Weights of Markov Traces on Cyclotomic Hecke Algebras

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1 Jounal of Algeba 238, do: jab , avalable onlne a hp: on Weghs of Maov Taces on Cycloomc Hece Algebas Hebng Ru 1 Depamen of Mahemacs, Eas Chna Nomal Unesy, Shangha, , Chna Communcaed by Godon James Receved May 10, 2000 In J, Jones used he Maov aces on he Hece algebas of ype A o consuc he no nvaans. Movaed by Jones s wo, Lambopoulou L noduced he Maov aces on he cycloomc Hece algebas of ype Gm,1, Žsee GL fo he case m 2.. Snce any lnea ace funcon can be expessed as a lnea combnaon of he educble chaaces, whee he coeffcens ae called weghs, s naual o as how o deemne he weghs of he Maov aces. In W1, Wenzl poved ha he weghs of he Maov aces on he Hece algebas of ype A Ž.e., m 1. can be expessed va Schu funcons Žsee Ž In O, Oellana poved ha hee s an epmophsm fom he Hece algeba of ype B Ž.e., m 2. wh specal paamees o some educed algeba of a ype-a Hece algeba. Ths enabled he o use Wenzl s esul o deemne he weghs of he Maov aces on ype-b Hece algebas. In I, Iancu gave a conjecue abou he geneal wegh fomulas of he ype-b Hece algebas. The man pupose of hs pape s o deemne he weghs of he Maov aces on he cycloomc Hece algebas of ype GŽ m,1,., genealzng he esuls n O. Ou agumens ae based on hose n loc. c. Howeve, we wll no use he esuls on Jones basc consucon. The conen of hs pape s oganzed as follows. In Secon 1, we collec some of he esuls on ype-a Hece algebas. We dscuss he cycloomc Hece algebas n Secon 2. The man esul of hs secon s he 1 Reseach was suppoed by he Naonal Naual Scence Foundaon , Foundaon fo Unvesy Key Teache by he Mnsy of Educaon, and he 973 Pojec of Chna. The pape was wen whle he auho was vsng he Inenaonal Cene fo Theoecal Physcs, Tese, Ialy. He hans ICTP fo s hospaly dung hs vs $35.00 Copygh 2001 by Academc Pess All ghs of epoducon n any fom eseved. 762

2 WEIGHTS OF MARKOV TRACES 763 exsence of he epmophsm fom a cycloomc Hece algeba wh specal paamees o some educed Hece algeba of ype A. We deemne he weghs of he Maov aces on he cycloomc Hece algebas n Secon 3. Afe he pape was compleed, he auho was ndly nfomed ha Gec e al. GIM ndependenly obaned esuls smla o Ž Moeove, hey exended hem o geneal choces of paamees. 1. HECKE ALGEBRA OF TYPE A 1 In hs secon, we collec some of he esuls on he Hece algebas of ype A. Le be he symmey goup on lees. Le be he complex feld wh nonzeo elemen q. The Hece algeba H assocaed o s an assocave algeba ove wh geneaos T subjec o he condons Ž 1. Ž T q.ž T 1. 0, 1 1, Ž 2. TT 1T T1TT 1, 1 2, Ž 1.1. Ž 3. TT j TT, j 1 j 1 2. Le eq be he mnmal nege n such ha 1 q q n1 0. Ž 1.2. If such an nege n does no exs, hen se eq. I s nown ha H s semsmple f and only f eq. Fom hee onwad, we always assume ha eq snce we ae neesed n he semsmple Hece algebas. The nonsomophc smple H -modules V ae ndexed by he se Ž. of paons of Žsay a wealy deceasng sequence of nonnegave neges Ž,,.... s a paon of f Ý Thee ae seveal ways o consuc he smple H -modules Žsee, e.g., KL, DJ, and M.. In hs pape, we ecall he consucon of V due o Wenzl n W1. Fo each Ž., one can denfy wh he Young dagam YŽ., whch consss of boxes aanged n a manne as llusaed by he example Ž 3, 2. fo whch we have Y Ž.. A ableau s obaned by eplacng each box wh one of he numbes 1, 2,...,, allowng no epeas. The -ableau s called sandad f he enes ae nceasng along each ow and each column. Le T s Ž. be he se of all sandad -ableaux.

3 764 HEBING RUI Fo each numbe, 1, le cž,. j f s n he jh column and h ow of T s Ž.. Le d Ž 1,. cž, 1. cž,.. Then V s a veco space ove wh bass, T s Ž., and he acon of T, 1 1, s gven as T a q c q, 1.3 d d s d d Ž d1 whee d d1,, a q q 1 q 1 q, c q 1 q. d d Ž d1 1 q.4 12 Ž1 q d. 1, and s s he ableau obaned fom by swch- ng and 1. I s easy o see ha s s no sandad only f, 1 ae ehe n he same column o n he same ow of. Howeve, d 1 and cd 0 n hs case. 2 Le T,1 T1 T 1. The full-ws elemen T,1 s n he cene 2 of H. In W2, 3.2.1, Wenzl poved ha acs on V as he scala c and c q Ž 1.Ý jž 1. j fo Ž 1, 2,.... Ž.. Ž 1.4. s Fo each sandad -ableau T, Wenzl W1, 2.7 noduced a mnmal dempoen p H, called he pah dempoen wh espec o. Le z be he mnmal cenal dempoen of H wh espec o V. Le be he sandad ableau obaned fom by emovng he box conanng. Then I s poved n W1, Coollay 2.3 ha p1 1 and p z p. Ž 1.5. ps p s ps fo any s, T s Ž., Ž.. Ž 1.6. Fo T s Ž., Ž., he subalgeba p Hf p of Hf s called he educed algeba of H wh espec o n O f. I s nown ha p H p s semsmple f H s semsmple Žsee, e.g., f f Ma, Chap. 1, Example 12.. Le V, Ž f., be he smple Hf-module defned as above. Usng he odnay banchng ule Žsee, e.g., W1, Ž fo V and Ž 1.5., one wll see mmedaely ha pv 0 unless. Ž 1.7. Recall ha, fo wo paons,, we we and say s conaned n f fo evey. Suppose. Say s T s Ž. conans T s Ž. f s obaned fom s by doppng he boxes conanng 1,...,. By Ž 1.3. and Ž 1.5., s, s, ps ½ Ž , ohewse.

4 WEIGHTS OF MARKOV TRACES 765 Thus, pv s spanned by, whee s ae sandad -ableaux conanng. s 1.9 The complee se of nonsomophc smple p Hf p-modules s pv, Ž f., wh. Ths follows decly fom Ma, Chap. 1, Example CYCLOTOMIC HECKE ALGEBRA OF TYPE Gm,1, Le W Ž m. be he weah poduc of he cyclc goup of ode m and he symmec goup. Then W s no a Coxee goup excep fo m 1, 2. Le be he complex feld wh nonzeo elemens q, u,..., u. The cycloomc Hece algeba H assocaed o W AK, BM 1 m s an assocave algeba ove wh geneaos T 0, T 1,...,T1 subjec o he condons Ž 1. Ž T0 u1. Ž T0 um. 0 2 TTTT TTTT, Ž 3. Ž T q.ž T 1. 0, 1 1, Ž TT T T TT, 1 2, TT TT, 1 j 1 2. j j I s nown ha H s he Hece algeba of ype A Ž see Secon 1. 1, esp. B,f m 1, esp. m 2, and u1 Q, u21. Also, when q 1 and u whee s he pmve mh oo of uny, H s he goup algeba of he complex eflecon goup of ype Gm,1, ove. Rema. In AK and BM, H was defned ove abay commuave ng R whou he assumpon u 1 R. Snce he man pupose of hs pape s o sudy he Maov aces on cycloomc Hece algebas ove 1 he complex feld, we have o add he assumpon u Žsee L, Secon 4.. We can assume um 1 whou loss of any genealy snce we ge an algeba wh paamees q, u1u 1 m,...,um1u 1 m, 1, whch s somophc o he ognal one f we eplace T wh u T n Ž Le 0 m 0 m1 m 1 fm, Ł Ł Ł Ž uq u j j1 1

5 766 HEBING RUI Le eq be defned as n Ž The followng esul s he specal case of he man heoem of A1. See also DR1, Ž 5.2. Ž 2.3. Ceon of Semsmplcy. Le H be he cycloomc Hece alge- 1 ba of ype Gm,1, ove he feld wh paamees q, q, u, 1,...,m. Then H s semsmple f and only f f 0 and eq. Fom hee onwads, we always assume f 0 and eq m,, whch says ha H s semsmple. Fx posve neges l, n wh l and n, 1 m 1. Le H be he cycloomc Hece algeba wh nonzeo paamees q, u,1m, such ha eq and m, u q Ž m.lý m 1 j n j, 1,...,m1, and u 1. Ž 2.4. If j, hen uq u 0 snce Ž j l. j n nj1 0 fo any 1 1 and eq. Thus fm, 0, and consequenly H s semsmple. The man esul of hs secon s ha hee s a sujecve algebac homomophsm fom H o a cean educed algeba of he Hece algeba of ype A. A by poduc s a consucon of smple H -modules. The esul fo m 2 s due o Oellana O, Secon 3. We pon ou ha he smple H -modules ove an abay feld have been classfed n AK, DJM fo he semsmple case, n DR2 unde he assumpon f 0, and n A2 m, n geneal. We need some noaon. Denoe ž / Ž m 1. l,..., Ž m 1. l,..., l,...,l. Ž 2.5. n1 n m1 m 1 Then f, f Ý n Ž m l.. Le Ž f. 1 wh. The se dffeence s nown as he sew dagam. One can denfy wh a sequence of paons Ž Ž1., Ž2.,..., Žm.., called a mulpaon o an m-paon of. Snce n, 1 m 1, he Ž. lengh of he paon s scly less han n Ž say n s he lengh of f n s he maxmal ndex wh 0.. So hee s a bjecon beween he se Ž. of all m-paons of and he se Ž f. m of all paons of f, whch conan defned as n Ž Moe explcly, fo Ž m f. hee s a unque Ž. deemned unquely by he sew dagam m m

6 WEIGHTS OF MARKOV TRACES 767, such ha ž / Ž1. Ž1. Žm. Žm. 1 n1 1 n m m 1 l,..., m 1 l,...,,...,. n1 nm 2.6 Hee nm s pa of he Ž m.. Recall ha can be denfed wh f s obaned fom by addng o deleng some zeoes a he end of. Fo example, Ž 2, 0. can be denfed wh Ž 2. and Ž 2, 0, 0., ec. So, Ž 2.6. s vald fo all Ž. m. Le be he sandad -ableau n whch he numbes 1,..., appea n ode along successve columns. Fo example, 1 2 3, f Ž Le 1 f. Snce, Ž.,1m, whee Ž. Ž 2.7. Ž. 1 and 1 0,..., 0, 1, 0,..., 0 m THEOREM. Keep he noaons aboe. Le, whee s defned as n Ž Ž. 1 Thee s an epmophsm : H f, p Hf p subjec o he cond- ons Ž 1. p, f, c 2 2 f, Ž 0. f f1 c Ž m. T p, f, Ž T. pt f, 1,..., 1. 2 Fo each, p V s a smple H -module and he se pv Ž.4 foms he complee se of nonsomophc H -modules. Ž. Ž. m m Poof. The poof of Ž. 2 wll be ncluded n he poof of Ž. 1. To show ha f, s a homomophsm of an algeba, we only need o vefy he map pesevng he defnng elaons Ž2.1Ž 15.. f,. Snce p H f, com- mues wh T fo 1, and he elaons Ž2.1Ž 35.. f follows. On he ohe hand, he full-ws elemen 2 f s n he cene of Hf and 2 f 2 f1 T T. So 2.1Ž. f1, 1 1, f1 2 follows easly fom he bad elaons gven n Ž1.1Ž We clam ha Ž T. p H p acs on he smple f, 0 f1

7 768 HEBING RUI p H p -modules pv Ž. f1 V Ž., 1 m, as scala u,1m. Be- cause p s he deny elemen n he educed algeba p Hf p, he clam mples Ł m Ž Ž T. up. 0, povng 2.1Ž 1. 1 f, 0. 2 By and he odnay banchng ule fo V Ž., acs on pv Ž. f as scala c. So, Ž T. acs on pv Ž. as he scala c Ž. c Žm. f, 0. A dec compuaon shows ha c ½ Ž. Ž m.lý m j 1 n j Ž m. m u q, f m, Ž 2.9. c u 1, f m, povng ou clam. Obvously, each p H p -module s an H f -module. In pacula, pv s an H -module, oo. If Ž. 2 holds, hen f, s sujecve by compang he dmensons of m f, and p Hf p va he Weddebun Theoem fo a fne-dmensonal semsmple algeba ove a feld. Now, we pove Ž. 2 by nducon on. Assume 1. Snce eq, u u, fo any 1 j m Žsee Ž 2.4.., and pv Ž. pv j j, j. Snce hee s only one sandad Ž. -ableau conanng, dm pv Ž. 1. By m 2 AK, 3.10, dm H m Ý dm pv Ž By he Weddebun Theoem, pv Ž. 1,...,m4 foms he complee se of nonsomophc H 1- modules. In ode o deal wh he case 1, we need he noon of a sandad -ableau fo Ž.. Say a sequence of ableaux u Ž u,...,u. m 1 m s a -ableau fo Ž Ž1.,..., Žm.. Ž. m f s obaned fom he se- quence of he Young dagams YŽ. ŽYŽ Ž1..,..., YŽ Ž m... by eplacm Ž. ng each box wh one of he numbes 1, 2,..., Ý 1, allowng no epeas. Such a ableau u s called sandad f he numbes appea o be nceasng along each column and each ow of each subableau u,1 m. sž. Le s T wh s. If we eplace he eny a n he sew ableau s by a f, one wll ge a sandad -ableau and vce vesa. In s s s s pacula, T Ž. T Ž., whee T Ž. s s T Ž.4. By AK, Ž o DJM, Ž 3.30., dm H T Ž dm pv Ý Ý s 2 2 m mž. Now, he case 1 follows fom Ž and he Weddebun Theoem ogehe wh he esuls Ž. a and Ž. b whch follow. Ž. a pv pv, fo any, Ž. 1 m wh 1. 1 Ž b. Fo each Ž., pv s smple.

8 WEIGHTS OF MARKOV TRACES 769 Usng he odnay banchng ule fo V, we have pv pv Ž whee anges ove all m-paons of f 1, whch ae obaned fom Ž Ž1.,..., Žm.. by deleng one box fom some YŽ Ž..,1 m. In hs case, we we. If,, Ž. 1 1 m, hen pv pv 1 snce hey have dffeen decomposons of H 1-modules. Le be an m-paon of 1 wh. If such a s unque, hen Ž foces pv o be a smple H 1-module. I mus be a smple H -module, oo. Suppose ha he m-paon s no unque. Le W be a H -submodule of pv. Then W conans a smple H f1-module V. Le Ž 1. wh and. Tae a sandad 1 m 1 1 -ableau such ha f 1, esp. f, s n he box, esp.. Obvously, f and f 1 ae nehe n he same ow no n he same column. So, d d Ž 1,. 1 and c 0. By Ž 1.3., d Ž T. a Ž q. c Ž q. W., f 1 d d s f1 So, W pv W, focng W pv. s f WEIGHTS OF THE MARKOV TRACES In hs secon we deemne he weghs of he Maov aces on he cycloomc Hece algebas H. Le H Ž esp. H. be he semsmple cycloomc Hece algeba of ype Gm,1, ove he complex feld Žesp., wh specal paamees q, u, 1,...,m gven n Ž Then he algeba H Ž esp. H. can be embedded naually no H Ž esp. H Le H H and H H. 1 1 Ž THEOREM L, Theoem 6. Gen z, s q, u,...,u. 1 m wh 0 m 1. Then hee s a lnea funcon : H Ž q, u,...,u, z, s.,0m, deemned unquely by he ules 1 m 1 ab ba, fo a, b H, 2 1 1, fo all H, Ž 3. Ž at. z Ž a., fo a H, Ž 4. a s Ž a., fo a H,0m1, whee T T T 1. 1, 1 0 1, 1

9 770 HEBING RUI As menoned n Secon 2, we can assume um 1. If one hopes o ge he fomula nvolvng um wh um 1, one should use u 1 m u o eplace u. The ace funcon defned above s nown as he Maov ace on H wh espec o he paamees z, s,0 m 1. Noe ha any lnea ace funcon on a fne-dmensonal algeba s a lnea combnaon of he educble chaaces, whee he coeffcens ae called he weghs. By AK, 3.10 o DJM, 3.30, Ý Ž x. Ž x., fo x H, Ž 3.2. H mž. whee s he chaace of he educble epesenaon of H ndexed by he m-paon Ž. m. In fac, snce he consucon of smple H -modules s ndependen of he value q, u 1,...,um unde he assumpon Ž 2.3. Žsee, e.g., DJM, Ž o DR2, Ž 2.1.., we can assume ha coesponds o pv when u u,1m. Now, we ecall some of he esuls on he Maov ace, whch s defned on he Hece algebas of ype A. In W1, Theoem 3.6, Wenzl compued he wegh w fo Ž. when H s he Hece algeba of ype A1 and z q n Ž 1 q. Ž1 q n., n. Moe explcly, s 1, q,...,q n1 s, n, s 1, q,...,q n1 1 jj lž. 1 q n1 Ý 1 Ž1. s 1, q,...,q q Ł, j 1 q 1jn Ž 3.3. whee s s he Schu funcon Mc and lž. s he lengh of Ž.e., he maxmal ndex wh 0.. I s poved n W1, Lemma 3.5 ha lž. s, n 0, fo lž. n. By Ž 1.8., p fo any s T s Ž.. Thus Ž p. s, s s, and Ž p. Ž p. w s. Ž 3.4. Hf, n In ode o ge he nonzeo elemen s, we have o assume n Ý m 1 n m 1 snce l Ý n Žsee Ž Consde he lnea funcon 1, n 1 Ž x. p Ž x., x p Hf p, Ž 3.5. Ž p. Ž. 1 n Ž n. whee s he Maov ace on H H wh espec o he paamee z q 1 q 1 q. Noe ha p s he deny elemen n

10 WEIGHTS OF MARKOV TRACES 771 p H p. Snce Ž. f s he Maov ace on H, one can vefy easly ha p defned as n 3.5 s he ace funcon on 1 p Hf p, whch sasfes Ž3.1Ž Usng Ž 3.3. Ž 3.5., he wegh wh espec o s s, n s, n. Fom hee onwad, we always assume ha n n n n wh n,1 m 1, n. 1 2 m m Snce s an epmophsm, he ace defned as n Ž 3.5. f, p esuls n a lnea funcon Ž. on H, Ž x. Ž x., x H. Ž 3.6. H p, f Ž 3.7. THEOREM. Keep he noaon aboe. Ž. 1 Ž. s he Mao ace on H wh espec o he paamees n Ž n. m z q 1 q 1 q, wh n Ý1 n, n, 1 m 1, nm, and s ŽT. 0,0m1. Ž. 2 Le be he paon defned as n Ž Then s, n H Ž x. Ý Ž x.. s Ž., n m Poof. As menoned below Ž 3.5., he ace funcon p defned on p H p s he Maov ace wh espec o he paamee z q n Ž f 1 n q. Ž 1 q.. So, defned on H sasfes he condons Ž3.1Ž By Ž2.8Ž 2.., Ý Ž x. Ž x. Ž x., H p, f Ž. m whee s he educble chaace wh espec o he smple module pv. So, s s. We clam ha, n, n 1 ž pt w pt f1, f1ž Tf1,1T1, f1. Tf1, f1/ p ž w f1,1 1, f1 / f Ž pt p. p T T, w. Ž 3.8. I s easy o see ha Ž 3.8. holds f w e, whee e s he deny elemen n. Suppose lw 1, whee lw s he lengh of w. We pove Ž 3.8. f by nducon on and lw. Suppose 1. Snce z s he mnmal cenal dempoen wh espec o he educble epesenaon of Hf ndexed by, z Hf z End Ž z H. and z H z z. By Ž 1.5., Hf f f p H p p. 3.9 f

11 772 HEBING RUI If lw 1, hen w s,1f. By W1, Ž 2.3e. and Ž 3.9., pt w p cp fo some c, focng Ž 3.8. o hold. Suppose lw 1. Then w s, f1 x, whee x wh lw lž s. lž x. f, f1. The case f 1 follows fom Ž 3.9. snce T H.If f 1, hen w f 1 ž pt, f1tx pt f1ž Tf1,1T1, f1. Tf1/ 1 ž, f f1 f f1 x Ž f1,1 1, f1. / 1 ž pt, ftt f f1tf Tx p Ž Tf1,1T1, f1. / by 2.1Ž 4. z ž pt, ftx p Ž Tf1,1T1, f1. / by 3.1Ž 3., 3.1Ž 1. z p Ž pt, ftx p. ž pž Tf1,1T1, f1. / by Ž 3.9. p Ž w. ž Ž f1,1 1, f1. /, f1 pt T TT T p T T by 3.1Ž 1., 2.1Ž 5. pt p p T T by 3.1 3, w s x. Ths complees he poof of Ž 3.8. fo 1. Suppose 1 and lw 1. Then w s x, whee x wh lw lž s. lž x., f f1, f.if f, hen T H and w f1 1 ž w f1, f1ž f1,1 1, f1. f1, f1/ 1 ž w f, f1ž f1,1 1, f1. f, f1/ p ž w f1,1 1, f1 / pt pt T T T pt pt T T T by 3.1Ž 1., 2.1Ž 5. pt p p T T by nducon assumpon. Suppose f. We have 1 1 ž pt, ftx pt ftf, f1ž Tf1,1T1, f1. Tf, f1tf / 1 1 ž pt, f1 TfTf1TfTx pt f, f1ž Tf1,1T1, f1. Tf, f1/ 1 1 ž pt, f1tf1 TfTf1Tx pt f, f1ž Tf1,1T1, f1. Tf, f1/ 1 z ž pt, f1tx pt f, f1ž Tf1,1T1, f1. Tf, f1/ z p Ž pt T p. ž p Ž T T., f1 x f1,1 1, f1 / p ž w f1,1 1, f1 / Ž pt p. p T T by 3.1Ž 3.. by he nducon assumpon

12 WEIGHTS OF MARKOV TRACES 773 Ths complees he poof of Ž Fo any h H, Ž h. f, p Hf p. Snce T w 4 foms a -base of H, by Ž 3.8., we have w f f 1 ž, fž h. Tf1, f1ž Tf1,1T1, f1. Tf1, f1/ pž, f Ž h.. ž p Ž Tf1,1T 1, f1. /. By 3.6, we have h s h fo any h H, povng Suppose uq j u j 0, fo 1 j m and 1 n j n j 1, and n, 1 m 1. Fo any Ž Ž1.,..., Žm.. Ž. m, le m lž Ý Ž.. Ž j 1. Ž. and n Ý m n. Le j1 j 1 Ž. Ž. m j j m 1 q 1 q Ý 1 m Ž 1 m. n Ł Ł j W q, u,...,u q ž / 1 q 1 q 1 1jn n Ž. Ž j. n j l l uq uq j Ł Ł Ł l. uq uq 1jm 1 l1 j I s easy o see ha W q, u 1,...,um s a aonal funcon wh uq uq j l 0 fo all 1 j m, 1 n, and 1 l n j. Ž THEOREM. Le n wh n, 1 m 1, and le n Ý n. Defne Ž. : H o be he lnea funcon wh Ý Ž x. W Ž x. fo any x H. H mž. Then Ž. s he Mao ace on H wh espec o he paamees z q n Ž1 q. Ž1 q n., and s ŽT.,0m1, whee 0 1 q u uq s u. Ž n n nj j Ý n Ł 1 1 q j,1jm uj u Poof. Snce s a ace funcon on H, s easy o see ha H sasfes 3.1Ž. 1. Howeve, when u u, 1,...,m Žsee Ž 2.4.., uns ou o be he Maov ace funcon on H wh espec o he paamees z, s ŽT.. A dec compuaon shows ha W Ž q, u,...,u. 0 1 m. Now we pove ha Ž3.1Ž 24.. holds. Le FŽ q, u,..., u. 1 m1 be a polynomal n ndeemnae q, u 1,..., u m. We clam ha FŽ q, u,...,u. 0f FŽ q, u,...,u. 1 m1 1 m1 0 fo all u, 1 m 1. In fac, we we Ý FŽ q, u,...,u. f Ž q, u,...,u. u, 1 m1 1 m2 m1

13 774 HEBING RUI whee f Ž q, u,...,u. 1 m2 s a polynomal n ndeemnaes q, u 1,...,u m2. We hope o pove ha f Ž q, u,...,u. 1 m2 0. Fx u u,1m2. Then FŽ q, u,...,u. Ý f Ž q, u,...,u. 1 m1 1 m2 u m1 0. By he deny heoem, FŽ q, u,...,u, u. 0. So, f Ž q, u,...,u. 1 m2 m1 1 m2 0. By n- ducon, we have f Ž q, u,...,u. 1 m2 0. Now, consde he funcon Ž. 1 q, u 1,...,um1 1. Snce W and ae analyc aonal funcons, Ž x. F Ž q, u,...,u. G Ž q, u,...,u. x 1 m x 1 m fo some polynomals F Ž q, u,..., u. and G Ž q, u,..., u. x 1 m x 1 m dependng on x such ha G Ž q, u,...,u. 0. Snce Ž x.ž q, u,...,u. s he Maov ace on H, x 1 m 1 m Ž 1.Ž q, u,...,u. 1 and F Ž q, u,...,u. G Ž q, u,...,u.. So, 1 m 1 1 m 1 1 m F1Ž q, u 1,...,um. G1Ž q, u 1,...,um. and Ž One can pove Ž3.1Ž 34.. smlaly. Ž. Le ŽŽ. 0,..., Ž.Ž.Ž. 0, 1, 0,..., Ž.. 0. Va Ž 3.11., 1 q n u uq n j j W Ž. Ł. n 1 q u u j,1jm j Ž. Ž. Ž. 0 Now, 3.13 follows snce T u see 2.9. ACKNOWLEDGMENT I s my pleasue o han Pofesso Lambopoulou fo sendng he epn L o me, whch movaed me o we hs pape. REFERENCES A1 S. A, On he sem-smplcy of he Hece algeba of Ž. n, J. Algeba 169 Ž 1994., A2 S. A, On he classfcaon of smple modules fo cycloomc Hece algebas of ype Gm, Ž 1, n. and Kleshchev mulpaons, pepn. AK S. A and K. Koe, A Hece algeba of Ž. n and he consucon of s educble epesenaons, Ad. Mah. 106 Ž 1994., BM M. Boue and G. Malle, Zyloomsche Hecealgeben, Asesque 212 Ž 1993., DJ R. Dppe and G. James, Repesenaons of Hece algebas of geneal lnea goups, Poc. London Mah. Soc. Ž Ž 1986., DJM R. Dppe, G. James, and A. Mahas, Cycloomc q-schu algebas, Mah. Z. 229 Ž 1998., DR1 J. Du and H. Ru, AKoe algebas wh sem-smple booms, Mah. Z. 234 Ž 2000., DR2 J. Du and H. Ru, Spech modules fo A-Koe algebas, Comm. Algeba, o appea.

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