Quiver Schur algebras and q-fock space

Transcription

1 Quver Schur algebras and q-fock space Catharna Stroppel Unverstät Bonn Bonn, Deutschland Ben Webster Unversty of Vrgna Charlottesvlle, VA, USA Abstract. We develop a graded verson of the theory of cyclotomc q-schur algebras, n the sprt of the work of Brundan-Kleshchev on Hecke algebras and of Ark on q-schur algebras. As an applcaton, we dentfy the coeffcents of the canoncal bass on a hgher level Fock space wth q-analogues of the decomposton numbers of cyclotomc q-schur algebras. We present cyclotomc q-schur algebras as a quotent of a convoluton algebra arsng n the geometry of quvers - we call t quver Schur algebra - and also dagrammatcally, smlar n flavor to a recent constructon of Khovanov and Lauda. They are manfestly graded and so equp the cyclotomc q-schur algebra wth a non-obvous gradng. On the way we construct a graded cellular bass of ths algebra, resemblng smlar constructons for cyclotomc Hecke algebras. The quver Schur algebra s also nterestng from the perspectve of hgher representaton theory. The sum of Grothendeck groups of certan cyclotomc quotents s known to agree wth a hgher level Fock space. We show that our graded verson defnes a hgher q-fock space (defned as a tensor product of level 1 q-deformed Fock spaces). Under ths dentfcaton, the ndecomposable projectve modules are dentfed wth the canoncal bass and the Weyl modules wth the standard bass. Ths allows us to prove the already descrbed relaton between decomposton numbers and canoncal bases. Contents 1. Introducton 1 2. The quver Schur algebra 6 3. Dagrams and Demazure operators Hgher level generalzatons A graded cellular bass Dpper-James-Mathas cellular deals and graded Weyl modules Graded multplctes and q-fock space 56 References Introducton Recent years have seen remarkable advances n hgher representaton theory; the most exctng from the perspectve of classcal representaton theory were probably the proof of Broué s conjecture for the symmetrc groups by Chuang and Rouquer [CR08] and the ntroducton and study of graded versons of Hecke algebras by Brundan and Kleshchev [BK09a] wth ts Le theoretc orgns ([BK08], [BS11]). At the same tme, the queston of fndng categorcal analogues of the usual structures of Le theory has proceeded n the work of Khovanov, Lauda, Rouquer, Vazran, the authors and others. 1

2 In ths paper, we address a queston of nterest from both perspectves, representaton theory and hgher categorcal structures. As classcal (or quantum) representaton theorsts, we ask Is there a natural graded verson of the q-schur algebra and ts hgher level analogues, the cyclotomc q-schur algebra? Ths queston has been addressed already n the specal case of level 1 by Ark [Ar09]. We gve here a more general constructon that both llumnates connectons to geometry and s more explct. Our constructon ncludes the case of ordnary Schur algebras, see Remark 6.4 and the explct example at the end of the paper. As hgher representaton theorsts, we ask Is there a natural categorfcaton of q-fock space and ts hgher level analogues wth a categorcal acton of ŝl n? We wll show that the above two questons not only have natural answers; they have the same answer. Our man theorem s a graded verson wth a graded cellular bass of the cyclotomc q-schur algebra of Dpper, James and Mathas, [DJM98] and a combnatorcs of graded decomposton numbers usng hgher Fock space. To descrbe the results more precsely, let k be an algebracally closed feld and n, l, e natural numbers wth e > 1 (we wll allow the possblty that e = ). Let q k be a scalar wth e the smallest nteger such that 1 + q + + q e 1 = 0 and (Q 1,..., Q l ) an l-tuple of elements of k satsfyng the same equaton. In partcular, Q = q z for some z Z/eZ (snce q e = 1), unless q = 1, n whch case k has characterstc e, and we set Q = z. The assocated cyclotomc Hecke algebra or Ark-Koke algebra H(n; q, Q 1,..., Q l ) = H(S n Z/lZ; q, Q 1,..., Q l ) s the assocatve untary k-algebra wth generators T, 1 n 1 modulo the followng relatons, for 1 < j 1 < n 1, (T 0 Q 1 ) (T 0 Q l ) = 1, (T q)(t + 1) = 1, T T j = T j T, T T +1 T = T +1 T T +1, T 0 T 1 T 0 T 1 = T 1 T 0 T 1 T 0. The cyclotomc q-schur algebra we consder s the endomorphsm rng (1.1) S(n; q, Q 1,..., Q l ) = End H(n;q,Q1 ˆµ Λ,...,Q l ) M(ˆµ) where the sum runs over all admssble l-mult-compostons ˆµ of n and M(ˆµ) denotes the (sgned) permutaton module assocated to ˆµ. The admssblty condton we choose (see Lemma 5.4) defnes a certan subset Λ of all l-mult-compostons; n ths way we pck va (1.1) a specfc representatve out of the famly of cyclotomc q-schur algebras from [DJM98, 6]. To make contact wth the theory of quver Hecke algebras, we encode the parameters (q, Q 1,..., Q l ) as a sequence ν = (w z1,..., w zl ) of fundamental weghts for the 2

3 affne Le algebra ŝl e. The correspondng cyclotomc Hecke algebra only depends on the multplctes of each charge, that s, only on the weght ν = ν. Thus, we wrte (1.2) S ν = n 0 S(n; q, Q 1,..., Q l ) resp. H ν := n 0 H(n; q, Q 1,..., Q l ) for the sums of correspondng cyclotomc q-schur and Ark-Koke algebras of all dfferent ranks. To ths data we ntroduce then a certan Z-graded algebra A ν whch we call the cyclotomc quver Schur algebra. The name stems from the fact that the algebra s related to the cyclotomc quver Hecke algebras R ν and ther tensor product analogues T ν (defned n [Web10]) lke the fnte Schur algebra s to the classcal Hecke algebra. Our man result (Theorem 6.2) says that A ν s a graded verson of S ν gven by an extenson of the Brundan-Kleshchev somorphsm Φ ν : R ν = H ν from [BK09a] between the dagrammatc cyclotomc quver Hecke algebras R ν ntroduced n [KL09] and the sum of cyclotomc Hecke algebra H ν. Theorem A. There s an somorphsm Φ ν from A ν to the cyclotomc q-schur algebra S ν, extendng the somorphsm Φ ν : R ν = H ν. In partcular, the cyclotomc q-schur algebra S ν from (1.2) nherts a Z-gradng. Lke the cyclotomc quver Hecke algebra, the algebra A ν can be realzed as a natural quotent of a geometrcally defned convoluton algebra. Ths constructon s based on the geometry of quvers usng a certan category of flagged nlpotent representatons (quver partal flag varetes) of the cyclc affne type A quver. It naturally extends the work of Varagnolo and Vasserot [VV11] and clarfes the orgn of the gradng. The constructon of A ν (and ts summand A ν n for fxed n) proceeds n three steps. We frst defne an nfnte dmensonal convoluton algebra, whch we call quver Schur algebra, usng flagged representatons of the cyclc quver Γ. Ths algebra only depends on e (and has summands dependng on n). The second step s to add some extra shadow vertces to the quver and defne an convoluton algebra workng wth flagged representatons of the extended quver Γ dependng on the parameters Q and l. Fnally the last step s to pass to a certan fnte dmensonal quotent whch s the desred algebra A ν (wth the drect summand An ν for fxed n). Although we focus here on the cyclotomc quotents, we want to stress that the quver Schur algebra appears naturally n representaton theory. It s, after completon, somorphc to Vgneras Schur algebra [Vg03] for the general lnear p-adc group, see [MS14]. To make explct calculatons we descrbe the quver Schur algebra algebracally by consderng a fathful representaton on a drect sum of polynomal rngs, extendng the correspondng result for quver Hecke algebras. Moreover, we gve a dagrammatcal descrpton of the algebra by extendng the dagram calculus of Khovanov and Lauda [KL09]. In contrast to ther work, however, we are not able to gve a complete lst of relatons dagrammatcally. Stll, we have enough nformaton to construct (sgned) permutaton modules for the cyclotomc Hecke algebra and show that our algebra A ν s somorphc to the cyclotomc q-schur algebra usng the known three dfferent descrpton (geometrc, algebrac and dagrammatcal) of the quver Hecke algebras. 3

4 Note that ths can also be vewed as an extenson of work of the second author [Web10, 5.31], whch showed that smlar dagrammatc algebras were the endomorphsm algebras of some, but not all, (sgned) permutaton modules. Independently, Ark [Ar09] ntroduced graded q-schur algebras and studed ther permutaton modules. Ths s a specal case of our results when l = 1 (and e and n are not too small) and ndeed, we show that our gradng concdes wth Ark s. In the course of the proof we also establsh, smlar n sprt to the arguments n [BS11], the exstence of a graded cellular bass (Theorem 5.8) n the sense of Hu and Mathas [HM10]: To avod case by case arguments for the combnatorcs n the specal case e = 2 we restrct ourselves n the combnatoral part (startng n Secton 5.1) of the paper and for the remanng results of the ntroducton to the case e > 2, although the man theorems extend to the case e = 2 as well. Theorem B. The cyclotomc quver Schur algebra A ν s a graded cellular algebra. Moreover, The cellular deals concde under Φ ν wth those of Dpper-James-Mathas. The cell modules defne graded lfts of the Weyl modules. The gradng on the bass vectors comes from a degree functon defned on semstandard multtableaux extendng known degree functons on standard tableaux. We show that ths gradng comes naturally from geometry and extends the gradng on the tensor algebras from [Web10]. After crculatng a draft of ths paper, we receved a preprnt of Hu and Mathas [HM11] whch gves a defnton of a dfferent, but closely related algebra whch they also call a quver Schur algebra. They prove Theorems B, C and D n ths context for the (specal) case of a lnear, rather than cyclc, quver (that s, when e = ). In [Web10, 5.31], the second author shows that Hu and Mathas s algebra s Morta equvalent to certan tensor product algebras T λ ; these algebras are, n turn, Morta equvalent to those defned here when e = by Proposton From both the geometrc and the dagrammatc sdes, our constructon fts qute snugly nsde Rouquer s program of categorcal representaton theory, [Rou08]: Theorem C. The category of graded A ν -modules carres a categorcal acton of U q (ŝl e) n the sense of Rouquer. As a U q (ŝl e)-module, ts complexfed graded Grothendeck group s canoncally somorphc to the l-fold tensor product F l = F 1 (z l ) F 1 (z 1 ), of level 1 (fermonc) Fock space wth central charge z = (z 1,..., z l ) gven by ν. Here the parameter q corresponds to the effect of gradng shft on the Grothendeck group, and s thus a formal varable not a complex number. The acton of the standard Chevalley generators E, F s gven by -nducton and -restrcton functors. In the ungraded case these functors and ther connecton to undeformed Fock space were ndependently studed by [Wad11]. Our constructons can also be descrbed usng an affne verson of the thck calculus ntroduced by Khovanov, Lauda, Mackaay and Stošć for the upper half of ŝl e. Unlke n 4

5 [KLMS10] however, our category has objects whch do not appear n the thn calculus and categorfes the upper half of U q (ĝl e) nstead of U q (ŝl e). It s temptng to thnk ths could easly be extended to a graphcal categorfcaton of the Le algebra ĝl e, whch s the drect sum of ŝl e and a Hesenberg Le algebra H modulo an dentfcaton of ther centers. Ths s especally ntrgung and promsng gven the varous nterestng categorfcaton of Hesenberg algebras ([CL12], [Kho10], [LS11], [LS12], [CLS12]) whch appeared recently n the lterature. However, the connecton cannot be as straghtforward as one mght hope at frst, snce the functors assocated to the standard generators n the categorfcaton of the upper half of U q (ĝl e) smply do not have badjonts (unlke those n U q (ŝl e)). Ths was already ponted out n [Sha11, 5.1,5.2]. In ther acton on the categores of A ν -modules, they send projectves to projectves, but not njectves to njectves, so ther rght adjonts are exact, but not ther left adjonts. The Grothendeck group of graded A ν -modules s naturally a Z[q, q 1 ]-module and comes also along wth several dstngushed lattces and bases. To descrbe them combnatorally we ntroduce a bar-nvoluton on the tensor product F l of Fock spaces appearng n Theorem C whch allows us to defne, apart from the standard bass, two other dstngushed bases: the canoncal and dual canoncal bases. Ths canoncal bass s a lmt of that for hgher level q-fock spaces defned by Uglov, [Ugl00], n a sense we descrbe later. Our canoncal somorphsm nduces correspondences bar nvoluton Serre-twsted dualty canoncal bass ndecomposable projectves (char(k) = 0) dual canoncal bass smple modules (char(k) = 0) standard bass Weyl modules As a consequence we get nformaton about (graded) decomposton numbers of cyclotomc q-schur algebras: Theorem D (Theorem 7.20). If k has characterstc 0, the graded decomposton numbers of the cyclotomc q-schur algebra are the coeffcents of the canoncal bass n terms of the standard bass on the hgher level q-fock space F l. Agan ths problem was studed ndependently by Ark [Ar09] for the level l = 1 case. The above theorem combnes and generalzes therefore results of Ark on Schur algebras and Brundan-Kleshchev-Wang [BKW11] on cyclotomc Hecke algebras. It s very smlar n sprt to a conjecture of Yvonne [Yvo06, 2.13]; however, there are several small dfferences between Yvonne s conjecture and our results. The most mportant s that Yvonne used Jantzen fltratons to defne a q-analogue of decomposton numbers nstead of a gradng. Ths approach has been worked out n level 1 by Ram-Tngley [RT10] and Shan [Sha11]. Ther results show that the same q-analogue of decomposton numbers arse from countng multplctes wth respect to depth n Jantzen fltratons. We expect that ths wll hold n hgher level as well; t should follow from the followng fact (whch was conjectured n a frst draft of ths paper and) proved n [Mak14]: Theorem E. The (underlyng basc algebra of) the cyclotomc quver Schur algebra A ν s Koszul. 5

6 We prefer workng wth gradngs nstead of fltratons, snce they are easer to handle n practce. (A smlar phenomenon appears for the classcal category O for sem-smple complex Le algebras, where the Jantzen fltraton can also be descrbed n terms of a gradng, [BGS96], [Str03]. Ths gradng s actually drectly connected wth the gradng on the algebras R ν = H ν n case e =, see [BS11], [HM10]). Agan, for level l = 1, the Theorem E was already known to be true, [CM12]; an elementary argument for e = s gven n [BS10]. We should emphasze that at the moment, ths approach only allows us to understand the hgher-level Fock spaces whch are constructed as tensor products of level 1 Fock spaces (or ther rreducble ĝl e consttuents). Ths does not nclude the twsted hgher level Fock spaces studed by Uglov, [Ugl00], whch wll requre a generalzaton of the algebras we consder here. The same Fock spaces are categorfed by category O of certan Cherednk algebras, see e.g. [Sha11], [GL14]. The acton s agan gven by nducton and restrcton functors as n [Wad11], but t remans to be clarfed how our work fts nto ths framework. Let us brefly summarze the paper. Secton 2 contans prelmnares of the geometry of quver representatons needed to defne the quver Schur algebra both as a geometrc convoluton algebra and n terms of an acton on a polynomal rng whch then s related to Demazure operators n Secton 3. We connect ts graded Grothendeck group wth the generc nlpotent Hall algebra of the cyclc quver. In Secton 4, we dscuss a generalzaton of ths algebra usng extended (or shadowed) quver representatons that wll allow us to deal wth hgher level Fock spaces. In Secton 5, we defne cyclotomc quotents, equp them wth a (graded) cellular structure and establsh the somorphsm to cyclotomc q-schur algebras n Secton 6. In Secton 7, we consder the connecton of these constructons to hgher representaton theory, descrbe the categorcal acton of ŝl e on these categores, and show that they categorfy q-fock spaces. In partcular, we consder the relatonshp between projectve modules, canoncal bases, and decomposton numbers. Acknowledgment: We thank Georde Wllamson, Mchela Varagnolo, Peter Tngley, Ian Gordon, Mchael Ehrg, Bernard Leclerc and Peter Lttelmann for useful nput and frutful dscussons. We are n partcularly grateful to Peng Shan, Vanessa Memtz and Danel Tubbenhauer for ther careful readng and probng questons. We also would lke to thank the Hausdorff Center for supportng B.W. s vst to Bonn at the geness of ths work, and the organzers of the Oporto Meetng on Geometry, Topology and Physcs 2010 for facltatng our collaboraton. B.W. was supported by an NSF Postdoctoral Research Fellowshp and by the NSA under Grant H The quver Schur algebra Throughout ths paper, we wll fx an nteger e > 1; as mentoned before, we also allow the possblty that e =. (We wll assume for smplcty e > 2 at some pont later n the text.) Let Γ be the Dynkn dagram for ŝl e; wth the fxed clockwse orentaton f 6

7 e s fnte and wth the fxed lnear orentaton f e =, Fgure 1. Let V = {1, 2,..., e} (or V = Z for e = ) be the set of vertces of Γ, dentfed wth the set of remanders of ntegers modulo e. Let h : + 1 be the arrow from the vertex to the vertex + 1 where here and n the followng all formulas should be read takng ndces correspondng to vertces of Γ modulo e when e <. Defnton 2.1. A (fnte-dmensonal) representaton (V, f) of Γ over a feld k s a collecton of k-vector spaces V, V such that dm V <, together wth k-lnear maps f : V V +1. A subrepresentaton s a collecton of vector subspaces W V such that f (W ) W +1. A representaton (V, f) of Γ s called nlpotent f the map f e f 2 f 1 : V 1 V 1 s nlpotent (when e =, all representatons are called nlpotent) Quver representatons and quver flag varetes. The dmenson vector of a representaton (V, f) s the tuple d = (d 1,..., d e ), where d = dm V. We let d = d and denote by α the specal dmenson vector where d j = δ j. Mappng t to the smple root α of ŝl e dentfes the set of dmenson vectors wth the postve cone n the root lattce of ŝl e, and wth sem-smple nlpotent representatons of Γ: Lemma 2.2. There s a unque rreducble nlpotent representaton S j of dmenson vector α j. Any sem-smple nlpotent representaton s of the form (V, f) wth f = 0 for all. Proof. Obvously S j equals (V, f), where f and V are zero except of V j = k. Assume (V, f) s a non-trval rreducble nlpotent representaton. If, for gven j, f j 0 then f j s njectve, snce otherwse W j = ker f j and W r = {0} for r j defnes a non-trval proper subrepresentaton. Not all f s are njectve, snce the representaton s nlpotent and non-trval. Pck such that f s not njectve wth V 0. Then f = 0 and hence (V, f) s somorphc to S, snce S s a subrepresentaton. Any representaton (V, f) wth f = 0 for all s obvously sem-smple. Conversely, assume (V, f) s sem-smple, hence somorphc to e =1 Sd. In partcular, Sd s a drect summand (for any ) whch mples that f = 0. Let Rep d be the affne space of representatons of Γ wth dmenson vector d,.e. (2.1) Rep d = V Hom(C d, C d +1 ). Ths space has a natural algebrac acton by conjugaton of the algebrac group G d = GL(d 1 ) GL(d e ), and we are nterested n the modul space of representatons,.e. the quotent GRep d = Rep d /G d parametrzng somorphsm classes of representatons. Snce the G d -acton s very far from beng free, we must nterpret ths quotent n an ntellgent way. One opton s to consder t as an Artn stack. Whle ths s perhaps the most elegant approach, t s more techncal than necessary for our purposes. Instead the reader s encouraged to nterpret ths quotent as a formal symbol where, by conventon for any complex algebrac G-varety X. H (X/G) s the G-equvarant cohomology HG (X), and HBM (X/G) s the G- equvarant Borel-Moore homology of X (for a dscusson of equvarant Borel- Moore homology, see [Ful98, secton 19], or [VV11, 1.2]). 7

8 e e 1 1 Fgure 1. The orented Dynkn quver Γ and the extended Dynkn quver Γ. D(X/G) (resp. D + (X/G)) s the bounded (resp. bounded below) equvarant derved category of Bernsten-Lunts [BL94], wth the usual sx functor formalsm descrbed theren. See also [WW09] as an addtonal reference for our purposes. Note that H G d (Rep d ) = H (BG d ), where BG d denotes the classfyng space of G d (or the classfyng space of ts C-ponts n the analytc topology for the topologcally mnded) snce Rep d s contractble. Thus, we can use the usual Borel somorphsm to dentfy the H (GRep d ) s wth polynomal rngs. We wll consder the drect sum H (GRep) = d H (GRep d ) whch corresponds to takng the unon of the quotents GRep = d GRep d as Artn stacks. One can thnk of ths as a quotent by the groupod G = d G d, and we wll speak of the G-equvarant cohomology of Rep = d Rep d, etc. We ll be nterested n spaces of quver representatons equpped wth compatble flags. These have appeared several tmes n the lterature, most mportantly n work of Lusztg, e.g. [Lus91], on the geometrc constructon of the canoncal bass; our work bulds on hs deas. Now, we ntroduce the combnatorcs underlyng these spaces. A composton of length r of n Z >0 s a tuple µ = (µ 1, µ 2,..., µ r ) Z r >0 such that r =1 µ = n. In contrast, a vector composton 1 of type m Z >0 and length r = r(ˆµ) s a tuple ˆµ = (µ (1), µ (2),..., µ (r) ) of nonzero elements from Z m 0. If m = e we call a vector composton also resdue data and denote ther set by Comp e. Alternatvely, ˆµ can be vewed as an r m matrx µ[, j] wth the th row µ (). We call the column sequence,.e. the result of readng the columns, the flag type sequence t( ˆµ), whereas the resdue sequence res( ˆµ) s the sequence (2.2) 1 µ(1) 1, 2 µ (1) 2,, e µ (1) e 1 µ(2) 1, 2 µ (2) 2,, e µ (2) e 1 µ(r) 1, 2 µ (r) 2,, e µ (r) e. The parts separated by the vertcal lnes are called blocks of res(ˆµ). We say that ˆµ has complete flag type f every µ () s a unt vector whch has exactly one non-zero entry. Hence blocks of res(ˆµ) contan n ths case at most one element. The transposed vector composton of ˆµ of type r(µ) and length m s defned to be ˇµ = (ˇµ (1), ˇµ (2),..., ˇµ (m) ) where ˇµ () j = µ[j, ] = ˆµ (j), whch we call n case m = e also 1 Ths dffers from the noton of an r-mult-composton where the tuples could be of dfferent lengths or an r-mult-partton where the tuples are parttons, but agan not necessarly of the same length m. 8

9 the flag data. Gven a composton µ of n of length r, we denote by F(µ) the varety of flags of type µ, that s the varety of flags F 1 (µ) F 2 (µ) F r (µ) = C n, where F (µ) s a subspace of dmenson j=1 µ j. For a vector composton ˆµ we let F(ˆµ) = F(ˇµ() ), a product of partal flag varetes nsde C d = e =1 Cd of type gven by the flag data. Here d = d(ˆµ) = (d 1,..., d e ) denotes the dmenson vector of the vector composton ˆµ whch s smply the sum d = r =1 ˆµ(), and d denotes the total dmenson. For a gven dmenson vector d denote VComp e (d) = {ˆµ VComp e d(ˆµ) = d}. Example 2.3. Let e = 2 and ˆµ = ((2, 1), (1, 1), (2, 3), (0, 1)), a resdue data of length r = 4 wth dmenson vector d = (5, 6). The flag data s ˇµ = ((2, 1, 2, 0), (1, 1, 3, 1)) and res(ˆµ) = 1, 1, 2 1, 2 1, 1, 2, 2, 2 2. There are ( 11 5 ) elements of complete flag type n VCompe (d). Defnton 2.4. For a gven vector composton ˆµ Comp e (of length r) a representaton wth compatble flags of type ˇµ s a nlpotent representaton (V, f) of Γ wth dmenson vector d = d(ˆµ) together wth a flag F () of type ˇµ () nsde V for each 1 e such that f (F () j ) F ( + 1) j 1 for 1 j r. We denote by Q(ˆµ) = { (V, f, F ) Rep d F(ˆµ) f (F () j ) F ( + 1) j 1, j } the subset of Rep d F(ˆµ) of representatons wth compatble flag. It comes equpped wth the obvous acton of the group G := G d by change of bass. Alternatvely (see Lemma 2.2), a representaton wth compatble flags s a tuple ((V, f), F ) consstng of a representaton (V, f) Rep d equpped wth a fltraton by subrepresentatons wth sem-smple successve quotents (wth dmenson vectors gven by ˆµ). In Example 5.7, we have a quver Γ wth two vertces and smple representatons S 1, S 2 and the resdue data or flag data gves flags of the form F 1 2 (1) F 2 3 (1) F 3 5 (1) F 4 5 (1) = C 5, F 1 1 (2) F 2 2 (2) F 3 5 (2) F 4 6 (2) = C 6, where the subndex denotes the dmenson of the subspaces. In partcular, f we set V = F (1) F (2), then the semsmple subquotents are the representatons gven by the dmenson vector ˆµ, namely V 4 = S2, V 3 /V 4 = S 2 1 S 3 2, V 2 /V 3 = S1 S 2, V 1 /V 2 = S 2 1 S 1 2. The flag data defnes a Young subgroup Sˇµ S d. Note that F(ˆµ) = G d /Pˇµ s the partal flag varety defned by the parabolc subgroup Pˇµ of G d, gven by upper trangular block matrces wth block szes determned by the flag sequence. The flags whch appear wll be complete f and only f ˆµ has complete flag type, as suggested by the name. Ths specal case s partcularly mportant; t played a key role n earler papers of Lusztg [Lus91] and Varagnolo and Vasserot [VV11]. Forgettng ether the flag or the representaton defnes two G d -equvarant morphsms (2.3) (2.4) p : Q(ˆµ) Rep d, ((V, f), F ) (V, f) Rep d, π : Q(ˆµ) F(ˆµ), ((V, f), F ) F F(ˆµ). 9

10 of algebrac varetes. Generalzng the noton of a quver Grassmannan we call the fbres of p the quver partal flag varetes. To each V assgn a polynomal rng over k n d varables x,1,..., x,d and set e (2.5) R(d) = k[x j,1,..., x j,dj ] = k[x 1,1,..., x 1,d1,..., x e,1,..., x e,de ]. j=1 Ths algebra carres an acton of the Coxeter group S d = S d1 S de by permutng the varables n the same tensor factor. Thus, the Borel presentaton dentfes the rngs (2.6) H (F(ˆµ)/G) = R(d) Sˆµ =: Λ(ˆµ), by sendng Chern classes of tautologcal bundles to elementary symmetrc functons, [Br98, Proposton 1]. In the extreme case where d = (r, 0,..., 0) we have ˆµ () = (1, 0,..., 0) for all and Sˆµ s trval. Hence we obtan the GL d -equvarant cohomology of the varety of full flags n C d. If ˆµ = (d) then ˆµ = ((d 1 ), (d 2 ),..., (d e )), the varety F(ˆµ) s just a pont and Λ(ˆµ) = R(d) S d. We call ths the rng of total nvarants. Lemma 2.5. The map π s a vector bundle wth affne fbre; n partcular, we have a natural somorphsm π : H (F(ˆµ)/G) = H (Q(ˆµ)/G), and thus H (Q(ˆµ)/G) = Λ(ˆµ). Gradng conventon. Throughout ths paper, we wll use a somewhat unusual gradng conventon on cohomology rngs: we shft the (equvarant) cohomology rng or equvarant Borel-Moore homology of a smooth varety X downward by the complex dmenson of X, so that the dentty class s n degree dm C X. Ths choce of gradng has the felctous effect that pull-back and push-forward maps n cohomology are more symmetrc. Usually, for a map f : X Y we have that pull-back has degree 0, and push-forward has degree 2 dm C Y 2 dm C X, whereas n our gradng conventon (2.7) both push-forward and pull-back have degree dm C Y dm C X. Those readers comfortable wth the theory of the constructble derved category wll recognze ths as replacng the usual constant sheaf wth the ntersecton cohomology sheaf, denoted k Q(ˆµ)/G D + (Q(ˆµ)/G), of Q(ˆµ)/G. Snce Q(ˆµ) s smooth, k Q(ˆµ)/G s smply a homologcal shft of the usual constant sheaf on each ndvdual component (but by dfferent amounts on each component). A more conceptual explanaton for the conventon s the nvarance of k Q(ˆµ)/G under Verder dualty. Ths gradng shft also provdes a straght-forward explanaton of the gradng on the representaton Pol of the quver Hecke algebra n [KL09], [KL11]. Snce p s proper and the constant sheaf of geometrc orgn, the Belnson-Bernsten- Delgne decomposton theorem from [BBD82] apples (n the formulaton [dcm09, Theorem 4.22]), and p sends the k Q(ˆµ)/G on Q(ˆµ)/G to a drect sum L of shfts of smple perverse sheaves on Rep d. Our frst object of study n ths paper wll be the algebra of extensons of these sheaves The convoluton algebra. Let ˆµ, ˆλ Comp e be vector compostons wth assocated dmenson vector d, and consder the correspondng Stenberg varety (2.8) Z(ˆµ, ˆλ) = Q(ˆµ) Repd Q(ˆλ). 10

11 Let G d act dagonally and abbrevate H(ˆµ, ˆλ) = Z(ˆµ, ˆλ)/G d. Note that Q(ˆµ) s smooth and p s proper. So, by [CG97, Theorem 8.6.7], we can dentfy the algebra of self-extensons of L (although not as a graded algebra) wth the equvarant Borel-Moore homology H BM,G of our Stenberg varety: we have the natural dentfcaton such that the Yoneda product Ext D b (GRep d ) (p k Q(ˆµ), p k Q(ˆλ) ) = H BM,G (Z(ˆµ, ˆλ)), Ext D b (GRep d ) (p k Q(ˆµ), p k Q(ˆλ) ) Ext D b (Rep d ) (p k Q(ˆλ), p k Q(ˆν) ) Ext D b (GRep d ) (p k Q(ˆµ), p k Q(ˆν) ) agrees wth the convoluton product. Ths defnes an assocatve non-untal graded algebra structure A d on Ext ( ) (2.9) D b (GRep d ) p k Q(ˆµ), p k Q(ˆµ) = H BM (H(ˆµ, ˆλ)). (ˆµ,ˆλ) Here the sums are over all elements n VComp e (d) VComp e (d). For reasons of dagrammatc algebra, we call ths product vertcal composton and we denote t lke the usual multplcaton (f, g) fg. We call the algebra A d the quver Schur algebra A fathful polynomal representaton. The quver Schur algebra A d acts, by [CG97, Proposton ], naturally on the sum (over all vector parttons of dmenson vector d) of cohomologes (2.10) V d := ˆµ VComp e (d) (ˆµ,ˆλ) H BM (Q(ˆµ)/G) = ˆµ VComp e (d) Note that ths s compatble wth the gradng conventon (2.7). Proposton 2.6. If char(k) = 0, the A d -module V d s fathful. Λ(ˆµ). In fact, the hypothess on characterstc s unnecessary, snce ths s a specal case of Proposton 4.7, whch s characterstc ndependent; however, there s a more geometrc proof n ths specal case, whch we gve here. Proof. Recall that the rng A d s just the Ext-algebra of ˆµ =d p k Q(ˆµ). The space V d s the hypercohomology of ˆµ =d p k Q(ˆµ)) up to shfts of gradng, snce Borel-Moore homology H BM (X) of a smooth space X equals hypercohomology H (X, D) of ts dualzng sheaf D whch s, up to a shft, the constant sheaf. Hence, f we let j be the map from Rep d to a pont, fathfulness s equvalent to j beng njectve, (2.11) j : Ext (p k Q(ˆµ), p k Q(ˆµ ) ) Ext (j p k Q(ˆµ), j p k Q(ˆµ ) ). We only need to check that the same property holds when p k Q(ˆµ) s replaced by a summand. The nlpotent orbts of G d n Rep d are equvarantly smply connected; thus every smple perverse sheaf supported on the nlpotent locus s the ntermedate extenson of a trval local system. Thus, by the decomposton theorem, the sheaves p k Q(ˆµ) decompose nto summands whch are shfts of the ntersecton cohomology sheaves of these orbts. We note that the spaces we deal wth have good party vanshng propertes. Each orbt has even equvarant cohomology, snce t has a transtve acton of G wth the stablzer of a pont gven by a connected algebrac group ([Lb07, Lemma 78, 11

12 Theorem 79]). Also, the stalks of ntersecton cohomology sheaves have even cohomology [Hen07, Theorem 5.2(3)]. Thus, by [BGS96, Theorem 3.4.2], j s fathful on sem-smple G d -equvarant perverse sheaves, and so the map (2.11) s njectve Monodal structure. The algebra A d comes along wth dstngushed dempotents eˆµ ndexed by vector compostons wth dmenson vector d. The A d -module (2.12) P (ˆµ) = ˆλ VComp e (d) eˆλa d eˆµ s a fntely generated ndecomposable graded projectve A d -module. Each fntely generated ndecomposable graded projectve A d -module s (up to a gradng shft) somorphc to one of ths form. We denote by A d pmod the category of graded fntely generated projectve A d -modules. Proposton 2.7. Assume char(k) = 0. The category A d pmod s equvalent to the addtve category of sums of shfts of sem-smple perverse sheaves n D + (GRep) whch are pure of weght 0 wth nlpotent support n GRep d. Agan, the characterstc 0 hypothess could be avoded, but at the cost of some dffcultes we prefer to avod. The correspondng perverse sheaves n the characterstc p case wll not be sem-smple, but rather party sheaves, n the sense of [JMW09]. Proof. There s( a functor sendng a fntely generated projectve A d -module M to the perverse sheaf d(ˆµ)=d p k Qˆµ) Ad M. Ths map s fully fathful, snce t nduces an somorphsm on the endomorphsms of A d tself. Thus, we only need to show that every smple perverse sheaf on GRep wth nlpotent support s a summand of p k Q(ˆµ) for some ˆµ. Every such smple perverse sheaf s IC( X) where X s the locus of modules somorphc to a fxed module N. Consder the socle fltraton on N. The dmenson vectors of the successve quotents defne a vector composton ˆµ N. The map Q(ˆµ N ) GRep s genercally an somorphsm over X, and thus for dmenson reasons, has mage X. In partcular, p k Q(ˆµN ) = IC( X) L where L s a fnte drect sum of shfts of sem-smple perverse sheaves supported on X \ X. Thus, every smple perverse sheaf wth nlpotent support s a summand of such a pushforward, and we are done. We are gong to defne a monodal structure on A d pmod usng correspondences. For ˆµ, ˆν VComp e the jon ˆµ ˆν s the vector composton ˆµ ˆν = (µ (1),..., µ (r(ˆµ)), ν (1),..., ν r(ˆν) ) obtaned by jonng the two tuples. Let ˆµ 1, ˆλ 1 and ˆµ 2, ˆλ 2 be vector compostons wth assocated dmenson vectors c and d respectvely. Let Q(ˆµ 1 ; ˆµ 2, ˆλ 1 ; ˆλ 2 ) Q(ˆµ 1 ˆµ 2 ) Repc Q(ˆλ 1 ˆλ 2 ) be the space of representatons wth dmenson vector c + d whch carry a par of compatble flags of type ˆµ 1 ˆµ 2 and ˆλ 1 ˆλ 2 respectvely, such that the subspaces of 12

13 dmenson vector c n the two flags concde. Let H(ˆµ 1 ; ˆµ 2, ˆλ 1 ; ˆλ 2 ) be the quotent by the dagonal G d -acton. Defnton 2.8. The horzontal multplcaton s the map (2.13) A c A d A c+d (a, b) a b nduced on equvarant Borel-Moore homology by the correspondence (.e. by pull-andpush on the followng dagram) (2.14) H(ˆµ 1, ˆλ 1 ) H(ˆµ 2, ˆλ 2 ) H(ˆµ 1 ; ˆµ 2, ˆλ 1 ; ˆλ 2 ) H(ˆµ 1 ˆµ 2, ˆλ 1 ˆλ 2 ). Here the rghtward map s the obvous ncluson, and the leftward s nduced from the map V (W, V/W ) of takng the common subrepresentaton W of dmenson vector c and the quotent by t. We let A = d (A d pmod) be the drect sum of the categores A d pmod over all dmenson vectors; that s ts objects are formal drect sums of fntely many objects from these categores, wth morphsm spaces gven by drect sums. Proposton 2.9. The assgnment : (P (ˆµ), P (ˆν)) P (ˆµ ˆν) extends to a monodal structure (A,, 1) wth unt element 1 = P ( ). Proof. In the ungraded case ths follows drectly from Lusztg s convoluton product [Lus91, 3], by Proposton 2.7. More explctly, we defne M N = A c+d Ac A d M N, meanng one frst takes the outer tensor product of the graded A c -module M and the graded A d -module N. The resultng A c A d -module s then nduced to a graded A c+d - module va the horzontal multplcaton (2.13). Ths s functoral n both entres and defnes the requred tensor product wth the asserted propertes Categorfed generc nlpotent Hall algebra. Let Kq 0 (A) be the splt Grothendeck group of the addtve Krull-Schmdt category A,.e. the free abelan group on somorphsm classes [M] of objects n A pmod modulo the relaton [M 1 ] + [M 2 ] = [M 1 M 2 ]. Ths s a free Z[q, q 1 ]-module where the acton of q s by gradng shft (and has nothng to do wth the parameter q from the ntroducton). For a graded vector space W = j W j, we defne ts gradng shfts W d, d Z, by (W d ) j = W d+j, and let q d [M] = [M d ]. The module Kq 0 (A) s of nfnte rank, but s naturally a drect sum of the Grothendeck groups Kq 0 (A d pmod), each of whch s fnte rank. Let VCompf e (d) VComp e be the set of vector compostons of d of complete flag type. For each d, there s a subalgebra, (2.15) R d = eˆλa d eˆµ. ˆλ,ˆµ VCompf e (d) There s also a correspondng monodal subcategory R of A generated by the ndecomposable projectves ndexed by the ˆµ VCompf e (d) for all d. Both A d and R d can be defned for any quver and the followng proposton holds n general, though n ths 13

14 paper we only use these categores for the affne type A quver. The algebra R d appears as quver Hecke algebra (assocated wth d) n the lterature: Proposton 2.10 (Vasserot-Varagnolo/Rouquer [VV11, 3.6]). As a graded algebra, R d s somorphc to the quver Hecke algebra R(d) assocated to the quver Γ n [Rou08]. In partcular, K 0 q (R) s naturally somorphc to the Lusztg ntegral form of U q (ŝl e) by mappng the somorphsm classes of ndecomposable projectve objects to Lusztg s canoncal bass. The dempotents n R d get dentfed wth those n R(d) by vewng the resdue sequence (2.2) as a sequence of smple roots α. We should note that Proposton 2.10 uses the sgned verson of the quver Hecke algebra appearng n [VV11], [BK09a] whch dffers from the frst paper of Khovanov and Lauda [KL09]. Remark For a Dynkn quver, the categores R and A are canoncally equvalent. In fact, both are equvalent to the full category of sem-smple perverse sheaves on GRep. However, n affne type A (the case of nterest n ths paper), they dffer. In terms of perverse sheaves, the IC-sheaves whch appear n p k Q(ˆµ) for ˆµ havng complete flag type are those whose Fourer transform has nlpotent support as well; for example, the constant sheaf on the trval representaton wth dmenson vector (1,..., 1) cannot appear. Thus, n ths case there are objects n A whch don t le n R. Recall that the nlpotent Hall algebra of the quver Γ s an algebra structure on the set of complex valued functons on the space of (somorphsm classes of) nlpotent representatons, typcally consdered over a fnte feld. The structure constants are polynomal n the cardnalty q of the feld. If [M] denotes the constant functon on the class of the representaton M wth dmenson vector d(m) then [M] [N] = q {d(m),d(n)} F Q M,N [Q], where {d, d } = e =1 d (d d +1 ) denotes the Euler form, q = q 2 and the F Q M,N are the Hall numbers. Hence t makes sense to consder q as a formal parameter and defne the generc Hall algebra over the rng of Laurent polynomals C[q, q 1 ]. Followng Vasserot and Varagnolo, [VV99], we denote ths algebra Ue. By work of Schffmann [Sch00, 2.2] t s somorphc as an algebra to Uq (ŝl e) Λ( ), where Λ( ) denotes the rng of symmetrc polynomals. Identfyng Λ( ) wth Uq (H), the lower half of a Hesenberg algebra, ths algebra can also be descrbed as Uq (ĝl e) as n work of Hubery [Hub05], [Hub10]. Ths generc Hall algebra has a bass gven by characterstc functons on the somorphsm classes of nlpotent representatons of Γ and s naturally generated as algebra by the characterstc functons f d on the classes of sem-smple representatons (whch we label by ther dmenson vectors d followng Lemma 2.2). Note that for nstance f := f α = [S ] and f α +α +1 = f +1 f = [S +1 S ], whereas f (1,...,1) s not n the subalgebra generated by the [S ] s, cf. Remark The ntegral form U e,z over Z[q, q 1 ] s gven here by the lattce generated by all f d s, analogous to Lusztg s ntegral form for quantum groups, see [Sch00]. Proposton If char(k) = 0, then there s an somorphsm K 0 q (A) = U e,z, [(d)] f d, of Z[q, q 1 ]-algebras from the graded Grothendeck rng of A to the ntegral form of the generc nlpotent Hall algebra of the cyclc quver. 14

15 The result s n fact also true for k of postve characterstc and can be proved usng the usual technque of deformng to a characterstc 0 dscrete valuaton rng. Snce the general result s not needed here, we omt t, but refer to [Mak13]. Proof. Fxng a prme p, there s a natural map from Kq 0 (A) to Ue q=p. Ths s gven by applyng the equvalence of Proposton 2.7, and then sendng the class of a sem-smple perverse sheaf to the functon gven by the super-trace of Frobenus on ts stalks. Ths s a functon on the ponts of Rep over the feld F p and hence defnes an element of the Hall algebra. By the defnton of the Hall multplcaton and the Grothendeck trace formula, ths s an algebra map. Snce these super-traces are polynomal n p (they are the Poncaré polynomals of the quver partal flag varetes), the coeffcents of the expanson of ths functon n terms of the characterstc functons of orbts are also polynomal, and ths assgnment can be lfted to an algebra map Kq 0 (A) U e,z. Ths map s obvously surjectve, snce the functon for each ntersecton cohomology sheaf on a nlpotent orbt, and thus the characterstc functon on the orbt, s n ts mage. It s also njectve, snce when we expand any non-zero class n the Grothendeck group n terms of the classes of ntersecton cohomology sheaves, we must have a nonzero value of the correspondng functon on the support of an ntersecton cohomology sheaf maxmal (n the closure orderng) amongst those wth non-zero coeffcent. Snce [(d)] corresponds to the skyscraper sheaf of the sem-smple representaton of dmenson d, t s sent to the characterstc functon of that pont. The monodal structure on A and the usual monodal structure (Vect k, k, k) on the category of vector spaces are compatble n the followng way: Lemma Let Φ d : A d End(V d ) be the representaton from (2.10). Then Φ c+d ((a b))(v) = Φ c (a)(v 1 ) Φ c (b)(v 2 ), where v s the mage of v 1 v 2 under the canoncal map Vˆµ Vˆλ Vˆµ ˆλ. That s, the functor V : A Vect k gven by ˆµ Vˆµ s monodal. Proof. Ths follows drectly from the defntons, see [CG97]. Thus, we can descrbe elements correspondng to vector compostons wth a large number of parts by lookng at (the acton) of the ones wth a small number of parts. 3. Dagrams and Demazure operators In ths secton we descrbe a bass of the algebras A d and elementary morphsms, called splts and merges. We gve a geometrc, algebrac and dagrammatcal descrpton of these maps. Let ˆλ, ˆλ Comp e be resdue data. Defnton 3.1. We say that ˆλ s a merge of ˆλ (and ˆλ a splt of ˆλ ) at the ndex k f ˆλ = (λ (1),, ˆλ (k) + ˆλ (k+1),..., ˆλ (r) ). 15

16 If ˆλ s a merge of ˆλ, then there s an assocated correspondence } Q(ˆλ, k) = {(V, f, F ) Q(ˆλ) f (F () k+1 ) F ( + 1) k 1 (3.1) Q(ˆλ) Q(ˆλ ) where the left map s just the obvous ncluson and the rght map s forgettng F k () for all vertces 1 e (and rendexng all subspaces n the flags wth hgher ndces). Obvously the same varety defnes also a correspondence n the opposte drecton (readng from rght to left) whch we assocate to the splt. We are nterested n the equvarant verson: Defnton 3.2. For ˆλ a merge (resp. splt) of ˆλ at k, we let ˆλ k ˆλ or just ˆλ ˆλ denote the element of A gven by multplcaton wth the equvarant fundamental class [Q(ˆλ, k)] (resp. [Q(ˆλ, k)]) pushed forward to H BM (H(ˆλ, ˆλ)). In the most obvous choce of gradng conventons, pull-back by a map s of degree 0, and pushforward has degree gven by mnus the relatve (real) dmenson of the map (.e. the dmenson of the target mnus the dmenson of the doman). Ths normalzaton has the dsadvantage of breakng the symmetry between splts and merges. It s, for example, carefully avoded n [KL09]. Instead, we use, (2.7), the perverse normalzaton of the constant sheaves whch averages the degrees of pull-back and pushforward. Then the degree of convolvng wth the fundamental class of a correspondence s mnus the sum of the relatve (complex) dmensons of the two projecton maps (note that for a correspondence over two copes of the same space, ths agrees wth the most obvous normalzaton). In partcular, we get the same answer n the splt and merge cases. Proposton 3.3. Let ˆλ be a merge or splt of ˆλ at ndex k. Then ˆλ ˆλ s homogeneous of degree e (3.2) λ (k) (λ (k+1) 1 λ (k+1) ) =: {λ (k+1), λ (k) }. =1 Proof. If ˆλ s a merge of ˆλ, then the map Q(ˆλ, k) Q(ˆλ ) s a smooth surjecton wth fber gven by the product of Grassmannans of λ (k) -dmensonal planes n λ (k) +λ (k+1) -dmensonal space, whch has dmenson λ (k) λ (k+1), and the map Q(ˆλ, k) Q(ˆλ) s a closed ncluson of codmenson e dm Hom(F ( 1) k+1 /F ( 1) k, F () k /F () k 1 ) = =1 The result follows for merges and hence also for splts. e =1 λ (k) λ (k+1) Explct formulas for merges and splts. We gve now explct formulas for elementary merges and splts. Consder the partcular choces for the vector compostons: H((c, d), (c + d)) = F(c, d)/g c+d. The varety Q(c + d) s just a pont, but equpped wth the acton of G = G c+d. We want to descrbe how the fundamental 16

17 classes of H((c, d), (c + d)) or H((c + d), (c, d)) (whch are somorphc as varetes, but dfferent as correspondences) act on V va Proposton 2.6 and determne n ths way the merge and splt map. They are gven by pullback followed by pushforward n equvarant cohomology va the dagram H BM (Q(c, d)/g) ι ι H BM (F(c, d)/g) q q H BM (Q(c + d)/g), where ι : F(c, d) Q(c, d) s the zero secton of the G-equvarant fbre bundle π : Q(c, d) F(c, d) and q : F(c, d) Q(c + d) s the proper G-equvarant map gven by forgettng the subspaces of dmenson c for any. Proposton 3.4. Let c, d Z e 0 non-zero. The followng dagram commutes (3.3) H (Q(c, d)/g) Borel Λ(c, d) q ι ι q nt E H (Q(c + d)/g) Borel Λ(c + d) where E s the ncluson map from the total nvarants Λ(c + d) nto the nvarants Λ(c, d) followed by multplcaton wth the Euler class (3.4) E := e c +1 d +c =1 j=1 k=c +1 (x +1,j x,k ), and nt s the ntegraton map whch sends an element f to the total nvarant ( ) w (x,j x,k ) (x,l x,m ) e (3.5) ( 1) l(w) 1 1 j<k c c <l<m c +d w(f) c w S c+d =1!d! (x,j x,k ) 1 j<k c +d where l denotes the usual length functon on the symmetrc group. Those readers who are nterested n the case where k s of small postve characterstc mght be worred about (3.5), snce t nvolves dvson by a scalar whch may not be nvertble n k; However, lke dvded dfference operators, these operatons preserve nteger valued polynomals, and so are well-defned maps modulo p for any prme p. Remark 3.5. By conventon, we set E = 1 f ether one of the products n (3.4) s empty or one of the varables x 1,k or x,j does not exst. The degrees of the maps q ι and ι q are agan not the degrees whch one would navely guess, but rather gven by the conventon (2.7), n partcular they are of the same degree. Proof. Snce π s an somorphsm and ι the ncluson of the zero secton, ι s also an somorphsm. On the other hand q s the map to a pont, hence q s just the ncluson of the total nvarants. By the usual adjuncton formula, ι ι (a) = e a, where e s the Euler class of the vector bundle π. To see that the map E s as asserted t s enough 17

18 to verfy the formula E = e for the Euler class. The map s the ncluson of the zero secton of the vector bundle Hom(V,d, V +1,c+1 ), V where V = e =1 V,d s the tautologcal vector bundle on the modul space of quver representaton wth dmenson vector d. As equvarant vector bundles over the maxmal torus of G d, we have a splttng nto lne bundles c +1 V +1,c = L +1,j, j=1 c +d V,d = k=c +1 L,k, where L,k s the tautologcal lne bundle for the correspondng weght space. Thus, c +1 Hom(V,d, V +1,c ) = c +d j=1 k=c +1 L +1,j L,k and the formula (3.4) for the Euler class follows. On the other hand, q s the projecton from the partal flag varety G a /P c,d to a pont, where a = c + d. The formula for equvarant ntegraton on the full flag varety s gven by ( 1) l(w) w f S (3.6) f = a. G a/b e (x,j x,k ) =1 j<k a Thus, we have that f = Q(c,d) F(c,d) P c,d /B c+d 1 e =1 c!d! fd = G c+d /B c+d 1 where D = j<k c (x,j x,k ) c <l<m c +d (x,l x,m ). follows then from (3.6). e =1 c!d! fd, The ntegraton formula 3.2. Demazure operators. The splttng and mergng maps can be descrbed algebracally va Demazure operators actng on polynomal rngs. The th Demazure operator or dfference operator = s acts on k[x 1,... x n ] by sendng f to f s (f) x x +1, where s = (, + 1) denotes the smple transposton actng by permutng the th and ( + 1)th varable. If w = s 1 s 2... s l s a reduced expresson of w S n we defne w = 1 2 s l. Ths s ndependent of the reduced expresson, see [Dem73]. If G s a product of symmetrc groups we denote by w 0 G the longest element. Demazure operators satsfy the twsted dervaton rule (fg) = (f)g + s (f) (g) and more generally for a reduced expresson w = s 1 s 2... s l the formula (3.7) 1 2 l (fg) = A 1 A 2 A l (f)b 1 B 2 B l (g), 18

19 where the sum runs over all possble choces of ether A j = j and B j = d or A j = s j and B j = j for each 1 r. Proposton 3.6. Let (c, d) be a vector composton. (1) Assume c + d = (c + d )α for some, then nt(f) = c w,d, 0 where w c,d 0 S c +d denotes coset representatve of w 0 n S c +d /(S c S d ) of mnmal length. In general, nt(f) s a product of parwse commutng Demazure operators w c,d 0, one for each. (2) Mergng successvely from a vector composton (α, α,..., α ) of length r to rα equals the Demazure operator for w 0 S r, n formulas (3.8) w0 (f) = ( 1) l(w) 1 w(f) w S r 1 <j r (x x j ). (3) The splt c+d nto c and d, where ether c +1 d = 0 for all or c+d = (c +d )α for some, s just the ncluson from Λ(c + d) to Λ(c, d). Proof. Part (3) s obvous, snce E = 1 by Remark 3.5. The second statement s clear for r = 1 and r = 2. The successve merge of the frst r 1 α s s gven by nducton hypothess, hence we only have to merge wth the last α and obtan f y S r 1 S 1 ( 1) l(y) y(f) 1 (r 1)! 1 <j r 1 (x x j ) =: P ( 1) l(z) z(p ) = ( 1) l(z)+l(y) 1 zy(f) z S r/s r 1 S 1 z,y 1 <j r (x x j ). Then (2) follows from the general formula for w0, see [Ful97, 10.12]. Assocatvty of the merges and formula (2) gves nt(f) w0 (c,d ) = w0 (c +d ) = w0 (c,d ) w0 (c,d ), where w 0 (c, d ) and w 0 (c + d ) are the longest elements n S c S d and S c +d respectvely. We have nt(f) = w0 (c,d ), snce w0 (c,d ) surjects to the S c S d -nvarants, and so (1) follows The pctoral nterpretaton. As shown n [VV11], the quver Hecke algebra R(d) s somorphc to the dagram algebra ntroduced by Khovanov and Lauda n [KL09] (modulo the mentoned small dfferences n sgns). Ths result allows to turn rather nvolved computatons n the convoluton algebra nto a beautful dagram calculus. Motvated by these deas, we present now a graphcal calculus for the algebra A where the splt and merge maps from Proposton 3.4 are dsplayed as trvalent graphs. We wll always read our dagrams from bottom to top. We represent the usual (vertcal) algebra multplcaton as vertcal stackng of dagrams, horzontal multplcaton as horzontal stackng of dagrams, 19

20 the dempotent eˆµ as a seres of lnes labeled wth the parts of the vector composton or equvalently the blocks of the resdue sequence (2.2), ˆµ (1) ˆµ (2) ˆµ (r 1) ˆµ (r) the morphsm (c, d) (c + d) as a jonng of two strands c, d, c + d c d the morphsm (c + d) (c, d) as ts mrror mage, c d c + d multplcaton by a polynomal s dsplayed by puttng a box contanng the polynomal. A typcal element of A s obtaned by horzontally and vertcally composng these morphsms. The composton of a merge followed by a splt of the form (c, d) (c + d) (d, c) s also abbrevated as a crossng and denoted (c, d) (d, c). Remark 3.7. Our calculus s an extenson of the graphcal calculus of [KL09]: gven a crossng as n the Khovanov-Lauda pcture, we nterpret t as merge-splt of the kth and (k + 1)th strands: α j α α j α (3.9) α + α j α α j α α j Assume t nvolves the ath α and bth α j n the resdue sequence. If j, + 1, then our map just flps the tensor factors k[x,a ] k[x j,b ] k[x j,b ] k[x,a ]. If = j, then we assocate the Demazure operator k as n [KL09]. For j = + 1, we multply by x +1,a x,b followed by flppng the tensor factors. In each case, ths agrees wth the acton on Pol defned n [KL11], though Khovanov and Lauda have a sngle alphabet of varables, whch they ndex by ther left-poston, as opposed to havng separate alphabets for each node of the Dynkn dagram. We beleve that when one ncorporates non-unt dmenson vectors, the latter conventon s 20