = C(3), E = C(2)2 , F = C(2)1 , G =
C(1), H = C(0). Thus
∼
Wa (B 3 ) = C(0) ∪ C(1) ∪ C(2)1 ∪ C(2)2 ∪ C(3) ∪ C(4) ∪ C(6) ∪ C(9)
∼
is the decomposition of Wa (B 3 ) into two-sided cells.
(b) The a-value on C(i) (resp. C(i)j ) is i, for i = 0, 1, 2, 3, 4, 6, 9.
∼
Remark 7.7. It seems likely that the number of right cells in Wa (B l ) is a square number for any
l ≥ 2.
§8 Other results on the cells of affine Weyl groups.
8.1 Although we have known how to describe all the cells of the affine Weyl groups of type
∼
Al , l ≥ 1, or of rank ≤ 3, it is still open for the same problem with respect to the other affine Weyl
groups. However, there are some partial results in this direction. Let (Wa , S) be an affine Weyl
group. For any w ∈ Wa , define δ(w) = deg P1,w . Then Lusztig showed that the inequality
a(w) ≤ l(w) − 2δ(w)
holds in general. Define
D = {w ∈ Wa | a(w) = l(w) − 2δ(w)}.
Lusztig showed that D is a finite set consisting of involutions which are called the distinguished
involutions. The following result is due to Lusztig
Theorem 8.2 [5]. (a) The number of left cells of Wa is finite.
(b) Each left cell of Wa contains a unique distinguished involution.
8.3 Define W(i) = {w ∈ Wa | a(w) = i} for i ≥ 0. Then W(i) is a union of some two-sided cells
of Wa . Lusztig showed that W(1) is a single two-sided cell of Wa consisting of all elements w 6= 1 of
Wa each of which has a unique reduced expression. He also showed that all the left cells of Wa in
W(1) are ST -classes of Wa [3]. Then Lawton described all the two-sided cells of Wa whose a-values
≤ 3 [16].
8.4 In 1986, the author showed that the set W(ν) is the unique lowest two-sided cell of Wa with
respect to the partial order ≤LR , where ν = 21 |Φ|. He gave an explicit description for the set W(ν)
and for all the left cells of Wa in W(ν) [10] [11]. Recently, Du found out all the two-sided cells of
∼
Wa (D4 ) [13]. Also, Xi got some results concerning with the relations among cells of Wa [14]. Now
we shall introduce the above results of Du, Xi and the author.
I. The lowest two-sided cell of Wa
In his paper [10], the author showed that the set W(ν) itself is a two-sided cell of Wa and he gave
the following alternative descriptions for this cell.
Cell theory of affine Weyl groups
17
Theorem 8.5 [10]. The following subsets of Wa are all the same.
(a) W(ν) .
(b) The lowest two-sided cell of Wa with respect to the partial order ≤LR .
(c) W (S) = {x · wJ · y | x, y ∈ Wa , J ∈ S}, where S is the set of all subsets J of S such that WJ is
isomorphic to the Weyl group of Φ.
(d) W = {w = (k(w, α))α∈Φ ∈ Wa | k(w, α) 6= 0 for α ∈ Φ}.
8.6 In his paper [10], the author showed the equality (a) = (b) in the above theorem by first
showing the equality
W = W (S) = W(ν)
which is the most hard part in the proof. Here we give a more direct proof for the equality (a) =
(b) as follows. Fix a set J ∈ S. Then wJ ∈ W(ν) . It is enough to show the relation x ∼LR wJ for
any x ∈ W(ν) . Write x = w · y with R(w) ∩ J = ∅ and y ∈ WJ . Let z = y −1 wJ . Then we have
x ∼R w · y · z = w · wJ ∼L wJ
by Theorem 6.20 (c) and by noting that w·wJ ≤R x, w·wJ ≤L wJ and a(w·wJ ) = a(wJ ) = a(x) = ν.
8.7 Concerning the left cells of Wa in W(ν) , the author gave the following results.
Theorem 8.8 [11]. (a) All the left cell of Wa in W(ν) are ST -classes of Wa and hence they are left
connected.
(b) Let N = {y −1 · wJ · y | J ∈ S, swJ y ∈
/ W(ν) for any s ∈ J}. Then N = W(ν) ∩ D. Hence N is a
set of representatives for the left cells of Wa in W(ν) .
(c) W(ν) consists of |W | left cells of Wa , where W is the Weyl group of Φ.
By Theorem 8.8, the key step for the proof of Theorem 8.8 is to show the equality N = D ∩ W(ν) .
The latter can be shown by making used of the structural properties of the corresponding affine
Hecke algebra [11].
8.9 Since the distinguished involutions of Wa play an important role in the cell theory of Wa , it
is desirable to give an explicit description for these elements. Let us make a conjecture for this. For
x, y ∈ Wa , let
X
Cx Cy =
hx,y,z Cz , hx,y,z ∈ Z[u, u−1 ].
z
It is known that there exists a unique element, written λ(x, y), in the set Λ(x, y) = {z ∈ Wa | hx,y,z 6=
0} satisfying the condition that
λ(x, y) ≥ w, for any w ∈ Λ(x, y) [10].
Conjecture 8.10. Let Γ be a left cell of Wa . Assume that D ∩ Γ = {d}. Then for any shortest
element x of Γ, we have d = λ(x−1 , x).
∼
II. The two-sided cells of Wa (D4 )
∼
Here we shall explain the description of Du for the two-sided cells of Wa (D4 ) [13].
18
Jian-yi Shi
∼
8.11 Let S = {si | 0 ≤ i ≤ 4} be the set of Coxeter generators of Wa (D4 ) with o(si s2 ) = 3 for all
i 6= 2. Let
S(i) = {I ⊂ S | l(wI ) = i}, for i ≥ 0.
S(2)1 = {{s0 , s1 }, {s3 , s4 }},
S(2)2 = {{s0 , s3 }, {s1 , s4 }},
S(2)3 = {{s0 , s4 }, {s1 , s3 }},
S(6)1 = {{s0 , s2 , s1 }, {s3 , s2 , s4 }},
S(6)2 = {{s0 , s2 , s3 }, {s1 , s2 , s4 }}, S(6)3 = {{s0 , s2 , s4 }, {s1 , s2 , s3 }}.
∼
For each i with S(i) 6= ∅, define by C(i) the set of all elements w ∈ Wa (D4 ) satisfying the following
conditions:
∼
(a) w = x · wI · z for some I ∈ S(i) and x, z ∈ Wa (D4 ).
∼
Si
(b) w 6= x · wJ · z for any J ∈
/ j=0 S(j) and x, z ∈ Wa (D4 ).
For each j, 1 ≤ j ≤ 3, and each i ∈ {2, 6}, define
½
¾
∼
C(i)j = w ∈ C(i) | w 6= x · wJ · z for any J ∈ S(i) − S(i)j and x, z ∈ Wa (D4 )
S3
and define C(7) = C(6) − j=1 C(6)j .
Then Du showed the following
∼
S3
S3
Theorem 8.12 [13]. (a) Wa (D4 ) = C(0) ∪ C(1) ∪ ( j=1 C(2)j ) ∪ C(3) ∪ C(4) ∪ ( j=1 C(6)j ) ∪
∼
C(7) ∪ C(12) is the decomposition of Wa (D4 ) into 12 two-sided cells.
(b) The a-value on C(i) (resp. C(i)j ) is i for any i ∈ {0, 1, 2, 3, 4, 6, 7, 12} and j = 1, 2, 3.
∼
Remark 8.13 (a) For the time being, let W0a denote an affine Weyl group of type Al , l ≥ 1, or
∼
of rank ≤ 3. Here we shall indicate two points concerning with the properties of cells for Wa (D4 )
which are different from those for W0a
Firstly, an a-value on W0a can always be realized on the subset {wJ | J ⊂ S} of W0a . In other
words, we have the equation
{a(x) | x ∈ W0a } = {a(wJ ) | J ⊂ S} = {l(wJ ) | J ⊂ S}.
∼
But this equation is no longer valid in the case Wa (D4 ) instead of W0a since
a(s1 s3 s2 s1 s3 s2 s4 s2 s1 ) = 7 ∈
/ {a(wJ ) | J ⊂ S} = {0, 1, 2, 3, 4, 6, 12}.
Secondly, the P L-classes all coincide with the left cells in W0a . But in the two-sided cell C(4) of
∼
Wa (D4 ), we can find a left cell which is a union of more than one P L-classes.
∼
(b) By detailed analysis of Du’s result, we see that in Wa (D4 ), any left cell is a union of ST -classes.
III. Some relations among cells of Wa
Recently, Xi Nanhua wrote a joint paper with Lusztig to be concerned with some relations between
the left cells and the two-sided cells of an affine Weyl group Wa [14].
For each s ∈ S, let Ys = {w ∈ Wa | R(w) ⊂ {s}}.
We say that s ∈ S is special if the subgroup WS−{s} of Wa generated by S − {s} is isomorphic to
the Weyl group of Φ.
Then the main result of Lusztig and Xi is as follows.
Cell theory of affine Weyl groups
19
Theorem 8.14 [14]. Let Ω be a two-sided cell of Wa and let s ∈ S. Then
(a) Ω ∩ Ys is non-empty and is a union of some left cells of Wa .
(b) If s is special, then Ω ∩ Ys is exactly one left cell of Wa .
(c) The number of left cells of Wa in W(ν) ∩ Ys is |WS−{s0 } |/|WS−{s} |.
The part (a) of the above theorem was showed by Xi. The crucial step in the proof of Xi is to
show the following result which is of its own interest.
Theorem 8.15 [14]. Suppose that Ω1 and Ω2 are two two-sided cells of Wa with Ω1 ≤LR Ω2 . Then
for any left cell Γ1 in Ω1 , there exists a left cell Γ2 in Ω2 with Γ1 ≤L Γ2 .
Remark 8.16 (a) If we are given a left cell Γ2 in Ω2 instead of Γ1 in Ω1 in Theorem 8.15, then by
the similar argument as that in the proof of Xi, we can find a left cell Γ1 in Ω1 such that Γ1 ≤LR Γ2
(b) Theorem 8.15 tells us that the direct graph associated with the partial ordered set of the twosided cells of Wa with respect to ≤LR is isomorphic to a subgraph of that associated with the partial
ordered set of left cells of Wa with respect to ≤L .
(c) It is well known that if x ≤L y then R(x) ⊇ R(y). In particular, if x ∼L y then R(x) = R(y) [1].
Recall the notation X ≤ Y for two sets X, Y of subsets of a set defined in §3. Then from Theorem
8.15 and Remark 8.16 (a), we see that if Ω1 and Ω2 are two two-sided cells of Wa with Ω1 ≤LR Ω2 ,
then
{R(x) | x ∈ Ω1 } ≥ {R(y) | y ∈ Ω2 }.
On the other hand, Xi made a detailed study on the properties of translation elements of Wa
in his paper [17]. He showed that the a-function of Wa reaches its next largest value on the set
of translation elements of Wa . He also showed that in Wa , any left cell consists of at most finitely
many left connected components [17]. The latter result is a progress in the direction of proving a
conjecture of Lusztig which says that any left cell of Wa is left connected [18].
References
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2. G.Lusztig, Cells in affine Weyl groups, Algebraic groups and related topics, Advanced Studies in Pure Math.,
vol. 6, Kinokunia and North-Holland, 1985, pp. 255–287.
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∼
, The two-sided cells of the affine Weyl group of type An , Inf. Dim. groups with appl. MSRI, vol. 4,
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, Cells in affine Weyl groups, II, J. Algebra 109 (1987), 536–548.
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Shi Jianyi, The Kazhdan-Lusztig cells in certain affine Weyl groups, vol. 1179, Springer-Verlag, Lecture Notes
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, Alcoves corresponding to an affine Weyl group, J. London Math. Soc. 35 (1987), 42–55.
, Sign types corresponding to an affine Weyl group, J. London Math. Soc. 35 (1987), 56–74.
, A two-sided cell in an affine Weyl group, J. London Math. Soc. 36 (1987), 407–420.
, A two-sided cell in an affine Weyl group, II, J. London Math. Soc. (1988).
∼
12. Du Jie, The decomposition into cells of the affine Weyl group of type B 3 , Comm. in Algebra (1988).
∼
13.
, Cells in the affine Weyl group of type D4 , J. Algebra (1988).
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15. R. Bédard, Cells for two Coxeter groups, Comm. in Algebra 14(7) (1987), 1253–1286.
20
Jian-yi Shi
16. G. M. Lawton, On cells in affine Weyl groups, Ph.D. Thesis, MIT (1986).
17. Xi Nanhua, Some results on the structure of the cells in affine Weyl groups, Ph.D. Thesis ECNU (1988).
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