Left Cells in the Affine Weyl Group of Type F 4

Transcription

1 Ž. JOURNA OF AGEBRA 200, ARTICE NO. JA eft Cells in the Affine Weyl Group of Type F 4 Jin-yi Shi* Mx-Plnck-Institut fur Mthemtik, Bonn, Germny; nd Deprtment of Mthemtics, Est Chin Norml Uniersity, Shnghi, , People s Republic of Chin Communicted by Gordon Jmes Received September 30, 1996 By pplying n lgorithm designed before, we complete the description for ll the left cells of the ffine Weyl group W of type F 4 by finding representtive set of its left cells together with ll its left cell grphs Žor with ll the ssocited essentil grphs. in ech of its two-sided cells. The generlized -invrints of left cells of W re exhibited grphiclly. A group-theoreticl interprettion is given on the numbers of left cells of W in some two-sided cells. Thus so fr the left cells of ll the ffine Weyl groups of rnks less thn or equl to 4 hve been known explicitly. Some techniques re developed in pplying the lgorithm. As consequence, we complete the verifiction of conjecture concerning the chrcteriztion of left cells of Weyl groups nd ffine Weyl groups Acdemic Press An lgorithm ws designed in 22 nd then improved in 25 for finding representtive set of left cells Ž n l.c.r. set for short. of W in two-sided cell, where W is Coxeter group belonging to certin fmily of crystllogrphic groups; the ltter includes ll the Weyl groups nd ll the ffine Weyl groups. By pplying this lgorithm, I described ll the left cells of the ffine Weyl groups of types C nd D 4 4, nd lso ll the left cells with Ž. 3, 4, 5 in the ffine Weyl group of type F Žsee 25, 23, Subsequently, three of my students, Zhng Xin-f, Rui He-bin, nd Tong Chng-qing, chieved some progress on this respect lso by pplying this lgorithm, where Zhng gve n explicit description for ll the left cells in the ffine Weyl group of type B 28 4, Rui for ll the left cells with Ž. 3 in ny irreducible ffine Weyl group 15, nd Tong for ll the left cells of the Weyl group of type E 27. In the present pper, we shll * Supported prtly by Mx-Plnck-Institut fur Mthemtik, Bonn, Germny nd prtly by the Ntionl Science Foundtion of Chin nd the Science Foundtion of the University Ph.D. Progrm of CNEC $25.00 Copyright 1998 by Acdemic Press All rights of reproduction in ny form reserved.

2 174 JIAN-YI SHI pply this lgorithm to complete the description of the left cells of the ffine Weyl group of type F 4. This, together with the erlier results of the others 16, 11, 2, 7, completes the description of the left cells for ll the ffine Weyl groups of rnks 4. Some techniques re developed in pplying the lgorithm Ž see Sects. 3 nd 4.. We find representtive set of left cells together with its left cell grphs Žor with ll the ssocited essentil grphs. in ech two-sided cell of W. The generlized -invrints of left cells of W re exhibited grphiclly. A group-theoreticl interpret- tion is given on the numbers of left cells of W in some two-sided cells, which involves both the usztig mp nd BlCrter correspondence mong two-sided cells of W, unipotent conjugcy clsses of the complex lgebric group G of type F, nd the G-clsses of pirs Ž, P. 4, where is evi subgroup of G, nd P is distinguished prbolic subgroup of the semisimple prt of. As consequence, we shll complete the verifiction of conjecture concerning the chrcteriztion of left cells of Weyl groups nd ffine Weyl groups which ws proposed in 22 nd prtly verified in 24. The content of the pper is orgnized s follows. Some known results on cells of Coxeter group, in prticulr of n ffine Weyl group W, re stted in Section 1. Then in Section 2, we recll the lgorithm of finding n l.c.r. set of W in two-sided cell nd lso stte some results nd terminologies which re needed in pplying the lgorithm. In Section 3, some techniques of pplying the lgorithm re developed, which will be frequently used for finding n l.c.r. set of the ffine Weyl group W Ž F. 4 in given two-sided cell. We illustrte them by severl exmples. We find n l.c.r. set together with ll the left cell grphs Žor with the corresponding essentil grphs. for W Ž F. 4 in Section 4. Finlly, in Section 5, we complete the verifiction of the bove-mentioned conjecture. 1. SOME RESUTS ON CES 1.1. et W Ž W, S. be Coxeter group with S its Coxeter genertor set. et be the Bruht order on W. For w W, we denote by lw Ž. the length of w. et A u be the ring of polynomils in n indeterminte u with integer coefficients. For ech ordered pir y, w W, there exists unique polynomil Py, w A, clled Kzhdnusztig polynomil, which stisfies the conditions: Py, w 0 if yw, Pw,w 1, nd deg Py,w 1 Ž.Žlw Ž. lž y. 1. if yw. et Ž w, y. Ž y, w. 2 be the coefficient of u Ž12.Ž lžw.lž y.1. in Py, w for y w. We denote yw if Ž y, w. 0. Checking the reltion yw for y, w W usully involves very complicted computtion of Kzhdnusztig polynomils. But it becomes esy in some specil cses: if x, y W stisfy y x nd lž y. lž x. 1, then

3 EFT CES 175 we hve yx. Another result concerning this reltion will be stted in Proposition The preorders,, on W nd the ssocited equivlence R R reltions,, on W re defined s in 8. The equivlence clsses R R for Ž resp.,. on W re clled left cells Žresp. right cells, two-sided R R cells An ffine Weyl group W is Coxeter group which cn be relized geometriclly s follows. et G be connected, djoint reductive lgebric group over. We fix mximl torus T G. et X be the chrcter group of T nd let X be the root set with,...,4 1 l choice of simple root system. Then E X is eucliden spce with n inner product ², : such tht the Weyl group Ž W, S. 0 0 of G with respect to T cts nturlly on E nd preserves its inner product, where S0 is the set of simple reflections si corresponding to the simple roots i,1il. We denote by N the group of ll trnsltions T Ž X. on E: T sends x to x. Then the semidirect product W W0 N is clled n ffine Weyl group. et K be the dul of the type of G. Then we define the type of W by K. Sometimes we denote W by W Ž K. to indicte its type K. There is cnonicl homomorphism from W to W 0: w w. et 0 be the highest short root in. We define s0 s T, where 0 0 s is the reflection corresponding to 0. Then the genertor set of W cn 0 be tken s S S s The lcove form of n element w W is, by definition, -tuple Žkw, Ž.. over subject to the following conditions. W ; Ž. kw, Ž. kw, Ž. for ny ; Ž b. ke, Ž. 0 for ny, where e is the identity element of Ž. Ž. c If wws 0 i l, then i Ž. kž w,. k w, Ž. s Ž,i. i with 0 if i; Ž,i. 1 if i ; 1 if i, where s s if 1 i l, nd s s Žsee 17, Proposition 4.2. i i 0 0. By condition Ž., we cn lso denote the lcove form of w W -tuple Žkw, Ž... by

4 176 JIAN-YI SHI 1.5. Condition 1.4Ž. c ctully defines set of opertors s 0 i l4 i on the lcove forms of elements of W : Ž. s : Ž k. k Ž,i.. i Ž.si These opertors could be described grphiclly. Assume tht W hs type F 4 nd tht the indices of simple roots re comptible with the following Dynkin digrm: o o o o 4 We denote root Ý by its coordinte form Ž,,,. i1 i i nd rrnge the entries of -tuple Ž k. in the following wy: Ž k k k k Ž2,4,3,2. Ž2,4,3,1. Ž2,4,2,1. Ž2,2,2,1. k k k k Ž2,3,2,1. Ž1,3,2,1. Ž1,2,2,1. Ž1,2,1,1. k k k k Ž2,2,1,1. Ž0,2,2,1. Ž1,2,1,0. Ž1,1,1,1. k k k k Ž0,2,1,1. Ž2,2,1,0. Ž0,1,1,1. Ž1,1,1,0. kž0,0,1,1. kž0,2,1,0. kž0,1,1,0. kž1,1,0,0. kž0,0,0,1. kž0,0,1,0. kž0,1,0,0. kž1,0,0,0. Then the ctions of s,0i4, on -tuple i m b n c p w e d q f r s g h t u i j w k l x y

5 EFT CES 177 re listed s in the following tble, s s s s s s h m1 n b e y m p c d w d n b q ws c u f c s e p r b s r g d t u q d g h q u q e j s i r f w j e s p h u x g l y k x u l g t n w y1 j x1 w i l1 k1 i where the entries in the positions remin unchnged To ech element x W, we ssocite two subsets of S s below: Ž x. sssx x4 nd RŽ x. s S xs x 4. For w, w W, we sy tht w is left extension of w if lwlw Ž. Ž. Ž 1 lww.. Then we hve the following results on the lcove form Žkw, Ž.. of w W. PROPOSITION 17, Propositions 4.1, 4.3. Ž. 1 lw Ž. Ý kw, Ž., where the nottion x stnds for the bsolute lue of x. Ž. 2 RŽ w. s kw, Ž i i Ž. 3 w is left extension of w if nd only if the inequlities kw,kw, Ž. Ž. 0nd kž w,. kw, Ž. hold for ny Now let W be either n ffine Weyl group or finite Weyl group. usztig defined function : W which stisfies the following properties: Ž. 1 z Ž. 2, for ny z W, where is the root system ssocited to W s in 1.3. Ž 2. xyx Ž. Ž y.. In prticulr, x y x Ž. Ž y..sowe R R my define the -vlue Ž. on Ž left, right or two-sided. cell of W by x Ž. for ny x. Ž 3. x Ž. Ž y. nd x y Ž resp. x y. x y Ž resp. x y.. R R Ž. 4 et wi be the longest element in the subgroup WI of W generted by I for ny I S with W finite. Then w Ž. lw Ž.. I I I The bove properties of function were shown by usztig in his ppers 11, 12. Now we stte some more properties of this function, the first two of which re simple consequences of properties Ž.Ž. 2, 3, nd Ž. 4.

6 178 JIAN-YI SHI et W w W w Ž. i4 Ži. for ny non-negtive integer i. Then by Ž. 2,W is union of some two-sided cells of W. Ži. Ž. 5 If WŽi. contins n element of the form wi for some I S, then ww RŽ w. I4 Ži. forms single left cell of W. Ž. 6 By the nottion x y z Ž x, y, zw., we men x yz nd lž x. lž y. lž z.. In this cse, we hve x z, x y nd hence x Ž. R Ž y., z. Ž. In prticulr, if I RŽ x. Žresp. I Ž x.., then x Ž. lw Ž. I. Ž. 7 W is single two-sided cell of W if i 0, 1, 4 Žsee Ž 1.. Ži.. As sets, W Ž i0, 1,. cn be described s below. W e 4 Ži. Ž0., e, the identity element of W. WŽ1. consists of ll the non-identity elements of W ech of which hs unique reduced expression Žsee. 9. WŽ. consists of ll the elements of W which hve no zero entry in their lcove forms Ž see WŽ. cn lso be described to be the lowest two-sided cell of W with respect to the prtil order Žsee 19, 20.. R Ž. 8 Now let W W be n ffine Weyl group. Cll n element s S specil, if the subgroup of W generted by S s4 is isomorphic to W0 Ž see Thus the element s is lwys specil. When W is of type F 0 4, there is no other specil element in S. It is known tht for ny two-sided cell e4 of W nd ny specil s S, the set Y w RŽ w. s44 s is non-empty nd is single left cell of W Žsee et G nd W be s in 1.3. Then the following result of usztig is importnt to our purpose. THEOREM 13, Theorem 4.8. There exists bijection u cu Ž. from the set of unipotent conjugcy clsses in G to the set of two-sided cells in W. This bijection stisfies the eqution ŽŽ.. cu dim u, where u is ny element in u, nd dim u is the dimension of the riety of Borel subgroups of G contin- ing u et G, W 0, nd W be s in 1.3. Then W0 is stndrd prbolic subgroup of W. It is known tht for ny w W, the vlue w Ž. 0 computed with respect to W0 is equl to tht computed with respect to W 12, Corollry 1.9. From the results of 3, 4, 10, 13, we know tht the bijection in Theorem 1.8 induces bijection between the set of specil unipotent clsses of G nd the set of two-sided cells of W 0. et w0 be the longest element of W 0. Then the permuttion x w0x of W0 induces n order-reversing bijective mp on the set of two-sided cells of W Ž 0 see 8, Under this mp, the two-sided cells W e4 nd W w 4 Ž0. Ž. 0 re trnsposed, nd the two-sided cell WŽ1. of W0 is sent to the second lowest one. Here lw Ž et W W Ž F. be the ffine Weyl group of type F 4 4. Then ccording to the knowledge of the unipotent clsses of the complex simple

7 EFT CES 179 lgebric group of type F 4, we see from Theorem 1.8 tht in W, the set W is non-empty if nd only if i 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 16, 244 Ži.. More precisely, W is single two-sided cell if i Ži. 0, 1, 2, 4, 5, 7, 10, 13, 16, 244, nd is union of two two-sided cells if i 3, 6, We see tht there exist some elements of the form w I, I S, in ll the sets W of W Ž F., i13 4. By the results of 22 Ži. 4 nd by 1.7Ž.Ž. 4 7, t this stge we cn find representtive set of the two-sided cells of W Ž F. with Ž. 4 6, 9 s follows. e, s 4, w 02, w 01, w 34, w 23, w 023, w 0124, w 0234, x w234 s1s2 s3s2 s1s4 s3s2 s3s 4, w 0123, nd w 1234, where we hve x WŽ13. by 1.9, 1.10, nd by the fct w0xsssssw Ž AN AGORITHM WITH SOME REATED RESUTS Here nd lter, the nottion W lwys stnds for n ffine Weyl group with S its Coxeter genertor set. The min purpose of the present pper is to describe the left cells of the ffine Weyl group W of type F 4 by finding its l.c.r. set together with ll its left cell grphs Žor with the corresponding essentil grphs.. We need n lgorithm to do this, which ws designed in 22 nd then improved in 25. This lgorithm is pplicble to certin fmily of crystllogrphic groups including ll the Weyl groups nd ll the ffine Weyl groups. In this section, we shll recll the lgorithm nd some relted results in 22, To ech element x W, we ssocite set Ž x. of ll left cells of W stisfying the condition tht there is some element y with yx, RŽ y. RŽ x., nd Ž y. x. Ž. We hve the following result. THEOREM 22, Theorem 2.1. If x yinw,then RŽ x. RŽ y. nd Ž x. Ž y To ech x W, we denote by MŽ x. the set of ll elements y such tht there is sequence of elements x0 x, x 1,..., xry in W with r0, where for every i, 1ir, the conditions x 1 i1 xi S nd Ž. R x RŽ x. i1 i re stisfied A subset K W is sid to be distinguished if K nd x y for ny x y in K. The following re three processes on non-empty set PW Žsee 25.. Ž A. Find lrgest possible subset Q from the set MŽ x. x P with Q distinguished. Ž. 1 B To ech x P, find elements y W such tht y xs, RŽ y. RŽ x., nd Ž y. x; Ž. dd these elements y on the set P to form set P nd then tke lrgest possible subset Q from P with Q distinguished.

8 180 JIAN-YI SHI Ž C. To ech x P, find elements y W such tht y x, yx, RŽ y. RŽ x., nd Ž y. x; Ž. dd these elements y on the set P to form set P nd then tke lrgest possible subset Q from P with Q distinguished. A subset P of W is clled A-sturted Ž resp. B-sturted, resp. C- sturted., if Process Ž A. Žresp. Ž B., resp. Ž C.. on P cnnot produce ny element z stisfying z x for ll x P Sy set of left cells of W is represented by set K W if is the set of ll left cells of W with K. K is clled representtie set for, if K represents with K distinguished. By 22, Theorem 3.1 nd 8, 2.3f, we see tht representtive set of left cells Ž n l.c.r. set for short. of W in two-sided cell is exctly distinguished subset of which is A-, B-, nd C-sturted. So to get such set, we my use the following lgorithm 25, 2.7. Ž. 1 Find non-empty subset P of Žusully we tke P to be distinguished to void unnecessry complictions whenever it is possible.. Ž. 2 Perform Processes Ž A.Ž, B.Ž, C. lterntely on P until the resulting distinguished set cnnot be further enlrged by these processes. In generl, Process Ž A. Žresp. Ž B.. is esier to perform thn Process Ž B. Žresp. Ž C.. in pplying the lgorithm. So we shll mke the priority first to Process Ž A. nd second to Process Ž B.. In other words, in pplying the lgorithm, we lwys first perform Process Ž A.; Process Ž B. is performed only when Process Ž A. lone cnnot mke ny further progress; finlly Process Ž C. is performed when no progress cn be mde only by Processes Ž A. nd Ž B.. In pplying Algorithm 2.4, we need some results nd terminologies. Note tht the terminologies concerning grphs re dopted from 25 which differ from those in 22. From 1.7, Ž. 3, nd Theorem 2.1, we hve the following result on set MŽ x.. PROPOSITION , Proposition 3.1. Ž. 1 For ny w W, the set MŽ x. is wholly contined in some right cell of W. Ž 2. If x yinw,then MŽ x. nd MŽ y. represent the sme set of left cells of W In Coxeter system Ž W, S., sequence of elements of the form Ž ys, yst, ysts,... m 1terms is clled n s, t4-string Ž or just cll it string. if s, t S nd y W stisfy the conditions tht the order ost Ž. of the product st is m nd RŽ y. s,t4.

9 EFT CES 181 It is esily seen tht string is wholly contined in some right cell of W. For ny x W, we cn redefine MŽ x. to be the miniml set contining x, subject to the requirement: ny string Ž regrded s set. meeting MŽ x. must be wholly contined in MŽ x.. Suppose tht we re given two s,t4-strings x, x,..., x nd y, y,..., y with ost Ž. 1 2 m1 1 2 m1 m. We denote the integers Ž x, y. Ž see 1.1. i j by ij for 1 i, j m 1. Then it is known tht PROPOSITION 2.7 9, hold. In the boe setup, the following ssertions Ž. When m 3, we he 12 21,11 22, Ž b. When m 4, we he ,11 33,13 31, nd We hve the following result corresponding to this. PROPOSITION 2.8 8, Corollry 4.3; 10, Proposition 10.3; 21, Proposition 4.6. Keep the setup of 2.6. Ž. 1 If m 3, then Ž. xyxy; Ž b. x y x y Ž. 2 If m 4, then Ž. Ž b. x y x y ; Ž. c Ž. xyxyxyxy; xy x y; d x y either x y or x y Two elements x, y W form primitie pir, if there exist two sequences of elements x0 x, x 1,..., xr nd y0 y, y 1,..., yr in W such tht the following conditions re stisfied. Ž. xyfor i i ll i,0ir. Ž b. For every i,1ir, there exist some s i, ti S such tht x i1, xi Ž nd lso y, y. re two neighboring terms in some s, t 4 i1 i i i -string. Ž c. Either RŽ x. RŽ y. nd RŽ y. RŽ x.,or Ry Ž. Rx Ž. r r nd RŽ x. RŽ y. hold. r r In this cse, we hve x y by Proposition 2.5. R Assume tht x, x Ž nd lso y, y. re two neighboring terms in some s,t4-string with xy nd tht t lest one of x, y is terminl term of the s,t4-string contining it. Then by Proposition 2.7, we hve xy. In

10 182 JIAN-YI SHI prticulr, it is lwys the cse when ost Ž. 3. Thus, if in Ž b., we hve in ddition tht t lest one of x, y is terminl term of the s, t 4 i i i i -string contining it for ll i,0ir, then we cn replce condition Ž. by the following weker one in the definition of primitive pir: Ž. x0y By grph, we men set of vertices M together with set of edges, where ech edge is two-elements subset of M, nd ech vertex is lbelled by subset of S. et nd be two grphs with their vertex sets M nd M. They re sid to be isomorphic, written, if there exists bijective mp from the set M to the set M stisfying the following two conditions. Ž. Ž. 1 The lbelling of w is the sme s tht of w for ny w M. Ž. 4 Ž. Ž.4 2 For w, z M, w, z is n edge of if nd only if w, z is n edge of. This is n equivlence reltion on grphs We define grph Ž x. ssocited to n element x W s follows. Its vertex set is MŽ x.. Its edge set consists of ll two-element subsets y, z4 MŽ x. with y, z two neighboring terms of string. Ech vertex y MŽ x. is lbelled by the set RŽ y.. A left cell grph ssocited to n element x W, written Ž x.,isby definition grph, whose vertex set M Ž x. consists of ll left cells of W with MŽ x.. Two vertices, M Ž x. re joined by n edge, if there re two elements y MŽ x. nd y MŽ x. such tht y, y4 is n edge of Ž x.. Ech vertex of Ž x. is lbelled by the common lbelling of elements of MŽ x. Žthis mkes sense by 8, Proposition Clerly, the grph Ž x. is lwys connected A subgrph of Ž x. Ž ww. is sid to be essentil, if there is n isomorphism from to Ž x. with y Ž y. for ech vertex y of. It is esily seen tht when subgrph of Ž x. is essentil, its vertex set must be distinguished. In prticulr, the grph Ž x. itself is essentil if nd only if its vertex set MŽ x. is distinguished. But it should be pointed out tht in generl there does not lwys exist n essentil subgrph in Ž x. Ž counterexmple could be found in the two-sided cell WŽ3. of the ffine Weyl group W Ž D. or in W of W Ž A., n1. 4 Ž1. n. However, we shll see tht for ny x W Ž F. 4, there lwys exists some essentil subgrph of Ž x. contining x s its vertex et nd be two grphs with N nd N the corresponding vertex sets. We sy tht is opposed to Ž up to isomorphism., written op, if there exists bijective mp from the set N to N stisfying

11 EFT CES 183 tht for ny x, y N, Ž. Rx Ž Ž.. S Rx; Ž. Ž b. x, y4 is n edge of if nd only if Ž x., Ž y.4 is n edge of. Clerly, the reltion of two grphs being opposed is mutul. So Ž op. op By pth in grph Ž x., we men sequence of vertices z, z,..., z in MŽ x. such tht z, z 4 is n edge of Ž x. 0 1 t i1 i for ny i,1it. Two elements x, x W hve the sme generlized -in- rint, if for ny pth z x, z,..., z in grph Ž x. 0 1 t, there is pth z x, z,..., z in Ž x. with RŽz. RŽ z. 0 1 t i i for every i,0it, nd if the sme condition holds when interchnging the roles of x with x It my hppen tht for two elements x, y W with x y, the grphs Ž x. nd Ž y. re not isomorphic Žtke x s0 nd y ss 1 0 in W Ž C. for exmple.. But we hve the following result. 4 PROPOSITION 25, Ž. The elements in the sme left cell of W he the sme generlized -inrint. Ž b. If x y in W, then the left cell grphs Ž x. nd Ž y. re isomorphic. 3. SOME TECHNIQUES IN APPYING THE AGORITHM 3.1. We shll pply the lgorithm to find n l.c.r. set, together with ll left cell grphs or with the corresponding essentil grphs, in ech two-sided cell of W W Ž F. 4. This hs been done for ll the two-sided cells with Ž. 3, 4, 54 in my previous pper 22. On the other hnd, n l.c.r. set in the two-sided cells W, i 0, 1, 244, hs been found before Žsee Ži. 6, 9, 19, 20.. Thus ctully we need only del with the two-sided cells of Ž. 6, 7, 9, 10, 13, 164. We shll use the nottion i for the simple reflection s Ž 0 i 4. i in the subsequent discussion First we choose the strting set P of the lgorithm. Write y wi y y with I Ž y. for ny y W. We prefer to Ž but not hve to. y choose the elements x for the set P to stisfy the following conditions. Ž. 1 x Ž. lw Ž. minž y. lw Ž. y. 4 Ix Iy Ž. 2 et A be the set of ll the elements y in which stisfy condition Ž. 1 for y insted of x. Then lž x. minž l y. ya 4. Thus the elements of the form w, I S, re the best cndidtes to be I chosen into P whenever they re contined in.

12 184 JIAN-YI SHI Ech W Ži7, 10, 164. Ži. consists of single two-sided cell, which contins unique element of the form w I, I S, i.e., w 0124, w 0234, nd w 0123, respectively. Thus in deling with the two-sided cells W Ži., i 7, 10, 164, we cn tke the strting set P of the lgorithm to be w , w 4, nd w 4, respectively. The set W Ž resp. W Ž6. Ž9. consists of two two-sided cells. There re two elements of the form wi in the set WŽ6. Ž resp. W., i.e., w nd w Ž resp. w nd w. Ž We do not know in dvnce whether or not these two elements re in the sme two-sided cell. Thus in deling with two-sided cell in W Ž res. W. Ž6. Ž9., the strting set P of the lgorithm will be tken s w 4 or w 4 Žres. w 4 or w rther thn w, w 4 Žresp. w, w et x w nd y w We hve y W by We cn show tht x 1, y Ž13. is primitive pir Ž see This implies x y nd hence x W Ž13.. So for the two-sided cell W, we shll tke x4 Ž13. s the strting set P of the lgorithm. For ny z W, we denote by Ž z. Žresp. Ž z.. the two-sided cell Ž resp. the left cell. of W contining z In pplying the lgorithm, we shll first del with the two-sided cell W, then W, W, Ž w., W Ž w., W, Ž w. Ž16. Ž10. Ž Ž Ž7. 012, WŽ6. Ž w. 012 in turn. The reson for tking such n order is to mke it esier to perform Processes Ž B. nd Ž C., in prticulr in the determintion of the -vlues of the elements occurring in the intermedite steps of these two processes Now we introduce some techniques which we shll use in Section 4: Ž. 1 to find n essentil subgrph from grph of the form Ž x.ž. ; 2 to determine the -vlue of n element; Ž. 3 to tell whether or not two sets of the form MŽ x. represent the sme set of left cells. We illustrte our methods by some exmples To exmine whether or not grph Ž. Ž W. is essentil, we should first consider the generlized -invrints of vertices of. If they re ll different, then is essentil. If there is some pir of vertices of, sy x, y, hving the sme generlized -invrint, then we should further compre the set Ž x. with Ž y.. If for ll such pirs x, y, we hve Ž x. Ž y., then is still essentil. For exmple, let y 0 w Then y W by 1.7Ž.Ž Ž , 1.10, nd by the lcove form of y. The grph Ž y. 0 0 is isomorphic to 18 with y0 the vertex lbelled by 0234 Žhere nd lter, the grphs denoted by i, i 1, re displyed t the end of Sect. 4.. We wnt to show tht the grph Ž y. 0 itself is essentil. By Proposition 2.15Ž., it is enough to show tht in Ž y. 0, where by buse of nottion, we identify vertex with its lbelling in the grph Ž y. 0, the numbers inside box represent the corresponding elements in S, nd the subscripts of boxes re used to

13 EFT CES 185 distinguish the positions of the vertices with the sme lbelling Žsuch n identifiction will be used quite often lter, which will not cuse ny confusion in the context.. Assume lž 012. lž for the ske of definiteness. We my check tht Ž Ž 012. Ž nd hence Ž 012. Ž This implies by Theorem In exmining whether or not grph Ž. is essentil, it my hppen tht there will be more thn one pir of vertices x, y4 in hving the sme generlized -invrints. In such cse, it is not lwys necessry to check the eqution Ž x. Ž y. for ech pir x, y 4. We need only to check the equtions on some pirs nd then pply Proposition 2.8 to get the conclusion on the remins. For exmple, let x w Then x W by 3.2. The grph Ž x. Ž13. is isomorphic to the one in Fig. 1 with the vertex x lbelled by We wnt to find n essentil subgrph Ž x. in Ž x. e. et y 0 x nd z 21 y. Then x y z by 3.2 nd by 1 the fct z MŽy 1.. We hve Ž z.. By Proposition 2.15Ž b. 6, we see tht essentil grphs Ž x. nd Ž z. e e should be isomorphic when- ever they exist. By compring Ž x. with Ž z. nd by Proposition 2.15Ž., Ž. Ž. op we cn tke x to be subgrph of x isomorphic to Ž e 25 the grph op i is obtined from i by replcing the number set I in ech box by 0, 1, 2, 3, 44 I nd with the subscripts of boxes unchnged whenever they re ttched.. Ž. FIG. 1. x.

14 186 JIAN-YI SHI Now we shll del with more complicted exmple, where in ddition to the bove tsk, we shll do two more things. One is to determine the -vlues of some elements in virtue of primitive pirs. The other is to conclude pir of elements, hving the reltion by observing tht one is left extension of the other with Ž. Ž.. et zw Then we clim z Ž For, let y z1. Then y is the vertex lbelled by 2 in Ž w , nd z is the vertex lbelled by 12 in Ž z. Ž see Fig. 2.. So y, z4 1 is primitive pir nd hence z Ž. Ž y. w Ž. 6. In the grph Ž z. 012, the vertex lbelled by 01 2 is left extension of tht by On the other hnd, let z be the vertex lbelled by 12 in Ž z.. Then we cn show tht Ž z4. Ž z. Ž z. 2 nd hence z z by Theorem 2.1. Thus we hve n essentil subgrph Ž z. of Ž z. e isomorphic to et x w Then we hve x WŽ13. by 3.2. The grph Ž x. is displyed in Fig. 1. Assume tht n l.c.r. set of W in the two-sided cell WŽ16. hs been found. Then we see tht there is no grph in W of the form Ž. which is isomorphic to Ž x. Ž16.. This fct will help us in finding n l.c.r. set of the two-sided cell W Ž13.. et y w Then the grph Ž y is isomorphic to 23 with y0 the vertex lbelled by We clim y0 W Ž13.. For, we hve y y 0 MŽ x. nd tht y, y is primitive pir. We wnt to find n essentil subgrph Ž y. in Ž y. e 0 0. et, be the vertices of Ž y. lbelled by 3 nd 3, respectively Ž see Then y0 123 nd y We hve ; denote it by y. Then y is common left extension of nd. We hve Ž y. Ž y. 0. Now we wnt to determine the -vlue of y. By 1.7Ž.Ž. 6 7, 1.10, nd by the lcove form of y, we hve Ž y. 13, 164. et y1 y32132 nd y2 y10. Then we hve y MŽ y., Ž y. Ž x., nd primitive pir y, y So by the bove observtion, we hve Ž y. Ž y This implies y by 1.7Ž 3.. Therefore Ž y. contins n essentil subgrph Ž y. 0 e 0 isomorphic to 22. Ž. FIG. 2. z.

15 EFT CES Suppose tht we re given two isomorphic grphs, sy nd 1. We wnt to exmine whether or not their vertex sets N, N represent the sme left cell set of W. It is known tht N1 nd N2 represent the sme left cell set if nd only if there exist some vertices of Ž i 1, 2. with Ž 1 2 hence 1 nd 2 must hve the sme generlized -in- vrint.. Thus the problem is reduced to checking whether or not the reltion holds for some N. To do this, we first choose 1 2 i i vertices N, i 1, 2, such tht nd hve the sme generlized i i 1 2 -invrint. Thus nd represent the sme set of left cells if 1 2. For exmple, let w , w , nd 3 w Then the grphs Ž. Ž i1, 2, 3. re ll essentil nd isomorphic to. We shll show i 2 Ž. i Ž tht W for i 1, 2, 3, tht the left cell set represented by M is disjoint with tht represented by M Ž., nd tht M Ž. nd M Ž. represent the sme set of left cells. et x1 13, y0 24, y y00, nd y 4. Then we cn check tht x, 4, y, 4, y, y 4, nd y, 4 re Ž. Ž ll primitive pirs. We lso see tht y M y nd x, y x, where xwž13. is defined s in 3.2. This implies i WŽ13. for i 1, 2, 3. et bj Ž j1, 2, 3. be the vertex lbelled by 03 in Ž. Ž see.. To show j 2 M Ž. nd M Ž. 1 2 representing disjoint left cell sets, it suffices to show b b. But this follows by Theorem 2.1 nd by the fct Ž b 4. Ž b Ž b 1.. Finlly, we wnt to show tht M Ž 2. nd M Ž 3. represent the sme set of left cells. It is enough to show b b. We hve b b 3; denote it by w. Then w is common left extension of b2 nd b 3. By 1.7Ž. 6, 1.10, nd the lcove form of w, we hve w Ž. 13, 164. et w w4, w 32, nd 0. Then it cn be checked tht both w, w nd, 4 re primitive pirs, tht w MŽ., nd tht Ž is isomorphic to Ž x. in Fig. 1. By the observtion t the beginning of 3.4.4, this implies w Ž. Ž. 13 nd hence b w b by 1.7Ž Our result follows. Remrk 3.5. Ž. 1 Note tht the elements y in nd w in re found in virtue of lcove forms nd by Proposition 1.6Ž. 3. Ž. 2 Besides the bove techniques, sometimes we use the property of distinguished involutions of W to determine whether or not two elements x, y in the sme s, t4-string Žost Ž. 4. with RŽ x. RŽ y. stisfy the reltion x y. An element x W is clled distinguished, if lž x. x Ž. 2Ž x. 0, where Ž x. deg P e, x. It is known tht ny distinguished element of W is n involution nd tht ech left cell of W contins unique distinguished involution. Denote by dž x. the distinguished involution in the left cell contining x. Suppose tht y s, y st, y sts is n i i

16 188 JIAN-YI SHI s,t4-string with ost Ž. 4. Written dž yst. z with, in the group generted by s, t nd s, t Ž z. RŽ z.. Then by 21, Proposition 5.12, we know tht ys ysts if nd only if sz, tz4 zs, zt 4. This result cn be used, for exmple, to conclude tht grph isomorphic to essentil in the two-sided cell Ž w is 4..C.R. SETS OF THE TWO-SIDED CES In the present section, we shll give n l.c.r. set, together with ll the left cell grphs Ž or ll the corresponding essentil grphs. for ech two-sided cell of W W Ž F. 4. Since the techniques pplied re more or less similr to those in 3.4, we shll only give very brief rguments in most cses et x w 0123, y x 4323, y0 y 4, y1 y , 1 y1 1, z y 1232, z z 1, w y 2, w w 3, y w 1234, y 0, w w 12340, w 1, w w 231, w 0, y 213, 4, u , u u 1, hz , h h 4, j u 210, j j 4, u j 34, u 3, j j , j 0, j j , j 0, k u , k k 1, k k , k 3, m j 234, b1m 1, nj2 210, n0 n 1, n1 n , b2 n1 1, p n1 21, nd p p 2. et I x, y, z, w,, u, h, j, k, n, p, Ž1 i i 8,. b Ž l1, 2.4 l. We see tht ech I hs some zero entries in its lcove form nd stisfies Ž. 0, 1, 2, 34. This implies I WŽ16. by By the techniques of 3.4, we cn show tht ll the grphs Ž. Ž I. re essentil. Thus we get Tble I. We cn show tht the set MŽ. I forms n l.c.r. set of W Ž Now consider the two-sided cell W Ž10.. et x w 0234, y x 1232, x x 12340, x x , y y 1, x 1, b x 4, y y , z y 014, y y 01234, w y , y 1, z z 2, y2 3, w0w 0, w , w2 w , w2 21, 4 w1 4, b w 1, nd 2. Then y, x, x MŽ x., y, y, z,wmž y., nd w, w, Mw Ž.. We cn check tht y, y 4, x, 4, x, b 4, y, , z, z 4, y, 4, w,w 4, w, 4, w, b 4, nd, re ll primitive pirs. et I x, y, z, w,, Ž 1 i 4,. b Ž l1, i l. Then I W Ž10..By the techniques of 3.4, we see tht Ž. Ž I. re ll essentil. So we get Tble II. It cn be shown tht the set MŽ. I forms n l.c.r. set of W Ž Next consider the two-sided cell W Ž13.. et x w , x x 0, x x , x 3, y x , y y 0, y 4, z x 23123, z z 4, z z , z 3, w z 42, w w 4, w w , w 3, z 1243, 1, 420, 1, u w 13210, u u 2, u u 32432, u 2, h x 1232, h h 1, , b 1, u u 1243, b u 4, j u 4121, nd j j 2. et

17 EFT CES 189 TABE I Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. op x y z op w u h op j k 0124 b n 0124 b p Ix, x, y, z,w,,u, h, j, Ž 1i6,. b Ž l1, i l. We shll show I W. It is known lredy tht x W Ž see 3.2. Ž13. Ž13.. We lso hve the reltions x, y, z MŽ x., w, z,w,,umž z., hmž x , 1,2 MŽ., nd u, u, j Mu Ž.. On the other hnd, x, x 4, x, 4, y, y , y, 4, z, z 4, z, 4, w,w 4, w, 4,, 4,, 4, u,u 4, u, , h,h 4,, b 4, u, b 4, nd j, j re ll primitive pirs. So I W Ž13..By the techniques of 3.4, we see tht for I, the grph Ž. is essentil if nd only if z, w,, u, j, Ž 1 i 6,b. Ž l1, i l. So for these, we hve Ž. Ž. e. On the other hnd, we cn find n essentil subgrph Ž. in ech Ž., x, x, y, h 4. We get Tble III. e TABE II Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. op x y z w 023 b b

18 190 JIAN-YI SHI TABE III Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. e e e e op x op x y z w op u b h 123 b j et M Ž. be the vertex set of the chosen essentil subgrph Ž. e e Ž such nottion will be used throughout the remining prt of the pper.. Then we cn show tht the set M Ž. I e forms n l.c.r. set of W Ž et us consider the two-sided cell Ž w et x w 123, y x 014, z x , y0 y 2, 1 y 3, z0 z 2, z1 z , 2 z14, wz 1213, w 21, w w 4, nd 4. Then y, z MŽ x nd z,w, MŽ z.. We cn check tht y, y 4, y, 4, z, z 4, z, 4, w, w , nd, 4 re ll primitive pirs. et I x, y, z, w,,, Then IŽ w.. It cn be shown tht the grph Ž. 123 is essentil if z,w,,, 4 nd is not essentil if x, y By choosing some essentil subgrphs from these grphs, we get Tble IV. We cn show tht the union M Ž. forms n l.c.r. set of Ž w.. I e 123 TABE IV Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. e e e e x y z w

19 EFT CES From the results in 4.4, we see tht w Ž w This implies Ž w. W Ž w. 234 Ž by Now we consider the two-sided cell Ž w et x w 234, x1 x 10213, y x1 234, z y , 1 x1 4, y0 y 3, z0 z 0, z1 z024, y1 y0 2123, 2 z12, b1 y1 4, w z , z2 z0 2123, z3 z2 2134, w0 w 2, b2 z2 4, nd b3 z 3. Then x, y, z MŽ x., y MŽ y., nd w, z, z, z MŽ z We cn check tht x, 4, y, y 4, z, z 4, z, 4, y, b 4, w, w 4, z, b , nd z, b 4 re ll primitive pirs. et I x, y, z, w, Ž i1, 2., b Ž i l 1 l 3.4. Then I Ž w.. We cn show tht Ž. Ž I. 234 re ll essentil nd hence get Tble V. The set MŽ. cn be shown to form n l.c.r. set of Ž w.. I Next consider the two-sided cell W. et x w, y x 3, Ž z x , w x 21, y0 y 4, z0 z 1, nd w0 w 2. Then y, z,wmž x.. We cn check tht y, y 4, z, z 4, nd w, w 4 re ll primitive pirs. et I x, y, z, w Then this implies I W Ž7.. It cn be shown tht ll the grphs Ž. Ž I. re essentil. Hence we hve Tble VI. We cn show tht the set MŽ. I forms n l.c.r. set of W Ž Now consider the two-sided cell Ž w et x w 012, y x , z x , y0 y 3, z0 z 4, w y0 14, nd w0 w 2. Then y, z MŽ x. nd w MŽ y.. One cn check tht y, y 4, z, z 4, nd w,w 4 re ll primitive pirs. et I x, y, z, w 4. Then I Ž w We cn show tht grph Ž. is essentil if z, w 4 0 0, nd is not essentil if x, y 4. Hence by choosing n essentil subgrph Ž. 0 e from Ž. for I, we get Tble VII. It cn be shown tht the set M Ž. forms n l.c.r. set of Ž w. I e Finlly consider the two-sided cell W Ž w. Ž By 1.10, it is equl to Ž w. since w Ž w by 4.7. et x w 0134, y x 23, y0 y 2, y1 y0 1423, b1 y1 4, y2 y , b2 y2 3, z y11421, TABE V Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. x y b z b w b

20 192 JIAN-YI SHI TABE VI Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. x 0124 z y 0134 w nd z z 2. Then y MŽ x. nd y, y, z MŽ y We cn check tht y, y 4, y, b 4, y, b 4, nd z, z re ll primitive pirs. et I x, y, z, b, b 4. Then I Ž w.. We cn show tht ll Ž. Ž I re essentil. So we hve Tble VIII. We cn show tht the set MŽ. forms n l.c.r. set of Ž w. I To be complete, we shll lso give n l.c.r. set for ech two-sided cell with Ž. 5or Ž. 24. These results re either obtined from my pper 22 Ž for the two-sided cells of -vlues 3, 4, 5. or by direct clcultion Ž for the two-sided cells of -vlues 0, 1, 2, 24.. WŽ0. is the l.c.r. set of itself. Ž 1. et x 4 W. Tke n essentil subgrph Ž x. in Ž x. Ž1. e such tht Ž x. e is isomorphic to 1 with x the vertex lbelled by 4. The set M Ž x. e forms n l.c.r. set of W Ž1.. Ž 2. et x 24 W. Tke n essentil subgrph Ž x. in Ž x. Ž2. e such tht Ž x. e is isomorphic to 4 with x the vertex lbelled by 24. The set M Ž x. forms n l.c.r. set of W. e Ž2. There re two two-sided cells with -vlue 3: Ž w. nd Ž w Ž. 3 et x w nd y w. Then the grph Ž x.žresp. Ž y is essentil, isomorphic to Ž resp.., nd hs x Ž resp. y. 5 8 s its vertex lbelled by 01 Ž resp The union MŽ x. MŽ y. forms n l.c.r. set of Ž w. 01. Ž. 4 et x w. Then the grph Ž x. 34 is essentil, isomorphic to, nd hs x s its vertex lbelled by 34. The set MŽ x. 20 forms n l.c.r. set of Ž w.. 34 TABE VII Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. e e e e x 012 z y 23 w

21 EFT CES 193 TABE VIII Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. x 0134 b y 023 b z Ž. 5 et x w 034, y w 014, nd z w23 be in W Ž4.. Then the grphs Ž x., Ž y., nd Ž z. re ll essentil, isomorphic to 9, 21, 10, respectively. The elements x, y, z re the vertices lbelled by 034, 014, nd 23 in,,, respectively. The set MŽ x. MŽ y. MŽ z forms n l.c.r. set of W Ž4.. Ž. 6 et x w 023, b1 x 12343, y0 x , z0 b1 21, nd b2 y 431. Then the grphs Ž b., Ž b., nd Ž z. re essentil but Ž x nd Ž y. re not. et I x, y, z, b, b By choosing n essentil subgrph Ž. from Ž. for I, we get Tble IX. e The set M Ž. I e forms n l.c.r. set of W Ž According to the results of 19, 20, we know tht there re W 0 left cells of W in the two-sided cell WŽ24. ech of which is represented by sign type over the symbol set, 4 Žsee 18 for the definition.. It is known tht there is unique shortest element in ech of such left cells nd tht the l.c.r. set consisting of such elements cn be described explicitly s 4 M w W sw W for ny s Ž w. Ž loc. cit... Ž24. Ž24. It cn be shown tht for ny x M, the grph Ž x. is essentil with MŽ x. M. Thus it remins to find ll the left cell grphs of W. Ž24. TABE IX Position of Isom. cls Position of Isom. cls in Ž. of Ž. in Ž. of Ž. e e e e x b y b2 134 p z

22 194 JIAN-YI SHI It is known tht if X is sign type over, 4 nd if Y is obtined from X by trnsposing the symbols nd, then Y is lso sign type 4 op over,. Cll Y the opposed sign type of X nd denote it by X. For given left cell grph in W, we cn replce ech vertex Ž Ž24. represented by sign type. of by its opposed sign type to get nother left cell grph of WŽ24. opposed to. Keeping this fct in mind, we cn describe the left cell grphs of W s below. Define the following sign types: Ž24. X01 X02 X03 X04 X05 X06 X07 X08 X11 X12 X13 X21 X22 X31 X32

23 EFT CES 195 X4 X5 X6 X7 X8 X9 Ž. Ž op Then X, X. with P P P 0h,1i,1j,2k,m1h8, 1 i 3, 1j, k2, 4 m 9 4 form complete set of left cell grphs of W Ž24.. The corresponding isomorphism clsses of the grphs Ž X. P re listed in Tble X It is worth mentioning tht owing to the priority we mde on the processes of the lgorithm Ž see 2.4., so fr ll the elements for our l.c.r. set of W Ž F. re only by Processes Ž A. nd Ž B. 4, nd none of them by Process Ž C et be two-sided cell of W W Ž F.. We denote by nž. 4 the number of left cells of W in, byu Ž. the corresponding unipotent clss of the complex lgebric group G of type F4 under usztig s mp in Theorem 1.8 Žpresented by its type with the nottion s in 5, Chp. 13 ; lso see 4.13., nd by AŽ. CuCu Ž. Ž. 0 the component group of the centrlizer of n element u už.; the ltter mkes sense since it is independent of the choice of u up to isomorphism. Then by the bove results, the results in 5, Chp. 13, nd by close nlysis on the usztig bijective mp in Theorem 1.8 Žsee 13., we hve Tble XI. TABE X Position of Isom. cls Position of Isom. cls X X in Ž X. of Ž X. X X in Ž X. of Ž X. X Ž 1h8. 0 h X X Ž 1i X i X Ž 1j X j Ž. X 1k2 1 X 234 3k X 0 X

24 196 JIAN-YI SHI TABE XI Ž. Ž. Ž. Ž. Ž. Ž. n u A n u A W 1 F 1 Ž w. 96 A Ž A1 1 W 5 F Ž. S W 96 A A Ž Ž W 14 F Ž. S Ž w. Ž A2 S2 Ž w. 24 C 1 Ž w. 192 A Ž w B 3 1 W Ž A 1 A 1 1 W 42 F Ž. S W 432 A Ž Ž13. 1 S2 W 96 C Ž. Ž S2 WŽ A1 1 Ž w. 96 B S W Ž24. Thus the totl number of left cells in W Ž F. 4 is According to the BlCrter Theorem, there is bijection between unipotent clsses of G nd G-clsses of pirs Ž, P., where is evi subgroup of G nd P is distinguished prbolic subgroup of the semisimple prt of. The unipotent clss corresponding to the pir Ž,P. contins the dense orbit of P on its unipotent rdicl Žsee 5, Theorem Now for two-sided cell of W, let Ž, P. be the pir ssocited to the unipotent clss už. of G. Then the type of už. listed in Tble XI just records such correspondence. et W be the Weyl group of. Then from Tble XI, we see tht the number nž. of left cells in is W W 0 if nd only if the corresponding component group AŽ. is trivil. Note tht this phenomenon does not lwys occur in n irreducible ffine Weyl group. By the existing dt, we see tht it occurs in the ffine Weyl groups of type A, n 1, of rnk 4 with its type D n 4, nd in the two-sided cells of W Ž B. Ž l3. or W Ž C. Ž m2. l m with Ž. 4, but not in the ffine Weyl groups of types D, n 4, E n 7, nd Ẽ 8. Note tht the bove phenomen were predicted implicitly by usztig in his conjecture on the number of left cells in two-sided cell of n ffine Weyl group Žsee It hs been shown in 26 tht the usztig bijective mp u cž u. from the set of unipotent conjugcy clsses of the complex lgebric group G of type F to the set of two-sided cells of W Ž F. 4 4 is order-preserving: u is contined in the closure of u Ž in the vriety of unipotent elements of G. if nd only if cž u. cž u. Ž see For two-sided cell of W, let TŽ. R be the set of ll subsets I of S such tht I Ž w. for some w. Then we cn find the following fct in the group W Ž F. 4 : two two-sided cells, e4 stisfy the reltion if nd only if TŽ. TŽ.. This R result my be expected to hold in ny ffine Weyl group The following re the grphs i, 1i26, mentioned in Sections 3 nd 4.

25 EFT CES

26 198 JIAN-YI SHI 6 7

27 EFT CES

28 200 JIAN-YI SHI 14 15

29 EFT CES

30 202 JIAN-YI SHI 18 19

31 EFT CES

32 204 JIAN-YI SHI ON A CONJECTURE 5.1. I proposed the following conjecture in the pper 22, Conjecture 2.3. Conjecture. et W be either Weyl group or n ffine Weyl group. For x, y W, x y if nd only if RŽ x. RŽ y. nd Ž x. Ž y..

33 EFT CES In my pper 24, I verified this conjecture but with the following cses in W Ž F. excluded: x Ž. 6, 7, 9, 10, 13, 164 nd RŽ x. RŽ y. 4 0, 1, 24, 3, 444. Now we cn del with these exceptionl cses. By Theorem 2.1, we need only to show the direction : if RŽ x. RŽ y. nd Ž x. Ž y. then x y. For the ske of definiteness, we my ssume lž x. lž y. without loss of generlity et us ssume x Ž.,x Ž. y, Ž. nd RŽ x. RŽ y.. By the condition Ž x. Ž y. Ž which is non-empty by our ssumption., we hve tht x y nd tht the elements x nd y hve the sme generlized R -invrint. We my ssume tht both x nd y belong to the l.c.r. set of Ž x. chosen in Section 4 by replcing x nd y by some elements in Ž x. nd Ž y., respectively, if necessry. Hence we hve y MŽ x.. We rgue by contrry. Assume x y. If RŽ x. RŽ y. 0, 1, 24, then by the results of Section 4, we hve x Ž. Ž y. 16, tht the grph Ž x. is isomorphic to 18 or 26, nd tht x nd y re the vertices lbelled by nd 012, respectively, in Ž x.. It cn be shown tht ŽŽ y 4. Ž y. Ž x. 2. If R Ž x. R Ž y. 3, 4 4, then one of the following cses must occur. Ž. 1 The grph Ž x. is isomorphic to, nd x Ž w Ž w Ž. 2 The grph Ž x. is isomorphic to, nd x Ž w Ž. 3 The grph Ž x. is isomorphic to, nd x Ž w Ž. Ž. op 4 The grph x is isomorphic to, nd x Ž. 18 is equl to 10, 13, or 16. It cn be shown tht Ž y 1. Ž y. Ž x. in cses Ž. 1, Ž. 3, nd Ž. 4, nd tht Ž y 0. Ž y. Ž x. in cse Ž 2.. Thus in ll the bove cses, we hve Ž x. Ž y., contrdiction. Therefore Conjecture 5.1 is verified without ny exception. REFERENCES 1. T. Asi et l., Open problems in lgebric groups, in Proc. Twelfth Interntionl Symposium, Tohoku Univ., Jpn, 1983, p R. Bedrd, Cells for two Coxeter groups, Comm. Algebr 14, No. 7 Ž 1986., D. Brbsch nd D. Vogn, Primitive idels nd orbitl integrls in complex clssicl groups, Mth. Ann. 259 Ž 1982., D. Brbsch nd D. Vogn, Primitive idels nd orbitl integrls in complex exceptionl groups, J. Algebr 80 Ž 1983., R. W. Crter, Finite Groups of ie Type: Conjugcy Clsses nd Complex Chrcters, Wiley Series in Pure nd Applied Mthemtics, Wiley, ondon, 1985.

34 206 JIAN-YI SHI 6. Chen Chengdong, Two-sided cells in ffine Weyl groups, Northest. Mth. J. 6 Ž 1990., Du Jie, The decomposition into cells of the ffine Weyl group of type B 3, Comm. Algebr 16, No. 7 Ž 1988., D. Kzhdn nd G. usztig, Representtions of Coxeter groups nd Hecke lgebrs, Inent. Mth. 53 Ž 1979., G. usztig, Some exmples in squre integrble representtions of semisimple p-dic groups, Trns. Amer. Mth. Soc. 277 Ž 1983., G. usztig, Chrcters of Reductive Groups over Finite Field, Mth. Studies, Vol. 107, Princeton Univ. Press, Princeton, NJ, G. usztig, Cells in ffine Weyl groups, in Algebric Groups nd Relted Topics, Advnced Studies in Pure. Mth., pp , Kinokuni nd North-Hollnd, G. usztig, Cells in ffine Weyl groups, II, J. Algebr 109 Ž 1987., G. usztig, Cells in ffine Weyl groups, IV, J. Fc. Sci. Uni. Tokyo Sect. IA Mth. 36, No. 2 Ž 1989., G. usztig nd Xi Nn-hu, Cnonicl left cells in ffine Weyl groups, Ad. in Mth. 72 Ž 1988., Rui Hebing, eft cells in ffine Weyl groups of types other thn A nd G n 2, J. Algebr 175 Ž 1995., Jin-yi Shi, The Kzhdnusztig cells in certin ffine Weyl groups, in ect. Notes in Mth., Vol. 1179, Springer-Verlg, Berlin, Jin-yi Shi, Alcoves corresponding to n ffine Weyl group, J. ondon Mth. Soc. () 2 35 Ž 1987., Jin-yi Shi, Sign types corresponding to n ffine Weyl group, J. ondon Mth. Soc. () 2 35 Ž 1987., Jin-yi Shi, A two-sided cell in n ffine Weyl group, J. ondon Mth. Soc. () 2 36 Ž 1987., Jin-yi Shi, A two-sided cell in n ffine Weyl group, II, J. ondon Mth. Soc. () 2 37 Ž 1988., Jin-yi Shi, The joint reltions nd the set D1 in certin crystllogrphic groups, Ad. in Mth. 81, No. 1 Ž 1990., Jin-yi Shi, eft cells in ffine Weyl groups, Tohoku ˆ J. Mth. 46, No. 1 Ž 1994., Jin-yi Shi, eft cells of the ffine Weyl group W Ž D., Osk J. Mth. 31, No. 1 Ž , Jin-yi Shi, The verifiction of conjecture on the left cells of certin Coxeter groups, Hiroshim J. Mth. 25, No. 1 Ž 1995., Jin-yi Shi, eft cells of the ffine Weyl group of type C 4, MPIM preprint Jin-yi Shi, The prtil order on two-sided cells of certin ffine Weyl groups, J. Algebr 179 Ž 1996., Tong Chng-qing, eft cells in the Weyl group of type E 6, Comm. Algebr 23, No. 13 Ž 1995., Zhng Xin-f, Cells decomposition in the ffine Weyl group W Ž B. 4, Comm. Algebr 22, No. 6 Ž 1994.,