TENSOR PRODUCT CATEGORIFICATIONS AND THE SUPER KAZHDAN-LUSZTIG CONJECTURE

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1 TENSOR PRODUCT CATEGORIFICATIONS AND THE SUPER KAZHDAN-LUSZTIG CONJECTURE JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER Abstract. We gve a new proof of the super Kazhdan-Lusztg conjecture for the Le superalgebra gl n m (C) as formulated orgnally by the frst author. We also prove for the frst tme that any ntegral block of category O for gl n m (C) (and also all of ts parabolc analogs) possesses a graded verson whch s Koszul. Our approach depends crucally on an applcaton of the unqueness of tensor product categorfcatons establshed recently by the second two authors. Contents 1. Introducton 2 2. Tensor product categorfcatons Schuran categores Combnatorcs Hecke algebras Categorfcaton Recollectons about hghest weght categores Tensor product categorfcatons Revew of the proof of Theorem 2.12 for fnte ntervals Truncaton Decomposton numbers and blocks Classfcaton of prnjectves The general lnear Le superalgebra Super category O Super parabolc category O Proof of the super Kazhdan-Lusztg conjecture Stable modules Tower of Hecke algebras Stable modules and the double centralzer property Categorcal acton on stable modules Proof of Theorem 2.12 for nfnte ntervals Graded tensor product categorfcatons Quantzed envelopng algebras Graded lfts Graded hghest weght categores Graded categorfcatons Graded tensor product categorfcatons Proof of Theorem 5.11 for fnte ntervals Proof of Theorem 5.11 for nfnte ntervals Koszulty Kazhdan-Lusztg polynomals 56 References Mathematcs Subject Classfcaton: 16E05, 16S38, 17B37. Authors supported n part by NSF grant nos. DMS , DMS and DMS , respectvely. 1

2 2 1. Introducton In ths paper we explan how the unqueness of tensor product categorfcatons establshed by the second two authors n [LW] yelds a quck proof of the Kazhdan-Lusztg conjecture for the general lnear Le superalgebra gl n m (C). Ths conjecture was formulated orgnally by the frst author n [B1] and has been proved already by a dfferent method by Cheng, Lam and Wang n [CLW]. Actually we prove here a substantally stronger result, namely, that the analog of the Bernsten-Gelfand-Gelfand category O for gl n m (C) possesses a Koszul graded lft, n the sprt of the classc work [BGS]. Roughly, the super Kazhdan-Lusztg conjecture asserts that combnatorcs n ntegral blocks of category O for gl n m (C) s controlled by varous canoncal bases n the sl -module V n W m, where V s the natural sl -module and W s ts dual. In fact, we prove a generalzaton of the conjecture whch s adapted to the hghest weght structure on O arsng from any choce of conjugacy class of Borel subalgebra; changng the Borel corresponds to shufflng the tensor factors n the mxed tensor space V n W m nto more general orders. Ths generalzaton was suggested n the ntroducton of [Ku], then precsely formulated and proved n [CLW]. (We pont out also the paper [CMW] whch establshes an equvalence of categores from an arbtrary non-ntegral block of category O for gl n m (C) to an ntegral block of a drect sum of other general lnear Le superalgebras of the same total rank.) The basc dea of our proof s as follows. For a fnte nterval I Z, let sl I be the specal lnear Le algebra consstng of (complex) trace zero matrces wth rows and columns ndexed by ntegers from the set I + := I (I +1). Let V I be the natural sl I -module of column vectors and W I := VI. We construct a subquotent O I of the super category O whch s an sl I -categorfcaton of the tensor product V n I W m I n the sense of Chuang and Rouquer [CR],[R]. Then, observng that V n I W m I = V n I ( I V I ) m, one can apply the unqueness of tensor product categorfcatons from [LW] to deduce that O I s equvalent to another well-known categorfcaton O I of ths tensor product arsng from the parabolc category O assocated to the Le algebra gl n+m I (C) and ts Lev subalgebra gl 1 (C) n gl I (C) m. The combnatorcs of the latter category s understood by the ordnary Kazhdan-Lusztg conjecture proved n [BB],[BrKa]. Snce the fnte nterval I can be chosen freely, ths gves enough nformaton to deduce the super Kazhdan-Lusztg conjecture. By the well-known results from [BGS] and [B], the category O I has a graded verson whch s Koszul. Hence so does the equvalent category O I. To construct a Koszul gradng on category O for gl n m (C), we show further that the Koszul gradngs on each O I can be chosen n a compatble way so that they lft to O tself. Agan we do ths also for all of the parabolc analogs of O, so that a very specal case recovers the Koszul gradng on the subcategory of O consstng of fnte dmensonal representatons that was constructed explctly n [BS]. Our man result here can be paraphrased as follows. Theorem A. Any block of parabolc category O for gl n m (C) wth ntegral central character has a graded lft whch s a standard Koszul hghest weght category.

3 SUPER KAZHDAN-LUSZTIG 3 Moreover ts graded decomposton numbers can be computed n terms of fnte type A parabolc Kazhdan-Lusztg polynomals, as predcted n [B1]. In the man body of the artcle we adopt a more axomatc approach n the sprt of [LW]. In Secton 2, we wrte down the formal defnton of an sl -tensor product categorfcaton of a tensor product of exteror powers of V and W. Ths s a category wth an sl -acton n the sense dscussed n Defnton 2.6, a hghest weght category structure as n Defnton 2.8, and some compatblty between these structures explaned n Defnton 2.10, such that the complexfed Grothendeck group of the underlyng category of - fltered objects s somorphc to the gven tensor product of exteror powers of V and W. Then the bulk of the artcle s taken up wth provng the followng fundamental result about such categorfcatons. Theorem B. There exsts a unque sl -tensor product categorfcaton assocated to any tensor product of exteror powers of V and W. Moreover such a category has a unque graded lft compatble wth all the above structures. The exstence part of ths theorem s proved n Secton 3, smply by verfyng that parabolc category O for the general lnear Le superalgebra satsfes the axoms; n fact ths s the only tme Le superalgebras enter nto the pcture. The unqueness (up to strongly equvarant equvalence) s proved n Secton 4. It s a non-trval extenson of the unqueness theorem for fnte sl I -tensor product categorfcatons establshed n [LW]. The proof for sl depends on the constructon of an nterestng new category of stable modules for a certan tower of quver Hecke algebras. Fnally n Secton 5, we ncorporate gradngs nto the pcture, defnng the noton of a U q sl -tensor product categorfcaton of a tensor product of q- deformed exteror powers of V and W ; see Defntons 5.4, 5.6 and 5.9. We prove the exstence and unqueness of these by explotng graded stable modules over our tower of quver Hecke algebras. Then we prove that any such category s standard Koszul and deduce the graded verson of the Kazhdan-Lusztg conjecture. Conventons. We fx an algebracally closed feld K of characterstc 0 throughout the artcle. All categores and functors wll be assumed to be K-lnear wthout further notce. Let Vec be the category of fnte dmensonal vector spaces. For a fnte dmensonal graded vector space V = n Z V n, we wrte dm q V for ts graded dmenson n Z (dm V n)q n Z[q, q 1 ]. 2. Tensor product categorfcatons In ths secton, we revew the defnton of tensor product categorfcaton from [LW, Defnton 3.2] n the specal case of tensor products of exteror powers of the natural and dual natural representatons of sl I. We nclude the possblty that the nterval I Z s nfnte, when these are not hghest weght modules. Then we state our frst man result assertng the exstence and unqueness of such tensor product categorfcatons, extendng the case of fnte ntervals from [LW]. After that, we make some preparatons for the proof (whch actually takes place n Sectons 3 and 4), and dscuss some frst applcatons.

4 4 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER 2.1. Schuran categores. By a Schuran category we mean an Abelan category C such that all objects are of fnte length, there are enough projectves and njectves, and the endomorphsm algebras of the rreducble objects are one dmensonal. For example, the category mod-a of fnte dmensonal rght modules over a fnte dmensonal K-algebra A s Schuran. Note throughout ths text we wll work n terms of projectves, but obvously C s Schuran f and only f C op s Schuran, so that everythng could be expressed equvalently n terms of njectves. We use the made-up word prnjectve for an object that s both projectve and njectve. Gven a Schuran category C, we let pc be the full subcategory consstng of all projectve objects. Let Fun f (pc, Vec op ) denote the category of all contravarant functors from pc to Vec whch are zero on all but fntely many somorphsm classes of ndecomposable projectves. The Yoneda functor C Fun f (pc, Vec op ), M Hom C (, M) (2.1) s an equvalence. Hence C can be recovered (up to equvalence) from pc. Ths asserton can be formulated n more algebrac terms as follows. Let {L(λ) λ Λ} be a complete set of parwse non-somorphc rreducble objects n C, and fx a choce of a projectve cover P (λ) of each L(λ). Let A := Hom C (P (λ), P (µ)) (2.2) λ,µ Λ vewed as an assocatve algebra wth multplcaton comng from composton n C. Let 1 λ A be the dentty endomorphsm of P (λ). If Λ s fnte then A s a untal algebra wth 1 = λ Λ 1 λ, ndeed, A s the endomorphsm algebra of the mnmal projectve generator λ Λ P (λ). However n general A s only locally untal, meanng that t s equpped wth the system {1 λ λ Λ} of mutually orthogonal dempotents such that A = λ,µ Λ 1 µa1 λ. Let mod-a denote the category of all fnte dmensonal locally untal rght A-modules, that s, fnte dmensonal rght A-modules M such that M = λ Λ M1 λ. Then our earler asserton about the Yoneda equvalence amounts to the statement that the functor H : C mod-a, M λ Λ Hom C (P (λ), M) (2.3) s an equvalence of categores. Of course ths functor sends P (λ) to the (necessarly fnte dmensonal) rght deal 1 λ A; these are the ndecomposable projectve modules n mod-a. The lnear duals of the ndecomposable njectve modules are somorphc to the left deals A1 λ, so that the latter are fnte dmensonal too. Conversely gven any locally untal K-algebra A wth dstngushed dempotents {1 λ λ Λ} such that all of the deals 1 λ A and A1 λ are fnte dmensonal, the category mod-a s Schuran. Let K 0 (C) (resp. G 0 (C)) be the splt Grothendeck group of the addtve category pc (resp. the Grothendeck group of the Abelan category C). Set [C] := C Z K 0 (C), [C] := C Z G 0 (C). So [C] s the complex vector space on bass {[P (λ)] λ Λ}, whle [C] has bass {[L(λ)] λ Λ}. These bases are dual wth respect to the blnear Cartan parng (, ) : [C] [C] C defned from ([P ], [L]) := dm Hom C (P, L).

5 SUPER KAZHDAN-LUSZTIG Combnatorcs. Let I Z be a (non-empty) nterval and set I + := I (I + 1). Let sl I be the Le algebra of (complex) trace zero matrces wth rows and columns ndexed by I +, all but fntely many of whose entres are zero. It s generated by the matrx unts f := e +1, and e := e,+1 for all I. The weght lattce of sl I s P I := I Zϖ where ϖ s the th fundamental weght. The root lattce s Q I := I Zα < P I where α s the th smple root defned from α := 2ϖ ϖ 1 ϖ +1, nterpretng ϖ as 0 f / I. Let P I Q I Z, (ϖ, α) ϖ α be the blnear parng defned from ϖ α j := δ,j, so that ( α α j s the Cartan matrx. Let ),j I P + I (resp. Q + I ) be the postve cone n P I (resp. Q I ) generated by the fundamental weghts (resp. the smple roots). The domnance order on P I s defned by β γ f β γ Q + I. For any I + we set ε := ϖ ϖ 1, agan nterpretng ϖ as 0 for / I. The followng lemma s well known. Lemma 2.1. For β = I + b ε and γ = I + c ε n P I wth b = c, we have that β γ f and only f h b h c for all h I. An sl I -module M s ntegrable f t decomposes nto weght spaces as M = ϖ P I M ϖ, and moreover each of the Chevalley generators f and e acts locally nlpotently. Basc examples are the natural sl I -module V I of column vectors wth standard bass {v I + } and ts dual W I wth bass {w I + }; the Chevalley generators act on these bass vectors by f v j = δ,j v +1, e v j = δ +1,j v, f w j = δ +1,j w, e w j = δ,j w +1. The vector v s of weght ε whle w s of weght ε. More generally we have the exteror powers n V I and n W I for n 0; henceforth we denote these nstead by n,0 V I and n,1 V I, respectvely. For c {0, 1} let Λ I;n,c denote the set of 01-tuples λ = (λ ) I+ such that { I + λ c} = n. Ths set parametrzes the natural monomal bass {v λ λ Λ I;n,c } of n,c V I defned from { v1 v v λ := n f c = 0, w 1 w n f c = 1, where 1 < < n are chosen so that λ j c for each j. The actons of the Chevalley generators are gven explctly by { vt (λ) f λ f v λ := = 1 and λ +1 = 0, (2.4) 0 otherwse, { vt (λ) f λ e v λ := = 0 and λ +1 = 1, (2.5) 0 otherwse,

6 6 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER where t (λ) denotes the tuple obtaned from λ by swtchng λ and λ +1. λ P I denote the weght of the vector v λ. We have that λ = I + λ ε, Let nterpretng the sum on the rght hand sde when I s nfnte and c = 1 usng the conventon that + ε 1 + ε = ϖ and ε +1 + ε +2 + = ϖ. We are also gong to be nterested n tensor products of the modules n,c V I. Suppose that we are gven n = (n 1,..., n l ) N l and c = (c 1,..., c l ) {0, 1} l ; we refer to the par (n, c) as a type of level l. Let n,c V I := n 1,c 1 V I n l,c l V I. (2.6) Ths module has the obvous bass of monomals v λ := v λ1 v λl ndexed by elements λ = (λ 1,..., λ l ) of the set The vector v λ s of weght Λ I;n,c := Λ I;n1,c 1 Λ I;nl,c l. (2.7) λ := λ λ l. It s often convenent to regard λ Λ I;n,c as a 01-matrx λ = (λ j ) 1 l,j I+ wth th row λ = (λ j ) j I+. (There are several other ndexng conventons possble; for example earler papers of the frst and thrd authors have used the conventon that λ s represented by a column-strct tableau wth l columns such that the th column s flled wth all j I + such that λ j = 1.) Assume for a moment that I s fnte and that Λ I;n,c s non-empty. Let κ = κ I;n,c be the 01-matrx n Λ I;n,c n whch all the entres 1 are as far to the left as possble wthn each row. Thus κ P I s the unque hghest weght of n,c V I wth respect to the domnance orderng. For any λ Λ I;n,c defne ts defect by def(λ) := 1 2 ( κ κ λ λ ) = κ α 1 2α α, (2.8) where α := κ λ. In combnatoral terms, ths s 1 2 j I + (kj 2 l2 j ) where k j (resp. l j ) counts the number of entres equal to 1 n the jth column of κ (resp. λ). The followng lemma extends ths defnton to nclude nfnte ntervals I. Lemma 2.2. Suppose that I s an nfnte nterval and λ Λ I;n,c. Let J I be a fnte subnterval such that J + 2 max(n) and λ,j = c for all 1 l and j I + \ J +. Let λ J Λ J;n,c be the submatrx (λ,j ) 1 l,j J+ of λ. Let κ J := κ J;n,c. Then the natural number def(λ) := 1 2 ( κ J κ J λ J λ J ) s ndependent of the partcular choce of J. Proof. Defne the trval column to be the column vector (c ) 1 l. Let J and J be two ntervals satsfyng the hypotheses of the lemma wth J J. The condtons mply that κ J (resp. λ J ) can be obtaned from κ J (resp. λ J ) by removng J J trval columns. The lemma follows easly from ths usng the combnatoral formulaton of the defnton of defect.

7 SUPER KAZHDAN-LUSZTIG Hecke algebras. To prepare for the defnton of an sl I -categorfcaton, we recall the defnton of certan assocatve untal K-algebras, namely, the (degenerate) affne Hecke algebra AH d, and the quver Hecke algebra QH I,d assocated to the lnear quver wth vertex set I and an edge j f = j + 1. The latter s also known as a Khovanov-Lauda-Rouquer algebra after [KL1] and [R]. Defnton 2.3. The affne Hecke algebra AH d s the vector space K[x 1,..., x d ] KS d wth multplcaton defned so that the polynomal algebra K[x 1,..., x d ] and the group algebra KS d of the symmetrc group S d are subalgebras, and also 1 f k = j + 1, (AH) t j x k x tj (k)t j = 1 f k = j, 0 otherwse. Here t j denotes the transposton (j j+1) S d. Defnton 2.4. The quver Hecke algebra QH I,d s defned by generators {1 I d } {ξ 1,..., ξ d } {τ 1,..., τ d 1 } subject to relatons: (QH1) the elements ξ 1,..., ξ d commute wth each other and all {1 I d }; (QH2) the elements { 1 I d} are mutually orthogonal dempotents whose sum s the dentty; (QH3) τ j 1 = 1 tj ()τ j where t j () s the tuple obtaned from = ( 1,..., d ) by flppng ts jth and (j + 1)th entres; 1 f k = j + 1 and j = j+1, (QH4) (τ j ξ k ξ tj (k)τ j )1 = 1 f k = j and j = j+1, 0 otherwse; (QH5) τ j τ k = τ k τ j f j k > 1; (QH6) τj 21 = 0 f j = j+1, (ξ j ξ j+1 )1 f j = j+1 1, (ξ j+1 ξ j )1 f j = j+1 + 1, 1 otherwse; (QH7) (τ j+1 τ j τ j+1 τ j τ j+1 τ j )1 = 1 f j = j+1 1 = j+2, 1 f j = j = j+2, 0 otherwse. An mportant feature of QH I,d s that t possesses a non-trval Z-gradng. Ths s defned by declarng that each dempotent 1 s n degree 0, each ξ j n degree 2, and fnally τ k 1 s n degree α k α k+1. The algebras AH d and QH I,d are closely related as explaned n [R, Proposton 3.15]. Ths result can also be formulated as an somorphsm between certan cyclotomc quotents of AH d and QH I,d as n [BK2]. Let ϖ P + I be a domnant weght. Defne AHd ϖ (resp. QHϖ I,d ) to be the quotent of AH d (resp. QH I,d ) by the two-sded deal generated by the polynomal I (x 1 ) ϖ α (resp. by the elements {ξ ϖ α I d }). These are fnte dmensonal algebras. The mage of the polynomal algebra K[x 1,..., x d ] n AHd ϖ s a fnte dmensonal commutatve algebra, hence t contans mutually orthogonal dempotents {1 K d } such

8 8 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER that 1 projects any module M onto ts -th word space M := { v M (x j j ) N v = 0 for each j = 1,..., d and N 0 }. Then let AHI,d ϖ := 1 AHd ϖ 1 j = AH ϖ / d 1 / I d,,j I d whch s a sum of blocks of the algebra AHd ϖ. The followng theorem gves an explct choce of somorphsm between QHI,d ϖ and AHϖ I,d ; any other reasonable choce of somorphsm such as the one from [R, Proposton 3.15] could be used nstead throughout ths artcle. Theorem 2.5 ([BK2], [R]). For ϖ P + I defned on generators by there s an somorphsm QH ϖ I,d AH ϖ I,d 1 1 ; (2.9) ξ j 1 (x j j )1 ; (2.10) (1 + t j )(1 x j + x j+1 ) 1 1 f j = j+1, τ j 1 (1 + t j x j t j x j+1 )1 f j = j+1 + 1, (2.11) (1 + t j x j t j x j+1 )(1 x j + x j+1 ) 1 1 otherwse. (In fact, ths somorphsm can be extended to the completons QH I,d and ÂH I,d wth respect to these systems of quotents as dscussed n [W2].) Proof. Ths follows by [BK2, Man Theorem]. To get exactly ths somorphsm one needs to choose the power seres q j () of [BK2, (3.27) (3.29)] so that q j () = p j () f j = j and q j () = 1 p j () f j / { j+1, j+1 + 1}. Note also that the opposte orentaton of the quver was used n [BK2] so that the elements ψ j e() n [BK2] are our τ j 1 f j { j+1, j+1 +1} and our τ j 1 otherwse; the elements y j e() n [BK2] are our elements ξ j 1. Henceforth we wll smply dentfy QHI,d ϖ and AHϖ I,d va the somorphsm from the theorem Categorfcaton. Followng ther work [CR], Chuang and Rouquer ntroduced the noton of an sl I -categorfcaton, also known as a categorcal sl I -acton. The followng s essentally [R, Defnton 5.32] (takng q = 1 and swtchng the roles of E and F ). Defnton 2.6. An sl I -categorfcaton s a Schuran category C together wth an endofunctor F, a rght adjont E to F (wth a specfed adjuncton), and natural transformatons x End(F ) and t End(F 2 ) satsfyng the axoms (SL1) (SL4) formulated below. For the frst axom, we let F be the subfunctor of F defned by the generalzed -egenspace of x,.e. F M = k 0 ker(x M ) k for each M C. (SL1) We have that F = I F,.e. F M = I F M for each M C. (SL2) For d 0 the endomorphsms x j := F d j xf j 1 and t k := F d k 1 tf k 1 of F d satsfy the relatons of the degenerate affne Hecke algebra AH d. (SL3) The functor F s somorphc to a rght adjont of E.

9 SUPER KAZHDAN-LUSZTIG 9 For the fnal axom, we let c : d EF and d : F E d be the unt and count of the gven adjuncton, respectvely. The endomorphsms x and t of F and F 2 nduce endomorphsms x and t of E and E 2 too: x : E ce EF E ExE EF E Ed E, (2.12) t : E 2 ce 2 EF E 2 EcF E 2 E 2 F 2 E 2 E 2 te 2 E 2 F 2 E 2 E 2 F de E 2 F E E2 d E 2. (2.13) (We remark that these satsfy slghtly dfferent relatons to the orgnal x and t: the sgns on the rght hand sde of the degenerate affne Hecke algebra relaton (AH) must be reversed.) Let E be the subfunctor of E defned by the generalzed -egenspace of x End(E). The axoms so far mply that E = I E and moreover F and E are badjont, so they are both exact and send projectves to projectves. (SL4) The endomorphsms f and e of [C] = C Z K 0 (C) nduced by F and E, respectvely, make [C] nto an ntegrable representaton of sl I. Moreover the classes of the ndecomposable projectve objects are weght vectors. The axom (SL4) has the followng equvalent dual formulaton. (SL4 ) The endomorphsms f and e of [C] = C Z G 0 (C) nduced by F and E, respectvely, make [C] nto an ntegrable representaton of sl I. Moreover the classes of the rreducble objects are weght vectors. The axom (SL1) mples that F d decomposes as I d F where F := F d F 1. Ths further shows that the acton of AH d factors through the completon ÂH I,d of the nverse system of cyclotomc quotents {AH ϖ I,d ϖ P + I }. Lettng 1 End(F d ) be the projecton onto F we can then use the somorphsm of completons gven by (2.9) (2.11) to convert the homomorphsm AH d ÂH I,d End(F d ) nto a homomorphsm QH I,d QH I,d End(F d ). Hence the defnton of an sl I -categorfcaton can be formulated equvalently usng the quver Hecke algebra QH I,d n place of the degenerate affne Hecke algebra AH d. In ths ncarnaton, C should be equpped wth adjont pars (F, E ) of endofunctors for all I (wth specfed adjunctons), together wth natural transformatons ξ End(F ) and τ End(F 2 ) where F := I F, satsfyng the axoms (SL1 ) (SL4 ). (SL1 ) The endomorphsm ξ s locally nlpotent,.e. F M = k 0 ker ξk M for each M C. (SL2 ) For d 0 the endomorphsms ξ j := F d j ξf j 1 and τ k := F d k 1 τf k 1 of F d plus the projectons 1 of F d onto ts summands F for each I d satsfy the relatons of the quver Hecke algebra QH I,d. (SL3 ) Each functor F s somorphc to a rght adjont of E. (SL4 ) Same as (SL4). In fact ths s just the frst of several alternate defntons of sl I -categorfcaton n the lterature. Notably n [R, Theorem 5.30] Rouquer proves that the data of an sl I -categorfcaton as above s equvalent to the data of an ntegrable 2- representaton of the Kac-Moody 2-category assocated to sl I n the sense of [R, Defnton 5.1]; see also [KL2] and [CaL] for closely related notons. (We pont out

10 10 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER also the recent artcle [B6], whch shows that the seemngly dfferent defntons n [R, KL2, CaL] actually yeld somorphc 2-categores.) Defnton 2.7. Gven two sl I -categorfcatons C and C, and denotng F, E, x, t for C nstead by F, E, x, t for clarty, a functor G : C C s strongly equvarant f there exsts an somorphsm of functors ζ : F G G F such that (E1) the natural transformaton E Gε E ζe η GE : G E E G s an somorphsm; (E2) ζ x G = Gx ζ n Hom(F G, G F ); (E3) ζf F ζ t G = Gt ζf F ζ n Hom(F 2 G, G F 2 ). If t happens that G s an equvalence of categores then the axom (E1) holds automatcally, and we call G a strongly equvarant equvalence. As usual, the defnton of strongly equvarant functor can be formulated n terms of quver Hecke algebras. In that settng, the somorphsm ζ s nduced by somorphsms ζ : F G G F for each, and the endomorphsms x and t n (E2) (E3) are replaced by ξ and τ Recollectons about hghest weght categores. We must also make a few remnders about (artnan) hghest weght categores n the sense of [CPS1]; see also [D, Appendx] whch s a good source for proofs of all the results stated n ths subsecton (although t only treats fnte weght posets). Defnton 2.8. A hghest weght category s a Schuran category C together wth an nterval-fnte poset (Λ, ) ndexng a complete set of parwse non-somorphc rreducble objects {L(λ) λ Λ} of C, such that the followng axom holds. (HW) Let P (λ) be a projectve cover of L(λ) n C. Defne the standard object (λ) to be the largest quotent of P (λ) such that [ (λ) : L(µ)] = δ λ,µ for µ λ. Then P (λ) has a fnte fltraton wth top secton somorphc to (λ) and other sectons of the form (µ) for µ > λ. It s well known that ths s equvalent to the axom (HW ) below; n other words C s hghest weght f and only f C op s hghest weght. (HW ) Let I(λ) be an njectve hull of L(λ) n C. Defne the costandard object (λ) to be the largest subobject of I(λ) such that [ (λ) : L(µ)] = δ λ,µ for µ λ. Then I(λ) has a fnte fltraton wth bottom secton somorphc to (λ) and other sectons of the form (µ) for µ > λ. If C s a hghest weght category, we wrte C and C for the exact subcategores consstng of objects wth a -flag and objects wth a -flag, respectvely. Ther complexfed Grothendeck groups wll be denoted [C ] and [C ]; they have dstngushed bases {[ (λ)] λ Λ} and {[ (λ)] λ Λ}, respectvely. The natural ncluson functors nduce lnear maps [C] [C ] [C] [C ]. When Λ s fnte all these maps are actually somorphsms so that all the Grothendeck groups are usually dentfed. There are a couple of well-known constructons whch wll be essental later on. Suppose that we are gven a decomposton Λ = Λ Λ such that Λ s an deal (lower set); equvalently Λ s a codeal (upper set). Let C be the Serre subcategory of C generated by {L(λ) λ Λ }. We wrte ι : C C for the natural ncluson, and ι! (resp. ι ) for the left (resp. rght) adjont to ι

11 SUPER KAZHDAN-LUSZTIG 11 whch sends an object M to ts largest quotent (resp. subobject) belongng to C. The category C s tself a hghest weght category wth weght poset Λ. Its rreducble, standard and costandard objects are the same as the ones n C ndexed by the set Λ. For λ Λ the projectve cover (resp. njectve hull) of L(λ) n C s ι! P (λ) (resp. ι I(λ)), whch wll n general be a proper quotent of P (λ) (resp. a proper subobject of I(λ)). For any M, N C we have that Ext n C(M, N) = Ext n C (M, N) (2.14) for all n 0. Ths s proved by a Grothendeck spectral sequence argument exactly lke n [D, A3.2 A3.3]. A key step s to check that the hgher rght derved functors R n ι vansh on objects from C ; dually the hgher left derved functors L n ι! vansh on objects from C. As well as the subcategory C, we can consder the Serre quotent category C := C/C ; we stress that accordng to the defnton of quotent category the objects of C are the same as the objects of C; morphsms M N n C are obtaned by takng a drect lmt of the morphsms M N/N n C over all subobjects M of M and N of N such that M/M and N belong to C. Let π : C C be the quotent functor, and fx a choce π! (resp. π ) of a left (resp. rght) adjont to π. Note that the unt (resp. count) of adjuncton gves a canoncal somorphsm d π π! (resp. π π d). The rreducble, standard, costandard, ndecomposable projectve and ndecomposable njectve objects n C are the same as the ones n C ndexed by weghts from Λ. Also for λ Λ we have that π! P (λ) = P (λ), π I(λ) = I(λ), π! (λ) = (λ) and π (λ) = (λ) n C; the frst two somorphsms here follow from propertes of adjunctons; see Lemma 2.9 below for justfcaton of the latter two. Fnally for M, N C such that ether M has a -flag wth sectons of the form (λ) ndexed by weghts λ Λ, or N has a -flag wth sectons (λ) for λ Λ, we have that for all n 0. Ths s [D, A3.13]. Ext n C(M, N) = Ext n C (M, N) (2.15) Lemma 2.9. Let π : C C be the quotent assocated to a codeal Λ Λ. For λ Λ there are canoncal somorphsms π! (λ) = (λ) and (λ) = π (λ) n C nduced by the count and unt of the fxed adjunctons. Proof. Let C λ (resp. C <λ ) be the hghest weght subcategory of C assocated to the deal {µ Λ µ λ} (resp. {µ Λ µ < λ}). Let C λ := C λ /C <λ. Ths category s a copy of Vec wth unque (up to somorphsm) rreducble object L(λ). Let π λ : C λ C λ be the quotent functor wth left adjont π λ!. The projectve cover of L(λ) n C λ s (λ), hence by propertes of adjunctons we have that (λ) = π λ! L(λ) n C. Smlarly, workng wth C n place of C, we defne subcategores C, λ and C,<λ. The quotent C, λ /C,<λ s another copy of Vec, hence s equvalent to C λ. Ths means that there s another quotent functor π,λ : C, λ C λ such that π λ = π,λ π, hence π λ! = π! π,λ!. Agan we have that (λ) = π,λ! L(λ) n C. Hence we get somorphsms n C: π! (λ) = π! (π!,λ L(λ)) = π! λ L(λ) = (λ).

12 12 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER It remans to observe that the count π! (λ) = π! (π (λ)) (λ) s an epmorphsm as (λ) has rreducble head L(λ) and λ Λ ; hence ths gves a canoncal choce for the somorphsm. The argument for s smlar Tensor product categorfcatons. Suppose we are gven a type (n, c) of level l. Recall the sl I -module n,c V I from (2.6). Defnton An sl I -tensor product categorfcaton of type (n, c) means a hghest weght category C together wth an endofunctor F of C, a rght adjont E to F (wth specfed adjuncton), and natural transformatons x End(F ) and t End(F 2 ) satsfyng axoms (SL1) (SL3) and (TP1) (TP3). (TP1) The weght poset Λ s the set Λ I;n,c from (2.7) partally ordered by λ µ f and only f λ = µ and λ λ k µ µ k for all k. (TP2) The exact functors F and E send objects wth -flags to objects wth -flags. (TP3) The lnear somorphsm [C ] n,c V I, [ (λ)] v λ ntertwnes the endomorphsms f and e of [C ] nduced by F and E wth the endomorphsms of n,c V I arsng from the actons of the Chevalley generators f and e of sl I. Snce [C] embeds nto [C ] = n,c V I, we deduce mmedately from the axoms that [C] s tself an ntegrable sl I -module,.e. the axom (SL4) holds automatcally. Thus tensor product categorfcatons are categorfcatons n the sense of Defnton 2.6 too. Remark Ths defnton s a slghtly modfed verson of [LW, Defnton 3.2], where a general noton of tensor product categorfcaton for arbtrary Kac- Moody algebras was ntroduced. The defnton n [LW] s expressed n terms of quver Hecke algebras rather than affne Hecke algebras; but of course the above defnton can be formulated equvalently wth the axoms (SL1 ) (SL3 ) replacng (SL1) (SL3); so ths s a superfcal dfference. More sgnfcantly, n our formulaton of the axoms (TP2) (TP3), we have ncorporated the explct monomal bass {v λ λ Λ} whch s only avalable n our specal mnuscule stuaton. The analogous axoms (TPC2) (TPC3) n [LW] are couched n terms of some commutng categorcal sl I -actons on the assocated graded category gr C := λ Λ C λ (where C λ s as n the proof of Lemma 2.9). The functors F j defnng these actons can be recovered by takng a sum of equvalences C λ C tj (λ) for all λ Λ such that λ j = 1 and λ (j+1) = 0, where t j (λ) s obtaned from λ by nterchangng λ j and λ (j+1). Such functors exst snce for a hghest weght category each C λ s equvalent to Vec. Any sl I -tensor product categorfcaton decomposes as C = ϖ P I C ϖ (2.16) where C ϖ s the Serre subcategory of C generated by the rreducble objects {L(λ) λ Λ, λ = ϖ}. In partcular, two rreducble objects L(λ) and L(µ) belong to the same block of C only f λ = µ ; see Theorem 2.22 for the converse. Gven another type (n, c ) of the same level, we say that (n, c) and (n, c ) are equvalent f one of the followng holds for each : ether c = c and n = n ;

13 SUPER KAZHDAN-LUSZTIG 13 or I s fnte, c c and n = I + n. Observe n that case that the posets of 01-matrces Λ I;n,c and Λ I;n,c are smply equal, and there s an sl I-module somorphsm n,c V I n,c V I, v λ v λ. We can now state the frst man result of the artcle. Theorem For any nterval I Z and type (n, c), there exsts an sl I -tensor product categorfcaton C of type (n, c). Moreover C s unque n the sense that f C s another tensor product categorfcaton of an equvalent type (n, c ) then there s a strongly equvarant equvalence G : C C wth GL(λ) = L (λ) for each weght λ. In the case that I s fnte, Theorem 2.12 s a specal case of the man result of [LW]; see 2.7 for some further dscusson of that. For nfnte ntervals, Theorem 2.12 s new and wll be proved later n the artcle. Specfcally, we wll establsh exstence for I = Z n 3.2, then exstence for the other nfnte but bounded above or below ntervals follows by the truncaton argument explaned n 2.8. The unqueness wll be establshed n 4.4. Corollary Any sl I -tensor product categorfcaton C admts a dualty such that F = F, E = E and L(λ) = L(λ) for each weght λ. Smlarly ts category of projectves has a dualty # such that F # = # F, E # = # E and P (λ) = P (λ) # for each λ. Proof. Usng the homologcal crtera for - and -flags, one checks that the axoms (TP2) (TP3) are equvalent to the axoms (TP2 ) (TP3 ) obtaned from them by replacng all occurrences of wth. In other words C s a tensor product categorfcaton f and only f C op s one; when I s fnte ths asserton s [LW, Proposton 3.9]. Now apply the unqueness from Theorem 2.12 wth C := C op to get. To obtan the dualty # on projectves, one can use (2.1) to reduce to the problem of defnng a dualty # on the subcategory of Fun f (pc, Vec op ) consstng of all exact functors; there one sets Hom C (P, ) # := Hom C (P, ) (where the fnal s the dualty on Vec). Transportng through the Yoneda equvalence ths yelds a dualty # on pc such that Hom C (P #, M) = Hom C (P, M ) (2.17) for all M C. It s clear from (2.17) that P (λ) # = P (λ), whle the fact that # commutes wth F and E follows by adjuncton as commutes wth E and F. (Alternatvely ths defnton can be understood va (2.3) n terms of the algebra A: t corresponds to the composton N : pmod-a pmod-a where N s the Nakayama functor Hom A (, A) : pmod-a mod-a, and pmod-a and mod-a denote the categores of projectve and njectve A-modules, respectvely.) 2.7. Revew of the proof of Theorem 2.12 for fnte ntervals. In ths subsecton, we assume that I s fnte and recall for future reference some of the key deas behnd the proof of Theorem 2.12 from [LW]. Suppose we are gven a type (n, c). To avod trvaltes we assume that n I + for each. There are two general approaches to the constructon of the tensor product categorfcaton C n Theorem Frst t can be realzed n terms of certan blocks of the parabolc category O assocated to the general lnear Le algebra; see [LW, Defnton 3.13].

14 14 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER Alternatvely, C can be constructed usng the tensor product algebras of [W4]; see [LW, Theorem 3.12]. Turnng our attenton to unqueness, let C be some gven sl I -tensor product categorfcaton wth weght poset Λ = Λ I,n,c. Let A be the algebra A := Hom C (P (λ), P (µ)) (2.18) λ,µ Λ from (2.2), and H : C mod-a be the canoncal equvalence of categores from (2.3). There s a formal way to transport the categorcal sl I -acton from C to mod-a n such a way that H : C mod-a becomes a strongly equvarant equvalence. The approprate functor F : mod-a mod-a s the functor defned by tensorng over A wth the (A, A)-bmodule B := Hom C (P (λ), F P (µ)). (2.19) λ,µ Λ The natural transformatons x End(F ) and t End(F 2 ) come from bmodule endomorphsms x : B B and t : B A B B A B defned as follows: let x : B B be defned on the summand Hom C (P (λ), F P (µ)) of B by composng wth x P (µ) : F P (µ) F P (µ); let t : B A B B A B be nduced smlarly by t P (µ) : F 2 P (µ) F 2 P (µ) usng also the followng canoncal somorphsm B A B = Hom C (P (λ), F 2 P (µ)). λ,µ Λ Then we may take E : mod-a mod-a to be the canoncal rght adjont to F gven by the functor λ Λ Hom A(1 λ B, ). In ths way we have made explct the categorcal sl I -acton on mod-a. The strategy for the proof of unqueness s as follows. Suppose that we are gven another sl I -categorfcaton C of an equvalent type (n, c ). We repeat all of the above, defnng ts assocated basc algebra A := Hom C (P (λ), P (µ)), (2.20) and an (A, A )-bmodule λ,µ Λ B := λ,µ Λ Hom C (P (λ), F P (µ)) (2.21) together wth endomorphsms x : B B and t : B A B leadng to a categorcal sl I -acton on mod-a too, such that the equvalence H : C mod-a s strongly equvarant. Then the pont s to construct an algebra somorphsm A = A, nducng an somorphsm of categores mod-a mod-a. To see that ths somorphsm of categores s strongly equvarant, we must also defne an somorphsm B = B that ntertwnes the actons of A, x and t wth A, x and t. Composng the somorphsm mod-a mod-a on one sde wth H and wth the canoncal adjont equvalence to H on the other, we obtan the desred strongly equvarant equvalence G : C C from the statement of Theorem Let us begn. Recall that κ = κ I;n,c s the 01-matrx ndexng the bass vector of maxmal weght n n,c V I. The rreducble object L(κ) s the only rreducble

15 SUPER KAZHDAN-LUSZTIG 15 n ts block,.e. L(κ) = (κ) = (κ) = P (κ) = I(κ). Snce the functor F has both a left and rght adjont t sends prnjectves to prnjectves, hence the object T = d 0 T d := d 0 F d L(κ) C (2.22) s prnjectve. The modules T d and T d for d d belong to dfferent sums of the blocks from (2.16), hence we have that Hom C (T d, T d ) = 0 for d d. We say that M C s homogeneous of degree d f t belongs to the same sum of blocks as T d ; then we have that Hom C (T, M) = Hom C (T d, M). Note further that T d = 0 for d 0. Let H = H d := AH κ I,d. (2.23) d 0 d 0 The followng theorem s at the heart of everythng; see [LW, Proposton 3.2] for the frst asserton, [LW, Theorem 5.1] for the second, and the proof of [LW, Theorem 6.1] for the fnal one; n the specal case that C s the tensor product categorfcaton arsng from parabolc category O from [LW, Defnton 3.13] the results here go back to [BK1]. Theorem 2.14 ([LW]). The acton of AH d on T d nduces a canoncal somorphsm between H d and End C (T d ); hence H = End C (T ). Moreover the exact functor U := Hom C (T, ) : C mod-h (2.24) s fully fathful on projectves. Fnally for each weght λ Λ the H-module Y (λ) := UP (λ) (2.25) s ndependent (up to somorphsm) of the partcular choce of C. Remark The proof of [LW, Theorem 5.1] establshes a slghtly stronger result: the map U : Hom C (M, P ) Hom H (UM, UP ) s an somorphsm for any M, P C wth P projectve. Thus the functor U has smlar propertes to Soergel s combnatoral functor V from [S]. The modules Y (λ) may be called Young modules by analogy wth the modular representaton theory of symmetrc groups. The second asserton of Theorem 2.14 s a verson of the double centralzer property, whch has already appeared n numerous related contexts n representaton theory; see e.g. [MS, Example 2.7] where several are lsted. It mples that the functor U defnes an algebra somorphsm A = Hom H (Y (λ), Y (µ)). (2.26) λ,µ Λ Smlarly for the prmed category we get that A = Hom H (Y (λ), Y (µ)) (2.27) λ,µ Λ where Y (λ) := U P (λ) for U defned analogously to U. Then, applyng the fnal asserton of Theorem 2.14, we choose H-module somorphsms Y (λ) = Y (λ) for each λ. These choces nduce the desred algebra somorphsm A = A.

16 16 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER It remans to construct the bmodule somorphsm B = B. Ths needs just a lttle more preparaton. The category mod-h s also equpped wth a categorcal sl I -acton. The endofunctors F : mod-h mod-h, E : mod-h mod-h (2.28) for ths are the nducton and restrcton functors assocated to the homomorphsms H d H d+1 nduced by the natural nclusons AH d AH d+1 for all d 0; so for a rght H d -module M we have that F M := nd H d+1 H d M = M Hd H d+1 and EM := res H d H d 1 M. The canoncal adjuncton makes (F, E) nto an adjont par. Left multplcaton by x d+1 defnes an (H d, H d+1 )-bmodule endomorphsm of H d+1, from whch we obtan the natural transformaton x End(F ). Also, by transtvty of nducton, F 2 s somorphc to the functor sendng a rght H d - module M to M Hd H d+2 ; then left multplcaton by t d+1 defnes an (H d, H d+2 )- bmodule endomorphsm of H d+2 nducng t End(F 2 ). Ths gves us the data of an sl I -categorfcaton n the sense of Defnton 2.6. The fact that the axoms (SL1) (SL4) hold goes back at least to [CR, Remark 7.13]; see also [K, Corollary 7.7.5] for the proof that F s somorphc to a rght adjont of E. The followng lemma was noted already n the frst paragraph of [LW, 5.1]; the alternatve proof gven below s a bt more explct. Lemma The quotent functor U : C mod-h s strongly equvarant. Proof. Frst we construct the requred natural transformaton ζ : F U U F. Take M C that s homogeneous of degree d. We need to produce a natural H d+1 -module homomorphsm ζ M : Hom C (T d, M) Hd H d+1 Hom C (T d+1, F M). The functor F defnes a natural H d -module homomorphsm from Hom C (T d, M) to the restrcton of the H d+1 -module Hom C (T d+1, F M). Then we use the adjuncton of nducton and restrcton to convert ths nto the desred homomorphsm ζ M. Next we show that ζ s an somorphsm of functors. It s certanly an somorphsm on T as for that t reduces to an dentty map. Hence t s an somorphsm on any drect sum of summands of T. By [LW, Lemma 5.3], any projectve object P C fts nto an exact sequence 0 P J K such that J and K are drect sums of summands of T. Note further that both F U and U F are exact functors. Hence when we apply ζ to our exact sequence we obtan a commutng dagram wth exact rows: 0 F U(P ) F U(J) F U(K) 0 U F (P ) U F (J) U F (K). We know already that the rght hand vertcal maps are somorphsms, hence so too s the frst one. Now we have proved that ζ defnes an somorphsm on every projectve object. For an arbtrary object M we pck a projectve resoluton Q P M 0 and make another argument wth the Fve Lemma. It remans to check the axoms (E1) (E3) from Defnton 2.7. For (E1), we observe on some homogeneous M C of degree (d + 1) that the natural transformaton EUε EζE ηue defnes the H d -module homomorphsm Hom C (T d, EM) Hom C (F T d, M)

17 SUPER KAZHDAN-LUSZTIG 17 defned by the gven adjuncton between F and E. Hence t s an somorphsm. For (E2), assume that M C s homogeneous of degree d. Then t suffces to show that the followng dagram commutes: Hom C (T d, M) Hd H d+1 ζ M (xu) M HomC (T d, M) Hd H d+1 ζ M Hom C (T d+1, F M) (Ux) M HomC (T d+1, F M). Take θ Hom C (T d, M) and h H d+1. Gong south then east, θ h maps to the homomorphsm T d+1 F M, v x M ((F θ)(hv)), whle gong east then south produces the homomorphsm v (F θ)(x d+1 hv). These are equal by the naturalty of x : F F wth respect to the homomorphsm θ : T d M. The proof of (E3) s smlar. Applyng Lemma 2.16, we deduce that the functor U defnes (A, A)-bmodule somorphsms B = Hom H (Y (λ), F Y (µ)), (2.29) λ,µ Λ B A B = λ,µ Λ Hom H (Y (λ), F 2 Y (µ)). (2.30) Under these somorphsms, x : B B and t : B A B B A B correspond to the endomorphsms of the bmodules on the rght nduced by all of the homomorphsms x Y (µ) : F Y (µ) F Y (µ) and t Y (µ) : F 2 Y (µ) F 2 Y (µ), respectvely. Smlarly B = Hom H (Y (λ), F Y (µ)), (2.31) λ,µ Λ B A B = λ,µ Λ Hom H (Y (λ), F 2 Y (µ)). (2.32) Then the H-module somorphsms Y (λ) = Y (λ) chosen earler nduce the desred somorphsm B = B. It s mmedate that t ntertwnes the actons of A, x and t wth A, x and t. Ths completes our sketch of the proof of unqueness n Theorem 2.12 for fnte ntervals Truncaton. In ths subsecton we ntroduce our key tool for provng results about tensor product categorfcatons when I s nfnte. Throughout we fx a type (n, c) and any nterval I, and set Λ := Λ I;n,c. Gven a subnterval J I, there s an obvous embeddng sl J sl I. Let Λ J be the subposet of Λ consstng of all 01-matrces λ such that λ j = c whenever j / J +. Ths s order-somorphc to the poset Λ J;n,c va the map sendng λ = (λ j ) 1 l,j I+ Λ J to ts submatrx λ J := (λ j ) 1 l,j J+ Λ J;n,c. In turn, the sl J -module n,c V J can be dentfed wth the sl J -submodule of n,c V I spanned by {v λ λ Λ J }. We then have that n,c V I = n,c J V J, takng the unon just over the fnte subntervals J I. We are gong to develop a categorcal analog of ths decomposton. Lemma For λ, µ Λ, the followng are equvalent:

18 18 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER () λ µ; () for all h I and 1 k l we have that wth equalty when k = l; () for all h I and 1 k l we have that wth equalty when k = l. k =1 j h λ j c ( 1) k =1 j>h λ j c ( 1) c c k =1 j h µ j c ( 1) k =1 j>h µ j c ( 1) Proof. The equvalence of () and () follows from Lemma 2.1. The equvalence of () and () s obvous. Agan let J I be any subnterval. Let Λ J denote the set of all λ Λ whch satsfy the condtons k c ( 1) 0 for all h < mn(j) and 1 k l, =1 j h λ j c (2.33) k c ( 1) 0 for all h > max(j) and 1 k l. =1 j>h λ j c Also let Λ <J denote the set of all λ Λ J such that at least one of the above nequaltes s strct. Lemma 2.17 mples that both Λ J and Λ <J are deals n the poset Λ. Moreover Λ J = Λ J \ Λ <J. Suppose next that we are gven an sl I -tensor product categorfcaton C of type (n, c). Let C J (resp. C <J ) be the hghest weght subcategory of C assocated to the deal Λ J (resp. Λ <J ). Let C J := C J /C <J. We denote the quotent functor by π J : C J C J. Lemma For j J the functors F j and E j preserve the subcategores C J and C <J of C. Proof. We just explan for C J ; the same argument works for C <J. Take any λ Λ J. We need to show that F j L(λ) and E j L(λ) both belong to C J. Snce L(λ) s a quotent of (λ), ths follows f we can show that F j (λ) and E j (λ) belong to C J. These objects have fltratons wth sectons of the form (µ) for weghts µ obtaned from λ by applyng the transposton t j to one of ts rows. The ntegers on the left hand sde of the nequaltes (2.33) are the same for each of these µ as they are for λ, so that each µ arsng s an element of Λ J and (µ) does ndeed belong to C J. Hence for j J the functors F j and E j nduce a well-defned badjont par of endofunctors of C J. Let F J := j J F j and E J := j J E j. The natural transformatons x and s restrct to endomorphsms of F J and FJ 2, respectvely, such that the assocated endomorphsms x j and t k of FJ d satsfy the degenerate affne Hecke algebra relatons as n (SL2). The axoms (TP1) (TP3) for C mply the analogous statements for C J. Thus we have proved: c, c,

19 SUPER KAZHDAN-LUSZTIG 19 Theorem The subquotent C J of C equpped wth the endofunctors F J and E J s an sl J -tensor product categorfcaton of type (n, c). We record one more techncal lemma for later use. Lemma Suppose that I s nfnte and J I s a fnte subnterval wth J + 2 max(n). Let κ Λ be the unque weght such that κ J = κ J;n,c Λ J;n,c. Then L(κ) s the unque ndecomposable object n ts block; n partcular t s prnjectve n C. Proof. In vew of (2.16) t suffces to show for λ Λ that λ = κ λ = κ. We proceed by nducton on l, the case l = 0 beng trval. For the nducton step assume frst that c = 0 for some. Amongst all the wth c = 0 choose one for whch n s mnmal. Thus n n j for all j wth c j = 0, and n 2 max(n) n j J + n j for all j wth c j = 1. Lettng s := mn(j + ) 1, t follows that the columns s + 1,..., s + n of the 01-matrx κ have all entres equal to 1. Snce λ = κ the number of entres 1 n each column of λ s the same as n κ. Hence all n of the entres 1 n the th row of λ appear n columns s + 1,..., s + n. Thus the th row of λ s the same as the th row of κ. Then we remove ths row and proceed by nducton. Ths just leaves us wth the case that c = 1 for all. Choose so that n s mnmal and let s := max(j + ) n. Then columns s + 1,..., s + n of κ, hence also of λ, have all entres equal to 0. So the th row of λ s the same as n κ, and then we can nduct as before Decomposton numbers and blocks. For any I and (n, c), let C be an sl I -tensor product categorfcaton of type (n, c). In the fnte case, Theorem 2.12 shows that C s equvalent to some blocks of parabolc category O for the general lnear Le algebra, hence we can explot the extensve lterature about parabolc category O to deduce results about C. In the nfnte case, many questons about C can be answered by pckng a suffcently large fnte subnterval J I, then passng to the subquotent C J and nvokng Theorem For example, the followng theorem shows that decomposton numbers n C can be computed by reducng to the Kazhdan-Lusztg conjecture (whch descrbes the decomposton numbers n parabolc category O); see 5.9 for more about the explct combnatorcs here. We wll appeal to ths observaton n the next secton to prove the super Kazhdan-Lusztg conjecture for gl n m (C). Theorem Gven λ, µ Λ, choose a fnte subnterval J I such that λ and µ both belong to Λ J. Then the composton multplcty [ (λ) : L(µ)] n C concdes wth the multplcty [ (λ J ) : L(µ J )] computed n C J. Hence, recallng that C J s equvalent to a sum of ntegral blocks of parabolc category O for the general lnear Le algebra, these multplctes can be computed va the Kazhdan- Lusztg conjecture. Proof. Ths s mmedate from the exactness of the quotent functor π J. As another llustraton of the truncaton technque, we classfy the blocks of C. Theorem For λ, µ Λ, the rreducble objects L(λ) and L(µ) le n the same block of C f and only f λ = µ n the weght lattce P I.

20 20 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER Proof. When I s fnte, the theorem has been proved already n [B4] (workng n the parabolc category O settng). Now suppose that I s nfnte. We observed already from (2.16) that λ and µ le n the same block only f λ = µ. Conversely suppose that λ = µ. Pck a fnte nterval J I such that λ, µ Λ J. Then λ J = µ J n P J, so by the fnte result there exsts a sequence of weghts λ = λ 0,..., λ n = µ n Λ J such that one of [ (λ ) : L(λ 1 )] or [ (λ 1 ) : L(λ )] s non-zero for each = 1,..., n. Snce these composton multplctes are the same n C or C J ths does the job Classfcaton of prnjectves. Let C be as n the prevous subsecton. To avod trvaltes assume moreover that Λ := Λ I;n,c s non-empty. The goal n ths subsecton s to classfy the ndecomposable prnjectve objects n C. For fnte I, ths s a generalzaton of an old result of Irvng [I] whch was establshed already n the context of parabolc category O n [MS, Theorem 5.1] or [BK1, Theorem 4.8]. The formulaton for nfnte I gven here s new; we prove t by usng truncaton to reduce to the fnte case. We start by notng that there s a crystal graph structure on Λ. Ths s a certan I-colored drected graph wth vertex set Λ, such that there s at most one edge of each color enterng and one edge of each color leavng any gven vertex. To determne the edges of color ncdent wth vertex λ one proceeds as follows. Frst label rows of the matrx λ by the sgn f the th and ( + 1)th entres of the row are 1 0, or by + f these entres are 0 1; leave all the other rows unlabeled. Then reduce the labels by repeatedly erasng + -pars of labels whenever the +-row s above the -row and all the rows n between are unlabeled. If at the end of ths process a -row (resp. a +-row) remans, then there s an edge λ µ (resp. λ µ) n the crystal graph, where µ s obtaned from λ by swtchng the th and ( + 1)th entres of the lowest -row (resp. the hghest +-row). Lemma For λ Λ and I, we have that F L(λ) = 0 (resp. E L(λ) = 0) unless there s an edge λ µ (resp. λ µ) n the crystal graph, n whch case F L(λ) (resp. E L(λ)) s ndecomposable wth rreducble head and socle somorphc to L(µ). Proof. Ths s already known for fnte I; see [LW, Theorem 7.2] for the most recent but also most conceptual proof (actually the arguments of [L] are suffcent here snce C s a hghest weght category). In the nfnte case we pck J I contanng such that λ and all the weghts µ ndexng the composton factors of E L(λ) and F L(λ) le n Λ J, and then pass to the subquotent C J. To formulate the man result, we need slghtly dfferent notaton accordng to whether I s fnte or nfnte: - In the fnte case, we let Λ be the vertex set of the connected component of the crystal graph contanng κ := κ I;n,c. Let T C denote the object from (2.22). - In the nfnte case, we fx fnte subntervals I 1 I 2 I such that I = r 1 I r, I max(n), and I r+1 = I r + 1 for each r. Let Λ r := Λ Ir Λ and C r := C Ir. Let κ r be the element of Λ r correspondng to κ Ir;n,c and Λ r be the vertex set of the connected subgraph of the crystal