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On the category O for rational Cherednik algebras
Ginzburg, V.; Guay, N.; Opdam, E.M.; Rouquier, R.
DOI
10.1007/s00222-003-0313-8
Publication date
2003
Published in
Inventiones Mathematicae
Link to publication
Citation for published version (APA):
Ginzburg, V., Guay, N., Opdam, E. M., & Rouquier, R. (2003). On the category O for rational
Cherednik algebras. Inventiones Mathematicae, 154(3), 617-651.
https://doi.org/10.1007/s00222-003-0313-8
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arXiv:math.RT/0212036 v4 26 Jun 2003
VICTORGINZBURG, NICOLASGUAY, ERIC OPDAMAND RAPHA
EL ROUQUIER
Abstra t. Westudythe ategoryOofrepresentationsoftherationalCherednikalgebraA W
atta hed to a omplex re e tion group W. We onstru t an exa t fun tor, alled
Knizhnik-Zamolod hikovfun tor: O!H W
-mo d,whereH W
isthe( nite)Iwahori-He kealgebraasso
i-atedtoW. WeprovethattheKnizhnik-Zamolo d hikovfun torindu esanequivalen eb etween
O =O tor
,thequotient ofObythesub ategory ofA W
-mo dulessupp ortedonthedis riminant,
andthe ategoryof nite-dimensionalH W
-mo dules. ThestandardA W
-mo dulesgo,underthis
equivalen e,to ertainmo dulesarisinginKazhdan-Lusztigtheoryof \ ells",providedW isa
Weylgroup and theHe ke algebraH W
hasequalparameters. Weprove that the ategory O
is equivalentto themo dule ategoryover a nitedimensionalalgebra,ageneralized "q -S hur
algebra"asso iatedtoW.
Contents
1. Intro du tion 1
2. CategoryO 3
2.1. Algebraswithtriangularde omp osition. 3
2.2. Lo allynilp otentmo dules 3
2.3. Standardmo dules 4
2.4. Gradedmo dules 5
2.5. Highest weighttheory 7
2.6. Prop erties of ategory O 8
3. RationalCherednikalgebras 9
3.1. Basi de nitions 9
3.2. Category OfortherationalCherednikalgebra 11
4. Duality,Tiltings,andProje tives 12
4.1. Ringelduality 12
4.2. NaivedualityforCherednikalgebras 13
4.3. Homologi alprop ertiesofCherednikalgebras 14
5. He ke algebrasviamono dromy 16
5.1. Lo alisation 16
5.2. Dunkl op erators 18
5.3. TheKnizhnik-Zamolo d hikovfun tor. 20
5.4. Mainresults 21
6. RelationtoKazhdan-Lusztigtheoryof ells 25
6.1. Lusztig'salgebraJ 25
6.2. Standardmo dulesfortheHe kealgebraviaKZ-fun tor 27
Referen es 27
1. Introdu tion
LetW b ea omplexre e tiongroupa tingon ave torspa eV. LetA W
denotethe rational
of W with the algebra of p olynomial di erentialop erators on V. The algebra A W
an b e also
realized as an algebra of op erators (Dunkl op erators) a ting on p olynomial fun tions on V.
When W is a Weyl group, A W
is a rational degeneration ofthe double aÆneHe kealgebra.
Ani e ategoryOofA W
-mo duleshasb eendis overedin[DuOp℄, f. also[BeEtGi℄. Itshares
manysimilaritieswiththeBernstein-Gelfand-Gelfand ategoryO fora nite-dimensional
semi-simpleLie algebra.
We develop a general approa h to the ategory O for a rational Cherednik algebra, similar
in spirit to So ergel's analysis, see[So1 ℄,of the ategoryO inthe Lie algebra ase. Sp e i ally,
in addition to the algebra A W
, we onsider an appropriate ( nite) He ke algebra H W
, and
onstru t an exa tfun tor KZ:O !H W
-mo d, thatmay b e thoughtof as aCherednikalgebra
analogue of the fun tor Vof [So1℄. One of our main results says that the fun tor KZ is fully
faithful on proje tives. Thus, the (non ommutative!) He ke algebra plays, in our ase, the
role similar to that the oinvariant algebra (= ohomology of the ag manifold) plays in the
Lie algebra ase. It is also interesting to note that, in b oth ases, the algebra in question is
Frob enius.
To proveour results, in x2 we develop somebasi representation theory over a ground ring
(whi h is not ne essarily a eld) of a general asso iative algebra with a triangular
de ompo-sition. This generalizes earlier work of the se ond author [Gu℄ and of the last two authors
(unpublished). Su h generality willb e essential for us in order to use deformationarguments
in x5 . The results of se tion 2are appliedto Cherednik algebras in x3.2 .
In x4 ,we explainhowto generalizesome lassi al onstru tionsfor D (V), the Weylalgebra,
(su h as hara teristi varieties, duality) to the rational Cherednik algebra. We study two
kinds of dualities. One of them is related to Fourier transform while the other, mu h more
imp ortant one, generalizes the usual (Verdier typ e) duality on D -mo dules. This enables usto
showthat the Ringel dualof ategoryO is a ategoryO forthe dual re e tiongroup. We also
giveaformula for thedimensionof the hara teristi varietyinvolving onlythe highestweight
stru ture of O .
Our most imp ortant results are on entrated in x5.4. We use the de Rham fun tor for
Knizhnik-Zamolo d hikovtyp e D -mo dules overthe omplementof the rami ation lo us inV.
This way, we relate the ategory O with a He kealgebra. We prove that the ategory O an
b e re overed from its quotient by the sub ategory of obje ts with non-maximal hara teristi
variety(Theorem5.3 and Corollary5.5).
Then,weobtaina\double entralizer"Theorem5.16,assertinginparti ularthatthe ategory
O isequivalentto the ategoryof mo dules overthe endomorphismring of someHe kealgebra
mo dule. A ru ial p oint is the pro of that the de Rham fun tor sends the D -mo dules oming
from obje ts of O to representationsof the braid group that fa tor through the He kealgebra
(Theorem5.13 ).
In a di erent p ersp e tive, our results provide a solution to the problem of asso iating a
generalized\q-S huralgebra"to an arbitrary nite omplexre e tiongroupW. Thisseemsto
b enew evenwhenW isaWeylgroup (ex eptfor typ esA;B). For instan e,letW b etheWeyl
groupofanirredu iblesimply-la edro otsystem. Then,thedatade ningtheCherednikalgebra
A W
redu es to a single omplexparameter 2 C. In this ase, H W
is the standard
Iwahori-He kealgebra ofW,sp e ialisedatthe parameterq=e 2 i
. If isarationalnumb er,thenq isa
ro ot ofunity,andthe orresp onding ategoryH W
-mo db e omesquite ompli ated. Ourresults
ategory H W
-mo d, whi h is not itself quasi-hereditary. As a onsequen e, the de omp osition
matri esofHe kealgebras(in hara teristi 0)aretriangular(Corollary5.19 ). Weremarkthat,
inviewof[CPS2℄,onemighthaveexp e tedongeneralgrounds thatthe ategoryH W
-mo donly
has a\strati ed over", whi his weakerthan having a\quasi-hereditary over".
The reader should b e reminded that, in typ e A, a well-known \quasi-hereditary over" of
H W
-mo d isprovided by the q-S huralgebra. We exp e t that the latter ategory is equivalent
to the ategoryO . Furthermore,foran arbitrary nite Weyl groupW, weproveinx6that the
KZ-fun tor sends the standard mo dules in O to mo dules over the He ke algebra (with equal
parameters)that anb edes rib edviaKazhdan-Lusztig'stheoryof ells. Itfollowsinparti ular
that, intyp e A,thestandard mo dulesin ategoryO go to Sp e ht(or `dualSp e ht',dep ending
on the sign of parameter ` ') H W
-mo dules,intro du edin[DJ℄.
A knowledgments. These ond namedauthor gratefullya knowledgesthe nan ialsupp ort of the
FondsNATEQ.Thethird namedauthorwaspartiallysupp ortedbyaPionergrantoftheNetherlands
Organi-zationforS ienti Resear h(NWO).
2. Category O
2.1. Algebras with triangular de omposition. In this se tion, we assume given an
asso- iative algebra A with a triangular de omp osition. We study a ategory O (A) of A-mo dules,
similar to the Bernstein-Gelfand-Gelfand ategory O for a omplex semi-simple Lie algebra.
The main result of this se tion is Theorem 2.19 b elow, saying that the ategory O (A) is a
highest weight ategory (in the sense of [CPS1 ℄).
Throughout this se tion 2 , letk 0
b e an algebrai ally losed eld and k a ommutativeno
e-therian k 0
-algebra.
Let A b e agraded k-algebra with three graded subalgebras B,
B and H su h that
A=
BH B as k-mo dules
B and
B are proje tiveoverk
BH =HB and H B = B H B = L i0 B i , B = L i0 B i ,and B 0 = B 0 =k and H A 0 . H =k k 0 H(k 0 )where H(k 0
) isa nite dimensionalsemi-simple splitk 0
-algebra
thegradingonAisinner,i.e.,thereexists 2A 0 su hthatA i =fu2Aju u =iug. WedenotebyBH and
BH thesubalgebrasBH and BH. WeputB i =B i . Wedenote by Irr(H(k 0
)) the set of isomorphism lasses of nite dimensional simple H(k 0 )-mo dules. We put = 0 0 with 0 2 B H B >0 and 0 2Z(H). For E 2Irr(H(k 0 )),we denoteby E
the s alarby whi h 0
a ts on k k0
E.
The theorydevelopp edhere is loselyrelatedto the onedevelopp edby So ergel[So2 , x3-6℄ in
the asewhereg isagraded Liealgebra withg 0 redu tive,A=U(g),B =U(g >0 ), B =U(g <0 ) and H =U(g 0 ). 1
2.2. Lo ally nilpotent modules. We denote by O ln
the full sub ategory of the ategory of
A-mo dules onsisting of those mo dules that are lo ally nilp otent for B, i.e., an A-mo dule M
is in O ln
if for every m 2 M, there exists n 0 su h that B >n
m = 0. This is a Serre
sub ategory of the ategory of A-mo dules.
1
IntheLiealgebra ase,thealgebraH=U(g 0
Remark2.1. The anoni al fun torD b (O ln )!D b
(A)isnotfaithfulingeneral. Nevertheless,
for i=0;1, and any M;M 0
2O
ln
, one stillhas Ext i O ln (M;M 0 ) !Ext i A (M;M 0 ). 2.3. Standard modules. 2.3.1. Let h 2 H. We denote by h : B !
B H A the map de ned by
h ( b) = h b. Similarly,we denote by h
:B !H B Athe map de nedby h
(b)=bh.
LetE b ean H-mo dule. TheaugmentationB !B=B >0
=k indu esamorphismofalgebras
BH ! H and we view E as a BH-mo dule by restri tion via this morphism. All simpleBH
-mo dules that are lo ally nilp otent overB are obtained by this onstru tion,starting withE a
simpleH-mo dule.
Weput (E)=Ind A BH E =A BH E:
The anoni al isomorphism(E) ! BE isan isomorphismof graded BH-mo dules (E is
viewedin degree0), where
B a ts by multipli ationon
B and the a tionof h 2H isgiven by
h H 1 E : BE ! BH H E = BE.
We now put r(E)= Homgr
B H (A;E)= L i Homgr i B H
(A;E) (this is also the submo dule of
elementsof Hom B H
(A;E)thatare lo ally niteforB). Here,E isviewedas a
BH-mo dulevia
the anoni al morphism BH ( B= B >0 )H =H.
We have an isomorphism of graded BH-mo dules r(E)
! Hom
k
(B;k)E where B a ts
by left multipli ation on Hom k
(B;k) and the a tion of h 2 H is given by f e 7! (be 7!
(1f)( h
(b))e).
The A-mo dule (E) is a graded mo dule, generated by its degree 0 omp onent. The
A-mo duler(E) is also graded. Both (E) and r(E) are on entratedin non-negative degrees,
hen e are lo ally nilp otentfor B.
2.3.2. We have Ext i A ((E);r(F))'Ext i B H (Res B H (E);F)'Ext i B H (Ind B H H E;F)'Ext i H (E;F):
It follows that, when k is a eld and E;F are simple,then
(1) Ext
i A
((E);r(F))=0if i6=0 or E 6'F and Hom A
((E);r(E))'k:
Let N b e any A-mo dule. We have
(2) Hom A ((E);N) !Hom BH (E;Res BH N)
2.3.3. A - ltration for a A-mo duleM is a ltration 0 = M 0 M 1 M n = M with M i+1 =M i ' (k k 0 E i ) for some E i 2 Irr(H(k 0 )). We denote by O
the full sub ategory of
O ln
of obje tswith a - ltration.
Givenan H-mo dule E and n 0,we also onsidermoregeneral mo dules
n (E)=Ind A BH (B=B >n ) k E The mo dules n (k k 0
F) havea - ltration,when F is a nite dimensional H(k 0
)-mo dule.
For N a A-mo dule,we have
Hom A ( n (E);N) !Hom BH (B=B >n ) k E;N :
N is in O
N is a quotient of a (possibily in nite) sum of n
(E)'s
N has anas ending ltration whose su essive quotients are quotients of (E)'s.
2.4. Graded modules.
2.4.1. Given 2k and M a A-mo dule,de negeneralized weight spa esin M by
W
(M)=fm2M j ( ) n
m =0 for n 0g:
LetOb ethefullsub ategoryofO ln
onsistingofthosemo dulesM su hthatM = P 2k W (M) where W
(M) is nitelygenerated overk, for every 2 k. This is a Serre sub ategory of the
ategory of A-mo dules.
Let ~
O b e the ategory of graded A-mo dules that are in O . This is a Serre sub ategory of
the ategoryof graded A-mo dules.
Let ~ O
b e thefullsub ategory of ~
O onsisting ofthose obje tsM su hthat M i
W
i (M)
for all i. Note that this amounts to requiring that 0
(i + F
) a ts nilp otently on
Homgr i H (k k 0 F ;M) for F 2Irr(H(k 0 )),sin e and 0 ommute.
Moregenerally,if I is asubset of k, wedenote by ~ O I
the full sub ategory of ~
O onsisting of
those obje tsM su h thatM i P 2I W i (M). Wedenoteby ~
(E)the gradedversionof(E)(itisgeneratedindegree0andhasno terms
in negative degrees). Further, write hr i for `grading shiftby r ' of a graded ve tor spa e.
Lemma 2.3. Let E 2Irr(H(k 0 )). We have ~ (k k 0 E)hr i2 ~ O E r .
Proof. Notethat 0 a tsas zeroon ~ (k k0 E) 0 ,sin eB >0
a tsas zero onit. So, a ts as E
on it. It follows that a ts by i E on B i ~ (k k 0 E) 0 = ~ (k k 0 E) i
and we are done.
2.4.2. LetP b e the quotient of S E2Irr(H(k 0 )) ( E
+Z) by the equivalen e relationgivenas the
transitive losureof the relation : if is not invertible.
Wemakethe following assumption untilthe end of x2.4 .
Hypothesis 1. We assume that E
E
+n for some n 2 Z impliesn = 0 (this holds for
examplewhen k is a lo alring of hara teristi zero).
Proposition 2.4. We have ~ O = L a2P ~ O a .
The image by the anoni al fun tor ~ O !O of ~ O a+n is a full sub ategory O a+Z independent of n2Z. We have O = L a2P=Z O a+Z
and the forgetful fun tor ~ O a !O a+Z isan equivalen e.
Proof. LetM b ean obje tof O . Leta 2P and M a = P 2 a+Z W (M). ByLemma2.3 and
Prop osition 2.2, we havea de omp ositionM = L a2P=Z M a as A-mo dules. Similarly,given ~ M 2 ~ O ,wehave ~ M = L a2P ~ M a where ~ M a = M i X 2a (W i (M)\M i )2 ~ O a : GivenM 2O a+Z
,weput a grading on M by settingM i = P 2i a W
(M) (here we use the
assumptiononk). Thisde nesanelementof ~ O a
and ompletesthepro ofoftheprop osition.
Wedenote byp : ~ O! ~ O a
2.4.3. We now give a onstru tion of proje tiveobje ts (underHyp othesis1).
Lemma 2.5. Let a2P and d 2Z. There is an integer r su h that the anoni al map
Hom( ~ m (H)h di;M) !M d
is an isomorphism forall mr and M 2 ~ O a
.
Proof. Repla ingM by Mhdi and a by a+d, we an assume that d=0.
There isan integer r su h that p a ( ~ (H)hr 0 i)=0 for r 0
r . The exa t sequen e
0! ~ (B m H)hmi! ~ m (H)! ~ m 1 (H)!0
shows that the anoni al map
Hom( ~ r (H);M) !Hom( ~ m (H);M)
is an isomorphismfor any M 2 ~ O a
and mr . Equivalently,the anoni al map
Hom B (B=B r ;M) !Hom B (B=B m ;M)
is an isomorphism. Sin e M is lo ally B-nilp otent, this givesan isomorphism
Hom( ~ m (H);M) !M 0 :
Corollary 2.6. LetE 2Irr(H(k 0
)) and a2 E
+Z. Then, the obje t p a ( ~ r (k k 0 E)ha E i) of ~ O a
is independent of r ,for r 0. It isproje tive, has a ltration bymodules ~
(k
k0 F)hr i
and has a quotient isomorphi to ~ (k k 0 E)ha E i.
Corollary 2.7. LetE 2Irr(H(k 0
)). Then, forr 0, themodule r
(k
k 0
E)has a proje tive
dire tsummand whi h is - ltered and has a quotient isomorphi to (k k
0 E).
Corollary 2.8. Thereisaninteger r su hthat r
(H) ontains aprogeneratorof O as adire t
summand.
Lemma 2.9. Let E;F 2 Irr(H(k
0
)) su h that Ext 1 O ((k k 0 E);(k k 0 F)) 6= 0. Then, F E is a positive integer.
Proof. ByLemma2.3andProp osition2.4,wehaveExt 1 O ((k k 0 E);(k k 0 F))=0if F E
is not an integer. Assumenow F E is an integer. Then Ext 1 O ((k k 0 E);(k k 0 F))' Ext 1 ~ O ( ~ (k k 0 E); ~ (k k 0 F)h F E i) ' Ext 1 A ( ~ (k k0 E); ~ (k k0 F)h F E i);
by Lemma2.3 and Prop osition 2.4 . Now,
Ext 1 A ( ~ (k k 0 E); ~ (k k 0 F)h F E i)'Ext 1 BH (k k 0 E;Res BH ~ (k k 0 F)h F E i):
If the last Ext 1
is non zero, then F
E
is ap ositiveinteger.
Corollary 2.10. Assume k is a eld. Let E 2 Irr(H). Then, L(E) has a proje tive over
P(E) with a ltration Q 0
=0 Q
1
Q d
=P(E) su h that Q i =Q i 1 ' (F i ) for some F i 2Irr(H), F E
is a positive integer for i6=d and F d
Proof. We know already that there is an inde omp osable proje tive mo dule P(E) as in the
statementsatisfying allassumptions but the one on F
i
E
, by Corollary2.7.
Take r 6= d maximal su h that Q r =Q r 1 ' (F) with F E
not a p ositive integer. By
Lemma 2.9 , the extension of P(E)=Q r 1
by (F) splits. So, we have a surje tive morphism
P(E)!(E)(F). This isimp ossible sin e P(E) is inde omp osableand proje tive.
2.5. Highest weight theory.
2.5.1. We assume here that k is a eld.
For E asimpleH-mo dule,allprop ersubmo dulesof (E)are graded submo dulesby Prop
o-sition 2.4, hen e are ontained in (E) >0
. Consequently, (E)has a unique maximalprop er
submo dule, hen ea unique simplequotient whi hwe denote by L(E).
Itfollows from(1 )thatL(E)istheuniquesimplesubmo duleofr(E)andthatL(E)6'L(F)
for E 6'F.
Proposition 2.11. The simple obje ts of O ln
are the L(E) forE 2Irr(H).
Proof. LetN 2O ln
. Then there is a simpleH-mo dule E su hthat Hom BH
(E;Res BH
N)6=0.
By(2 ),itfollowsthateverysimpleobje tofO ln
isaquotientof(E)forsomesimpleH-mo dule
E.
2.5.2. LetM b eaA-mo dule. Letp(M)b e thesetofelementsofM annihilatedbyB >0
. This
is an H-submo duleof M.
Lemma 2.12. Let M be a A-module and E an H-module. Then,
M isaquotient of (E)if andonlyifthere isa morphismofH-modules':E !p(M)
su h that M =A'(E) ;
Ifk isa eldandEissimple,thenM 'L(E)ifandonlyifM =Ap(M)andp(M)'E.
In parti ular, Ap(M) is the largest submodule of M that is a quotient of (F) for some H
-module F.
Proof. The rst assertion follows from (2) and the isomorphism
Hom BH (E;Res BH M)'Hom H (E;p(M)):
Now,we assumek is a eld and E is simple.
Assume p(M) ' E and M = Ap(M). Then, M is in O ln
. Let N b e a non-zero submo dule
of M. We have 0 6= p(N) p(M), hen e p(N) = p(M) and N = M. So, M is simple and
isomorphi to L(E) sin e M is aquotientof (E).
Assume M ' L(E). Sin e dim
k
Hom((F);L(E)) = 1 if E ' F, and this Hom-spa e
vanishesotherwise, it follows from(2) that p(M) 'E.
Let Mf0g =0 and de ne by indu tion Nfig =M=Mfig, Lfig =Ap(Nfig) and Mfi+1g
as the inverse image of Lfig in M. We have obtained a sequen e of submo dules of M, 0 =
Mf0gMf1gM.
Sin e(E)islo allynilp otentforB,the followingprop osition is lear. Itdes rib es howthe
obje tsof O ln
are onstru tedfrom (E)'s ( f Prop osition 2.2 ).
Proposition 2.13. A A-module M islo ally nilpotent for B ifand only if S
i
Lemma 2.14. Assume k is a eld. Every A-module quotient M of (E) has a nite Jordan-Holder series 0 = M 0 M 1 M d = M with quotients M i =M i 1 ' L(F i ) su h that F i 2Irr(kW), E F i
is a positive integer for i6=d and F d
=E.
Proof. By Prop osition 2.4, we an assume F
E
2 Z. FromProp osition 2.4, it follows that
M inherits a grading from (E) (with M 0
6=0 and M <0
=0). Note that sin e 0 a ts as zero on p(M), we havep(M) L F2Irr(H) M E F .
Wewill rst showthat M has a simplesubmo dule.
Take i maximalsu h that p(M) i
6=0 and F a simpleH-submo duleof p(M) i . Let L=AF. Then, p(L) p(M)\M i and p(L)F L >i
, hen e p(L) =F. It follows fromLemma2.12
that L'L(F) andweare done.
Let d(M)= P F2Irr(H) dimM E F . We put M 0 = M=L. We have d(M 0
) < d(M). So, the lemma follows by indu tion on
d(M).
2.6. Properties of ategory O . We assume here in x2.6 that k is a eld. We now derive
stru tural prop erties of our ategories.
2.6.1.
Corollary 2.15. Every obje t of O ln
has an as ending ltrationwhose su essivequotients are
semi-simple.
Proof. Follows from Lemma2.14 and Prop osition 2.2.
Corollary 2.16. Every obje t of O has a nite Jordan-Holder series.
Proof. Themultipli ityofL(E)ina ltration ofM 2O givenbyCorollary2.15isb ounded by
dimW
E
(M), hen e the ltration must b e nite.
Corollary 2.17. The ategory ~ O a
is generated by the L(E)hr i, with r = E
a.
Proof. Follows from Lemma2.3 and Prop osition 2.11.
Corollary 2.18. Given a 2k, the full abelian Serre sub ategory of the ategory of A-modules
generated by the L(E) with E
2a+Z is O a+Z
.
2.6.2.
Theorem 2.19. The ategory O is a highest weight ategory (in the sense of [CPS1 ℄) with
respe tto the relation: E <F if F
E
isa positive integer.
Proof. Follows from Corollary2.10 and Lemma2.14 .
Thestandard and ostandardobje tsare the(E)andr(E). Thereareproje tivemo dules
P(E),inje tivemo dulesI(E), tiltingmo dulesT(E). Wehavere ipro ityformulas, f. [CPS1 ,
Theorem 3.11℄:
[I(E):r(F)℄=[(F):L(E)℄ and [P(E):(F)℄=[r(F):L(E)℄:
Corollary 2.20. If E
F
62Z f0g for all E;F 2Irr(H(k 0
)), then O is semi-simple.
Ext O (M;r(H))=0 for i>0 Ext 1 O (M;r(H))=0 the restri tion of M to B is free.
Proof. The equivalen e b etween the rst three assertions is lassi al. The remaining
equiva-len es followfromthe isomorphismExt 1 O (M;r(H)) !Ext 1 A (M;r(H)) !Ext 1 B (M;k).
2.6.3. FromProp osition 2.2, we dedu e
Lemma 2.22. Let M 2O
ln
. The following onditions are equivalent
M 2O
M is nitely generated as a A-module
M is nitely generated as a
B-module.
Lemma 2.23. There is r 0 su h that for all M 2 O , a 2 k and m in the generalized
eigenspa e for 0
for the eigenvalue a, then ( 0
a) r
m=0.
Proof. The a tion of 0
on (H) is semi-simple. It follows that, given r 0, a 2 k and
m 2
r
(H)in the generalized eigenspa e for 0
for the eigenvalue a, then ( 0
a) r
m=0.
Now, by Corollary 2.7 , there is some integer r su h that everyobje t of O is a quotient of
r
(H) l
for somel .
Proposition 2.24. There is r 0 su h that every module in O ln
is generated by the kernel
of B r
. Further, there is an integer r >0 su h that for M 2O ln
, we have Mfig=Mfr g for
ir .
Proof. Let r 0 su h that every proje tive inde omp osable obje t in O is a quotient of
r 1
(H). This meansthat everyobje tinO is generated bythe kernelofB r
. Now, onsider
M 2O
ln
and m 2M. LetN b e the A-submo dule of M generated by m. This is inO , hen e
m is inthe submo duleof N generated by the kernelof B r
.
Proposition 2.25. Every obje t in O ln
is generated by the 0-generalized eigenspa e of 0
.
Proof. It isenough to provethe prop osition forproje tiveinde omp osableobje ts inO ,hen e
for r
's,where itis obvious.
2.6.4. Let Q b e a progenerator for O ( f Corollary 2.8 ) and = (End A
Q) opp
. Then, is a
nitelygenerated proje tiveO -mo dule. Wehave mutuallyinversestandard equivalen es
(3) Hom(Q; ):O
! -mo d ; Q ( ): -mo d
!O :
Let now X b e a (non-ne essarily nitelygenerated) -mo dule. Then, Q X is a quotient
of Q (I)
for some set I, where X is a quotient of (I)
. Now, Q
(I)
is in O ln
. So, the fun tor
Q ( ): -Mo d!A-Mo d takes values in O ln
and we haveequivalen es
Hom(Q; ):O ln ! -Mo d; Q : -Mo d !O ln :
3. Rational Cherednik algebras
3.1. Basi de nitions. Let V b e a nite dimensional ve tor spa e and W GL (V) a nite
omplex re e tion group. Let A b e the set of re e ting hyp erplanes of W. Given H 2 A;
let W H
W b e the subgroup formed by the elementsof W that x H p ointwise. We ho ose
v H 2V su h that Cv H is a W H
-stable omplementto H. Also, let H
2V
Let k b e a no etherian ommutativeC-algebra. The group W a ts naturally on A and on
the group algebra kW, by onjugation. Let : A ! kW; H 7! H ; b e a W-equivariant map su hthat H isan elementof kW H
kW with tra e zero, forea h H 2A.
Given as ab ove, one intro du es an asso iative k-algebra A(V; ); the rational Cherednik
algebra. It is de ned as the quotient of k C
T(V V
)oW, the ross-pro du t of W with
k-tensoralgebra, by the relations
[;℄=0 for ;2V; [x;y℄=0for x;y2V [;x℄=h;xi+ X H2A h; H ihv H ;xi hv H ; H i H
Remark 3.1. Let Re W denote the set of (pseudo)-re e tions. Clearly Re is an AdW
-stable subset. Giving as ab ove is equivalent to giving a W-invariant fun tion : Re !
k; g 7! g su h that H = P g 2W H rf1g g
g. One may use the fun tion instead of , and
write v g 2 V, resp. g 2 V , instead of v H , resp. H , for any g 2 W H
rf1g. Then the last
ommutationrelationin the algebra A(V; ) reads:
[;x℄=h;xi+ X g 2Re g h; g ihv g ;xi hv g ; g i g;
whi h is essentially the ommutation relation used in [EtGi℄. In ase of a Weyl group W,
in [EtGi, BeEtGi, Gu℄, the o eÆ ients
( a ro ot) were used instead of the g 's. Then, H = 2
g for H the kernel of and g the asso iated re e tion.
Remark 3.2. Put e H = jW H j. Denote by " H ;j = 1 e H P w 2W H det(w ) j
w the idemp otent of
CW H
asso iated to the hara ter det j jW H
. Given as ab ove, there is a unique familyfk H ;i = k H ;i ( )g H2A=W;0ie H
of elementsof k su h that k H ;0 =k H ;e H =0 and H =e H e H 1 X j=0 (k H ;j+1 ( ) k H ;j ( ))" H ;j :
We observethat an b e re overedfrom the k H ;i ( )'s by the formula H = X w 2W H f1g e H 1 X j=0 det(w ) j (k H ;j+1 ( ) k H ;j ( )) ! w :
This way,we get ba k to the de nition of [DuOp℄.
Intro du e free ommutative p ositively graded k-algebras P = k C S(V ) = L i0 P i and =k C S(V)= L i0 i
. Wehaveatriangularde omp ositionA=P k kW k ask-mo dules [EtGi,Theorem1.3℄. For H 2A,weput a H ( )= e H 1 X i=1 e H k H ;i ( )" H ;i 2k[W H ℄ and z( )= X H2A a H ( )2Z(kW):
For E 2Irr(CW), we denote by E
= E
( ) the s alar by whi h z( ) a ts on k C E. The elements a H ( ); z( ); and E
( ); may b e thought of as fun tions of the o eÆ ients k H ;i
=
k H ;i
( ) (through their dep enden e on ). In parti ular, it was shown in [DuOp, Lemma 2.5℄
that E
, expressed as a fun tion of the k H ;i
's, is a linear fun tion with non-negative integer
o eÆ ients.
Below, we will often use simpli ed notations and write A for A( ), k H ;i
for k H ;i
( ), and z
for z( ); et .
We intro du e a grading on A by putting V
in degree 1, V in degree 1 and W in degree
0 (thus, the indu ed grading on the subalgebra P A oin ides with the standard one on P,
while the indu ed grading on the subalgebra A di ers by a sign from the standard one
on ). 2 Leteu k = P b2B b _
bb ethe\deformedEulerve tor eld",whereBisabasisofV andfb _
g b2B
is the dual basis. Wealso put eu=eu k
z. The elementseu k
and eu ommute with W. Note
that P b [b;b _ ℄=dimV + P H H . Wehave
(4) [eu;℄ = and [eu;x℄=xfor 2V and x2V
:
This shows the grading on Ais \inner",i.e., A i
=fa 2Aj[eu;a℄=iag.
3.2. Category O for the rational Cherednik algebra.
3.2.1. We apply now the results of x2 in the sp e ial ase: A = A = A(V; ), B = , B = P, H = kW, k 0 = C, H(k 0 ) = CW, = eu, 0 = eu k and 0
= z. In parti ular, we have the
ategory O ( ) := O (A(V; )), whi h was rst onsidered, in the setup of Cherednik algebras,
in [DuOp℄.
For any ( ommutative)algebra map : k ! k 0
, there is a base extension fun tor O ( ) !
O ( ( ))givenbyA( ( )) A
( ).
3.2.2. Assume k is a eld. Sin e O and ~
O have nite global dimension(Theorem 2.19 ), the
Grothendie k group of the ategory of mo dules oin ides with the Grothendie k group K 0
of
proje tivemo dules.
We havea morphism of Z-mo dules f : K 0 ( ~ O )!Z[[q℄℄[q 1 ℄K 0
(CW) given by taking the
graded hara ter of the restri tion of the mo duleto W :
M 7! X E2Irr(k W) X i q i dimHom k W (E;M i )[E℄: Set[P℄:= P E2Irr(k W) P i q i dimHom k W (E;P i
)[E℄. ThisisaninvertibleelementofZ[[q℄℄[q 1
℄
K 0
(CW),andforanyF 2Irr(kW),wehavef([(F)℄)=[P℄[F℄:Sin ethe lasses ofstandard
mo dulesgeneratetheK 0
-group,weobtainanisomorphism 1 [P℄ f :K 0 ( ~ O ) !Z[q;q 1 ℄K 0 (CW). Let k[(k H ;i ) 1ie H 1
℄ b e the p olynomial ring in the indeterminates k H ;i
with k w (H);i
= k
H ;i
for w 2 W. We have a anoni al evaluation morphism k[(k H ;i
)℄ ! k given by the hoi e of
parameters. Letm b ethe kernel ofthat morphism,R the ompletionof k[(k H ;i
)℄at m, and K
the eldof fra tionsof R.
Wehave ade omp osition map K 0 (O K ) ! K 0 (O ). It sends [(K C E)℄ to [(k C E)℄. 2
We use sup ers ripts to indi ate the standard (non-negative) grading on , and subs ripts to denote the
gradingsonAandP. Thus,putting formally := i
Proposition 3.3. Assume k is a eld. Then, [(E)℄= [r(E)℄ and [P(E)℄ = [I(E)℄ for any
kW-module E.
Proof. We rst onsider the equality[(E)℄=[r(E)℄. The orresp onding statement for K is
true, sin e the ategory is semi-simplein that ase. Hen ethe mo dules are determined,up to
isomorphism,by their so le (resp. by their head).
The statementfor k follows by using the de omp ositionmap.
Now,the equality [P(E)℄=[I(E)℄follows, using the re ipro ityformulas (x2.6.2 ).
4. Duality, Tiltings, and Proje tives
4.1. Ringel duality. We keepthe setup ofx2.1 , with k b eing a eld. We makethe following
two additionalassumptions
We have
B H B =B H
B =A;
The subalgebra B A is Gorenstein (with parameter n), i.e.,there existsan integern
su h that Ext i B (k;B)= ( k if i=n 0 if i6=n:
The Gorenstein ondition implies that, for any E 2 Irr(H), viewed as a BH-mo dule via the
proje tion BH ! H, we have Ext i BH
(E;BH) = 0;for all i 6=n; moreover, Ext n BH (E;BH) = E [ ; whereE [
is aright BH-mo dule su hthat dimE [
=dimE.
Assume further that the algebra A has nite homologi al dimension. Thus (see[Bj℄), there
is awell-de ned duality fun tor
RHom A ( ;A):D b (A-mo d) !D b (A opp -mo d) opp :
Furthermore, this fun tor is an equivalen e with inverseRHom A
opp( ;A).
The triangular de omp osition A=BH
B givesa similarde omp osition A opp = B opp H opp B opp
,for the opp ositealgebras. Therefore, wemay onsiderthe ategoryO (A opp
)and,
for any simplerightH-mo dule E 0
,intro du ethe standard A opp -mo dule opp (E 0 ):=Ind A opp (BH) opp E 0 =E 0 BH A;
and also the proje tiveobje tP opp
(E 0
)2O (A opp
), the tilting obje t T opp (E 0 )2O (A opp ), et .
Lemma4.1.Thefun torRHom A
( ;A[n℄)sends(E)to opp
(E [
);forE a nite-dimensional
H-module.
Proof. Usingthat A is free as a left BH-mo dule,we ompute
Ext i A ((E);A) !Ext i BH (E;A) !Ext i BH (E;BH) BH A:
We see that this spa e vanishes for i 6= n, and for i = n we get RHom A (E); A[n℄ ' E [ BH A= opp (E [ ).
Wewouldlike to use the duality fun tor RHom A
( ;A[n℄)to obtaina fun tor D b (O (A))! D b (O (A opp )) opp
. Tothis end, wewill exploita general result b elow valid for arbitraryhighest
weight ategories (a ontravariant versionof Ringel duality [Ri, x6℄).
Givenanadditive ategoryC,letK b
Proposition4.2. LetA andA betwo quasi-hereditary algebrasand C =A-mo d, C =A-mo d
the asso iatedhighest weight ategories. LetF bea ontravariant equivalen e between theexa t
ategories of - ltered obje ts F :C !(C 0 ) opp . Then, F restri tsto equivalen es C-proj !(C 0 -tilt) opp and C-tilt !(C 0 -proj) opp
The anoni al equivalen es K b (C-proj) !D b (C) andK b (C 0 -tilt) !D b (C 0 ), yield an
equiv-alen e of derived ategories
D :D b (C) ! D b (C 0 ) opp : Let T = F(A), an (A A 0
)-module. Then, we have D = RHom
A ( ;T) and D 1 = RHom A 0
( ;T). Via duality (A-mo d)
!(A
opp )-mo d
opp
, the fun tor D identi es C opp with the Ringel dual of C 0 . Proof. LetM 2C
. Then,M isproje tiveifandonlyif Ext 1
(M;(E))=0foreverystandard
obje t (E) of C (indeed, if 0!M 0
! P ! M ! 0 is an exa t sequen e with P proje tive,
then M 0
is - ltered, hen e the sequen e splits). The mo dule F(M) is tilting if and only if
Ext 1
((E 0
);F(M))=0for everystandard obje t(E 0
)ofC 0
. Wededu ethat M isproje tive
if and only if F(M)is tilting.
So, F restri ts to equivalen es C-proj !(C 0 -tilt) opp and C-tilt !(C 0 -proj) opp .
The last assertions of the Prop osition are lear.
We an now apply this onstru tion to the ategory O (A). Sp e i ally, Lemma 4.1
im-plies that the fun tor RHom A
( ;A[n℄) restri ts to an equivalen e O (A) ! (O (A opp ) ) opp .
Therefore, using Prop osition 4.2 weimmediatelyobtain the following
Proposition 4.3. The fun tor RHom A ( ;A[n℄) opp restri ts to equivalen es O (A)-proj !(O (A opp )-tilt) opp and O (A)-tilt !(O (A opp )-proj ) opp :
The anoni alequivalen es K b (O (A)-proj) !D b (O (A))andK b (O (A opp )-tilt) ! D b (O (A opp )) indu ean equivalen e D:D b (O (A)) !D b (O (A opp )) opp
;su h that(E)7! opp (E [ ); P(E)7! T opp (E [ ); and T(E)7!P opp (E [ ):
Corollary 4.4. The ategory O (A opp
) opp
isthe Ringel dual of O (A).
4.2. Naive duality for Cherednik algebras. Re allthe setup of x3.2.
Denote by ( ) y
:CW
!CW the anti-involutiongiven by w7!w y
:=w 1
forw2W.
In this se tion, we ompare the algebras A = A(V; ) and A(V
; y
). This will provide us
with means to swit h b etween left and right mo dules, b etween -lo ally nite and P-lo ally
nite mo dules. The anti-involution( ) y :CW !CW extends to an isomorphism (5) ( ) y :A( ) !A( y ) opp ; V 37! ; V 3 x7!x; W 3w7!w 1 :
Remark 4.5. Ifallpseudo-re e tions of W have order2, then y
= .
Further, we de nean isomorphismof k-algebras reversingthe gradings by
':A(V; ) !A(V ; y ) opp 1
Remark 4.6. When V is self-dual, an isomorphismof CW-mo dules F : V ! V
extends to
an algebra isomorphism(Fouriertransform)
F :A(V; ) !A(V ; ) V 3 7!F(); V 3x 7! F 1 (x); W 3w7!w : The fun tor F restri tsto an equivalen e O (V; ) !O (V ; y ). 4.2.1. Given M 2 O ln , denote by M _
the k-submo dule of P-lo ally nilp otent elements of
Hom k
(M;k). This is a right A-mo dule. Via'
, this b e omes aleft A(V ; y )-mo dule. IfM is graded, then M _ =Homgr k (M;k).
Thuswehavede ned afun tor (analogous of the standard duality on the ategoryO in the
Lie algebra ase):
(6) ( ) _ :O ln (V; )!O ln (V ; y ) opp
When k is a eld, this fun tor is an equivalen e.
Givena kW-mo dule E, we use the notation E _
=Hom
k
(E;k) for the dual kW-mo dule.
Proposition 4.7. We have (E) _
!r(E
_
) for any kW-module E. If k is a eld, then
L(E) _ !L(E _ ); P(E) _ !I(E _ ); I(E) _ !P(E _ ); r(E) _ !(E _ ); T(E) _ !T(E _ ): Proof. Wehave Homgr k (A W E;k) !Homgr (W) opp(A;Hom k (E;k))
and the rst part of the prop osition follows.
The se ond assertion follows fromthe hara terization of L(E) (resp. L(E _
))as the unique
simplequotient(resp. submo dule)of(E)(resp. r(E _
)). Theotherassertions areimmediate
onsequen esofthehomologi al hara terizationsoftheobje tsand/ortheexisten eofsuitable
ltrations.
Notethat the fun tor ( ) _ restri tsto afun tor O (V; )!O (V ; y ) opp . When k is a eld,
itisan equivalen e. A ompatible hoi eofprogenerators forO (V; )andO (V
; y
)givesthen
an isomorphismb etween the algebra (V) for O (V; ) and the oppp osite algebra (V ) opp for O (V ; y ) ( f x2.6.4 ).
Corollary 4.8. Let E and F be two simple kW-modules. Then, the multipli ity of (E) in a
- ltration of P(F), for O (V; ), is equal to the multipli ity of L(F _ ) in a omposition series of (E _ ), for O (V ; y ).
Proof. Byx2.19 ,themultipli ityof r(E _
) inar- ltrationof I(F _
) isequaltothemultipli ity
of L(F _
) ina omp ositionseries of (E _ ). The fun tor( ) _ sends P(F)toI(F _
)and(E)to r(E _
)(Prop osition 4.7) andthe result
follows.
Remark4.9.Whenkisa eldandW isreal,weobtain,viaFouriertransform,adualityonO ln
and on O . Sin e all omplexrepresentationsof W are self-dual,we havethen(E) _
4.3.1. The rational Cherednik algebra is a deformation of the ross-pro du t of W with the
Weyl algebra of p olynomial di erential op erators on V. In parti ular, there is a standard
in reasing ltration on A with W pla ed in degree 0 and V V
in degree 1. The asso iated
graded ring, grA,is isomorphi to S(V V
)oW [EtGi,x1℄. It follows (see[Bj℄, [Bo,xV.2.2℄),
that Ais leftand rightno etherian, providedk isno etherian. Sin e V
V is asmo oth variety
of dimension2dimV,the algebra Ahashomologi aldimensionat most 2dimV. Furthermore,
the usual results and on epts on D -mo dules ( hara teristi variety, duality) also make sense
for A,eventhough the algebra gr A isnot ommutative.
4.3.2. Weassumekisa eld,and putn=dimV. ThealgebrasandPare learlyGorenstein
with parameter n. Moreover,wehave Ext n (k;) ' n V . Hen e,E [ = n V Hom k (E;k)= n V E _
; for any nite dimensional W-mo duleE.
It will b e useful to omp ose the fun tor RHom A( )
( ;A( )) with the anti-involution ( ) y
,
see(5 ), to get the following omp osite equivalen e
(7) RHom A( ) ( ;A( )) y : D b (A( )-mo d) ! D b (A( ) opp -mo d) opp ! D b (A( y )-mo d) opp :
FromProp osition 4.2 we immediatelyobtain the following
Proposition 4.10. The fun tor RHom A
( ;A[n℄) y
gives rise to an equivalen e
D :D b (O ( )) !D b (O ( y )) opp :
Wefurther intro du ean equivalen e
( ) _ ÆD : D b (O (V; )) !D b (O (V ; )) su hthat (E)7!r( n V C E) T(E)7!I( n V C E) P(E)7!T( n V C E)
In parti ular, we obtain( f. Corollary 4.4)
Corollary 4.11. The ategory O (V
; ) is the Ringel dual of O (V; ).
Remark 4.12. Notethat if W is real, then O is itsown Ringel dual.
4.3.3. Semiregularbimodule. WriteP ~
=
i
Hom(P i
;k)forthegraded dualofP, andformthe
ve tor spa e R := P ~
k
W. Let us x an isomorphismof C-ve tor spa es n
V
! C. We
have the following anoni al isomorphisms:
Homgr W (A; n V C W) !Homgr k (P;W) P ~ k W !P ~ P A:
The rst two isomorphisms de ne a left A-mo dule stru ture on R, and the last one de nes a
right A-mo dule stru ture on R. It is p ossible to he k by expli it al ulations that the left
and right A-mo dulestru tures ommute,so that R b e omes an A-bimo dule. It is aCherednik
algebra analogue of the semiregular bimo dule, onsidered in [A℄,[So2 ℄ inthe Liealgebra ase.
Fromthe isomorphisms of A-mo dules
R A (E)=R W E !Homgr W (A; n V E) wededu e
Proposition 4.13. The fun tor M 7! Homgr k (R A M;k) y (= left A(V; y )-module) sends (E) to ( n V E _ ).
4.3.4. GivenM a nitelygenerated A-mo dule,ago o d ltration ofM isastru ture of ltered
A-mo dule on M su h that grM is a nitelygenerated grA-mo dule. The hara teristi variety
Ch(M)isthe supp ortofgrM,viewedas aW-equivariantsheafon V
V (a losedsubvariety).
Itiswellde ned,i.e.,isindep endentofthe hoi eofthego o d ltration(every nitelygenerated
A-mo duleadmitsago o d ltration). NotethatBernstein'sinequality: dimCh (M)dimV do es
not hold in general. Further, for M in O , the omplexD (M) has zero homology outside the
degrees 0;:::;n.
Let T = L
E
T(E) where E runs overthe simplekW-mo dules.
Corollary 4.14. Let M 2O . Then, dimCh(M)=dimV minfij Ext i O
(T;M)6=0g.
Proof. Let R = End (T) opp
. The fun tor RHom O (T; ) : D b (O ) ! D b (R-mo d) is an
equiv-alen e. Comp osing with the inverse of ( ) _ ÆD we obtain an equivalen e D b (O (V ; )) ! D b
(R-mo d) that restri ts to an equivalen e O (V
; )
! R-mo d . We see that minfi j
Ext i O
(T;M)6=0g = minfij H i
(D M) 6=0g; where the RHS is equalto minfi j Ext i A
(M;A) 6=
0g dimV bythe de nitionofD . Theresult nowfollows fromthewell-knownformula,seee.g
[Bj℄: dimCh(M)=2dimV minfij Ext i A
(M;A) 6=0g:
5. He ke algebrasvia monodromy
5.1. Lo alisation. 5.1.1. LetV reg =V S H2A H and P reg =k[V reg ℄=P[( 1 H )℄ H2A
. The algebra stru ture on A
extends to an algebra stru ture on A reg =P reg k k kW. Wedenote by M 7!M reg =A reg A M :A-Mo d !A reg -Mo d
the lo alisation fun tor. Note that Res Preg M reg =P reg P
M. Notealso that every elementof
M reg
an b e written as r
m for somer 0,m 2M, where = Q
H2A
H
. This makesthe
lo alisation fun torhave sp e iallygo o d prop erties.
The restri tion fun tor A reg
-Mo d ! A-Mo d is a right adjoint to the lo alisation fun tor. It
is fully faithful. The adjun tion morphism oin ides with the natural lo alisation morphism
M !M
reg
ofA-mo dules. Itskernel is M tor
, the submo dule ofM of elementswhosesupp ort is
ontained inV V reg
. Denote by (A-Mo d ) tor
the full sub ategory of A-Mo d of obje ts M su h
that M reg
=0. The followingis lear.
Lemma 5.1. The lo alisation fun tor indu es an equivalen e
A-Mo d=(A-Mo d) tor !A reg -Mo d:
The ategory O is a Serre sub ategory of A-Mo d . Let O tor = O\(A-Mo d ) tor . Then, the anoni al fun tor O =O tor ! A-Mo d=(A-Mo d ) tor
is fullyfaithful. Consequently, the anoni al
fun tor O =O tor !A reg -Mo d
5.1.2. When k isa eld, wehave a ommutativediagram D b (A( )-mo d) RHom A ( ;A) y
//
D b (A( y )-mo d) opp D b (A( ) reg -mo d) RHom Areg ( ;A reg ) y//
D b (A( y ) reg -mo d) oppwhere the verti al arrows are givenby lo alisation.
5.1.3.
Lemma 5.2. Assume k is a eld. Then, ( )
_ restri ts to an equivalen e O tor (V; ) ! O tor (V ; y ) opp .
Proof. LetM 2O . We put a grading on M (Prop osition 2.4). Sin e M isa nitelygenerated
graded P-mo dule(Lemma 2.22 ),the dimensionof Ch(M),the hara teristi variety ofM, an
b e obtained from the growth of the fun tion i7! dimM i . In parti ular, M 2O tor if and only if lim i!1 i 1 dimV dimM i
=0. Su haprop erty is preservedby ( ) _ . Denote by V : O ! O = O =O tor ; M 7!
M; the quotient fun tor (the notation V has
b een used by So ergel [So1 ℄ for an analogous fun tor in the Lie algebra setup). The fun tor
Vadmits, by the standard `abstra t nonsense', b oth a left adjoint and right adjoint fun tors > V;V > : O !O .
Theorem5.3. Assumek isa eld, andQisaproje tivein O . Then, the anoni aladjun tion
morphism a : Q ! V
> (
Q) is an isomorphism. In parti ular, for any obje t M in O , the
following anoni al morphism isan isomorphism
(8) V : Hom O (M;Q) !Hom O ( M; Q):
Proof. Byx5.1.1 , for any twoobje tsM;Q; of O ,we have a anoni al isomorphism
Hom O ( M; Q) !Hom A reg (M reg ;Q reg ):
Assume Q has a - ltration. Then it is free over P and thus has no non-zero submo dule
lying inO tor
, hen e V
isinje tive.
Assume furthermore that M has a r- ltration. Then M _
has a - ltration (Prop osition
4.7), hen e has no non-zero submo dule lying in O tor . Sin e ( ) _ restri ts to an equivalen e O tor (V; ) !O tor (V ; y ) opp
(Lemma 5.2), it follows that M has no non-zero quotient lying in
O tor
. This shows that V
in(8) is an isomorphism.
Fromnowon, weassumethat Q isproje tive. It followsthat Q 0
=D (Q) istilting (Prop
osi-tion 4.10 ), hen e r- ltered.
Nowlet M b e a - lteredobje t. Then, M 0
=D (M) is - ltered. Weapply the result on
V
, that we have already proved, to O ( y
). This yields, by duality ( f x5.1.2), that (8) is an
isomorphism,for any - lteredobje t M.
Sin e any proje tiveis - ltered,for any two proje tive obje ts P ;Q in O , we have
estab-lished the isomorphisms
(9) Hom O (P ;Q) V !Hom ( P; Q) adjun tion =====Hom O (P ;V > ( Q)):
The ab ove isomorphisms imply, in parti ular, that, for any inde omp osable proje tive P,
we have dimHom O (P ;Q) = dimHom O (P ;V > (
Q)). It follows readily that the obje ts Q and
V >
(
Q)have the same omp ositionfa tors with the samemultipli ities.
We an nallyprove that the anoni al adjun tion map a:Q !V >
(
Q) is an isomorphism.
Bythe previousparagraph, itsuÆ esto showthata isinje tive. Tothisend,put K :=ker(a),
and assume K 6=0. LetL(E) b e asimple submo dulein K, and P(E) L(E); itsproje tive
over. By onstru tion,the omp ositemapg :P(E)L(E),!K ,!Qisnonzero. Thismap
g 2 Hom
O
(P(E);Q) go es, under the isomorphismb etween the left-hand and right-hand sides
of (9), to the map a Æ g : P(E)!K ,!Q a !V > (
Q). But thelatter mapis the zeromap sin e
K =ker(a), whi h ontradi tsthe fa t that (9 ) isan isomorphism. Thus, ker(a)=0, and the
Theorem isproved.
Remark5.4. Ingeneral,the assumptionthatQisproje tive annot b erepla edbytheweaker
assumption that it is- ltered(already for W =Z=2Z). Nevertheless,see Prop osition 5.9 .
Corollary 5.5. Let X be a progenerator of O and E := (End O X) opp . Then there is an equivalen e O ! (E-mo d ) opp :
Proof. The pre eding theorem implies that (End O
X) opp
! E sin e proje tive mo dules are
- ltered. Hen ewe an use ategoryequivalen es(3 ).
5.2. Dunkl operators.
5.2.1. One has an A-a tion on the ve tor spa e P, hen e an -a tion, arising via the
identi - ation P=(k). One nds, inparti ular, thatthe a tion of 2V on P isgivenby the Dunkl
op erator T = + X H2A h; H i H a H 2 D (V reg )oW; whereD (V reg
)standsfor the algebra of regulardi erentialop erators on V reg
, a tedup on by W
inanaturalway,anda H
2kW isviewedasanelementofD (V reg
)oW. ItfollowsthatT
(P)P
(as partofA-a tiononP =(k));furthermore,this A-a tiononPisknown (Cherednik,[EtGi,
Prop osition 4.5℄) to b e faithful:
Theorem 5.6. The A-representation (k) is faithful. Thus, the natural a tion of PW on P
extends to an inje tive algebra morphismi :A,!k C D (V reg )oW whi h maps 2V to T .
The map i indu es an algebra isomorphismA reg !k C D (V reg )oW. 5.2.2. We onsiderM =Ind A W
X =PX,whereX islo allynilp otentand nitelygenerated
as an -mo dule,free overk. The a tion of 2V on pv,p2P and v2X is given by
(pv)=pv+ (p)v+ X H X 0i;je H 1 e H (k H ;i+j k H ;j ) H () H " H ;i (p)" H ;j (v):
Using Dunkl op erators, i.e., via the isomorphism of Theorem 5.6 , we have a stru ture of W
-equivariant(k C D (V reg ))-mo dule on M reg
. The orresp onding onne tion isgivenby
= X H H () H X 0i;je 1 e H k H ;i+j " H ;i " H ;j :
Hen e (pv)=pv+ (p)v X H X i e H k H ;i H () H p" H ;i (v):
For the rest of x5 , we assumethat k =C.
The followingresultiswell-knownto exp erts,butwe ouldnot nd anappropriatereferen e
in the literature.
Proposition 5.7. The above formula for
de nes a W-equivariant integrable algebrai
on-ne tion on M with regular singularities.
Proof. Allthe laimsfollowfromthe onstru tion,with the ex eptionofthe assertion that the
singularities ofthe onne tionare regular. The onne tion has visiblyonly simplep oles at the
re e tionhyp erplanes,hen e itsuÆ es to provethe regularity at in nity with resp e t to some
(hen e any, see[De℄) ompa ti ationof V.
Consider the W-equivariant ompa ti ationY =P(C+V)of V,and extend M to the free
O Y -mo duleM Y :=O Y
X. Usinga ltration ofX we anredu e to the ase whereX =E is
simple.
A straightforward omputationshows that with resp e tto theextensionM Y
of M and with
resp e t to any standard o ordinate pat h on Y, the p oles at in nity are also simple in this
ase.
5.2.3. We de ne a morphism of ab elian groups r : K 0
(O ) ! Z by r ([(E)℄) = dimE; for
E 2Irr(CW).
Lemma 5.8. Let M 2O . Then, M reg
is a ve tor bundle of rankr ([M℄) on V reg . Proof. Sin eM reg isa nitelygeneratedC[V reg
℄-mo dulewitha onne tion,itisave torbundle.
Now, taking the rank of that ve tor bundle indu es a morphism K 0
(O ) ! Z, whi h takes
the orre tvalue on (E).
5.2.4. Proposition5.9. Assumek H ;i k H ;j + i j e H
62Z,for all H 2A andall 0i6=j e H
1. Let
N be a - ltered obje t in O . Then, for any M 2O , we have Hom O (M;N) !Hom O ( M; N).
Proof. Assume rst that M is also a - ltered obje t. Then, we an write M = Ind A W X and N = Ind A W
Y with X ;Y nite dimensional W-mo dules, nilp otent over . The spa e
Hom A
(M;N) is the interse tionof Hom P (M;N) =PHom k (X ;Y) withHom A reg (M reg ;N reg ).
As in the pro of of Theorem 5.3, we have to show that any element of Hom P reg (M reg ;N reg )
that ommutes with the a tion of A reg
extends to a P-morphism M ! N. Observe that is
nothing but a at, W-invariantse tion of the onne tion on Hom Areg (M reg ;N reg ).
The residue of this onne tion on a hyp erplane H 2 A is onstant, and has eigenvalue
e H (k H ;i k H ;j ) on Hom k (X i ;Y j ), where X i is the summand of X of W H -typ e det i jW H (and likewise forY j ).
Lo ally near ageneri p oint p of H weexpand = P ll 0 l H l with l holomorphi on H near p,ofW H -typ e det l jW H ,andwith l 0
notidenti allyzeroon H. Fromthelowestorderterm
of theequation v
H
( ) =0weseethat thereexisti;j su hthat i j =l 0 mo d(e H Z), andsu h that l 0 +e H (k H ;i k H ;j )=0. Thus l 0
The general ase follows fromthe sp e ial ase ab ove by rep eating the part of the argument
from the pro of of Theorem5.3, starting with formula (9 ).
Remark 5.10. The ondition of the Prop osition is equivalent to the semi-simpli ity of the
He ke algebra H(W H
) of W H
. One ould onje ture that this assumption an b e repla ed by
theassumptionthatKZ(N)isaproje tiveH(W H
)-mo dule(thiswouldstillnot over ompletely
Theorem 5.3).
Remark 5.11. Ife H
=2 for allH, then the ondition of the Prop osition reads: k H 62 1 2 +Z. 5.2.5. LetC[(k H ;i ) 1ie H 1
℄ b e the p olynomial ring inthe indeterminates k H ;i with k w (H);i = k H ;i
forw2W. Wehavea anoni almorphismofC-algebras C[(k H ;i )℄!C; k H ;i 7!k H ;i . Let
m b e the kernel of that morphismand R the ompletionof C[(k H ;i
)℄at the maximalideal m.
Fix x 0 2V reg ,and letB W = 1 (V reg =W;x 0
) b e the Artinbraid group asso iated to W.
Let H R
=H
R
(W;V; ) b e the He kealgebra of W overR, that isthe quotient ofR[B W ℄ by the relations (T 1) e H 1 Y j=1 (T det(s) j e 2i k H;j )=0
for H 2 A, s 2 W the re e tion around H with non-trivial eigenvalue e 2i =e
H
and T an
s-generator of the mono dromy around H, f [BrMaRou, x4.C℄. Note that the parameters di er
from[BrMaRou℄b e ausewewillb eusingthehorizontalse tionsfun torinsteadofthesolution
fun tor. Weput H K =H R R
K,where K is the eld of fra tions of R and H =H R
R
(R=m).
Remark 5.12. It is known that H R
is free of rank jWj over R for all W that do not have
an irredu ible omp onent of typ e G 17:::19 , G 24:::27 , G 29 , G 31:::34
in Shephard-To dd notation (in
these ases, the statementis onje tural) [Mu℄.
5.3. The Knizhnik-Zamolod hikov fun tor. Let M b e a (C[V
reg
℄oW)-mo dule, free of
nite rank over P reg
= C[V
reg
℄. Let r : M ! M
C
R b e an R-linear integrable onne tion.
Then,the horizontalse tionsof rde ne,viathe mono dromyrepresentation,an RB W
-mo dule
L, free overR.
Let r
0
: M ! M b e the sp e ial b er of r. Then, the horizontal se tions of r 0 is the CB W -mo dule L R (R=m). Let r K :M ! K C
M b e the generi b er of r. Then, the horizontal se tions of r K is the KB W -mo duleL R K.
Takinghorizontalse tionsde nesanexa tfun torfromthe ategoryofW-equivariantve tor
bundles on R C
V reg
with an integrable onne tion to the ategory ofRB W
-mo dules that are
free of nite rankoverR.
Sin ethe onne tionon (R C
E) reg
hasregularsingularitiesitfollows that the onne tion
on M
reg
has regularsingularities for any M 2O R .
Comp osing with the lo alisation fun tor, we obtain an exa t fun tor KZ R from O R to the ategory of RB W
-mo dules that are free of nite rankoverR.
Similarly,we obtain exa tfun tors KZ:O !CB W -mo d and KZ K :O K !KB W -mo d.
It is well-known ( f. e.g. [BrMaRou, Theorem 4.12℄) that the representation of KB W on KZ K ((K C
E)) fa tors through H K
to givea representation orresp onding (via Tits'
Theorem5.13(He ke algebra a tion). Thefun tor KZ:O !CB W
-mo dfa tors througha
fun tor KZ:O =O tor
!H-mo d. Similarly, the fun tor KZ K
:O K
!KB
W
-mo d fa tors through
a fun tor KZ K :O K =(O K ) tor !H K -mo d. For M 2O R , thea tion of RB W on KZ R (M) fa tors through H R .
We have a ommutative diagram
O K KZ K
//
H K -mo d O R KZ R//
C R K ROO
H R -mo d C R K ROO
O KZ//
H-mo d Proof. First, O tor (and (O K ) tor) are the kernels of lo alisation.
When M = (K
C
E), then, we have the Knizhnik-Zamolo d hikov onne tion and the
representationKZ K (M)fa torsthroughH K . Sin eO K
issemi-simple(Corollary2.20 ),itfollows
that the a tion on KZ K (M) fa tors through H K forany M in O K .
Wenow onsiderthe aseofa- lteredmo duleM ofO R
. Weknowthatthea tionofKB W on K R KZ R (M)'KZ K (K R M) fa torsthrough H K . Sin eKZ R
(M) isfreeoverR, itfollows
that the a tion of RB W on KZ R (M) fa tors through H R .
Fromthis result,wededu ethatthea tionofCB W on KZ( r (CW)) !C R KZ R ( r (RW))
fa tors through H. Sin e everyinde omp osable proje tive obje t of O isa dire t summandof
r
(CW)forappropriater (Corollary 2.7), itfollowsthat the a tionofCB W
on KZ(M)fa tors
through H for every proje tiveM, hen e forevery M inO .
5.4. Main results. In this subse tion weassume that dimH =jWj, f. Remark5.12.
The fun tor KZ:O ! H-mo dis exa t. Hen e,it is represented by a proje tiveP KZ
2O . In
other words, there exists an algebra morphism : H ! (End O
P KZ
) opp
su h that the fun tor
KZ isisomorphi to Hom O
(P KZ
; ).
Weknowalso, see x5.1.1,that the fun tor KZfa tors through O =O tor
!H-mo d.
Theorem 5.14. The fun tor KZindu es an equivalen e: O =O tor
!H-mo d.
This Theorem isequivalentto
Theorem 5.15. The morphism :H!(End O P KZ ) opp is an algebra isomorphism.
Proof of Theorems5.14-5.15. Re all that the horizontal se tions fun tor gives an equivalen e
from the ategory of ve tor bundles over V reg
=W with a regular integrable onne tion to the
ategoryof nite-dimensionalCB W
-mo dules(Riemann-Hilb ert orresp onden e,[De,Theorems
I.2.17 and I I.5.9℄).
We dedu e from x5.1.1 that KZ :O =O tor
! H is a fully faithfulexa t fun tor with image a
full sub ategory losed undertaking sub obje ts and quotients. Furthermore,P KZ , theimage of P KZ in O =O tor , is aprogenerator of O =O tor
. Thus, Theorem5.14 follows from Theorem5.15 .
To prove Theorem5.15 , observe that the morphism is surje tive. Indeed, let C 0
b e a full
sub ategory of an ab elian ategory C, losed under taking quotients, and >
F a left adjoint to
the in lusionF :C 0
,!C. Then the adjun tion morphism:Id !F Æ
( >
(X) is the anoni al map from X to its largestquotient inC 0
. Thisproves surje tivity of the
morphism ab ove. Further, we have P KZ = M E2Irr(CW)
(dimKZ(L(E)))P(E):
Hen e,we ompute dim(End O P KZ )= M E;F
dimKZ(L(E))dimKZ(L(F))dimHom(P(E);P(F))
= M
E;F ;G
dimKZ(L(E))dimKZ(L(F))[P(E):(G)℄[(G):L(F)℄
= M
E;F ;G
dimKZ(L(E))dimKZ(L(F))[r(G):L(E)℄[(G):L(F)℄
= M
G
dimKZ(r(G))dimKZ((G))
Now, the restri tions of r(G) and (G) to V reg
are ve tor bundles of rank dimG (Prop
osi-tion 3.3 and Lemma 5.8), hen e dim(End O
P KZ
) = jWj = dimH. This shows that is an
isomorphism. Notethat this rank omputation an also b e a hievedby deformationto R.
The following result shows that the ategory O an b e ompletelyre overed from H and a
ertain H-mo dule :
Theorem 5.16 (Double- entralizer property). Let Q be a proje tive in O . Then, the
anoni al map Hom O
(M;Q)!Hom
H
KZ(M); KZ(Q)
isan isomorphism,for any M 2O .
Furthermore, if X is a progenerator for O , then, we have an equivalen e
End H KZ(X) opp -mo d !O :
Proof. The rstpartfollowsfromTheorems5.3and5.15andthese ondfromCorollary5.5.
Remark5.17. We onje turethat,if W =S n
, thenO isequivalenttothe ategoryof
nitely-generatedmo dulesovertheasso iated q-S huralgebra. Thatwouldimply,inparti ular,thatif
k H ;1
=k 1
<0 is a negativereal onstant, then the Cherednikalgebras A(S n ) with parameters k 1 and k 1
1;resp e tively,are Morita equivalent.
Let Z(H) denote the enter of the algebra H and Z(O ) the enter of ategory O (i.e. the
algebra of endomorphismsof the identity fun tor Id O
).
Corollary5.18. The anoni almorphismZ(O ) !End O P KZ indu es anisomorphismZ(O ) !
Z(H). In parti ular, the fun tor KZindu es a bije tion between blo ks of O andblo ks of H.
Proof. This follows immediatelyfrom Theorem5.16 : given two rings B and C and a(B;
C)-bimo dule M su h that the anoni al morphisms B !End C opp (M) and C !(End B M) opp are
isomorphisms, then wehave a anoni al isomorphismZ(B)
!Z(C).
The de omp osition matrix K 0
(O K
) !K
0
(O ) is triangular. We dedu e the triangularity of
de omp osition matri esof He kealgebras, in hara teristi 0:
5.4.1. KZ-fun tor and Twist. Let b e a one-dimensional hara ter of W and
: CW !CW
the automorphismgiven by w7!(w )w for w2W. This extends to an isomorphism
:A( ) !A( ( )); V 3 7!; V 3x7!x; W 3w7! (w )w : Weobtain an equivalen eO ( ) !O ( ( )),sending V(E)to V(E 1
),where V stands for
any ofthe symb ols: L;;r;P ;I;T.
For H 2 A, let d H 2 f1;:::;e H g su h that jW H = det d H jW H . De ne an automorphism of D (V reg )oW by P 3f 7! f; W 3w7!(w )w and 7! X H H () H " H ;e H e H k H ;e H d H for 2V:
(for notation, see Remark3.2). We havea ommutativediagram
A( ) i
//
D (V reg )oW A( ( )) i//
D (V reg )oW GivenM a(D (V reg)oW)-mo dule,then ( ) M !M O V reg ( 1 ) reg .
This self-equivalen e of the ategory of W-equivariant bundles with a regular singular
on-ne tion on V r eg
orresp onds, viathe horizontalse tionsfun tor, to the automorphismof CB W givenby T 7!e 2i k H;e H d H (s) 1 T
for H 2 A, s 2 W the re e tion around H with non-trivial eigenvalue e 2i =e
H
and T an
s-generator of the mono dromy around H. This indu es an isomorphism H() : H(W; ) !
H(W;
( )) and the following diagramis ommutative:
O ( ) ( )
//
KZ O ( ( )) KZ H(W; )-mo d H()//
H(W; ( ))-mo d5.4.2. KZ-fun tor and Duality. Wehave a ommutativediagram
D b (A( )-mo d) ( det )ÆRHom A( ) ( ;A( )) y
//
D b (A( det ( y ))-mo d) opp D b ((D (V reg )oW)- oh )//
D b ((D (V reg )oW)- oh) oppwherethe verti alarrowsare givenbylo alisation followedbytheDunkl op eratorisomorphism
i of Theorem5.6 and the b ottom horizontalarrowis the lassi al D -mo dule duality.
Consider the isomorphismCB !(CB ) opp givenby T 7!det(s) 1 e 2i k H;1 T 1
for H 2 A, s 2 W the re e tion around H with non-trivial eigenvalue e 2i =e
H
and T an
s-generator ofthe mono dromy around H. It indu esan isomorphism
H( y ): H(W; ) !H(W; y ) opp :
We on lude that wehave a ommutativediagram
D b (O ( )) D
//
KZ D b (O ( y )) opp KZ D b (H(W; )) H( y )//
D b (H(W; y )) oppOn the other hand, by Lemma 5.2, we know that ( ) _
preserves O tor
, hen e des ends to
the quotient ategory O =O tor
, i.e., there is an equivalen e making the following diagram
ommute: O (V; ) _
//
KZ O (V ; y ) opp KZ H(W;V; )-mo d//
H(W;V ; y )-mo d oppFurther, ho ose a W-invariant hermitian form on V, i.e., a semi-linear W-equivariant
iso-morphism :V
!V
. Then, we get an isomorphism 1 (V reg =W;x 0 ) ! 1 (V reg =W;(x 0 )). It indu es an isomorphismH() : H(W;V; ) ! H(W;V
; ). Comp osing with H( y ), we obtain an isomorphismH( Æ ( ) y ):H(W;V; ) !H(W;V ; y ) opp
, whi hwe denoteb elowby .
Remark 5.20. One ould onje ture that the two fun tors and
are isomorphi (they
indu e the samemaps at the levelof Grothendie k groups).
5.4.3. TheA-moduleP KZ
andDuality. LetIrr(W;V; )Irr(W)denotethesubsetformedbyall
E 2Irr(W)su hthatL(E) reg
6=0. Wehaveabije tionIrr(W;V; ) !Irr(W;V ; y ); E 7!E _
(Prop osition 4.7 and Lemma5.2 ). Thus,P KZ
= L
E2Irr(W;V; )
(dimKZ(L(E)))P(E).
To makethe dep enden eon V and expli it,wewill write P KZ
=P
KZ (V; ).
Proposition 5.21. (i) We have D (P KZ (V; )) ' P KZ (V; y ) and P KZ (V; ) _ ' P KZ (V ; ). In parti ular, P KZ
is both proje tive and inje tive.
(ii) ForE 2Irr(W), the following are equivalent
E 2Irr(W;V; )
L(E) is a submodule of a standardmodule
P(E) is a submodule of P KZ P(E) is inje tive P(E) is tilting I(E) is proje tive I(E) is tilting
Proof. The rst laimfollows fromx5.4.2. Prop osition 4.7 then impliesthat P KZ
is inje tive.
The onsiderations ab ove implythat if E 2 Irr(W;V; ), then P(E) is inje tiveand tilting.
The assertions ab out I(E) followby applying( ) _
.
This shows thatany of the assertionsab out P(E)or I(E)impliesthat E 2Irr(W;V; ):
6. Relation to Kazhdan-Lusztig theory of ells
WereviewsomepartsofKazhdan-LusztigandLusztig'stheoryofWeylgrouprepresentations.
6.1. Lusztig's algebra J.
6.1.1. Let(W;S)b ea niteWeylgroup,H b eitsHe kealgebra, aZ[v;v 1
℄-algebrawithbasis
fT w g w 2W and relations T w T w 0 =T w w 0 if l (w w 0 )=l (w )+l (w 0 ) and (T s +1)(T s v 2 )=0 for s2S:
Lusztig asso iated to W a Z-ring J, usually referred to as asymptoti He ke algebra, [Lu3 ,
x2.3℄. Let $ :H !Z[v;v 1 ℄ Z J b e Lusztig'smorphismof Z[v;v 1 ℄-algebras [Lu3, x2.4℄. The ring Q Z
J is semi-simpleand the morphismId Q(v ) $ is an isomorphism. For any ommutativeQ[v;v 1 ℄-algebra R weput H R :=R Z[v ;v 1 ℄ H. De nition 6.1. TheH R -modules S(M)=$ (R Q M), for M a simple Q Z J-module, will be referred to as standard H R -mo dules. 3
When R =Q(v),then the standard H R
-mo dules are simple and this givesa bije tion from
the setof simple(Q Z
J)-mo dulesto the setof simple(Q(v) Z[v ;v 1 ℄ H)-mo dules. Similarly,takingK =Q[v;v 1
℄=(v 1),weobtainabije tionfromthesetofsimple(Q Z
J
)-mo dules to the setof simpleQW-mo dules.
Wewillidentifythese sets of simplemo dulesvia these bije tions.
We have an order LR
on W onstru ted in [KaLu, p.167℄. We denote by C the set of
two-sided ellsofW and by the order on C oming from LR . LetfC w g w 2W
b e the Kazhdan-Lusztigbasis forH. LetI b e anideal of C,i.e., asubsetsu h
that given 0 , then 0 2I ) 2I. We putH I = L 2I;w 2 Z[v;v 1 ℄C w . This is atwo-sided ideal of H [Lu1 , p.137℄.
The ring J omes with a Z-basis ft w g w 2W and we put J = L w 2 Zt w . This is a blo k of J and J = L 2C J
. The orresp onding partition of the set of simple (Q Z
J)-mo dules is
alled the partition into families.
GivenI an ideal ofC, wedenote by I Æ
the setof 2I su hthat thereis 0
2I with < 0
.
The following isa slightreformulationof [Lu3 ,x1.4℄ :
Proposition 6.2. Let I be an ideal of C. Then, the assignment t w 7!C w indu es an isomor-phism of H-modules M 2I I Æ $ Z[v;v 1 ℄ Z J !H I =H I Æ : In parti ular, the (Q[v;v 1 ℄ Z[v ;v 1 ℄ H)-module Q[v;v 1 ℄ Z[v ;v 1 ℄ (H I =H I Æ ) is a dire t sum of standard H Q[v ;v 1 ℄ -modules.
Thisprop ositiongivesa hara terizationofstandardH Q[v ;v
1 ℄
-mo dulesviatheHe kealgebra
ltration oming fromtwo-sided ells.
6.1.2. Next,we onsider ltrations omingfrom ertainfun tionson thesetoftwo-sided ells.
De nition 6.3. A sorting fun tion f :W ! Z is a fun tion onstant on two-sided ells and
su h that 0
< )f( 0
)>f( ).
Givenasortingfun tionf,weputH i R := L w 2W;f(w )i R C w andH >i R := L w 2W;f(w )>i RC w : Then,H i R isa two-sided idealofH R
,sin e I =f 2C jf( )igisan ideal. Similarly,H >i R is a two-sidedideal of H R . Furthermore, I Æ
f 2 C jf( )>ig. Consequently, we dedu e from
Prop osition 6.2 :
Corollary 6.4. We have an isomorphism of H-modules M 2C;f( )=i $ Z[v;v 1 ℄ Z J !H i =H >i : In parti ular, the (Q[v;v 1 ℄ Z[v ;v 1 ℄ H)-module Q[v;v 1 ℄ Z[v ;v 1 ℄ (H i =H >i ) is adire tsum of standard H Q[v ;v 1 ℄ -modules.
Thus,wehaveanother hara terizationof standard H Q[v ;v
1 ℄
-mo dulesvia the He kealgebra
ltration oming fromf.
LetFb ethesetoffamiliesofirredu ible hara tersofW. Wetransferthe on eptsasso iated
with C to F viathe anoni al bije tionb etweenC andF.
In parti ular, we havea fun tion f :Irr(W)!Z onstant on families.
WehaveH i =H \ ( L f(E)i e E Q(v) Z[v ;v 1 ℄ H),wheree E
istheprimitive entralidemp otent
of Q(v) Z[v ;v
1 ℄
H that a ts as 1 on the simple(Q(v) Z[v ;v
1 ℄
H)-mo dule orresp onding to E.
This shows that, if R is alo alisation of Q[v;v 1
℄, thenthe ltration on H R =R Z[v ;v 1 ℄ H
givenby f an b e re overed without using the Kazhdan-Lusztig basis. Weobtain
Proposition 6.5. Let R be a lo alisation of Q[v;v 1 ℄ and P be a proje tive H R -module. Let Q i (resp. Q >i
)bethesum of thesimple submodulesE ofQ(v) R
P su h thatf(E)i (resp.
f(E)>i).
Then, (P \Q i
)=(P \Q >i
) is a dire t sum of standard H R
-modules.
Thus, any sorting fun tion yields a hara terization of the standard H R
-mo dules without
using the Kazhdan-Lusztig basis.
6.1.3. Given E 2 Irr(W), we denote by a E
(resp. A E
) the lowest (resp. highest) p ower of q
in the generi degreeof E [Lu1 ,x4.1.1℄.
By[Lu2 ,Theorem5.4 and Corollary 6.3 (b)℄,Lusztig's a-fun tionis asortingfun tion. The
orresp onding ltrations on proje tivemo dules haveb een onsidered in[GeRou℄.
WriteE <E 0
forthe orderonF arisingfrom< KL
viathe anoni albije tionb etweenC and
F. The following Lemmaisa lassi alresult :
Lemma 6.6. Let E;E 0 2F. If E <E 0 , then a E >a E 0 and A E >A E 0.
Proof. By [KaLu, Remark3.3(a)℄, we havev LR w if and only if w 0 w LR w 0 v, where w 0 is
the element of maximallength. Left multipli ation by w 0
indu es a automorphism of C. The
orresp onding automorphismofF istensor pro du tbydet [Lu1, Lemma5.14℄. It follows that
E <E 0
if and only if E 0
det <Edet ( f also [BaVo, Prop osition 2.25℄).
Wehave A
E
=N a
Edet
, whereN is the numb erof p ositive ro ots of W [Lu1, 5.11.5℄.
The Lemmais nowa onsequen eof the fa t that E <E 0
