The Partial Order on Two-Sided Cells of Certain Affine Weyl Groups

Ž .JOURNAL OF ALGEBRA 176, 607]621 1996ARTICLE NO. 0027

The Partial Order on Two-Sided Cells of Certain AffineWeyl Groups

Jian-yi Shi*

Department of Mathematics, East China Normal Uni ersity, Shanghai, 200062,People’s Republic of China

Communicated by Gordon James

Received October 10, 1994

w xIn their famous paper 6 , Kazhdan and Lusztig introduced the conceptof equivalence classes such as left cell, right cell, and two-sided cell in aCoxeter group W. We inherit the notations F , F , F , ; , ; ,L R LR L R

w x Ž .and ; in 6 . Thus w ; y resp. w ; y, resp. w ; y means thatLR LR L RŽthe elements w, y g W are in the same two-sided cell resp. left cell, resp.

.right cell of W, etc. Concerning an affine Weyl group W , Lusztig showedaŽ .that the set Cell W of two-sided cells of W is in a natural 1]1a a

Ž .correspondence with the set U G of unipotent classes in the correspond-w x Ž .ing algebraic group G 11 . We know that Cell W is a poset under theaŽ . Ž .relation F . Also, U G is a poset under the relation v F u in U G mLR

u ; v, where v is the closure of the conjugacy class v in the variety ofunipotent elements of G. Under the Lusztig correspondence, the two-sided

� 4cell c s 1 ; W is associated to the regular unipotent class of G, andW aaŽ w x.the lowest two-sided cell W see 11 of W is associated to the trivialŽ¨ . a

� 4class 1 ; G. Thus it is natural to formulate the following conjectureGŽ w x.which was suggested by Lusztig see 8, Conjecture D .

Conjecture A. There exists an order-preserving bijection between theŽ . Ž .set Cell W of two-sided cells of W and the set U G of unipotenta a

conjugacy classes of the corresponding algebraic group G.

The above conjecture has been verified in the cases when W has rankaŽ w x .F 3 see 1, 3 or by directly checking . In the present paper, we shall give

an affirmative answer of Conjecture A in the cases when W is of typeaA and when W has rank 4. That is, we haveny1 a

* Supported by the National Science Foundation of China and by the Science Foundationof the University Doctoral Program of CNEC.

607

0021-8693r96 $12.00Copyright Q 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

JIAN-YI SHI608

˜THEOREM B. If W is an affine Weyl group which has either type A ,a ny1n ) 1, or rank F 4, then Conjecture A holds in W .a

The proof of Theorem B is based mainly on two of our results, i.e.,Ž .Theorem 1.11 and Proposition 1.12 and on the knowledge of the sets T V

Ž .defined in 1.10 for all the two-sided cells V of the concerned affine Weylgroups.

The content of the paper is organized as follows. In Section 1, weestablish two results concerning the partially ordered relation on thetwo-sided cells of an affine Weyl group, i.e., Theorem 1.11 and Proposition1.12. They are crucial in the proof of our main result, Theorem B. Then we

˜ Ž .show Theorem B in the type A n ) 1 case in Section 2 and in theny1rank 4 cases in Section 3. Finally, in Section 4, we make some comments

Ž .on the set T V , where V is a two-sided cell of an affine Weyl group.

Ž .1. SOME RESULTS ON THE POSET Cell Wa

Ž .1.1. Let W s W, S be a Coxeter system; i.e., W is a Coxeter groupwith S its Coxeter generator set. We denote by l the length function of W.Let F be the Bruhat order on W. To any w g W, we associate two

Ž . � 4 Ž . � 4subsets of S: LL w s s g S ¬ sw - w and RR w s s g S ¬ ws - w .w y1 xLet A s Z u, u be the ring of Laurent polynomials in an indetermi-

Ž .nate u with integer coefficients. Then the Hecke algebra HH s HH W withrespect to W is by definition an associative algebra over A with an A-basis� 4T ¬ w g W whose multiplication rule is given byw

T T s T , if l ww9 s l w q l w9 ;Ž . Ž . Ž .w w 9 w w 9

T y uy1 T q u s 0, for s g S.Ž .Ž .s s

lŽw .y lŽ y . Ž y2 .1.2. For w g W, define C s Ý u P u T , where thew y F w y, w y� 4P ’s are Kazhdan]Lusztig polynomials. It is known that C ¬ w g Wy, w w

also forms an A-basis of HH. Thus for any x, y g W, we can write

C C s h C , with h g A.Ýx y x , y , z z x , y , zz

w xIn 8 , Lusztig defined a function a: W ª N such that for any z g W,

aŽ z . w xu h g Z u , for any x , y g W ,x , y , z

aŽ z .y1 w xu h f Z u , for some x , y g W .x , y , z

PARTIAL ORDER ON TWO-SIDED CELLS 609

Ž .We state the following property for a Coxeter system W, S :

There exists an integer N G 0 such that)Ž .N w xu h g Z u for all x , y , z g W .x , y , z

Ž .Obviously, any finite Coxeter group satisfies the property ) . LusztigŽ . Ž w x.showed that the property ) also holds for any affine Weyl group see 8 .

1.3. Kazhdan]Lusztig polynomials P , y, w g W, satisfy the propertiesy, w1 Ž Ž . Ž . .that P s 1, P s 0 if y F w, and deg P F l w y l y y 1 ifuw , w y, w y, w 2

Ž . Ž . Ž1r2.Ž lŽw .y lŽ y .y1.y - w. We denote by m y, w or m w, y the coefficient of uŽ . Ž . Ž .in P when l y F l w . We write y}w if m y, w / 0.y, w

1.4. The following results on the cells of W are well known: if x F yLŽ . Ž . Ž . Ž Ž . Ž ..resp. x F y in W, then RR x = RR y resp. LL x = LL y . In particu-R

Ž . Ž . Ž . Ž Ž .lar, the relation x ; y resp. x ; y implies RR x s RR y resp. LL xL RŽ .. Ž w x.s LL y see 6 .

Ž .1.5. A Coxeter system W, S is crystallographic if it arises from someŽ .Kac]Moody Lie algebra or group . In that case, we have that for any

Ž .s, t g S, the order o st of the product st is 1, 2, 3, 4, 6, or `. Weyl groupsand affine Weyl groups are all crystallographic. The following results wereshown by Lusztig and Springer.

Ž .PROPOSITION 1.6. Let W, S be a crystallographic Coxeter system. In theŽ . Ž . Ž . Ž .following 2 and 3 , we further assume W, S , satisfying the property ) .

Ž .1 For any x, y, z g W, all the coefficients of the Laurent polynomialw xh are non-negati e 8 .x, y, z

Ž . Ž .2 The function a is constant on a left resp. right, resp. two-sided cellŽ Ž . Ž . .G of W thus we may denote by a G the älue a w for any w g G . The

Ž . Ž . w xrelation y © w implies the inequality a w - a y for any y, w g W 10 .LR

Ž . Ž . Ž . w x3 If x, y g W satisfy y F x and a y s a x , then y ; x 10 .L L

Ž .4 If x, y g W satisfy x}y and x © y, then both relations x F yLR Lw xand x F y hold 19 .R

Ž .LEMMA 1.7. Let W, S be a crystallographic Coxeter system with theŽ .property ) . Let z, z9 g W be such that z}z9 and let s g S be such that

Ž . Ž .s g RR z9 _ RR z . Let y g W be such that y F z9. Then there exists1 1 Lx g W such that x F z and x ; y .1 1 L 1 R 1

y1 Ž .y1Proof. Let y s y . Then y F z9 . So C appears with non-zero1 R ycoefficient in C y1 C and hence in C C y1 C for some x g W byŽ z 9. x s z x

Ž . Ž .Proposition 1.6 1 . Again by Proposition 1.6 1 , this implies that thereexists x9 g W such that C appears with non-zero coefficient in C y1 Cx 9 z x

JIAN-YI SHI610

and that C appears with non-zero coefficient in C C . We have x9 F zy1y s x 9 RŽ .y1and hence x9 F z.L

Ž .y1If sx9 - x9, then y s x9. We can take x s x9 s y . Now assume1 1Ž .y1sx9 ) x9. Then y}x9 and y F x9. If y ; x9, then y ; x9 F z.L L 1 R L

Ž .y1 Ž . Ž .We can take x s x9 . If y © x9, then by Proposition 1.6 2 and 31 LŽ .we have y © x9. By Proposition 1.6 4 , this implies y F x9 and henceLR R

y1Ž .y F x9 F z. We can take x s y .1 L L 1 1

Ž .THEOREM 1.8. Let W, S be a crystallographic Coxeter system with theŽ .property ) . Let x, y g W be such that x F y. Then there exists z g WLR

such that z ; x and z F y.R L

Proof. Consider all the sequences of elements x s x, x , . . . , x s y0 1 rfrom x to y such that the relation x © x holds for every i,iy1 LR i

Ž .1 F i F r. Then by Proposition 1.6 2 , we see that the upper boundary ofthe lengths r of these sequences is finite, which must be less than or equal

Ž . Ž .to a x y a y . So we can assume that x ¤ y and that the result is trueLRwhen x, y are replaced by x, y9 with x F y9 F y and y9 ¤ y. FromLR LR LRthe definition of the relation x F y, we see that there exist ¨ , ¨ 9 suchLRthat x F ¨ © ¨ 9 ; y and that either ¨ F ¨ 9 or ¨ F ¨ 9 holds.LR LR LR L R

y1 Ž .y1Replacing if necessary ¨ , ¨ 9 by ¨ , ¨ 9 , we can assume that ¨ F ¨ 9.LŽLet w g W be such that ¨ 9 ; w, w ; y such an element w alwaysa R L

exists, since in a two-sided cell W , the intersection of a left cell with aa. X Xright cell is non-empty . Since w ; ¨ 9, we can find ¨ 9 s ¨ , . . . , ¨ s wR 1 r

X X Ž X . Ž X .such that for every i, 1 - i F r, ¨ y ¨ and RR ¨ RR ¨ . Applyingiy1 i iy1 irepeatedly Lemma 1.7, we find a sequence ¨ s ¨ , . . . , ¨ such that ¨ F ¨ X

1 r i L iŽ .1 F i F r , ¨ ; ¨ ; ??? ; ¨ . In particular, we have ¨ ; ¨ F w.1 R 2 R R r R r LSince w ; y, we have x F ¨ ; ¨ F y. Since y ¤ ¨ , the theoremL LR R r L LR r

Ž . Ž .is known to hold for x, ¨ instead of x, y . Thus there exists z g W suchr athat x ; z, z F ¨ and the theorem follows.R L r

wRemark 1.9. Lemma 1.7 and Theorem 1.8 are the counterparts of 12,xLemma 3.1 and Theorem 3.2 . So the proofs of the former are analogous

to those of the latter. But in our proof of Theorem 1.8, we do not assumeŽ .that the number of two-sided cells of W, S is finite. So the conclusion of

Ž .Theorem 1.8 is valid for any crystallographic group with the property ) ,not only for an affine Weyl group. By applying the argument analogous to

w xours, we can show that the conclusion of 12, Theorem 3.2 is also valid forŽ .any crystallographic group with the property ) .

1.10. Let S be the set of all the subsets of S. For any I, J : S, writeI < J, if for any I g I, there exists some J g J such that J = I.

Ž . � Ž .For any two-sided cell V of W, let L V s I ; S ¬ I s LL w for some4 Ž . � Ž . 4w g V and R V s I ; S ¬ I s RR w for some w g V . Then by the fact


y1 Ž . Ž .that x ; x for any x g W, we have L V s R V and hence we canLRŽ .denote this set by T V .

Ž .THEOREM 1.11. Assume that W , S is an affine Weyl group. AssumeaŽ .that V and V9 are two-sided cells of W with V9 F V. Then T V <a LR

Ž .T V9 .

Proof. This follows from 1.4 and Theorem 1.8.

This theorem will be useful in checking whether or not some two-sidedcells V and V9 of W have the relation V9 F V. Now we give one morea LRresult on this respect. Given I : S, if the subgroup W of W generated byI aI is finite, then we denote by w the longest element of W .I I

PROPOSITION 1.12. Let V and V9 be two-sided cells of an affine WeylŽ .group W , S . Assume that w g V for some I : S. Then V9 F V if anda I LR

Ž .only if there exists some J g T V9 with J = I.

Proof. The implication ‘‘« ’’ follows by Theorem 1.11. Now we showthe implication ‘‘¥.’’ By the assumption, there exists some element

Ž .w g V9 with LL w s J, J = I. Then we have an expression w s w ? x forIŽ . Ž . Ž .some x g W with l w s l w q l x . This implies that w F w andI LR I

hence V9 F V.LR

˜2. THE PROOF OF THEOREM B IN TYPE A CASEny1

Ž .In the present section, we always assume that the Coxeter system W , Sa˜has type A .ny1

Ž . Ž .2.1. A partition of n n g N is any sequence l s l , l , . . . , l of1 2 rnon-negative integers in decreasing order: l G l G ??? G l and1 2 rÝr l s n. We shall not distinguish between two such sequences whichis1 idiffer only by a string of zeros at the end. Sometimes it is convenient touse a notation which indicates the number of times each integer occurs as

Ž m1 m2 m r .a part: l s 1 2 ??? r ??? means that exactly m of the parts of liequal to i. Let L be the set of all the partitions of n.n

Ž . Ž .For l s l , l , . . . , l , m s m , m , . . . , m g L , we write l F m, if1 2 r 1 2 t nfor any i G 1, the inequality Ýi l F Ýi m holds.js1 j js1 j

We write l -? m, if l - m, and there does not exist any n g L ,nsatisfying l z n z m. It is well known that l -? m if and only if there existtwo integers i, j, j ) i G 1, satisfying the following conditions:

Ž .1 l s m for any h G 1 with h / i, j.h h

Ž .2 l s m y 1, l s m q 1.i i j j

Ž .3 l s l s ??? s l G l .i iq1 jy1 j

JIAN-YI SHI612

Ž .For l, m g L , we say that m s m , m , . . . , m is conjugate to l sn 1 2 tŽ .l , l , . . . , l , if for any i, 1 F i F t, m is equal to the number of parts1 2 r il with l G i. In this case, we denote m by l9. In general, we have thatj jŽ .l9 9 s l and that l F m if and only if m9 F l9.

Ž .2.2. Let G s GL n, C be the general linear group over the complexfield C of rank n. Then a unipotent conjugacy class u of G can be

Ž X X X .parametrized by a partition l g L with l9 s l , l , . . . , l such thatn 1 2 rlX ,lX , . . . , lX are the sizes of the Jordan blocks of a standard form of the1 2 r

Ž .class u counting multiplicities . We denote such a class u by u .l

It is well known that for l, m g L , we have u F u if and only ifn l m

w xl F m 18 .

� 42.3. Let S s s ¬ 0 F i - n be a Coxeter generator set of the affineiŽ .Weyl group W such that the order o s s of the product s s is 3 fora i iq1 i iq1

0 F i - n with the convention that s s s , i g Z. It is known that theiqn igroup W can be identified with the following permutation group on theainteger set Z:

n n

AA s w : Z ¬ Z i q n w s i w q n , for all i g Z; i w s i ;Ž . Ž . Ž .Ý Ýn ½ 5is1 is1

� 4where the Coxeter generator set S s s ¬ 0 F t F n y 1 of AA is given byt n

¡i , if i k t , t q 1 mod n ,Ž .~i q 1, if i ' t mod n ,Ž .i s sŽ . t ¢i y 1, if i ' t q 1 mod n .Ž .

To each element w g AA , we associate a sequence of integers d F d Fn 1 2??? F d s n ast

k

< <d s max X X s X ; Z; u k ¨ mod n , for all u / ¨ in X ;Ž .Dk i½is1

and u - ¨ in some X implies u w ) ¨ w ,Ž . Ž .i 5< <where the notation X stands for the cardinality of a set X. Greene

Ž . Žshowed that d , d y d , d y d , . . . , d y d is a partition of n see1 2 1 3 2 t ty1w x. Ž .5 . We denote this by s w . This defines a map s : AA ª L . It is knownn nthat the map s induces a bijection from the set of two-sided cells of AA ton

Ž w x.the set L see 13 .n


Ž .2.4. To each J ; S, we associate a partition p J of n as below.Decompose the set J into a disjoint union J s J j J j ??? j J satisfying1 2 rthe following conditions:

Ž . < < < < < <1 J G J G ??? G J ) 0.1 2 r

Ž . � 42 For any i, 1 F i F r, we have J s s ¬ 1 F j F h for somei k qj ii

k ,h g Z with h G 1 and s , s f J.i i i k k qh q1i i i

Let l s h q 1 for 1 F i F r and l s 1 for r - k F t s n q r yi i kr Ž . Ž . Ž .Ý l . Then we define p J s l , . . . , l . Clearly, we have p J g L .js1 j 1 t n

Ž .Also, we see that p J depends only on J itself, not on the choice of theŽ .decomposition of J. So the partition p J is well defined.

Ž . Ž .LEMMA. If J : I in S, then p J F p I .

Proof. We may assume J n I since otherwise there is nothing to prove.< < < <We need only to deal with the case of J s I y 1. Then the result can be

shown easily by comparing standard decompositions of I and J.

2.5. Recall the definition of the map s : AA ª L given above. It isn nŽ . Ž .easily seen that the equality s w s p J holds for any J ; S. Moreover,J

w xwe have the following result by 13 .

LEMMA. Let l g L .n

Ž . Ž .1 There exists some J ; S such that p J s l.Ž . y1Ž .2 The two-sided cell s l of AA contains all the elements of then

Ž .form w , J ; S, with p J s l.J

Ž . Ž y1Ž .. Ž .3 For any I g T s l , we haë p I F l.

LEMMA 2.6. Let I s I j I and J s J j J be standard decompositions1 2 1 2� 4 � 4of I, J ; S such that I s s ¬ 1 F h F k , I s s ¬ 1 F h F k ,1 h 1 2 k q1qh 21

� 4 � 4J s I j s , and J s I _ s . Then w © w .1 1 k q1 2 2 k q2 J LR I1 1

Proof.

w ; s s ??? s s wI L k qk k qk y1 k q2 k q1 I1 2 1 2 1 1

; s s ??? s s w s s ??? sR k qk k qk y1 k q2 k q1 I k q1 k 21 2 1 2 1 1 1 1

ª s s ??? s s w s s ??? sLR k qk q1 k qk k q2 k q1 I k q1 k 21 2 1 2 1 1 1 1

s s s ??? s s w s s ??? s s .k qk k qk y1 k q2 k q1 I k q1 k 2 11 2 1 2 1 1 1 1

Ž . Ž .Denote the last element of the above by z. Then we have s z s s wJand hence w ª z ; w .I LR LR J

To understand the above proof, we illustrate it by an example withk s 5 and k s 3 as below. Notice that an element w g AA can be1 2 n

JIAN-YI SHI614

identified with a Z = Z matrix whose entries are all zero except for those�Ž Ž . . 4 Ž w x.in the positions i, i w ¬ i g Z which are 1 see 13 :

1

... ...

......

...

11

1

11

11

...1

1 11

1

11

... ...1

11

11

... ...

...

...

11

1

11

11

1

11

... ...

...

...

11

1

11

11

1

11

... ...

...

...

11

1

11

1

11

11

Ž . Ž .COROLLARY 2.7. Assume that I, J ; S satisfy p I -? p J . Thenw © w .J LR I


Proof. Let I s I j ??? j I and J s J j ??? j J be standard decom-1 r 1 rpositions of I and J. By Lemma 2.5, we may assume that there exists some

w xi, j, 1 F i - j F r, such that I s J for any h g 1, r , h / i, j, and I sh h i� 4 � 4 � 4s ¬ 1 F h F k , I s s ¬ 1 F h F k , J s I j s , J s I _h 1 j k q1qh 2 i i k q1 j j1 1� 4s . Then by Lemma 2.6, we have w ª w . This impliesk q2 I j I LR J j J1 i j i j

immediately that w © w .J LR I

LEMMA 2.8. Let V and V9 be two-sided cells of W with V9 F V.a LRŽ . Ž .Then s V9 G s V .

Proof. By Lemma 2.5, there exists some I ; S such that w g V andIŽ . Ž . Ž . Ž .p I s s V . Also, we have T V < T V9 by Theorem 1.11. This im-

Ž .plies that there exists some J g T V9 with J = I. By Lemmas 2.4 and 2.5,Ž . Ž . Ž . Ž .we have s V9 G p J G p I s s V . Our proof is completed.

˜2.9. Proof of Theorem B in type A case. By Lemma 2.5, this isny1equivalent to show the following assertions. Let l, m g L .n

Ž . Ž .1 If l F m, then we have w F w for some and hence allJ LR Iy1Ž . y1Ž .I g p l and J g p m .Ž . y1Ž . y1Ž .2 If s l F s m , then l G m.LR

Ž . Ž .Assertion 2 is just the conclusion of Lemma 2.8. For assertion 1 , wemay assume l / m since otherwise there is nothing to do. On the otherhand, for any l - m in L , there exists a sequence of partitions n sn 0l, n , . . . , n s m in L such that the relation n -? n holds for every i,2 p n iy1 i1 F i F p. Thus in the proof of this assertion, we may assume l -? m

Ž .without loss of generality. But in this case, assertion 1 follows directlyfrom Lemma 2.5 and Corollary 2.7.

3. THE PROOF OF THEOREM B IN RANK 4 CASES

˜Besides A , there are four other types for the affine Weyl groups of4˜ ˜ ˜ ˜rank 4, i.e., B , C , D , and F . We shall prove Theorem B in these four4 4 4 4

cases in the present section.The unipotent classes of a classical algebraic group G can be

parametrized by partitions of a certain integer. We denote by u thel

unipotent class of G parametrized by a partition l. We have u F u ifl m

and only if l G m.

Ž .3.1. The unipotent classes of the projective symplectic group PSp C8Ž .which is of type C are parametrized by partitions of 8 in which each odd4

Ž . Ž . Ž 2 .part occurs with even multiplicity. These partitions of 8 are 8 , 62 , 61 ,Ž 2 . Ž 2 . Ž 2 . Ž 4. Ž 2 . Ž 2 2 . Ž 4. Ž 3 2 . Ž 2 4. Ž 6. Ž 8.4 , 42 , 421 , 41 , 3 2 , 3 1 , 2 , 2 1 , 2 1 , 21 , 1 . Thus

Ž .the unipotent classes of PSp C have the partial order as in Fig. 1a, where8

JIAN-YI SHI616

FIG. 1.

a partition l joins with a partition m from top to bottom by an edge if andŽonly if l G m if and only if u F u the same for the subsequentl m

.diagrams .

Ž .3.2. The unipotent classes of the special orthogonal group SO C9Ž .which has type B are parametrized by partitions of 9 in which each even4

Ž . Ž 2 . Ž .part occurs with even multiplicity. These partitions of 9 are 9 , 71 , 531 ,Ž 2 . Ž 4. Ž 2 . Ž 3. Ž 2 3. Ž 2 2 . Ž 6. Ž 4 . Ž 2 5. Ž 9.52 , 51 , 4 1 , 3 , 3 1 , 32 1 , 31 , 2 1 , 2 1 , 1 . Thus the

Ž .unipotent classes of SO C have the partial order as in Fig. 1b.9

3.3. The unipotent classes of the projective special orthogonal groupŽ . Ž .PSO C which has type D are parametrized by partitions of 8 in which8 4

each even part occurs with even multiplicity, except that each such parti-tion in which all parts are even gives rise to two unipotent classes. These

Ž . Ž . Ž 3. Ž 2 . Ž 2 2 . Ž 2 . Ž 5. Ž 4. Ž 2 4. Ž 8.partitions are 71 , 53 , 51 , 4 , 3 1 , 32 1 , 31 , 2 , 2 1 , 1 .Ž .Thus the unipotent classes of PSO C have the partial order as in Fig. 2a.8

3.4. The partial ordering on the unipotent classes of the reductiveŽ . Žcomplex algebraic group G F , C of type F , using Carter’s notation see4 4

w x.2 , is as in Fig. 2b.


FIG. 2.

3.5. According to Lusztig, the unipotent classes of the algebraic groupŽ . Ž Ž . Ž . Ž ..PSp C resp. SO C , resp. PSO C , resp. G F , C are in 1]1 corre-8 9 8 4

˜spondence with the two-sided cells of the affine Weyl group of type B4˜ ˜ ˜Ž .resp. C , resp. D , resp. F . The following tables give these correspon-4 4 4

dences explicitly, where the indexes i of the simple reflections s of thei˜ ˜ ˜ ˜Ž . Ž . Ž . Ž .affine Weyl groups W B , W C , W D and W F are compatiblea 4 a 4 a 4 a 4

Ž .with the respective extended Dynkin diagrams see Fig. 3 . We denote areflection s simply by i for brevity. A unipotent class u of G is repre-i

Žsented by its corresponding partition when G is of classical type i.e., of.type B , C or D or by a Carter notation when G has type F . The4 4 4 4

Ž .corresponding two-sided cell c u is represented by a subset J ; S such˜Ž . Ž .that w g c u , but with two exceptions: to the two-sided cell V in W DJ a 4

Ž .with a V s 7, we associate an element w s 121321432 g V; to the

JIAN-YI SHI618

FIG. 3.

˜Ž . Ž .two-sided cell V9 in W F with a V9 s 13, we associate an elementa 4z s 1213213234321324321 g V9.

Ž . Ž .1 The correspondence between the unipotent classes of PSp C , and8˜Ž .the two-sided cells of W B :a 4

Ž . Ž . Ž 2 . Ž 2 . Ž 2 . Ž 2 . Ž 2 .unip. class u: 8 62 61 4 42 421 3 2Ž .c u : B 1 1, 3 0, 1 0, 1, 3 3, 4 1, 2, 4

2 2 4 4 3 2 2 4 6 8Ž . Ž . Ž . Ž . Ž . Ž . Ž .unip. class u: 3 1 41 2 2 1 2 1 21 1Ž .c u : 1, 3, 4 0, 1, 2 0, 1, 3, 4 0, 1, 2, 4 2, 3, 4 0, 1, 2, 3 1, 2, 3, 4

Ž . Ž .2 The correspondence between the unipotent classes of SO C and9˜Ž .the two-sided cells of W C :a 4

Ž . Ž 2 . Ž . Ž 2 . Ž 2 . Ž 4 . Ž 3.unip. class u: 9 71 531 52 4 1 51 3Ž .c u : B 1 0, 4 1, 2 0, 2, 4 0, 1 1, 2, 4

2 3 2 2 6 4 2 5 9Ž . Ž . Ž . Ž . Ž . Ž .unip. class u: 3 1 32 1 31 2 1 2 1 1Ž .c u : 0, 1, 3 1, 2, 3 0, 1, 2 0, 1, 3, 4 0, 1, 2, 4 0, 1, 2, 3

Ž . Ž .3 The correspondence between the unipotent classes of PSO C and8˜Ž .the two-sided cells of W D :a 4

Ž . Ž . Ž 2 . Ž 3. Ž 2 . Ž 2 2 .unip. class u: 71 53 4 51 4 9 3 1Ž .c u : B 1 0, 1 0, 3 0, 4 0, 1, 3

2 4 5 4 2 4 8Ž . Ž . Ž . Ž . Ž . Ž .unip. class u: 32 1 2 31 2 9 2 1 1Ž .c u : 0, 1, 3, 4 0, 1, 2 0, 2, 3 0, 2, 4 w 1, 2, 3, 4


Ž . Ž .4 The correspondence between the unipotent classes of G F , C and4˜Ž .the two-sided cells of W F :a 4

˜Ž . Ž . Ž . Ž .unip. class u: F F a F a B C F a C a A q A4 4 1 4 2 3 3 4 3 3 1 2 1Ž .c u : B 1 0, 2 1, 2 0, 2, 4 2, 3 0, 2, 3 0, 1, 2

˜ ˜ ˜ ˜unip. class u: B A q A A A A q A A A 12 2 1 2 2 1 1 1 1Ž .c u : 0, 1, 3, 4 0, 1, 2, 4 2, 3, 4 1, 2, 3 0, 2, 3, 4 z 0, 1, 2, 3 1, 2, 3, 4

� Ž . Ž .3.6. Proof of Theorem B in rank 4 cases. Let G g PSp C , SO C ,8 9Ž . Ž .4PSO C , G F , C and let W be the affine Weyl group corresponding to8 4 a

G as above. We need only to prove the following assertions:

Ž .1 If u and v are two unipotent classes of G joining by an edge fromŽ . Ž .top to bottom in the above poset diagram, then c v F c u .LR

Ž . Ž . Ž . Ž .4 22 If G s PSp C , then c u F c u .u8 41 LR 3 2

Ž . Ž . Ž . Ž .4 23 If G s SO C , then c u F c u .u9 51 LR 4 1

˜Ž . Ž . Ž Ž .. Ž Ž ..4 If G F , C , then c u A F c u B .u4 2 LR 2

Ž .By Theorem 1.11, assertion 2 can be shown by noting that w g�1, 2, 44Ž . � 4 Ž Ž .. Ž . Ž Ž ..2 4 4c u , 1, 2, 4 f T c u , and a w ) 6 s a c u for any I ; S3 2 41 I 41

� 4 Ž w x. Ž .with I q 1, 2, 4 see 20 ; assertion 3 can be shown by noting thatŽ . � 4 Ž Ž .. Ž . Ž Ž ..2 4 4w g c u , 0, 2, 4 f T c u and a w ) 4 s a c u for any�0, 2, 44 4 1 51 I 51

� 4 Ž w x. Ž .I ; S with I q 0, 2, 4 see 16 ; assertion 4 follows by noting that˜Ž Ž .. � 4 Ž Ž Ž ... � 4w g c u B , 0, 1, 3, 4 f T c u A and that 0, 1, 3, 4 is a maxi-�0, 1, 3, 44 2 2

Ž w x.mal proper subset of S see 17 .Ž .On the other hand, assertion 1 follows by Proposition 1.12 and by the

w xresults in 15]17, 20 except for the following cases:

˜Ž . Ž . Ž .2 4 8a W has type D and u, v s u , u .a 4 2 1 1

˜ ˜Ž . Ž . Ž Ž . Ž ..b W has type F and u, v s u A , u A .a 4 1 1

Ž .In each of the above cases, the two-sided cell c u contains no elementŽ . Ž .8of the form w , I ; S. But in case a , c u is the lowest two-sided cell ofI 1

˜Ž . Ž . Ž . Ž .8 2 4W D . So the relation c u F c u holds in this case. In case b ,a 4 1 LR 2 1let x s 121321432132343213234 and y s x ? 4012321. Then by a result of

Ž w x.Geck and Lux see 4 , x is a distinguished involution of the two-sided cell˜Ž Ž ..V s c u A . We see that the element y can be obtained from x by1

Ž w xsuccessively applying right star operations on x see 13 for the definition.of a right star operation and hence y ; x. Let y9 s y ? 0. Then by theR

Ž . � 4 � 4 Ž . Ž . Ž .fact RR y9 s 0, 1, 2, 3 1, 2, 3 s RR y , we see that a y9 G a w�0, 1, 2, 34w xs 16 and hence y9 © y ; x. By a result of Lusztig 11, Theorem 4.8 ,LR R

this implies that the element y9 must belong to one of the two-sided cells˜Ž Ž .. Ž Ž .. Ž .V9 s c u A , V0 s c u 1 of W F . But we see that there are some1 a 4

zero entries occurring in the alcove form of the element y9 and so y9 does

JIAN-YI SHI620

˜Ž . w xnot belong to the lowest two-sided cell V0 of W F by 14, Theorem 2.4 .a 4This implies y9 g V9, i.e. V9 F V. The proof is completed.LR

4. COMMENTS

The proof of Theorem B relies heavily on the knowledge of the setsŽ .T V for all the two-sided cells of the concerned affine Weyl groups. Thus

it is interesting to give an explicit description of such sets. Comparing withLemma 2.5, we have an even stronger result which explicitly describes the

˜Ž . Ž .set T V for any two-sided cell V of the group W A .a ny1

Ž n.THEOREM 4.1. For any l g L withl / 1 , we haën

T sy1 l s I ; S ¬ I / B, p I F l .� 4Ž . Ž .Ž .The proof of this result is somewhat lengthy. So we prefer not to include

it here and will present it elsewhere. Empiricism encourages us to extendthis result to more general cases. Thus we suggest the following

Ž .Conjecture 4.2. Let W , S be an arbitrary irreducible affine Weyla� 4 Ž .group. Then for any two-sided cell V / 1 of W , the set T V consistsW aa

of all the non-empty subsets I of S such that the two-sided cell V9 of Wacontaining w satisfies the relation V F V9.I LR

This conjecture has been supported by all the irreducible affine Weylgroups of ranks not greater than 4. Notice that this conjecture is not validif the concerned Coxeter system is replaced by a Weyl group W of rankgreater than two. The lowest two-sided cell of such a Weyl group providesa counterexample.

ACKNOWLEDGMENT

I express my deep gratitude to the referee for hisrher invaluable comments.

REFERENCES

Ž .1. R. Bedard, Cells for two Coxeter groups, Comm. Algebra 14 1986 , 1253]1286.´2. R. W. Carter, ‘‘Finite Groups of Lie Type: Conjugacy Classes and Complex Characters,’’

Wiley Series in Pure and Applied Mathematics, Wiley, London, 1985.˜3. Du Jie, The decomposition into cells of the affine Weyl group of type B , Comm. Algebra3

Ž .16 1988 , 1383]1409.4. M. Geck and K. Lux, The decomposition numbers of the Hecke algebra of F , Manuscripta4

Ž .Math. 70 1991 , 285]306.


5. C. Greene, Some partitions associated with a partially ordered set, J. Combin. Theory Ser.Ž .A 20 1976 , 69]79.

6. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,Ž .Inënt. Math. 53 1979 , 165]184.

7. D. Kazhdan and G. Lusztig, Schubert varieties and Poincare duality, in ‘‘Proceedings of´Symposium in Pure Mathematics,’’ Vol. 36, pp. 185]203, Amer. Math. Soc., Providence,RI, 1980.

Ž8. G. Lusztig, Cells in affine Weyl groups, in ‘‘Algebraic Groups and Related Topics’’ R..Hotta, Ed. , pp. 255]287, Advanced Studies in Pure Mathematics, KinokuniyarNorth-

Holland, TokyorAmsterdam, 1985.˜9. G. Lusztig, The two-sided cells of the affine Weyl group of type A , in ‘‘Infiniten

Ž .Dimensional Groups with Applications’’ V. Kac, Ed. , pp. 275]283, MSRI Publications,Vol. 4, Springer-Verlag, BerlinrNew York, 1985.

Ž .10. G. Lusztig, Cells in affine Weyl groups, II, J. Algebra 109 1987 , 536]548.11. G. Lusztig, Cells in affine Weyl groups, IV, J. Fac. Sci. Uni . Tokyo Sect. IA Math. 36

Ž .1989 , 297]328.12. G. Lusztig and Xi Nan-hua, Canonical left cells in affine Weyl groups, Ad¨. in Math. 72

Ž .1988 , 284]288.13. Jian-yi Shi, ‘‘The Kazhdan]Lusztig Cells in Certain Affine Weyl Groups,’’ Lecture Notes

in Mathematics, Vol. 1179, Springer-Verlag, BerlinrNew York, 1986.Ž .14. Jian-yi Shi, A two-sided cell in an affine Weyl group, J. London Math. Soc. 36 1987 ,

407]420.˜Ž . Ž .15. Jian-yi Shi, Left cells of the affine Weyl group W D , Osaka J. Math. 31 1994 , 27]50.a 4

˜16. Jian-yi Shi, Left cells of the affine Weyl group of type C , in preparation.4˜17. Jian-yi Shi, Left cells of the affine Weyl group of type F , in preparation.4

18. N. Spaltenstein, ‘‘Classes Unipotentes et Sous-groupes de Borel,’’ Lecture Notes inMathematics, Vol. 946, Springer-Verlag, BerlinrNew York, 1982.

19. T. A. Springer, unpublished letter to G. Lusztig, 1987.˜Ž .20. Zhang Xin-fa, Cells decomposition in the affine Weyl group W B , Comm. Algebra 22,a 4

Ž .No. 6 1994 , 1955]1974.

Documents

The Partial Order on Two-Sided Cells of Certain Affine Weyl Groups