Representations of the general linear Lie superalgebra in the BGG category O

Transcription

1 Representatons of the general lnear Le superalgebra n the BGG category O Jonathan Brundan Abstract Ths s a survey of some recent developments n the hghest weght repesentaton theory of the general lnear Le superalgebra gl n m (C). The man focus s on the analog of the Kazhdan-Lusztg conjecture as formulated by the author n 2002, whch was fnally proved n 2011 by Cheng, Lam and Wang. Recently another proof has been obtaned by the author jont wth Losev and Webster, by a method whch leads moreover to the constructon of a Koszul-graded lft of category O for ths Le superalgebra. Keywords: General lnear Le superalgebra, category O. Mathematcs Subject Classfcaton (2010): 17B10, 17B37. 1 Introducton The representaton theory of the general lnear Le superalgebra (as well as the other classcal famles) was frst nvestgated serously by Vctor Kac [30, 31] around Kac classfed the fnte dmensonal rreducble representatons and proved character formulae for the typcal ones. Then n the 1980s work of Sergeev [44] and Berele-Regev [5] exploted the superalgebra analog of Schur-Weyl dualty to work out character formulae for the rreducble polynomal representatons. It took another decade before Serganova [43] explaned how the characters of arbtrary fnte dmensonal rreducble representatons could be approached. Subsequent work of the author and others [10, 47, 17, 16] means that by now the category of fnte dmensonal representatons s well understood (although there reman nterestng questons regardng the tensor structure). One can also ask about the representaton theory of the general lnear Le superalgebra n the analog of the Bernsten-Gelfand-Gelfand category O from [7]. Ths s the natural home for the rreducble hghest weght representatons. The classcal theory of category O for a semsmple Le algebra, as n for example Humphreys book [27] whch nspred ths artcle, sts at the heart of modern geometrc representaton theory. Its combnatorcs s controlled by the underlyng Weyl group, and many beautful results are deduced from the geometry of the Jonathan Brundan Department of Mathematcs, Unversty of Oregon, Eugene, OR 07403, e-mal: brundan@uoregon.edu Research supported n part by NSF grant no. DMS

2 2 Jonathan Brundan assocated flag varety va the Belnson-Bernsten localzaton theorem [3]. There stll seems to be no satsfactory substtute for ths geometrc part of the story for gl n m (C) but at least the combnatorcs has now been worked out: n [10] t was proposed that the combnatorcs of the Weyl group (specfcally the Kazhdan-Lusztg polynomals arsng from the assocated Iwahor-Hecke algebra) should smply be replaced by the combnatorcs of a canoncal bass n a certan U q sl -module V n W m. Ths dea led n partcular to the formulaton of a superalgebra analog of the Kazhdan-Lusztg conjecture. The super Kazhdan-Lusztg conjecture s now a theorem. In fact there are two proofs, frst by Cheng, Lam and Wang [18], then more recently n jont work of the author wth Losev and Webster [15]. In some sense both proofs nvolve a reducton to the ordnary Kazhdan-Lusztg conjecture for the general lnear Le algebra. Cheng, Lam and Wang go va some nfnte dmensonal lmtng versons of the underlyng Le (super)algebras usng the technque of super dualty, whch orgnated n [22, 17]. On the other hand the proof n [15] nvolves passng from category O for gl n m (C) to some subquotents whch, thanks to results of Losev and Webster from [36], are equvalent to sums of blocks of parabolc category O for some other general lnear Le algebra. The approach of [15] allows also for the constructon of a graded lft of O whch s Koszul, n the sprt of the famous results of Belnson, Gnzburg and Soergel [4] n the classcal settng. The theory of categorfcaton developed by Rouquer [42] and others, and the dea of Schur-Weyl dualty for hgher levels from [14], both play a role n ths work. Ths artcle s an attempt to gve a bref overvew of these results. It mght serve as a useful startng pont for someone tryng to learn about the combnatorcs of category O for the general lnear Le superalgebra for the frst tme. We begn wth the defnton of O and the basc propertes of Verma supermodules and ther projectve covers. Then we formulate the super Kazhdan-Lusztg conjecture precsely and gve some examples, before fttng t nto the general framework of tensor product categorfcatons. Fnally we hghlght one of the man deas from [15] nvolvng a double centralzer property (an analog of Soergel s Struktursatz from [45]), and suggest a related queston whch we beleve should be nvestgated further. In an attempt to maxmze the readablty of the artcle, precse references to the lterature have been deferred to notes at the end of each secton. We pont out n concluson that there s also an attractve Kazhdan-Lusztg conjecture for the Le superalgebra q n (C) formulated n [11], whch remans qute untouched. One can also ponder Kazhdan-Lusztg combnatorcs for the other classcal famles of Le superalgebra. Dramatc progress n the case of osp n 2m (C) has been made recently n [2]; see also [25]. Acknowledgements. Specal thanks go to Catharna Stroppel for several dscussons whch nfluenced ths exposton. I also thank Geoff Mason, Ivan Penkov and Joe Wolf for provdng me the opportunty to wrte a survey artcle of ths nature. In fact I gave a talk on exactly ths topc at the West Coast Le Theory Semnar at Rversde n November 2002, when the super Kazhdan-Lusztg conjecture was newborn. 2 Super category O and ts blocks Fx n,m 0 and let g denote the general lnear Le superalgebra gl n m (C). As a vector ( ) superspace ths conssts of (n + m) (n + m) complex matrces wth Z/2-gradng defned so that the j-matrx unt e, j s even for 1, j n or n + 1, j n + m, and e, j s odd otherwse. It s a Le superalgebra va the supercommutator

3 Super category O 3 [x,y] := xy ( 1) xȳ yx for homogeneous x, y g of partes x, ȳ Z/2, respectvely. By a g-supermodule we mean a vector superspace M = M 0 M 1 equpped wth a graded lnear left acton of g, such that [x,y]v = x(yv) ( 1) xȳ y(xv) for all homogeneous x,y g and v M. For example we have the natural representaton U of g, whch s just the superspace of column vectors on standard bass u 1,...,u n+m, where ū = 0 for 1 n and ū = 1 for n + 1 n + m. We wrte g-smod for the category of all g-supermodules. A morphsm f : M N n ths category means a lnear map such that f (M ) N for Z/2 and f (xv) = x f (v) for x g,v M. Ths s obvously a C-lnear abelan category. It s also a supercategory, that s, t s equpped wth the addtonal data of an endofunctor Π : g-smod g-smod wth Π 2 = d. The functor Π here s the party swtchng functor, whch s defned on a supermodule M by declarng that ΠM s the same underlyng vector space as M but wth the opposte Z/2-gradng, vewed as a g-supermodule wth the new acton x v := ( 1) x xv. On a morphsm f : M N we take Π f : ΠM ΠN to be the same underlyng lnear map as f. Clearly Π 2 = d. Remark 2.1 Gven any C-lnear supercategory C, one can form the enrched category C. Ths s a category enrched n the monodal category of vector superspaces. It has the same objects as n C, and ts morphsms are defned from Hom C (M,N) := Hom C (M,N) 0 Hom C (M,N) 1 where Hom C (M,N) 0 := Hom C (M,N), Hom C (M,N) 1 := Hom C (M,ΠN). The composton law s obvous (but nvolves the somorphsm Π 2 = d whch s gven as part of the data of C ). Ths means one can talk about even and odd morphsms between objects of C. In the case of g-smod, an odd homomorphsm f : M N s a lnear map such that f (M ) N + 1 for Z/2 and f (xv) = ( 1) x x f (v) for homogeneous x g,v M. Let b be the standard Borel subalgebra consstng of all upper trangular matrces n g. It s the stablzer of the standard flag u 1 < u 1,u 2 < < u 1,...,u n+m n the natural representaton V. More generally a Borel subalgebra of g s the stablzer of an arbtrary homogeneous flag n V. Unlke n the purely even settng, t s not true that all Borel subalgebras are conjugate under the approprate acton of the general lnear supergroup G = GL n m. Ths leads to some combnatorally nterestng varants of the theory whch are also well understood, but our focus n ths artcle wll just be on the standard choce of Borel. Let t be the Cartan subalgebra of g consstng of dagonal matrces. Let δ 1,...,δ n+m be the bass for t such that δ pcks out the th dagonal entry of a dagonal matrx. Defne a non-degenerate symmetrc blnear form (?,?) on t by settng (δ,δ j ) := ( 1)ūδ, j. The root system of g s R := {δ δ j 1, j n + m, j}, whch decomposes nto even and odd roots R = R 0 R 1 so that δ δ j s of party ū +ū j. Let R + = R + 0 R+ 1 denote the postve roots assocated to the Borel subalgebra b,.e. δ δ j s postve f and only f < j. The domnance order on t s defned so that λ µ f λ µ s a sum of postve roots. Let ρ := δ 2 2δ 3 (n 1)δ n + (n 1)δ n+1 + (n 2)δ n (n m)δ n+m. One can check that 2ρ s congruent to the sum of the postve even roots mnus the sum of the postve odd roots modulo δ := δ δ n δ n+1 δ n+m.

4 4 Jonathan Brundan Let so be the full subcategory of g-smod consstng of all fntely generated g-supermodules whch are locally fnte dmensonal over b and satsfy M = λ t M λ, where for λ t we wrte M λ = M M λ, 0 λ, 1 for the λ-weght space of M wth respect to t defned n the standard way. Ths s an abelan subcategory of g-smod closed under Π. It s the analog for gl n m (C) of the Bernsten-Gelfand-Gelfand category O for a semsmple Le algebra. All of the famlar basc propertes from the purely even settng generalze rather easly to the super case. For example all supermodules n so have fnte length, there are enough projectves, and so on. An easy way to prove these statements s to compare so to the classcal BGG category O ev for the even part g 0 = gl n (C) gl m (C) of g. One can restrct any supermodule n so to g 0 to get a module n O ev; conversely for any M O ev we can vew t as a supermodule concentrated n a sngle party then nduce to get U(g) U(g 0 ) M O. Ths reles on the fact that U(g) s free of fnte rank as a U(g 0 )-module, thanks to the PBW theorem for Le superalgebras. Then the fact that so has enough projectves follows because O ev does, and nducton sends projectves to projectves as t s left adjont to an exact functor. In fact t s possble to elmnate the super n the supercategory so entrely by passng to a certan subcategory O. To explan ths let Ĉ be some set of representatves for the cosets of C modulo Z such that 0 Ĉ. Then defne p z+n := n Z/2 for each z Ĉ and n Z. Fnally for λ t let p(λ) := p (λ,δn+1 + +δ n+m ). Ths defnes a party functon p : t Z/2 wth the key property that p(λ + δ ) = p(λ) + ū. If M so then M decomposes as a drect sum of g-supermodules as M = M + M where M + := λ t M λ,p(λ), M := λ t M λ,p(λ)+ 1. Let O (resp. ΠO) be the full subcategory of so consstng of all supermodules M such that M = M + (resp. M = M ). Both are Serre subcategores of so, hence they are abelan, and the functor Π defnes an equvalence between O and ΠO. Moreover there are no non-zero odd homomorphsms between objects of O; equvalently there are no non-zero even homomorphsms between an object of O and an object of ΠO. Hence: Lemma 2.2 so = O ΠO. Remark 2.3 Let sô be the enrched category arsng from the supercategory so as n Remark 2.1. Lemma 2.2 mples that the natural ncluson functor O sô s fully fathful and essentally surjectve, hence t defnes an equvalence between O and sô. In partcular sô s tself abelan, although the explct constructon of kernels and cokernels of nhomogeneous morphsms n sô s a bt awkward. Henceforth we wll work just wth the category O rather than the supercategory so. Note n partcular that O contans the natural supermodule U and ts dual U, and t s closed under tensorng wth these objects. For each λ t we have the Verma supermodule M(λ) := U(g) U(b) C λ,p(λ) O, where C λ,p(λ) s a one-dmensonal b-supermodule of weght λ concentrated n party p(λ). The usual argument shows that M(λ) has a unque rreducble quotent, whch we denote by L(λ). The supermodules {L(λ) λ t } gve a complete set of parwse non-somorphc rreducbles n O. We say that λ t s domnant f

5 Super category O 5 { (λ,δ δ +1 ) Z 0 for = 1,...,n 1, (λ,δ δ +1 ) Z 0 for = n + 1,...,n + m 1. Then the supermodules {L(λ) for all domnant λ t } gve a complete set of parwse nonsomorphc fnte dmensonal rreducble g-supermodules (up to party swtch). Ths s an mmedate consequence of the followng elementary but mportant result. Theorem 2.4 (Kac) For λ t the rreducble supermodule L(λ) s fnte dmensonal f and only f λ s domnant. Proof. Let L ev (λ) be the rreducble hghest weght module for g 0 of hghest weght λ. Classcal theory tells us that L ev (λ) s fnte dmensonal f and only f λ s domnant. Snce L(λ) contans a hghest weght vector of weght λ, ts restrcton to g 0 has L ev(λ) as a composton factor, hence f L(λ) s fnte dmensonal then λ s domnant. Conversely, let p be the maxmal ( ) parabolc subalgebra of g consstng of block upper trangular matrces of the form. There s an obvous projecton p 0 g 0, allowng us to vew L ev(λ) as a p-supermodule concentrated n party p(λ). Then for any λ t we can form the Kac supermodule K(λ) := U(g) U(p) L ev (λ) O. Snce K(λ) s a quotent of M(λ), t has rreducble head L(λ). Moreover the PBW theorem mples that K(λ) s fnte dmensonal f and only f L ev (λ) s fnte dmensonal. Hence f λ s domnant we deduce that L(λ) s fnte dmensonal. The degree of atypcalty of λ t s defned to be the maxmal number of mutually orthogonal odd roots β R such that (λ +ρ,β) = 0. In partcular λ s typcal f (λ +ρ,β) for all β R + 1. For typcal λ t, Kac showed further that the Kac supermodules K(λ) defned n the proof of Theorem 2.4 are actually rreducble. Thus most questons about typcal rreducble supermodules n O reduce to the purely even case. For example usng the Weyl character formula one can deduce n ths way a smple formula for the character of an arbtrary typcal fnte dmensonal rreducble g-supermodule. It s not so easy to compute the characters of atypcal fnte dmensonal rreducble supermodules, but ths has turned out stll to be combnatorally qute tractable. We wll say more about the much harder problem of fndng characters of arbtrary (not necessarly typcal or fnte dmensonal) rreducble supermodules n O n the next secton; nevtably ths nvolves some Kazhdan-Lusztg polynomals. Let P(µ) be a projectve cover of L(µ) n O. We have the usual statement of BGG recprocty: each P(µ) has a Verma flag,.e. a fnte fltraton whose sectons are Verma supermodules, and the multplcty (P(µ) : M(λ)) of M(λ) as a secton of a Verma flag of P(µ) s gven by (P(µ) : M(λ)) = [M(λ) : L(µ)], where the rght hand sde denotes composton multplcty. Of course [M(λ) : L(µ)] s zero unless µ λ n the domnance orderng, whle [M(λ) : L(λ)] = 1. Thus O s a hghest weght category n the formal sense of Clne, Parshall and Scott, wth weght poset (t, ). The partal order on t beng used here s rather crude. It can be replaced wth a more ntellgent order, called the Bruhat order. To defne ths, gven λ t, let A(λ) := {α R + 0 (λ + ρ,α ) Z >0 }, B(λ) := {β R (λ + ρ,β) = 0}, + 1 where α denotes 2α/(α,α). Then ntroduce a relaton on t by declarng that µ λ f we ether have that µ = s α λ for some α A(λ) or we have that µ = λ β for some

6 6 Jonathan Brundan β B(λ); here, for α = δ δ j R + 0 and λ t, we wrte s α λ for s α (λ + ρ) ρ, where s α : t t s the reflecton transposng δ and δ j and fxng all other δ k. Fnally defne to be the transtve closure of the relaton,.e. we have that µ λ f there exsts r 0 and weghts ν 0,...,ν r t wth µ = ν 0 ν 1 µ r = λ. Lemma 2.5 If [M(λ) : L(µ)] 0 then µ λ n the Bruhat order. Proof. Ths s a consequence of the super analog of the Jantzen sum formula from [39, 10.3]; see also [26]. In more detal, the Jantzen fltraton on M(λ) s a certan exhaustve descendng fltraton M(λ) = M(λ) 0 M(λ) 1 M(λ) 2 such that M(λ) 0 /M(λ) 1 = L(λ), and the sum formula shows that k 1 chm(λ) k = α A(λ) chm(s α λ) + β B(λ) k 1 ( 1) k 1 chm(λ kβ). To deduce the lemma from ths, suppose that [M(λ) : L(µ)] 0. Then µ λ, so that λ µ s a sum of N smple roots δ δ +1 for some N 0. We proceed by nducton on N, the case N = 0 beng vacuous. If N > 0 then L(µ) s a composton factor of M(λ) 1 and the sum formula mples that L(µ) s a composton factor ether of M(s α λ) for some α A(λ) or that L(µ) s a composton factor of M(λ kβ) for some odd k 1 and β B(λ). It remans to apply the nducton hypothess and the defnton of. Let be the equvalence relaton on t generated by the Bruhat order. We refer to the -equvalence classes as lnkage classes. For a lnkage class ξ t /, let O ξ be the Serre subcategory of O generated by the rreducble supermodules {L(λ) λ ξ }. Then, as a purely formal consequence of Lemma 2.5, we get that the category O decomposes as O = O ξ. ξ t / In fact ths s the fnest possble such drect sum decomposton,.e. each O ξ s an ndecomposable subcategory of O. In other words, ths s precsely the decomposton of O nto blocks. An nterestng open problem here s to classfy the blocks O ξ up to equvalence. Let us descrbe the lnkage class ξ of λ t more explctly. Let k be the degree of atypcalty of λ and β 1,...,β k R + 1 be dstnct mutually orthogonal odd roots such that (λ + ρ,β ) = 0 for each = 1,...,k. Also let W λ be the ntegral Weyl group correspondng to λ, that s, the subgroup of GL(t ) generated by the reflectons s α for α R such that + 0 (λ + ρ,α) Z. Then ξ = { w (λ + n 1 β n k β k ) n1,...,n k Z,w W λ }, where w ν = w(ν + ρ) ρ as before. Note n partcular that all µ λ have the same degree of atypcalty k as λ. The followng useful result reduces many questons about O to the case of ntegral blocks, that s, blocks correspondng to lnkage classes of ntegral weghts belongng to the set t Z := Zδ 1 Zδ n+m. Theorem 2.6 (Cheng, Mazorchuk, Wang) Every block O ξ of O s equvalent to a tensor product of ntegral blocks of general lnear Le superalgebras of the same total rank as g. If λ s atypcal then the lnkage class ξ contanng λ s nfnte. Ths s a key dfference between the representaton theory of Le superalgebras and the classcal representaton theory

7 Super category O 7 of a semsmple Le algebra, n whch all blocks are fnte (bounded by the order of the Weyl group). It means that the hghest weght category O ξ cannot be vewed as a category of modules over a fnte dmensonal quas-heredtary algebra. Nevertheless one can stll consder the underlyng basc algebra A ξ := Hom g (P(λ),P(µ)) λ,µ ξ wth multplcaton comng from composton. Ths s a locally untal algebra, meanng that t s equpped wth the system of mutually orthogonal dempotents {1 λ λ ξ } such that A ξ = λ,µ ξ 1 µ A ξ 1 λ, where 1 λ denotes the dentty endomorphsm of P(λ). Wrtng mof-a ξ for the category of fnte dmensonal locally untal rght A ξ -modules,.e. modules M wth M = λ ξ M1 λ, the functor O ξ mof-a ξ, M λ ξ Hom g (P(λ), ) s an equvalence of categores. Note moreover that each rght deal 1 λ A ξ and each left deal A ξ 1 λ s fnte dmensonal; these are the ndecomposable projectves and the lnear duals of the ndecomposable njectves n mof-a ξ, respectvely. Remark 2.7 It s also natural to vew A ξ as a superalgebra concentrated n party 0. Then the block so ξ = O ξ ΠO ξ of the supercategory so assocated to the lnkage class ξ s equvalent to the category of fnte dmensonal locally untal rght A ξ -supermodules. Ths gves another pont of vew on Lemma 2.2. Example 2.8 Let us work out n detal the example of gl 1 1 (C). Ths s easy but nevertheless very mportant: often gl 1 1 (C) plays a role parallel to that of sl 2 (C) n the classcal theory. So now ρ = 0 and the only postve root s α = δ 1 δ 2 R. The Verma supermodules M(λ) + 1 are the same as the Kac supermodules K(λ) from the proof of Theorem 2.4; they are twodmensonal wth weghts λ and λ α. Moreover M(λ) s rreducble for typcal λ. If λ s atypcal then λ = cα for some c C, and the rreducble supermodule L(λ) comes from the one-dmensonal representaton g C, x c str x where str denotes supertrace. Fnally let us restrct attenton just to the prncpal block O 0 contanng the rreducble supermodules L() := L(α) for each Z. We have shown that M() := M(α) has length two wth composton factors L() and L( 1); hence by BGG recprocty the projectve ndecomposable supermodule P() := P(α) has a two-step Verma flag wth sectons M() and M( + 1). We deduce that the Loewy seres of P() looks lke P() = P 0 () > P 1 () > P 2 () > 0 wth P 0 ()/P 1 () = L(), P 1 ()/P 2 () = L( 1) L( + 1), P 2 () = L(). From ths one obtans the followng presentaton for the underlyng basc algebra A 0 : t s the path algebra of the quver e 1 f 1 e f e +1 f +1 e +2 f +2

8 8 Jonathan Brundan wth vertex set Z, modulo the relatons e f + f +1 e +1 = 0,e +1 e = f f +1 = 0 for all Z. We stress the smlarty between these and the relatons e f + f e = c,e 2 = f 2 = 0 n U(g) tself (where c = e 1,1 + e 2,2 z(g), e = e 1,2 and f = e 2,1 ). One should also observe at ths pont that these relatons are homogeneous, so that A 0 can be vewed as a postvely graded algebra, wth gradng comng from path length. In fact ths gradng makes A 0 nto a (locally untal) Koszul algebra. To conclude the secton, we offer one pece of justfcaton for focussng so much attenton on category O. The study of prmtve deals of unversal envelopng algebras of Le algebras, especally semsmple ones, has classcally proved to be very rch and nspred many mportant dscoveres. So t s natural to ask about the space of all prmtve deals PrmU(g) n our settng too. It turns out for gl n m (C) that all prmtve deals are automatcally homogeneous. In fact one just needs to consder annhlators of rreducble supermodules n O: Theorem 2.9 (Musson) PrmU(g) = {Ann U(g) L(λ) λ t }. Ths s the analog of a famous theorem of Duflo n the context of semsmple Le algebras. Letzter showed subsequently that there s a bjecton PrmU(g 0 ) PrmU(g), Ann U(g 0 ) L ev (λ) Ann U(g) L(λ). Combned wth classcal results of Joseph, ths means that the fbers of the map t PrmU(g), λ Ann U(g) L(λ) can be descrbed n terms of the Robnson-Schensted algorthm. Hence we get an explct descrpton of the set PrmU(g). Notes. For the basc facts about super category O for basc classcal Le superalgebras, see 8.2 of Musson s book [39]. Lemma 2.2 was ponted out orgnally n [10, 4-e]. The observaton that sô s abelan from Remark 2.3 s due to Cheng and Lam [17]; n fact these authors work entrely wth the equvalent category sô n place of our O. The classfcaton of fnte dmensonal rreducble supermodules from Theorem 2.4 s due to Kac [30]. The rreducblty of the typcal Kac supermodules was establshed soon after n [31]. Kac only consdered fnte dmensonal representatons at the tme but the same argument works n general. Composton multplctes of atypcal Kac supermodules were frst computed as a certan alternatng sum by Serganova n [43]. In fact, all Kac supermodules are multplcty-free, so that Serganova s formula smplfes to 0 or 1. Ths was proved n [10] by a surprsngly drect representaton theoretc argument, confrmng a conjecture from [29]; see also [41] for a combnatoral proof of the equvalence of the formulae for composton multplces n [43] and [10]. Another approach to the fnte dmensonal representatons va super dualty was developed n [22, 17], showng n partcular that the Kazhdan-Lusztg polynomals appearng n [43, 10] are the same as certan Kazhdan-Lusztg polynomals for Grassmannans as computed orgnally by Lascoux and Schützenberger [33]. Subsequently Su and Zhang [47] were able to use the explct formula for these Kazhdan-Lusztg polynomals to extract some closed character and dmenson formulae for the fnte dmensonal rreducbles. There s also an elegant dagrammatc descrpton of the basc algebra that s Morta equvalent to the subcategory F of O consstng of all ts fnte dmensonal supermodules n terms of Khovanov s arc algebra; see [16]. The analog of BGG recprocty for gl n m (C) as stated here was frst establshed by Zou [48]; see also [12]. For the classfcaton of blocks of O and proof of Theorem 2.6, see [19, Theorems ]. A related problem s to determne when two rreducble hghest weght

9 Super category O 9 supermodules have the same central character. Ths s solved va the explct descrpton of the center Z(g) of U(g) n terms the Harsh-Chandra homomorphsm and supersymmetrc polynomals, whch s due to Kac; see [39, 13.1] or [21, 2.2] for recent expostons. Lemma 2.5 s slghtly more subtle and cannot be deduced just from central character consderatons. Musson has recently proved a refnement of the sum formula recorded n the proof of Lemma 2.5, n whch the rght hand sde s rewrtten as a fnte sum of characters of hghest weght modules; detals wll appear n [40]. The results of Musson, Letzter and Joseph classfyng prmtve deals of U(g) are n [38, 35, 28]; see also [39, Ch. 15]. The recent preprnt [24] makes some further progress towards determnng all nclusons between prmtve deals. 3 Kazhdan-Lusztg combnatorcs and categorfcaton In ths secton we restrct attenton just to the hghest weght subcategory O Z of O consstng of supermodules M such that M = λ t Z M λ,p(λ). In other words we only consder ntegral blocks. Ths s justfed by Theorem 2.6. The goal s to understand the composton multplctes [M(λ) : L(µ)] of the Verma supermodules n O Z. It wll be convenent as we explan ths to represent λ t Z nstead by the n m-tuple (λ 1,...,λ n λ n+1,...,λ n+m ) of ntegers defned from λ := (λ +ρ,δ ). Let P denote the free abelan group Z Zε and Q P be the subgroup generated by the smple roots α := ε ε +1. Thus Q s the root lattce of the Le algebra sl. Let be the usual domnance orderng on P defned by ξ ϖ f ϖ ξ s a sum of smple roots. For λ = (λ 1,...,λ n λ n+1,...,λ n+m ) t Z we let λ := ε λ1 + + ε λn ε λn+1 ε λn+m P. Then t s clear that two weghts λ,µ t Z are lnked f and only f λ = µ,.e. the fbers of the map t Z P,λ λ are exactly the lnkage classes. The Bruhat order on t Z can also be nterpreted n these terms: let { ελ for 1 n, λ := ε λ for n + 1 n + m, so that λ = λ λ n+m. Then one can show that λ µ n the Bruhat order f and only f λ λ µ µ n the domnance orderng on P for all = 1,...,n + m, wth equalty when = n + m. Let V be the natural sl -module on bass {v Z} and W be ts dual on bass {w Z}. The Chevalley generators { f,e Z} of sl act 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 tensor space V n W m has the obvous bass of monomals v λ := v λ1 v λn w λn+1 w λn+m ndexed by n m-tuples λ = (λ 1,...,λ n λ n+1,...,λ n+m ) of ntegers. In other words the monomal bass of V n W m s parametrzed by the set t Z of ntegral weghts for g = gl n m(c).

10 10 Jonathan Brundan Ths prompts us to brng category O back nto the pcture. Let OZ be the exact subcategory of O Z consstng of all supermodules wth a Verma flag, and denote ts complexfed Grothendeck group by K(OZ ). Thus K(O Z ) s the complex vector space on bass {[M(λ)] λ t Z }. Henceforth we dentfy K(O Z ) V n W m, [M(λ)] v λ. Snce projectves have Verma flags we have that P(µ) O Z ; let b µ V n W m be the correspondng tensor under the above dentfcaton,.e. By BGG recprocty we have that [P(µ)] b µ. b µ = [M(λ) : L(µ)]v λ. λ t Z Now the punchlne s that the vectors {b µ µ t Z } turn out to concde wth Lusztg s canoncal bass for the tensor space V n W m. The defnton of the latter goes va some quantum algebra ntroduced n the next few paragraphs. Let U q sl be the quantzed envelopng algebra assocated to sl. Ths s the Q(q)-algebra on generators { ḟ,ė, k, k 1 Z} 1 subject to well-known relatons. We vew U q sl as a Hopf algebra wth comultplcaton ( ḟ ) = 1 ḟ + ḟ k, (ė ) = k 1 ė + ė 1, ( k ) = k k. We have the natural U q sl -module V on bass { v Z} and ts dual Ẇ on bass {ẇ Z}. The Chevalley generators ḟ and ė of U q sl act on these bass vectors by exactly the same formulae as at q = 1, and also k v j = q δ, j δ +1, j v j and k ẇ j = q δ +1, j δ, j ẇ j. There s also an R-matrx gvng some dstngushed ntertwners V V V V and Ẇ Ẇ Ẇ Ẇ, from whch we produce the followng U q sl -module homomorphsms: v j v + q 1 v v j f < j, ċ : V V V V, v v j (q + q 1 ) v j v f = j, v j v + q v v j f > j; ċ : Ẇ Ẇ Ẇ Ẇ, ẇ j ẇ + qẇ ẇ j f < j, ẇ ẇ j (q + q 1 )ẇ j ẇ f = j, ẇ j ẇ + q 1 ẇ ẇ j f > j. Then we form the tensor space V n Ẇ m, whch s a U q sl -module wth ts monomal bass { v λ λ t Z } defned just lke above. Let ċ k := 1 (k 1) ċ 1 n+m 1 k for k n, whch s a U q sl -module endomorphsm of V n Ẇ m. Next we must pass to a formal completon V n Ẇ m of our q-tensor space. Let I Z be a fnte subnterval and I + := I (I +1). Let V I and Ẇ I be the subspaces of V and Ẇ spanned by the bass vectors { v I + } and {ẇ I + }, respectvely. Then V I n ẆI m s a subspace of V n Ẇ m. For J I there s an obvous projecton π J : V I n ẆI m V J n ẆJ m mappng v λ to v λ f all the entres of the tuple λ le n J +, or to zero otherwse. Then we set 1 We follow the conventon of addng a dot to all q-analogs to dstngush them from ther classcal counterparts.

11 Super category O 11 V n Ẇ m := lm V n I ẆI m, takng the nverse lmt over all fnte subntervals I Z wth respect to the projectons π J just defned. The acton of U q sl and of each ċ k extend naturally to the completon. Lemma 3.1 There s a unque contnuous antlnear nvoluton such that ψ : V n Ẇ m V n Ẇ m ψ commutes wth the actons of ḟ and ė for all Z and wth the endomorphsms ċ k for all k n; each ψ( v λ ) s equal to v λ plus a (possbly nfnte) Z[q,q 1 ]-lnear combnaton of v µ for µ > λ n the Bruhat order. Proof. For each fnte subnterval I Z, let U q sl I be the subalgebra of U q sl generated by { ḟ,ė, k ±1 I}. A constructon of Lusztg [37, 27.3] nvolvng the quas-r-matrx Θ I for U q sl I gves an antlnear nvoluton ψ I : V I n ẆI m V I n ẆI m commutng wth the actons of ḟ and ė for I. Moreover for J I the nvolutons ψ I and ψ J are ntertwned by the projecton π J : V I n ẆI m V J n ẆJ m, as follows easly from the explct form of the quas-r-matrx. Hence the nvolutons ψ I for all I nduce a well-defned nvoluton ψ on the nverse lmt. The fact that the resultng nvoluton commutes wth each ċ k can be deduced from the formal defnton of the latter n terms of the R-matrx. Fnally the unqueness s a consequence of the exstence of an algorthm to unquely compute the canoncal bass usng just the gven two propertes (as sketched below). Ths puts us n poston to apply Lusztg s lemma to deduce for each µ t Z that there s a unque vector ḃ µ V n Ẇ m such that ψ(ḃ µ ) = ḃ µ ; ḃ µ s equal to v µ plus a (possbly nfnte) qz[q]-lnear combnaton of v λ for λ > µ. We refer to the resultng topologcal bass {ḃ µ µ t Z } for V n Ẇ m as the canoncal bass. In fact, but ths s n no way obvous from the above defnton, each ḃ µ s always a fnte sum of v λ s,.e. ḃ µ V n Ẇ m before completon. Moreover the polynomals d λ,µ (q) arsng from the expanson ḃ µ = d λ,µ (q) v λ λ t Z are known always to be some fnte type A parabolc Kazhdan-Lusztg polynomals (sutably normalzed). In partcular d λ,µ (q) N[q]. Now we can state the followng fundamental theorem, formerly known as the super Kazhdan-Lusztg conjecture. Theorem 3.2 (Cheng, Lam, Wang) For any λ,µ t Z we have that [M(λ) : L(µ)] = d λ,µ(1). In other words, the vectors {b µ µ t Z } arsng from the projectve ndecomposable supermodules n O Z va the dentfcaton K(OZ ) V n W m concde wth the specalzaton of Lusztg s canoncal bass {ḃ µ µ t Z } at q = 1. We are gong to do two more thngs n ths secton. Frst we sketch brefly how one can compute the canoncal bass algorthmcally. Then we wll explan how Theorem 3.2 should

12 12 Jonathan Brundan really be understood n terms of a certan graded lft O Z of O Z, usng the language of categorfcaton. The algorthm to compute the canoncal bass goes by nducton on the degree of atypcalty. Recall that a weght µ t Z s typcal f {µ 1,..., µ n } {µ n+1,..., µ n+m } =. We also say t s weakly domnant f µ 1 µ n and µ n+1 µ n+m (equvalently µ +ρ s domnant n the earler sense). The weghts that are both typcal and weakly domnant are maxmal n the Bruhat orderng, so that ḃ µ = v µ. Then to compute ḃ µ for an arbtrary typcal but not weakly domnant µ we just have to follow the usual algorthm to compute Kazhdan-Lusztg polynomals. Pck k n such that ether k < n and µ k < µ k+1 or k > n and µ k > µ k+1. Let λ be the weght obtaned from µ by nterchangng µ k and µ k+1. By nducton on the Bruhat orderng we may assume that ḃ λ s already computed. Then ċ k ḃ λ s ψ-nvarant and has v µ as ts leadng term wth coeffcent 1,.e. t equals v µ plus a Z[q,q 1 ]-lnear combnaton of v ν for ν > µ. It just remans to adjust ths vector by subtractng bar-nvarant multples of nductvely computed canoncal bass vectors ḃ ν for ν > µ to obtan a vector that s both ψ-nvarant and les n v µ + λ>µ qz[q] v λ. Ths must equal ḃ µ by the unqueness. Now suppose that µ t Z s not typcal. The dea to compute ḃ µ then s to apply a certan bumpng procedure to produce from µ another weght λ of strctly smaller atypcalty, together wth a monomal ẋ of quantum dvded powers of Chevalley generators of U q sl, such that ẋḃ λ has v µ as ts leadng term wth coeffcent 1. Then we can adjust ths ψ-nvarant vector by subtractng bar-nvarant multples of recursvely computed canoncal bass vectors ḃ ν for ν > µ, to obtan ḃ µ as before. The catch s that (unlke the stuaton n the prevous paragraph) there are nfntely many weghts ν > µ so that t s not clear that the recurson always termnates n fntely many steps. Examples computed usng a GAP mplementaton of the algorthm suggest that t always does; our source code s avalable at [13]. (In any case one can always fnd a fnte nterval I such that ẋḃ λ V n Ẇ m ; then by some non-trval but I I ẆI m known postvty of structure constants we get that ḃ µ V I n too; hence one can apply π I pror to makng any subsequent adjustments to guarantee that the algorthm termnates n fntely many steps.) Example 3.3 Wth ths example we outlne the bumpng procedure. Gven an atypcal µ we let be the largest nteger that appears both to the left and to the rght of the separator n the tuple µ. Pck one of the two sdes of the separator and let j be maxmal such that all of, + 1,..., j appear on ths sde of µ. Add 1 to all occurrences of, + 1,..., j on the chosen sde. Then f j + 1 also appears on the other sde of µ we repeat the bumpng procedure on that sde wth replaced by j + 1. We contnue n ths way untl j + 1 s not repeated on the other sde. Ths produces the desred output weght λ of strctly smaller atypcalty. For example f µ = (0, 5, 2, 2 0, 1, 3, 4) of atypcalty one we bump as follows: ė 5 ḟ (1,6,3,3 0,2,4,5) (1,5,3,3 0,2,4,5) 4 ḟ 3 (1,5,3,3 0,2,3,4) ė (2) 2 ḟ 1 ė (1,5,2,2 0,2,3,4) (1,5,2,2 0,1,3,4) 0 (0,5,2,2 0,1,3,4). The labels on the edges here are the approprate monomals that reverse the bumpng procedure; then the fnal monomal ẋ output by the bumpng procedure s the product ė 5 ḟ 4 ḟ 3 ė (2) 2 ḟ1ė 0 of all of these labels. Thus we should compute ė 5 ḟ 4 ḟ 3 ė (2) 2 ḟ1ė 0 ḃ (1,6,3,3 0,2,4,5), where ḃ (1,6,3,3 0,2,4,5) can be worked out usng the typcal algorthm. The result s a ψ-nvarant vector equal to ḃ (0,5,2,2 0,1,3,4) plus some hgher terms whch can be computed recursvely (specfcally one fnds that (q + q 1 )ḃ (2,5,2,2 1,2,3,4) needs to be subtracted).

13 Super category O 13 Example 3.4 Here we work out the combnatorcs n the prncpal block for gl 2 1 (C). The weghts are {(0, ),(,0 ) Z}. The correspondng canoncal bass vectors ḃ µ are represented n the followng dagram whch s arranged accordng to the Bruhat graph; we show just enough vertces for the generc pattern to be apparent For example the center node of ths dagram encodes ḃ (0,0 0) = v (0,0 0) + q v (0,1 1) + q 2 v (1,0 1) ; the node to the rght of that encodes ḃ (0,1 1) = v (0,1 1) + q v (1,0 1) + q v (0,2 2) + q 2 v (2,0 2). Let us explan n more detal how we computed ḃ ( 1,0 1) here. The bumpng procedure tells us to look at ė 0 ė 1 ḃ (0,1 1). As (0,1 1) s typcal we get easly from the typcal algorthm that ḃ (0,1 1) = ċ 1 v (1,0 1) = v (0,1 1) + q v (1,0 1). Hence ė 0 ė 1 ḃ (0,1 1) = v ( 1,0 1) + (1 + q 2 ) v (0,0 0) + q v (0, 1 1) + q v (0,1 1) + q 2 v (1,0 1). Ths vector s ψ-nvarant wth the rght leadng term v ( 1,0 1), but we must make one correcton to remove a term v (0,0 0),.e. we must subtract ḃ (0,0 0) as already computed, to obtan that ḃ ( 1,0 1) = v ( 1,0 1) + q v (0, 1 1) + q 2 v (0,0 0). Returnng to more theoretcal consderatons, the key pont s that the category O Z s an example of an sl -tensor product categorfcaton of V n W m. Ths means n partcular that there exst some exact endofunctors F and E of O Z whose nduced actons on K(O Z ) match the actons of the Chevalley generators f and e on V n W m under our dentfcaton. To defne these functors, recall that U denotes the natural g-module of column vectors. Let U be ts dual. Introduce the badjont projectve functors The acton of the Casmr tensor F := U : O Z O Z, E := U : O Z O Z. Ω := n+m, j=1 ( 1)ū j e, j e j, g g defnes an endomorphsm of FM = M U for each M O Z. Let F be the summand of the functor F defned so that F M s the generalzed -egenspace of Ω for each Z. We then have that F = Z F. Smlarly the functor E decomposes as E = Z E where each E s badjont to F ; explctly one can check that E M s the generalzed (m n )-egenspace of Ω on EM = M U. Now t s an nstructve exercse to prove: Lemma 3.5 The exact functors F and E send supermodules wth Verma flags to supermodules wth Verma flags. Moreover the nduced endomorphsms [F ] and [E ] of K(O Z ) agree under the above dentfcaton wth the endomorphsms f and e of V n W m defned by the acton of the Chevalley generators of sl.

14 14 Jonathan Brundan In fact much more s true here. The acton of Ω on each FM defnes a natural transformaton x End(F). Also let t End(F 2 ) be such that t M : F 2 M F 2 M s the endomorphsm t M : M U U M U U, v u u j ( 1)ūū j v u j u. From x and t one obtans x := F d xf 1 End(F d ) and t j := F d j 1 tf j 1 End(F d ) for each d 0,1 d and 1 j d 1. It s straghtforward to check that these natural transformatons satsfy the defnng relatons of the degenerate affne Hecke algebra H d. Ths shows that the category O Z equpped wth the badjont par of endofunctors F and E, plus the endomorphsms x End(F) and t End(F 2 ), s an sl -categorfcaton n the sense of Chuang and Rouquer. In addton O Z s a hghest weght category, and Lemma 3.5 checks some approprate compatblty of the categorcal acton wth ths hghest weght structure. The concluson s that O Z s actually an sl -tensor product categorfcaton of V n W m n a formal sense ntroduced by Losev and Webster. We are ready to state the followng extenson of the super Kazhdan-Lusztg conjecture, whch ncorporates a Z-gradng n the sprt of the classc work of Belnson, Gnzburg and Soergel on Koszulty of category O n the purely even settng. Theorem 3.6 (Brundan, Losev, Webster) There exsts a unque (up to equvalence) graded lft O Z of O Z that s a U q sl -tensor product categorfcaton of V n Ẇ m. Moreover the category O Z s a standard Koszul hghest weght category, and ts graded decomposton numbers [Ṁ(λ) : L(µ)] q are gven by the parabolc Kazhdan-Lusztg polynomals d λ,µ (q) as defned above. A few more explanatons are n order. To start wth we should clarfy what t means to say that O Z s a graded lft of O Z. The easest way to understand ths s to remember as dscussed n the prevous secton that O Z s equvalent to the category mof-a of fnte dmensonal locally untal rght A-modules, where A s the locally untal algebra A := Hom g (P(λ),P(µ)). λ,µ t Z To gve a graded lft O Z of O Z amounts to exhbtng some Z-gradng on the algebra A wth respect to whch each of ts dstngushed dempotents 1 λ are homogeneous; then the category grmof-a of graded fnte dmensonal locally untal rght A-modules gves a graded lft of O Z. Of course there can be many ways to do ths, ncludng the trval way that puts all of A n degree zero! Theorem 3.6 asserts n partcular that the algebra A admts a postve gradng makng t nto a (locally untal) Koszul algebra; as s well known such a gradng (f t exsts) s unque up to automorphsm. For ths choce of gradng, the category O Z := grmof-a s a graded hghest weght category wth dstngushed rreducble objects { L(λ) λ t Z }, standard objects {Ṁ(λ) λ t Z } and ndecomposable projectve objects {Ṗ(λ) := 1 λ A λ t Z }; these are graded lfts of the modules L(λ),M(λ) and P(λ), respectvely, such that the canoncal maps Ṗ(λ) Ṁ(λ) L(λ) are all homogeneous of degree zero. Then the asserton from Theorem 3.6 that O Z s standard Koszul means that each Ṁ(λ) possesses a lnear projectve resoluton, that s, there s an exact sequence Ṗ 2 (λ) Ṗ 1 (λ) Ṗ(λ) Ṁ(λ) 0 such that for each 1 the module Ṗ (λ) s a drect sum of graded modules q Ṗ(µ) for µ > λ. Here q denotes the degree shft functor defned on a graded module M by lettng qm be the same underlyng module wth new gradng defned from (qm) := M 1.

15 Super category O 15 Let O Z be the exact subcategory of O Z consstng of modules wth a graded -flag. Its Grothendeck group s a Z[q,q 1 ]-module wth q actng by degree shft. Let K q ( O Z ) be the Q(q)-vector space obtaned by extendng scalars,.e. t s the Q(q)-vector space on bass {[Ṁ(λ)] λ t Z }. Then agan we dentfy K q ( O Z ) V n Ẇ m, [Ṁ(λ)] v λ. The asserton about graded decomposton numbers n Theorem 3.6 means under ths dentfcaton that [Ṗ(µ)] ḃ µ. The asserton that O Z s a U q sl -tensor product categorfcaton means n partcular that the badjont endofunctors F and E of O Z admt graded lfts Ḟ and Ė, whch are also badjont up to approrate degree shfts. Moreover these graded functors preserve modules wth a graded Verma flag, and ther nduced actons on K q ( O Z ) agree wth the actons of ḟ,ė U q sl under our dentfcaton. We wll say more about the proof of Theorem 3.6 n the next secton. Notes. The dentfcaton of the Bruhat order on t Z wth the reverse domnance orderng s justfed n [10, Lemma 2.5]. Our Lemma 3.1 s a varaton on [10, Theorem 2.14]; the latter theorem was used n [10] to defne a twsted verson of the canoncal bass whch corresponds to the ndecomposable tltng supermodules rather than the ndecomposable projectves n O Z. The super Kazhdan-Lusztg conjecture as formulated here s equvalent to [10, Conjecture 4.32]; agan the latter was expressed n terms of tltng supermodules. The equvalence of the two versons of the conjecture can be deduced from the Rngel dualty establshed n [12, (7.4)]; see also [15, Remark 5.30]. The algorthm for computng the canoncal bass sketched here s a varaton on an algorthm descrbed n detal n [10, 2-h]; the latter algorthm computes the twsted canoncal bass rather than the canoncal bass. Example 3.4 was worked out already n [20, 9.5]. Theorems 3.2 and 3.6 are proved n [18] and [15], respectvely. In fact both of these artcles also prove a more general form of the super Kazhdan-Lusztg conjecture whch s adapted to arbtrary Borel subalgebras b of g; at the level of combnatorcs ths amounts to shufflng the tensor factors n the tensor product V n Ẇ m nto more general orders. The artcle [15] also consders parabolc analogs. The dea that blocks of category O should possess Koszul graded lfts goes back to the semnal work of Belnson, Gnzburg and Soergel [4] n the context of semsmple Le algebras. The noton of sl-categorfcaton was ntroduced by Chuang and Rouquer followng ther jont work [23]. The defnton was recorded for the frst tme n the lterature n [42, Defnton 5.29]. For the defnton of tensor product categorfcaton, see [36, Defnton 3.2] and also [15, Defnton 2.9]. A full proof of Lemma 3.5 (and ts generalzaton to the parabolc settng) can be found n [15, Theorem 3.9]. 4 Prncpal W-algebras and the double centralzer property By a prnjectve object we mean an object that s both projectve and njectve. To set the scene for ths secton we recall a couple of classcal results. Let O 0 be the prncpal block of category O for a semsmple Le algebra g, and recall that the rreducble modules n O 0 are the modules {L(w 0) w W} parametrzed by the Weyl group W. There s a unque ndecomposable prnjectve module n O 0 up to somorphsm, namely, the projectve cover

16 16 Jonathan Brundan P(w 0 0) of the antdomnant Verma module L(w 0 0); here w 0 s the longest element of the Weyl group. Theorem 4.1 (Soergel s Endomorphsmensatz) The endomorphsm algebra C 0 := End g (P(w 0 0)) s generated by the center Z(g) of the unversal envelopng algebra of g. Moreover C 0 s canoncally somorphc to the convarant algebra,.e. the cohomology algebra H (G/B,C) of the flag varety assocated to g. Theorem 4.2 (Soergel s Struktursatz) The functor s fully fathful on projectves. V 0 := Hom g (P(w 0 0), ) : O 0 mof-c 0 Wth these two theorems n hand, we can explan Soergel s approach to the constructon of the Koszul graded lft of the category O 0. Introduce the Soergel modules Q(w) := V 0 P(w 0) mof-c 0 for each w W. The Struktursatz mples that the fnte dmensonal algebra A 0 := Hom C0 (Q(x),Q(y)) x,y W s somorphc to the endomorphsm algebra of a mnmal projectve generator for O 0. The algebra C 0 s naturally graded as t s a cohomology algebra. It turns out that each Soergel module Q(w) also admts a unque graded lft Q(w) that s a self-dual graded C 0 -module. Hence we get nduced a gradng on the algebra A 0. Ths s the gradng makng A 0 nto a Koszul algebra. The resultng category grmof-a 0 s the approprate graded lft O 0 of O 0. Now we return to the stuaton of the prevous secton, so O Z s the ntegral part of category O for g = gl n m (C) and we represent ntegral weghts λ t Z as n m-tuples of ntegers. The proof of Theorem 3.6 stated above follows a smlar strategy to Soergel s constructon n the classcal case but there are several complcatons. To start wth, n any atypcal block, there turn out to be nfntely many somorphsm classes of ndecomposable prnjectve supermodules: Lemma 4.3 For λ t Z, the projectve supermodule P(λ) O Z s njectve f and only f λ s antdomnant,.e. λ 1 λ n and λ n+1 λ n+m. (Recall λ denotes (λ + ρ,δ ) Z.) Proof. Ths follows by a specal case of [15, Theorem 2.22]. More precsely, there s an sl - crystal wth vertex set t Z, namely, Kashwara s crystal assocated to the sl -module V n W m. Then [15, Theorem 2.22] shows that the set of λ t Z such that P(λ) s njectve s the vertex set of the connected component of ths crystal contanng any weght (,..., j,..., j) for < j. Now t s a smple combnatoral exercse to see that the vertces n ths connected component are exactly the antdomnant λ t Z. Remark 4.4 More generally, for λ t, the projectve P(λ) O s njectve f and only f λ s antdomnant n the sense that (λ,δ δ j ) / Z 0 for 1 < j n and (λ,δ δ j ) / Z 0 for n + 1 < j n + m. Ths follows from Lemma 4.3 and Theorem 2.6. In other words, the projectve P(λ) s njectve f and only f the rreducble supermodule L(λ) s of maxmal Gelfand-Krllov dmenson amongst all supermodules n O.

17 Super category O 17 Then, fxng ξ t Z /, the approprate analog of the convarant algebra for the block O ξ s the locally untal algebra C ξ := Hom g (P(λ),P(µ)). Antdomnant λ,µ ξ For atypcal blocks ths algebra s nfnte dmensonal and no longer commutatve. Stll there s an analog of the Struktursatz: Theorem 4.5 (Brundan, Losev, Webster) The functor V ξ : O ξ mof-c ξ sendng M O ξ to V ξ M := Hom g (P(λ),M) s fully fathful on projectves. Antdomnant λ ξ However we do not at present know of any explct descrpton of the algebra C ξ. Instead the proof of Theorem 3.6 nvolves another abelan category mod-h ξ. Ths notaton s strange because actually there s no sngle algebra H ξ here, rather, there s an nfnte tower of cyclotomc quver Hecke algebras Hξ 1 H2 ξ H3, whch arse as the endomorphsm algebras ξ of larger and larger fnte drect sums of ndecomposable prnjectve supermodules (wth multplctes). Then the category mod-h ξ conssts of sequences of fnte dmensonal modules over ths tower of Hecke algebras subject to some stablty condton. Moreover there s an explctly constructed exact functor U ξ : O ξ mod-h ξ. The connecton between ths and the functor V ξ comes from the followng lemma. Lemma 4.6 There s a unque (up to somorphsm) equvalence of categores I ξ : mod-h ξ mof-cξ such that V ξ = Iξ U ξ. Proof. Ths follows because both of the functors U ξ and V ξ are quotent functors,.e. they satsfy the unversal property of the Serre quotent of O ξ by the subcategory generated by {L(λ) λ ξ such that λ s not antdomnant}. For U ξ ths unversal property s establshed n [15, Theorem 4.9]. It s automatc for V ξ. Each of the algebras Hξ r n the tower of Hecke algebras s naturally graded, so that we are able to defne a correspondng graded category grmod-h ξ. Then we prove that the modules Y (λ) := U ξ P(λ) mod-h ξ admt unque graded lfts Ẏ (λ) grmod-h ξ whch are self-dual n an approprate sense. Snce the functor U ξ s also fully fathful on projectves (e.g. by Theorem 4.5 and Lemma 4.6), we thus obtan a Z-gradng on the basc algebra A ξ := Hom g (P(λ),P(µ)) = Hom Hξ (Ẏ (λ),ẏ (µ)). λ,µ ξ λ,µ ξ that s Morta equvalent to O ξ. Ths gradng turns out to be Koszul, and grmof-a ξ gves the desred graded lft O ξ of the block O ξ from Theorem 3.6. The results just descrbed provde a substtute for Soergel s Endomorphsmensatz for gl n m (C), wth the tower of cyclotomc quver Hecke algebras replacng the convarant algebra. However we stll do not fnd ths completely satsfactory, and actually beleve that t should be possble to gve an explct (graded!) descrpton of the basc algebra C ξ tself.

18 18 Jonathan Brundan Ths seems lke a tractable problem whose soluton could suggest some more satsfactory geometrc pcture underpnnng the rch structure of super category O. Example 4.7 Here we gve explct generators and relatons for the algebra C 0 for the prncpal block of O for gl 2 1 (C). The prnjectves are ndexed by Z and ther Verma flags are as dsplayed on the bottom row of the dagram n Example 3.4. The algebra C 0 s somorphc to the path algebra of the same nfnte lnear quver as n Example 2.8 modulo the relatons e +1 e = f f +1 = 0 for all Z, f +1 e +1 f +1 e +1 + e f e f = 0 for 2 or 1, f 0 e 0 + e 1 f 1 e 1 f 1 = 0, f 1 e 1 f 1 e 1 + e 0 f 0 = 0. Moreover the approprate gradng on C 0 s defned by settng deg(e ) = deg( f ) = 1 + δ,0. Here s a bref sketch of how one can see ths. The man pont s to explot Theorem 3.6: the gradng on O 0 nduces a postve gradng on C 0 wth degree zero component Z C1. Let D() be the one-dmensonal rreducble C 0 -module correspondng to Z and let Q() be ts projectve cover (equvalently, njectve hull). The proof of Theorem 3.6 mples further that these modules possess self-dual graded lfts Ḋ() and Q(). A straghtforward calculaton usng the graded verson of BGG recprocty and the nformaton n Example 3.4 gves the graded composton multplctes of each Q(). From ths one deduces for each Z that there are unque (up to scalars) non-zero homomorphsms e : Q( 1) Q() and f : Q() Q( 1) that are homogeneous of degree 1+δ,0. By consderng mages and kernels of these homomorphsms and usng self-dualty, t follows that each Q() has rreducble head q 2 Ḋ(), rreducble socle q 2 Ḋ(), and heart rad Q()/soc Q() = Q () Q + (), where Q (0) := Ḋ( 1), Q + ( 1) := Ḋ(0) and all other Q ± () are unseral wth layers q 1 Ḋ( ± 1),Ḋ(),qḊ(±1) n order from top to bottom. Hence (e f ) 2 δ,0 0 ( f +1 e +1 ) 2 δ +1,0 for each Z. Snce each End C0 ( Q()) s one-dmensonal n degree 4, t s then elementary to see that e and f can be scaled to ensure that the gven relatons hold, and the result follows. Remark 4.8 Wth a smlar analyss, one can show for any n 1 that the algebra C ξ assocated to the block ξ of gl n 1 (C) contanng the weght ρ = (0,...,0 0) s descrbed by the same quver as n Examples 2.8 and 4.7 subject nstead to the relatons e +1 e = f f +1 = ( f +1 e +1 ) n δ +1,0(n 1) + (e f ) n δ,0(n 1) = 0 for all Z. Ths tme deg(e ) = deg( f ) = 1 + δ,0 (n 1). To fnsh the artcle we draw attenton to one more pece of ths puzzle. Frst we need to ntroduce the prncpal W-superalgebra W n m assocated to g = gl n m (C). Let π be a tworowed array of boxes wth a connected strp of mn(n,m) boxes n ts frst (top) row and a connected strp of max(n,m) boxes n ts second (bottom) row; each box n the frst row should be mmedately above a box n the second row but the boxes n the rows need not be left-justfed. We wrte the numbers 1,...,n n order nto the boxes on a row of length n and the numbers n + 1,...,n + m n order nto the boxes on the other row. Also let s (resp. s + ) be the number of boxes overhangng on the left hand sde (resp. the rght hand sde) of ths dagram. For example here s a choce of the dagram π for gl 5,2 (C):