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EXTENDING HECKE ENDOMORPHISM ALGEBRAS 5 JIEDU,BRIANJ.PARSHALL,ANDLEONARDL.SCOTT 1 0 WededicatethispapertothememoryofRobertSteinberg. 2 p ABSTRACT. The (Iwahori-)Heckealgebra in the title is a q-deformation of the group e H S algebra of a finite Weyl groupW. The algebra has a natural enlargementto an endo- H morphismalgebra = EndH( )where isa q-permutationmodule. Intype An (i.e., 6 A T T 2 W ∼= Sn+1), the algebra A is a q-Schur algebra which is quasi-hereditaryand plays an importantroleinthemodularrepresentationofthefinitegroupsofLietype.Inothertypes, ] is not always quasi-hereditary, but the authors conjectured 20 year ago that can be T A T enlargedtoan -module + sothat + = EndH( +)isatleaststandardlystratified,a R H T A T weakerconditionthanbeingquasi-hereditary,butwith“strata"correspondingtoKazhdan- . h Lusztigtwo-sidedcells. t Themainresultofthispaperisa“local"versionofthisconjectureintheequalparameter a m case,viewing asdefinedoverZ[t,t−1],withthelocalizationataprimeidealgenerated H [ balygaebcryacslo(atolsmoikcnpoowlynnaosmRiaDlAΦH2eA(ts)),oev6=er2si.mTihlaerplroocoafliuzsaetisotnhseothfeCo[rty,to−f1ra].tiIonnafultCuhreerpeadpneirk, 2 theauthorsexpecttoapplytheseresultstoproveglobalversionsoftheconjecture,atleast v intheequalparametercasewithbadprimesexcluded. 1 8 4 6 0 1. INTRODUCTION . 1 0 Let G = G(q) be a family of finite groups of Lie type having irreducible (finite) { } 5 Coxeter system (W,S) [CR87, (68.22)]. The pair (W,S) remains fixed throughout this 1 paper. Let B(q) be a Borel subgroup of G(q). There are index parameters c Z, s S, : s v ∈ ∈ defined by i X [B(q) : sB(q) B(q)] = qcs, s S. ∩ ∈ ar Thegeneric Hecke algebra overthe ring Z = Z[t,t−1]of Laurent polynomialsasso- ciated toG has basisT , w HW,subjectto relations w ∈ T , sw > w; (1.0.1) T T = sw s w (t2csTsw +(t2cs 1)Tw, sw < w − for s S,w W. This algebra is defined just using t2, but it is convenient to have its square∈root t a∈vailable. We call a Hecke algebra of Lie type over Z. It is related to the representation theory of the gHroups in G as follows: for any prime power q, let R be any field (we will shortly allow R to be a ring) in which the integer q is invertibleand has 1991MathematicsSubjectClassification. 20C08,20C33,16S50,16S80. ResearchsupportedinpartbytheAustralianResearchCouncilandNationalScienceFoundation. 1 2 JIEDU,BRIANJ.PARSHALL,ANDLEONARDL.SCOTT a square root √q. Let R = Z R be the algebra obtained by base change through H H ⊗ the map Z R, t √q. Then = End (indG(q)R). Thus, the generic Hecke → 7→ HR ∼ G(q) B(q) algebra isthequantumization(inthesenseof[DDPW08,§0.4])ofan infinitefamilyof H importantendomorphismalgebras. In type A , i.e., when G = GL (q) , one can also consider the q-Schur algebras, n n+1 { } viz.,algebras Moritaequivalentto (1.0.2) S := End ind R IND . R HR HHJ,R J! J S M⊆ Inthiscase, S isaquasi-hereditaryalgebrawhoserepresentationtheoryiscloselyrelated R tothatofthequantumgenerallineargroups. Theq-Schuralgebrashavehistoricallyplayed an important role in representation theory of the finite general linear groups, thanks to the work of Dipper, James, and others. More generally, although the definition (1.0.2) makes sense in all types, less is known about its properties or the precise role it plays in the representation theory or homological algebra of the corresponding groups in G. The purposeofthispaper, andits sequels,isto enhanceS inaway described below,sothat it R does becomerelevanttothesequestions. 1.1 Stratifying systems. At this point, it will be useful to review the notion of a strict stratifyingsystemforanalgebra. Thesesystemsprovideaframeworkforstudyingalgebras similartoquasi-hereditaryalgebras. TheyappearinthestatementofthefirstmainTheorem 5.6. Although the algebras in the Theorem 5.6 are shown later to be quasi-hereditary, the theory of stratifying systems is useful both in providing a framework and as a tool in obtainingthefinal results. First, recall that a preorder on a set X is a transitive and reflexive relation . The ≤ associatedequivalencerelation onX isdefined bysetting,forx,y X, ∼ ∈ x y x y&y x. ∼ ⇐⇒ ≤ ≤ A preorder induces an evident partial order, still denoted , on the set X of equivalence ≤ classes of . In thispaper, a set X with apreorder is called a quasi-poset. Also, ifx X, ∼ ∈ let x¯ X beitsassociated equivalenceclass. ∈ Now let R be a Noetherian commutative ring, and let A be an R-algebra, finitely generated and projective as an R-module. Let Λ be a finite quasi-poset. For each λ Λ, it ∈ is required that there is given a finitely generated A-module ∆(λ) and a finitely generated projective A-module P(λ) together with a fixed surjective P(λ) ։ ∆(λ) morphism of A-modules. Thefollowingconditionsare required: (1) Forλ,µ Λ, ∈ Hom (P(λ),∆(µ)) = 0 = λ µ. A 6 ⇒ ≤ (2) EveryirreducibleA-moduleLis ahomomorphicimageofsome∆(λ). EXTENDINGHECKEENDOMORPHISMALGEBRAS 3 (3) For λ Λ, the A-module P(λ) has a finite filtration by A-submodules with top section∈∆(λ) and othersectionsoftheform ∆(µ)withµ¯ > λ¯. Whentheseconditionsallhold,thedata ∆(λ) isastrictstratifyingsystemforA–mod. λ Λ { } ∈ It is also clear that ∆(λ) ,P(λ) ,... is a strict stratifying system for A -mod for any R′ R′ R′ basechangeR R,providedR isaNoetheriancommutativering. (Noticethatcondition ′ ′ → (2) is redundant, if it is known that the direct sum of the projective modules in (3) is a progenerator—apropertypreserved bybasechange.) An ideal J in an R-algebra A as above is called a stratifying ideal provided that J is an R-direct summand of A (or equivalently, the inclusion J ֒ A is R-split), and for → A/J-modulesM,N inflationdefines an isomorphism (1.1.1) Extn (M,N) ∼ Extn(M,N), n 0. A/J −→ A ∀ ≥ of Ext-groups. A standard stratification of length n of A is a sequence 0 = J ( J ( 0 1 ( J = A of stratifying ideals of A such that each J /J is a projective A/J - n i i 1 i 1 ··· − − module. If A–modhas astrictstratifyingsystemwithquasi-posetΛ,then ithas astandard stratificationoflengthn = Λ¯ ;see[DPS98a, Thm. 1.2.8]. | | InthecaseofafinitedimensionalalgebraAoverafieldk,thenotionofastrictstratifying system ∆(λ) for A–mod simplifies somewhat. In this case, it can be assumed that λ Λ { } ∈ each ∆(λ) has an irreducible head L(λ), that λ = µ = L(λ) = L(µ), and that P(λ) is ∼ 6 ⇒ 6 indecomposable. Two caveats are in order, however: (i) it may be necessary to enlarge the base set Λ to be able to index all the irreducible modules, though Λ can remain the same; (ii) it may be easier to verify (1), (2) and (3) over a larger ring and then base change. The irreducible head versions of the ∆(λ)’s can then be obtained as direct summands of the base-changed versions. WhenthealgebraAarises asanendomorphismalgebraA = End (T),thereisauseful B theory for obtaining a strict stratifying system for A–mod. In fact, this is how such stratifying systems initially arose (see [CPS96] and [DPS98a]). This approach is followed in the proof ofthe main theorem in this paper. For convenience, we summarizethe sufficient conditionsthatwillbeused, alltaken from[DPS98a,Thm. 1.2.10]. Theorem 1.1. Let B be a finitely generated projective R-algebra and let T be a finitely generatedrightB-modulewhichisprojectiveoverR. DefineA := End (T). Assumethat B T = T , where Λ is a finite quasi-poset. For λ Λ, assume there is given a fixed λ Λ λ R-sub⊕mo∈duleS T andanincreasingfiltrationF :∈0 = F0 F1 Ft(λ) = T λ ⊆ λ λ• λ ⊆ λ ⊆ ··· ⊆ λ λ satisfyingthefollowingconditions: (1) For λ Λ, F has bottom section F1/F0 = S , and, higher sections Fi+1/Fi (1 i ∈ t(λ) λ• 1)oftheformS withλν¯ >λλ¯∼. λ λ λ ν ≤ ≤ − (2) Forλ,µ Λ, Hom (S ,T ) = 0 = λ µ. B µ λ ∈ 6 ⇒ ≤ (3) Forλ Λ,Ext1(T /Fi,T) = 0 foralli. ∈ B λ λ Let A = End (T), and, for λ Λ, define ∆(λ) := Hom (S ,T) A-mod. Assume that B B λ ∈ ∈ each ∆(λ)is R-projective. Then ∆(λ) is astrictstratifyingsystem forA–mod. λ Λ { } ∈ 4 JIEDU,BRIANJ.PARSHALL,ANDLEONARDL.SCOTT Itisinterestingtonotethatthesesufficientconditionsarenot,ingeneral,preservedunder base change, though the resulting strict stratifying systems are preserved (becoming strict stratifyingsystemsforthebase-changed versionofthealgebraA). 1.2 Cells and q-permutation modules. We assume familiarity with Kazhdan-Lusztig cell theory for the Coxeter systems (W,S). See, for instance, [DDPW08] and [Lus03]. In Conjecture 1.2 below and in the main Theorem 5.6, the set Λ will be the set Ω of left Kazhdan-Lusztigcells for(W,S). Foreach ω Ω, let ∈ (1.2.1) S(ω) := Lω/ <Lω –mod ≤ H H ∈ H be the corresponding left cell module. It is known that S(ω) is a free Z-module with basis corresponding to certain Kazhdan-Lusztig basis elements C , x ω; see §2. The x′ ∈ correspondingdualleft cell moduleisdefined (1.2.2) Sω := HomZ(S(ω),Z) mod– . ∈ H It is regarded as a right -module. Because S(ω) and hence S are free over Z, if R is a ω commutativeZ-moduleH,wecan define SR(ω) := S(ω) Z R ⊗ (Sω,R := Sω Z R = HomR(SR(ω),R). ⊗ Forthespecial choiceR = Q—see(1.3.1)belowforthedefinitionofQ—wealso usethe notations S(ω) := SQ(ω), (1.2.3) (Sω := Sω,Q, ω Ω. ∈ e In addition, for λ S, let W be the parabolic subgroup of W generated by the s λ, λ ⊆ e ∈ and put x = T , with T as in (1.0.1) above. The induced modules x (also called q-pλermutatwio∈nWλmowdules) hawve an increasing filtration with sections S forλHvarious ω ω Ω(precisePly, thoseleft cellsω whoserightsetR(ω)containsλ). ∈ Let = x , and := End ( ). For ω Ω, put ∆(ω) := Hom (S , ) - T λ λH A H T ∈ H ω T ∈ A mod. The algebra is very well behaved in type A, a q-Schur algebra, and a theme of L A [DPS98a] was that suitable enlargements, appropriately compatible with two-sided cell theory,shouldhavesimilargoodpropertiesforall types. Eachtwo-sidedcellmaybeidentifiedwiththesetofleftcellsitcontains,andtheresult- ing collection Ω of sets of left cells is a partition of Ω. There are various natural preorders on Ω, but we will be mainly interested in those whose associated equivalence relation has precisely the set Ω as its associated partition. We call such a preorder strictly compatible withΩ. 1.3 Aconjecture. Nowwearereadytostatethefollowingconjecture,whichisavariation (see the Appendix) on [DPS98a, Conj. 2.5.2]. We informally think of the algebra + in A EXTENDINGHECKEENDOMORPHISMALGEBRAS 5 the conjecture as an extension of as a Hecke endomorphism algebra (justifying the title A ofthepaper). Conjecture 1.2. There exists a preorder on the set Ω of left cells in W, strictly com- ≤ patible with its partition Ω into two-sided cells, and a right -module such that the H X followingstatementshold: (1) hasanfinitefiltrationwithsectionsof theformS , ω Ω. ω X ∈ (2) Let + := andput T T ⊕X + := End ( +), A H T ∆+(ω) := Hom (S , +), forany ω Ω. ( ω H T ∈ Then, for any commutative, NoetherianZ-algebraR, the set ∆+(ω) is a R ω Ω strictstratifyingsystemfor +-mod relativeto thequasi-poset({Ω, ). } ∈ AR ≤ The main result of this paper, given in Theorem 5.6, establishes a special “local case" of this conjecture. A more detailed description of this theorem requires some preliminary notation. Throughout this paper, e is positiveinteger (= 2 in our main results). Let Φ (t) 2e 6 denote the (cyclotomic) minimum polynomial for a primitive 2eth root of unity √ζ = exp(2πi/2e) C. Fix amodularsystem(K,Q,k)byletting ∈ Q := Q[t,t 1]] ,where p = (Φ (t)); − p 2e (1.3.1) K := Q(t), thefraction field of Q;  k := Q/m = Q(√ζ), theresiduefield of Q. ∼ Here m denotes the maximal ideal of the the DVR Q. With some abuse of notation, we  sometimesidentify√ζ withitsimageink. (Withoutpassingtoanextensionorcompletion, theringQ mightnothavesucharootofunityinit.) Thealgebra issplitsemisimple, Q(t) H with irreducible modules corresponding to the irreducible modules of the group algebra QW. TheQ-algebra (1.3.2) := Z Q H H⊗ hasapresentationbyelementsT 1(whichwillstillbedenotedT ,w W)completely w w ⊗ e ∈ analogous to (1.0.1). Similar remarks apply to , replacing t2 by ζ. Then Theorem 5.6 k H establishes that there exists a -module which is filtered by dual left cell modules S ω such that the analogues of coHnditions (1X) and (2) over Q in Conjecture 1.2 hold. The preorderusedinTheorem5.6iseconstructeedasin[GGOR03]froma“sortingfunction"ef, and isdiscussedindetail inthenextsection. With more work, it can be shown, when e = 2, that the Q-algebra + := + is quasi- Q 6 A A hereditary. This isdoneinTheorem 6.4. ThenTheorem 6.5 identifiesthemodulecategory forabase-changedversionofthisalgebrawithaRDAHA-category ine[GGOR03]. Such O anidentificationintypeAwasconjecturedin[GGOR03],andthenprovedbyRouquierin [Ro08](when e = 2). 6 6 JIEDU,BRIANJ.PARSHALL,ANDLEONARDL.SCOTT Generally speaking, this paper focuses on the “equal parameter" case (i.e., all c = 1 in s (1.0.1)),whichcoverstheHeckealgebrasrelevanttoalluntwistedfiniteChevalleygroups. We will assume this condition unless explicitly stated otherwise, avoiding a number of complications involving Kazhdan-Lusztig basis elements and Lusztig’s algebra . In this J context,thecritical Proposition3.1 depends on results of[GGOR03]which, in part, were only determined in the equal parameter case. Nevertheless, much of our discussion ap- plies in the unequal parameter cases. In particular, we mention that the elementary, but important, Lemma 4.3 is stated and proved using unequal parameter notation. This en- courages theauthorstobelievethemainresultsarealso provablein theunequal parameter case, though this has not yet been carried out. Note that all the rank 2 cases are treated in [DPS98a], leaving the quasi-split cases with rank > 2. All these quasi-split cases have parameters confined totheset 1,2,3 . { } 2. SOME PRELIMINARIES Thissection recalls somemostlywell-knownfacts and fixes notationregardingcell theory. Let W be a finite Weyl group associated to a finite root system Φ with a fixed set of simple roots Π. Let S := s α Π . Let is a Hecke algebra over Z defined by α { | ∈ } H (1.0.1). We assume (unless explicitly noted otherwise) that each c = 1 for s S. Thus, s ∈ (W,S) corresponds, in the language of the introduction, to some types of split Chevalley groups,thoughwewillhaveno furtherneed ofthatcontext. Let C = t l(w) P T , w′ − y,w y y w X≤ wheretheP isaKazhdan-Lusztigpolynomialinq := t2. Then C isaKazhdan– Lusztig (or cy,awnonical) basis for . The element C is denoted c {inw′[}Lwu∈sW03], a reference wefrequentlyquote. Leth HZ denotethestrux′ctureconstanxts. In otherwords, x,y,z ∈ C C = h C . x′ y′ x,y,z z′ z W X∈ Using the preorders and on W, the positivity(see [DDPW08, §7.8])of thecoeffi- L R ≤ ≤ cientsoftheh implies x,y,z (2.0.3) h = 0 = z y,z x x,y,z L R 6 ⇒ ≤ ≤ The Lusztig function a : W N is defined as follows. For z W, let a(z) be the −→ ∈ smallest nonnegative integer such that ta(z)h N[t] for all x,y W. It may equally x,y,z ∈ ∈ be defined as the smallest nonnegative integer such that t a(x)h N[t 1], as used in − x,y,z − ∈ [Lus03](orsee[DDPW08,§7.8]). Infact,eachh isinvariantundertheautomorphism x,y,z Z Z sending tto t 1. It isnot difficultto see thata(z) = a(z 1). Forx,y,z W, let − − → ∈ γ bethecoefficient oft a(z) inh . Also,by [Lus03, Conjs. 14.2(P8),15.6], x,y,z − x,y,z−1 (2.0.4) γ = 0 = x y 1,y z 1,z x 1. x,y,z L − L − L − 6 ⇒ ∼ ∼ ∼ EXTENDINGHECKEENDOMORPHISMALGEBRAS 7 Thefunctionaisconstantontwo-sidedcellsinW,andsocan beregardedas afunction (with values in N on (a) the set of two-sided cells; (b) the set of left (or right) cells; and (c) the set Irr(QW) of irreducible QW-modules.1 In addition, a is related to the generic degrees d , E Irr(QW). For E Irr(QW), let d = btaE + +ctAE,with a A E E E E ∈ ∈ ··· ≤ and bc = 0, so that taE (resp., tAE) is the lowest (resp., largest) power of t appearing 6 nontriviallyind . Thena = a(E);cf. [Lus03,Prop. 20.6]. Also,asnotedin[GGOR03, E E §6], A = N a(E det), where N is the number of positive roots in Φ. Following E − ⊗ [GGOR03, §6], wewillusethe“sortingfunction"f : Irr(QW) N defined by → (2.0.5) f(E) = a +A = a(E)+N a(E det). E E − ⊗ The function f is also constant on two-sided cells: if E is an irreducible QW-module associatedatwo-sidedcell c,then E det isan irreduciblemoduleassociatedto thetwo- sidedcell w c. See [Lus84, Lem. 5.1⊗4(iii)] 0 The function f is used in [GGOR03] to define various order structures on the set Irr(QW) of irreducible QW-modules. Put E < E (our notation) provided f(E) < f ′ f(E ). Thereareatleasttwonaturalwaystoextend< toapreorder. Thefirstway,which ′ f is only in the background for us, is to set E E E = E or E < E . This gives f ′ ∼ ′ f ′ (cid:22) ⇐⇒ a poset structure, and is used, in effect, by [GGOR03] for defining a highest category ; O see[GGOR03,§2.5, 6.2.1]. We use < here to define a preorder on the set Ω of left cells: First, observe that f f ≤ thefunctionf aboveis constant onirreduciblemodulesassociated to thesameleft cell (or eventhesametwo-sidedcell)andsomaybeviewedasafunctiononΩ. Wecannowdefine the (somewhat subtle) preorder on Ω by setting ω ω (for ω,ω Ω) if and only if f f ′ ′ ≤ ≤ ∈ eitherf(ω) < f(ω ), orω and ω lieinthesametwo-sidedcell. Notethatthe“equivalence ′ ′ classes" of the preorder identify with the set of two-sided cells—thus, is strictly f f ≤ ≤ compatiblewiththeset oftwo-sidedcellsin thesenseof§1. Also, (2.0.6) E < E = E < E; LR ′ ′ f ⇒ see [GGOR03, Lem. 6.6]. Here E,E are in Irr(QW), and thenotationE < E means ′ LR ′ that the two-sided cell associated with E is strictly smaller than that associated with E , ′ withrespecttotheKazhdan-Lusztigorderontwo-sidedcells. Apropertysimilarto(2.0.6) holdsif< isreplaced with< , defined similarly,but usingleft cells. In termsof Ω, this LR L left cellversionreads: (2.0.7) ω,ω Ω,ω < ω = f(ω) > f(ω ). ′ L ′ ′ ∈ ⇒ Notice that (2.0.7) follows from (2.0.6) using [Lus87b, Cor. 1.9(c)]. (The latter result impliesthatω,ω ontheleftin(2.0.7)cannotbelongtothesametwo-sidedcell.) Thus,the ′ preorder is a refinement of the preorder op on Ω, and induces on the set of two- sidedcell≤saf refinementofthepartialorder ≤opL. Forfurther≤dfiscussion,seetheAppendix. ≤LR 1Itiswell-knownthatQisasplittingfieldforW [B71]. 8 JIEDU,BRIANJ.PARSHALL,ANDLEONARDL.SCOTT 3. (DUAL) SPECHT MODULES OF GINZBURG–GUAY–OPDAM–ROUQUIER Theasymptoticform of isaring withZ-basis j x W and multiplication x J H { | ∈ } j j = γ j . x y x,y,z−1 z z X Thisring was originallyintroduced in[Lus87b], thoughwe follow[Lus03, 18.3], usinga slightlydifferentnotation. 3.1 The mapping ̟ and its properties. As per [Lus03, 18.9], define a Z-algebra homomorphism (3.1.1) ̟ : H → JZ = J ⊗Z, Cw′ 7−→ hw,d,zjz, z W d∈D X∈ a(dX)=a(z) where D is the set of distinguishedinvolutions in W. Also, for any Z-algebra R, there is also an algebra homomorphism ̟R : R = Z R R = Z Z R, obtained by H H ⊗ → J J ⊗ basechange. In obviouscases, weoftendrop thesubscriptR from ̟ . R In particular,̟ becomesan isomorphism Q(t) (3.1.2) ̟ = ̟Q(t) : Q(t) ∼ Q(t). H −→ J See [Lus87b]. Also,̟ inducesamonomorphism (3.1.3) ̟ = ̟ : ֒ = Q[t,t 1]. Q[t,t−1] Q[t,t−1] Q[t,t−1] Q − H → J J ⊗ Moreover,basechangetoQ[t,t 1]/(t 1) inducesan isomorphism − − (3.1.4) ̟ = ̟Q : QW ∼ Q −→ J (cf., [Lus87c, Prop. 1.7]). This allows us to identify irreducible QW-modules with irreducible -modules.2 Q J For the irreducible (left) -module identified with E Irr(QW), the (left) - Q Q[t,t−1] J ∈ H module S(E) := ̟ (E Q[t,t 1]) = ̟ (E ) ∗ − ∗ Q[t,t−1] ⊗ is called here a dual Specht module for ; cf. [GGOR03, Cor. 6.10].3 Note that Q[t,t−1] H S(E) ∼= EQ[t,t−1]asaQ[t,t−1]-module. Therefore,S(E)isafreeQ[t,t−1]-module. Putting 2The map ̟ is the composition φ , where φ and are defined in [Lus03, 18.9] and [Lus87c, 3.5], ◦† † respectively. Thenumbersnz appearingthere(whichare 1bydefinitionin[Lus03, §18.8])areallequal ± to 1, because of the positivity (see [Lus03, §7.8]) of the structure constants appearing in [Lus03, 14.1]. This ̟ is not the same one as defined in [GGOR03, p.647], where the C-basis was used. Nevertheless, b the arguments of [GGOR03, §6] go through, using the C′-basis and our ̟ (see Remark 5.2 below), so [GGOR03,Thm. 6.8]guaranteesthemodulesS (E)definedbelowusingourset-uparethesame,atleast C uptoa(two-sidedcellpreserving)permutationoftheisomorphismtypeslabeledbytheE’s,asthemodules S(E)definedin[GGOR03, Defn. 6.1]withR = C. Theproofof[GGOR03, Thm. 6.8]alsoestablishes suchanidentificationofthevariousmodulesSR(E)whenRisacompletionofC[t,t−1]. 3In[GGOR03,Defn.6.1],themoduleS(E)thereiscalledastandardmodule.Ourchoiceofterminology isjustifiedbythediscussionfollowingtheproofofLemma5.1below. EXTENDINGHECKEENDOMORPHISMALGEBRAS 9 S = Hom (S(E),Q[t,t 1]),define E Q[t,t−1] − S(E) := SQ(E), (3.1.5) (SE := SE,Q, e where, in general, forbasechangetoacommutative,Noetherian Q[t,t 1]-algebraR, − e S (E) := S(E) R, R Q[t,t−1] ⊗ (SE,R := SE Q]t,t−1] R ∼= HomR(SR(E),R). ⊗ The followingpropositionis proved using RDAHAs, and it is the only ingredient in the proofofTheorem5.6 wherethesealgebrasare used. Proposition 3.1. Assume that e = 2. Suppose E,E are irreducible QW-modules. If ′ 6 E = E and ∼ ′ 6 Hom (S (E),S (E )) = 0, Hk k k ′ 6 thenf(E) < f(E ). Also,Hom (S (E),S (E)) = k. ′ Hk k k ∼ Proof. Withoutloss, wereplace k in thestatementofthepropositionbyC, usingtheanal- ogousdefinitionsofS (E). In addition,thestatementofthepropositionis invariantunder C anytwo-sidedcellpreservingpermutationofthelabelingoftheirreduciblemodules. After applying such a permutation on the right (say) we may assume, by [GGOR03, Thm. 6.8] and takingintoaccountftn. 1, that KZ(∆(E)) = S (E), ∼ C where (1) ∆(E)isthestandardmoduleforahighestweightcategory givenin[GGOR03], O havingpartialorder (see[GGOR03,Lem.2.9,6.2.1])onitssetofisomorphism f ≤ classes of irreducible modules, which are indexed by isomorphism classes of irreducible QW-modules . We take k = 1/e > 0 in [GGOR03] above Thm. 6.8 H,1 and inRem. 3.2 there. (2) The functor KZ : is naturally isomorphic to the quotient map M M O −→ O 7→ in [GGOR03, Prop. 5.9, Thm. 5.14], the quotient category there identifying with -mod. C H Using [GGOR03, Prop. 5.9], which requires e = 2, we have, for any irreducible CW- 6 modulesE,E , ′ Hom (∆(E),∆(E )) = Hom (∆(E),∆(E )) = Hom (S (E),S (E )). ′ ∼ ′ ∼ C C C ′ O O H If E = E , then ∆(E) = ∆(E )) and Hom (∆(E),∆(E )) = 0 implies that E < E , ∼ ′ ∼ ′ ′ f ′ 6 6 O 6 i.e., f(E) < f(E ). ′ Ontheotherhand,ifE = E , thenHom (∆(E),∆(E )) = C. This implies ∼ ′ ′ ∼ O Hom (S (E),S (E )) = C. C C C ′ ∼ H Returning to the original k = Q(√ζ), we may conclude the same isomorphism holds in theoriginalsettingas well. (cid:3) 10 JIEDU,BRIANJ.PARSHALL,ANDLEONARDL.SCOTT Corollary3.2. Assumee = 2. Let E,E beirreducibleQW-modules. Then ′ 6 Ext1 (S(E),S(E )) = 0 = f(E) < f(E ). e ′ ′ 6 ⇒ H In particular,Ext1 (S(E),S(E)) = 0. e e e H Proof. In(1.3.1)letπ = Φ (t)bethegeneratorofthemaximalidealmofQ,andconsider e 2ee theshortexactsequence π 0 S(E ) S(E ) S (E ) 0. ′ ′ k ′ → −→ → → By thelongexactsequence ofExt, thereisan exact sequence e e π 0 Home(S(E),S(E′)) Home(S(E),S(E′)) Home (Sk(E),Sk(E′)) → H → H → Hk Ext1 (S(E),S(E )) π Ext1 (S(E),S(E )) e ′ e ′ e e → e e → H H Ext1 (S (E),S (E )). e k k ′ → Hke e e e Because = issemisimple, Q(t) Q(t) H H Ext1 (S(E),S(E )) = Ext1 (S(E) ,S(E ) ) = 0. Hee ′ Q(t) ∼ HQ(t) Q(t) ′ Q(t) In other words, if it is nonzero, Ext1 (S(E),S(E )) is a torsion module, so that the map e e e ′ Ext1 (S(E),S(E )) π Ext1 (S(E),HS(E )) is not injective. Thus, it suffices to prove that e ′ e ′ → wheHn f(E) < f(E ),themHap e e ′ 6 e e e e (3.1.6) Home(S(E),S(E′)) Home (Sk(E),Sk(E′)) H → Hk is surjective. If E ∼= E′, Proposition 3.1 gives Home (Sk(E),Sk(E′)) = 0 implying the surjectivity of (36.1.6) trievially.eOn the other hand,Hikf E = E , the proposition gives ∼ ′ Hom (S (E),S (E )) = k. This also gives surjectivity of the map in (3.1.6), since it becomHkesskurjectivkeup′on∼restrictiontoQ Home(S(E),S(E′)) (takingE′ = E). (cid:3) ⊆ H 4. TWO PRELIMINARYeLEMMeAS Let R be a commutative ring and let C be an abelian R-category. For A,B C, ∈ let Ext1(A,B) denote the Yoneda groups of extensions of A by B. (We do not require C the higher Ext-groups in this section.) Let M,Y C, and suppose that Ext1(M,Y) C ∈ is generated as an R-module by elements ǫ , ,ǫ . Let χ := ǫ Ext1(M m,Y) 1 m i i C ⊕ ··· ⊕ ∈ correspond totheshortexactsequence0 Y X M m 0. ⊕ → → → → Lemma 4.1. The map Ext1(M,Y) Ext1(M,X), induced by the inclusion Y X, is C C → → thezeromap. Proof. Using the “long" exact sequence of Ext , it suffices to show that the map δ in the • sequence HomC(M,X) HomC(M,M⊕m) δ Ext1C(M,Y) −→ −→