SABIN CAUTIS AND CLEMENS KOPPENSTEINER

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1 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS arxv: v3 [math.rt] 10 Aug 2018 SABIN CAUTIS AND CLEMENS KOPPENSTEINER Abstract. We explan how quantum affne algebra actons can be used to systematcally construct exotc t-structures. The man dea, roughly speakng, s to take advantage of the two dfferent descrptons of quantum affne algebras, the Drnfeld Jmbo and the Kac Moody realzatons. Our man applcaton s to obtan exotc t-structures on certan convoluton varetes defned usng the Belnson Drnfeld and affne Grassmannans. These varetes play an mportant role n the geometrc Langlands program, knot homology constructons, K-theoretc geometrc Satake and the coherent Satake category. As a specal case we also recover the exotc t-structures of Bezrukavnkov Mrkovć [BM] on the (Grothendeck )Sprnger resoluton n type A. Contents 1. Introducton 1 2. Prelmnares 7 3. Structure results t-structures on the symmetrc sde t-structures on the skew sde Combnng the skew and symmetrc sdes Examples of categorcal actons Induced t-structures Some fnal remarks 33 Appendx A. From the Drnfeld Jmbo to the Kac Moody presentaton 36 Appendx B. Some deformaton theory 38 References Introducton Consder a semsmple complex group G and let Rep(G) denote the usual monodal category of fnte-dmensonal representatons of G. Consder further the Langlands dual group G and ts affne Grassmannan Gr = G ((t))/g [[z]]. The abelan category P(Gr) of G [[z]]-equvarant perverse sheaves on Gr s monodal wth respect to the convoluton product. The geometrc Satake correspondence of Mrkovć Vlonen [MV] states that there s an equvalence of monodal categores Rep(G) = P(Gr). 1

2 2 SABIN CAUTIS AND CLEMENS KOPPENSTEINER Subsequently P(Gr) s called the Satake category. There s a parallel story one can try to pursue on the coherent sheaf sde namely f one replaces constructble sheaves wth coherent sheaves. The analogue of perverse sheaves n then the noton of perverse coherent sheaves, n the sense of [AB]. Ths noton s n general not as well behaved. One of the obvous ssues s that over R (.e. on the constructble sheaf sde) one can always use the mddle perversty functon p(x) = 1 2 dm R(x). On the coherent sheaf sde ths choce s not always possble snce dm C (x) may not always be even. However, when t s, choosng p(x) = 1 2 dm C(x) leads to a nce theory theory whch ncludes, for nstance, the noton of ntermedate extenson [AB, Secton 4]. In the case of Gr the G [[z]]-orbts have the same party on any connected component. Ths allows us to use the mddle perversty functon to obtan a well behaved abelan category PCoh(Gr). Moreover, as shown n [BFM, Secton 8], the convoluton product on Gr preserves PCoh(Gr). Hence, PCoh(Gr) s monodal and we call ths the coherent Satake category. The coherent Satake category P Coh(Gr) behaves qute dfferently from P(Gr) and s nterestng n ts own way. For example, t s closely ted to 4d N = 2 gauge theory. Although not semsmple t carres the structure of a monodal cluster category [CW]. In order to understand P Coh(Gr) one needs to understand the convoluton product and subsequently the convoluton varetes, as we now explan Convoluton varetes. The G [[z]]-orbts of Gr are ndexed by domnant weghts of G. For such a weght λ we denote by Gr λ the closure of the correspondng orbt n Gr. For a sequence λ = (λ 1,...,λ n ) of such weghts we can consder the convoluton product varety Gr λ = Gr λ 1 Gr λn. Ths varety comes equpped wth a natural map to Gr whose mage s Gr λ. The resultng map m : Gr λ Gr λ, whch s sem-small, s used to defne the convoluton product on PCoh(Gr). Example. To llustrate the type of geometry nvolved let us consder the smplest nontrval case. We take G = SL 2 and λ 1 = λ 2 = ω where ω s the fundamental weght. Then Gr (ω,ω) s somorphc to the natural bundle compactfcaton of T P 1 (the cotagent bundle of P 1 ) whle Gr 2ω s somorphc to the quadrc cone n P 3. The map m : Gr (ω,ω) Gr 2ω s then the proecton whch collapses the zero secton (.e. the 2 curve) n T P 1 to a pont. Note that Gr (ω,ω) s smooth whle Gr 2ω has one sngular pont. These convoluton spaces are nterestng n themselves. For nstance, n [CK1] a plan was lad out for usng these varetes to categorfy the Reshetkhn Turaev lnk nvarants. Ths program was subsequently carred out for G = SL m n a seres of papers. Meanwhle n [CK5] these spaces are used to suggest a (quantum) K-theoretc analogue of the geometrc Satake equvalence. One of the goals of ths paper s to study natural t-structures on the derved category of coherent sheaves of these convoluton varetes. As mentoned above, the varetes Gr λ carry a natural t-structure correspondng to perverse coherent sheaves.

3 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 3 We would lke to fnd a lft of these t-structures to Gr λ. Snce the dmensons of the G [[z]]-orbts n Gr λ do not have the same party, the category of perverse coherent sheaves on Gr λ s not a good choce. In ths paper we wll acheve ths goal n the caseg=sl m andλ = (ω k1,...,ω kn ) where ω 1,...,ω m 1 are the fundamental weghts of SL m. Followng our notaton from earler papers we wll wrte k = (k 1,...,k n ) and denote Gr λ n ths case as Y(k). One of our man results, Theorem 8.6, defnes and characterzes a certan famly of natural t-structure on Y(k). The argument behnd Theorem 8.6 nvolves a general approach to constructng t-structures usng categorcal actons of quantum affne algebras. Before explanng ths n more detal (see Secton 1.3) we descrbe further motvaton for ths paper comng from an approach to provng some conectures by Lusztg Lusztg s conectures and the work of Bezrukavnkov, Mrkovć and Rumynn. Let G be a semsmple group wth Le algebra g. Lusztg [L] conectured the exstence and propertes of certan canoncal bases n the Grothendeck group of Sprnger fbers. These bases can be used to derve precse nformaton about numercal nvarants of g-modules. Recently, Bezrukavnkov and Mrkovć [BM] proved most of these conectures. A key part of ther argument s the constructon of two exotc t-structures on the Sprnger resoluton π: Ñ N of the nlpotent cone N (as well as on the Grothendeck Sprnger resoluton π: g g). The frst t-structure s called the representaton theoretc (RT) exotc t-structure. It s defned by a locally free tltng bundle E on g, whose pullback to Ñ s also tltng. The bundle E s constructed by repeatedly applyng certan reflecton functors R for Î (the affne Dynkn dagram of g) to the structure sheaf O g. The most dffcult part of ths argument s provng the necessary vanshng condton Ext >0 (E,E) = 0 (see [BM, Secton 2.4]). Ths s done n two steps. Frst one reduces to the correspondng vanshng on the formal neghborhood of the zero secton n g [BM, Secton 2.5.1]. Then one uses the derved equvalence, studed n [BMR], between the category of coherent sheaves on ths formal neghbourhood and the derved category of g-modules wth a fxed generalzed central character. Ths requres passng to postve characterstc and then lftng back up. The second t-structure, called the perversely exotc t-structure, s defned usng the same tltng bundle E together wth the category of perverse coherent sheaves, n the sense of [AB], on N. Ths t-structure can also be descrbed as follows. From [ArkB] there exsts an equvalence (1) Φ: D G (Ñ) D(Perv Fl ) between the derved category of G-equvarant coherent sheaves on Ñ and the derved category of ant-sphercal perverse sheaves on the affne flag varety Fl of the dual group. Under ths equvalence the perversely exotc t-structure on D G (Ñ) corresponds to the perverse t-structure on D(Perv Fl ) [BM, Lemma 6.2.4]. The equvalence Φ s ultmately used to reduce a key step n the proof of Lusztg s conectures to a purty statement n D(Perv Fl ).

4 4 SABIN CAUTIS AND CLEMENS KOPPENSTEINER The perversely exotc t-structure was already studed n [B]. That paper consdered a functor Ψ: D G C (Ñ) D(U q mod 0 ) (whch s almost an equvalence) and showed that rreducble perverse exotc sheaves on D G C (Ñ) correspond to ndecomposable tltng obects n U q mod 0. Here U q s Lusztg s quantum envelopng algebra at a prmtve root of unty q C and U q mod 0 s the block contanng the trval module nsde the category U q mod of fnte dmensonal graded U q -modules. The current paper s also motvated by our wsh to better understand these t- structures and to fnd an alternatve constructon (see Secton 9). In partcular, we wanted to avod havng to pass to postve characterstc or havng to use a deep result such as the derved equvalence from [BMR]. Ths s because n other cases, such as our convoluton varetes Y(k), no such equvalences are known (or expected to exst) Quantum affne algebra actons. We refer to Secton 2.1 for the precse defnton of a categorcal U q (Lgl n ) acton. One of the man results of ths paper, Theorems 4.1 and 5.1, explans how one can use an acton of U q (Lgl n ) to construct exotc t-structures. More precsely, suppose we start start wth an acton of U q (Lgl n ) on a graded trangulated category K = K(k) k=(k 1,...,k n) where multplcaton of q acts as an nternal shft {1} (rather than a cohomologcal shft [1]). Then, havng fxed a t-structure on the hghest weght category K(η) where η = (0,...,0,N), Theorem 4.1 says that under some mld assumpton on the t-structure on K(η), there exsts a unque t-structure on k K(k) such that (1) t matches the chosen t-structure on K(η) and (2) the generators {E,F : Î} of U q(lgl n ) act by t-exact functors. Then Theorem 5.1 explans what to do f we have an acton on a graded trangulated category K = k K(k) where q acts as a cohomologcal shft. We can apply these results to our convoluton varetes as follows. Frst we consder certan varetes Y(k), whch are analogous to Y(k), but occur n the geometry of the Belnson Drnfeld Grassmannan rather than the affne Grassmannan. In [CK4] an acton of U q (Lgl n ) on k D(Y(k)) s constructed where q acts as a C - equvarance shft {1}. Ths allows us to apply Theorem 4.1 wth K(k) := D(Y(k)) to obtan exotc t-structures on k D(Y(k)) (cf. Theorem 8.5). Then we take K(k) := D(Y(k)). In the papers [CKL1, CKL2, C2] one defned an acton of U q (gl n ) on K(k), whch we extend to an acton of U q (Lgl n ) n Secton 7.2. We then apply Theorem 5.1 to obtan exotc t-structures on k D(Y(k)) (cf. Theorem 8.6). As a consequence of these results one mmedately recovers the RT exotc t- structure on D( g) (Corollary 9.3) as well as the RT and perversely exotc t-structures on D(Ñ) (Corollary 9.5) n the case when g = gl m.

5 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS Drnfeld Jmbo vs. Kac Moody. The quantum group U q (Lgl n ) has two realzatons: the Drnfeld Jmbo (DJ) realzaton (sometmes refered to as the loop realzaton) and the Kac Moody (KM) realzaton. The DJ realzaton s generated by {E t,f t : I and Z} whereas the KM one s generated by E,F where Î (the vertex set of the affne Dynkn dagram). When t s mportant to dstngush between these two we wll wrte U q (Lgl n ) DJ or U q (Lgl n ) KM. At the categorcal level there s also a dstncton between these two. In a categorcal U q (Lgl n ) KM the relatons between functors are gven by somorphsms. For example, an mportant one s the sl 2 relaton E F 1 λ = F E 1 λ [ λ,α ] 1 λ where 1 λ denotes the dentty functor on the weght space ndexed by λ. In partcular, t makes sense for U q (Lgl n ) KM to act on addtve (graded) categores. Such actons have been studed n much detal [CR, KL, Ro, C3]. On the other hand, the defnton of a categorcal acton ofu q (Lgl n ) DJ s less clear (see [CK4] for a workng defnton). What s apparent however, s that wth such an acton one needs some sort of trangulated structure to express all the relatons between functors. For example, f α,α = 1, then one requres an somorphsm of cones Cone(E E,1 1 α E,1 E ) = Cone(E E,1 1 β E,1 E ) where α and β are the unque nonzero maps (here E,1,E,1 are the functors correspondng to E t,e t). In partcular, t makes sense that U q (Lgl n ) DJ should act on trangulated (graded) categores. To pass between the KM and DJ realzatons one wrtes E 0 and F 0 as a lnear combnatons of compostons of E t and F t. The analogous result at the categorcal level of ths s explaned n Appendx A. It turns out that E 0,F 0 are gven by complexes of functors correspondng to E t,f t. Ths suggests that, at the categorcal level, the equvalence of these two realzatons s acheved as an equvalence between homotopy categores: roughly speakng an equvalence between an U q (Lgl n ) DJ acton and the homotopy category of a U q (Lgl n ) KM acton. Now consder an acton of U q (Lgl n ) DJ on a trangulated (graded) category K. Swtchng to the Kac Moody realzaton one obtans an acton of U q (Lgl n ) KM on K. In deal crcumstances, the trangulated category K would be the homotopy category of an addtve category on whch U q (Lgl n ) KM acts. A weaker expectaton s that K should be equpped wth a t-structure where the generators ofu q (Lgl n ) KM s t-exact. The purpose of ths paper s to make ths expectaton precse. Our man applcatons are geometrc (Sectons 7 and 8). Ths s not surprsng snce, gong back to work of Nakama [N] on quver varetes, actons of U q (g) on categores of coherent sheaves often extend to an acton of U q (Lg) actons the extra loop structure usually comng by way of lne bundles. When g s of fnte type t makes sense to expect that the generators of U q (Lg) KM are exact wth respect to some exotc t-structure.

6 6 SABIN CAUTIS AND CLEMENS KOPPENSTEINER 1.5. Further remarks. The approach n ths paper should gve an explct way to understand the rreducble obects n the heart of these t-structures. We expect that these are gven as the mage of the rreducble obects n the hghest weght category under affne forest-lke maps. Such maps are morphsms n U q (Lgl n ) KM assocated to a collecton of affne trees (trees are dscussed n Secton 3.1). In many cases the hghest weght category s the category of graded vector spaces so there s only one rreducble (up to gradng shft). Ths not only gves a combnatoral way to dentfy the rreducbles but also an algebrac/algorthmc way to compute the Ext s between them namely these Ext-spaces should be expressble n terms of quver Hecke algebras (whch act by natural transformatons n a categorcal acton of U q (Lgl n ) KM ). One case where such rreducbles have been studed n more detal s that of two-block Sprnger fbers [AN]. In that paper, Anno and Nandakumar show that rreducbles are ndexed n a natural way by affne crossngless matchngs. In our language, these affne crossngless matchngs correspond to affne forests where each strand s labeled 1. More precsely, strands are generally labeled from {1,...,m 1} for some m N, but the fact that we are dealng wth two-block Sprnger fbers mples that m = 2. When m = 2 affne forests are the same as affne crossngless matchngs. In order to drectly extend our constructon of exotc t-structures to arbtrary Gr λ we need an affne brad group acton on such varetes. Although such an acton s not known n general t s possble to defne (buldng on some recent results of the frst author) n the specal case λ = (k 1 ω 1,...,k n ω 1 ). Ths wll allow us to also defne exotc t-structures for these spaces n an upcomng paper. One of the nterestng results from [BM] s Property ( ) from Secton Ths s a vanshng statement for certan Exts whch nvolves the nternal gradng { } on D G (Ñ). Ths gradng, whch s nduced by the natural C acton on Ñ, corresponds under the equvalence Φ from (1) to the gradng by Frobenus weghts. Bezrukavnkov and Mrkovć then apply the Purty Theorem of [BBD] to obtan ths vanshng. A smlar vanshng result, namely the Postvty Lemma [B, Lemma 9], s one of the man tools n that paper. The proof of ths Lemma also uses (1) and the Purty Theorem. It would be nterestng to obtan these result drectly (and n greater generalty) from the acton of U q (Lgl n ) wthout havng to turn to the equvalence Φ or the Purty Theorem. Ths would nvolve defnng weght structures (a.k.a. co-t-structures) n the sense of [Bon, Pa]. Fnally, perverse exotc sheaves on the convoluton varetes Y(k) should be thought of analogues of perverse coherent sheaves on the affne Grassmannan (the coherent Satake category). Although there are some concrete statements one can make (for example, the pushforward of a perverse exotc sheaf to the affne Grassmannan s perverse coherent) the relatonshp of perverse exotc sheaves to the coherent Satake category s somethng that begs further study. Acknowledgements. We were supported by NSERC through a dscovery/accelerator grant. S.C. would lke to thank Ivan Mrkovć for sharng, back n 2009, a prelmnary verson of [BM] whch turned out to be nsprng (and a bt amusng).

7 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 7 2. Prelmnares We wll always work over the base feld C although much (f not everythng) of what we do should work n arbtrary characterstc. For n 1 we denote by [n] the quantum nteger q n 1 +q n 3 + +q n+3 +q n+1. By conventon [ n] = [n]. If f = a f aq a N[q,q 1 ] and A s an obect nsde a graded category K we wrte f A for the drect sum a Z A fa a. For example, n 1 A = A n 1 2k. [n] k= Categorcal actons. By a graded 2-category K we mean a 2-category whose 1-morphsms are equpped wth an auto-equvalence 1. K s dempotent complete f for any 2-morphsm α wth α 2 = α the mage of α s contaned n K. We now recall the noton of a (Lgl n,θ) acton based on [C3]. We wll work wth the weght lattce of gl n, whch we can dentfy wth Z n equpped wth the standard blnear form, : Z n Z n Z. Elements of Z n are denoted k = (k 1,...,k n ). The affne root sublattce n ths case s generated by α = (0,..., 1,1,...,0) for I := {1,...,n 1} where the 1 appears n poston. We wrte α 0 = (1,0,...,0, 1) and Î := I {0}. We also shorten α,α as, for, Î. The type of (Lgl n,θ) actons we consder consst of a target graded, addtve, C-lnear dempotent complete 2-category K where the obects K(k) are ndexed by k Z n and equpped wth: (1) 1-morphsms: E 1 k = 1 k+α E and F 1 k+α = 1 k F where Î and 1 k s the dentty 1-morphsm of K(k), (2) 2-morphsms: for each k Z n, a lnear map span{α : Î} End2 (1 k ). On ths data we mpose the followng condtons. (1) Hom(1 k,1 k l ) s zero f l < 0 and one-dmensonal f l = 0 and 1 k 0. Moreover, the space of maps between any two 1-morphsms s fnte dmensonal. (2) E and F are left and rght adonts of each other up to specfed shfts. More precsely (a) (E 1 k ) R = 1 k F k,α +1 (b) (E 1 k ) L = 1 k F k,α 1. (3) We have E F 1 k = F E 1 k 1 k f k,α 0 [ k,α ] F E 1 k = E F 1 k [ k,α ] 1 k f k,α 0 (4) If I then F E 1 k = E F 1 k. (5) The composton E E decomposes as E (2) 1 E (2) 1 for some 1-morphsm E (2). Moreover, f θ span{α : Î} where θ,α 0 (resp. θ,α =

8 8 SABIN CAUTIS AND CLEMENS KOPPENSTEINER 0) then IθI End 2 (E 1 k E ) nduces a nonzero map (resp. the zero map) between the summands E (2) 1 on ether sde. (6) If α = α or α = α +α for some, Î wth, = 1 then 1 k+rα = 0 for r 0 or r 0. (7) Suppose Î. If 1 k+α and 1 k+α are nonzero then 1 k and 1 k+α +α are also nonzero. In ths paper we wll always assume that K(k) s zero (.e. 1 k = 0) f some k < 0. Thus condton (6) wll be obvous. Condton (7) wll lkewse be obvous. Remark 2.1. Snce K s graded a (Lgl n,θ) acton nduces an acton of U q (Lgl n ) at the level of Grothendeck groups. We chose to wrte (Lgl n,θ) rather than perhaps (U q (Lgl n ),θ) n order to smplfy notaton. In the dscusson above ths acton of U q (Lgl n ) appears n ts Kac Moody (KM) presentaton. Ths n contrast to the acton constructed n [CK4] whch appears n ts Drnfeld Jmbo (DJ) realzaton. When t s not clear from context and we need to dfferentate between the two realzatons we wll wrte (Lgl n,θ) KM or(lgl n,θ) DJ as the case may be (cf. Secton 7) Trangulated categores and convoluton. An addtve 2-category K, such as the ones from the last secton, s sad to be trangulated f for any two obects k,k K the category Hom(k,k ) s trangulated. For example, the homotopy 2- category Kom(K) of any addtve 2-category K s trangulated for the same reasons that the homotopy category of an addtve category s trangulated. Consder now a trangulated category C and a complex of obects d (A,d ) = [A n d n 2 d 1 A1 A0 ] wth A n cohomologcal degree. In the lterature there are varous conventons about how to defne the convoluton of ths complex as an terated cone. We wll convolve from the rght as follows. The convoluton of (A,d) s any obect B such that there exst (1) obects A 0 = B 0,B 1,...,B n = B and (2) morphsm f : A [ ] B 1, g : B 1 B and h : B A [ +1] (wth h 0 = d[1]) such that A [ ] f g h B 1 B s a dstngushed trangle for each and h 1 f = d. Note that n general convolutons may not exst and they may not be unque. But under some reasonable condtons they both exst and are unque, cf. [CK1, Proposton 8.3]. As a partcular case one fnds that the convoluton of a two term complex[a f B] wth B n degree zero s ust Cone(f). On the other hand, f the A belong to the heart of a t-structure then the convoluton s unque and somorphc (n the derved category) to the class of the complex where, as before, A 0 s n degree zero. d A n d n 2 d 1 A1 A0

9 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS t-structures. Suppose C and D are trangulated categores. We say that a set of obects {X s : s S} weakly generates C f Hom(X s [n],y) = 0 for all s S and n Z mples Y = 0. A functor Φ: C D s conservatve f Φ(X) = 0 mples X = 0. Recall that a t-structure on C conssts of two full subcategores C 0, C 0 such that: C 1 C 0 and C 1 C 0, where C n := C 0 [ n] and C n := C 0 [ n]; Hom(C 0,C 1 ) = 0; Every obect X can be embedded n a dstngushed trangle X 0 X X 1 wth X 0 C 0, X 1 C 1. From these axoms t follows that the ncluson C n C has a rght adont, denoted τ n : C C n, whle C n C has a left adont denoted τ n : C C n. Moreover the heart of the t-structure, C = C 0 C 0 s an abelan category and the functor H n = τ n τ n = τ n τ n : C C s a cohomologcal functor. The followng result of Polshchuk wll be our man tool for defnng t-structures. Theorem 2.2 ([Po, Theorem 2.1.2]). Let Φ: C D be a functor of cocomplete trangulated categores whch commutes wth all small coproducts and that admts a left adont functor Φ L : D C. Let C C and D D be full subcategores such that Φ(C) D, Φ L (D) C and Φ(X) D X C for any X C. Assume that D has a t-structure (D 0,D 0 ) such that Φ Φ L : D D s rght t-exact. Then there exsts a (unque) t-structure on C wth C 0 = { X C : Φ(X) D 0}. Moreover the functor Φ s t-exact wth respect to these t-structures. From hereon we wll always assume that whenever we want to apply Theorem 2.2 to a functor of trangulated categores φ: C D we can extend t to a functor Φ: C D between cocomplete trangulated categores whch satsfes all the condtons n the statement of Theorem 2.2. Let us explan why we can do ths. In all the examples we consder the categores C and D wll be bounded derved categores D(X) and D(Y) of coherent sheaves on smooth varetes X,Y. We take C and D to be the correspondng (cocomplete) unbounded derved categores of quas-coherent sheaves D qcoh (X) and D qcoh (Y). Any functor φ: C D wll be gven by a kernel and hence naturally extends to quas-coherent sheaves. Furthermore, all such functors wll have rght adonts and hence commute wth small coproducts by the Adont Functor Theorem. Fnally, φ wll always be conservatve and by Lemma 2.3 below ths means that Φ(X) D X C. Lemma 2.3. Let X and Y be smooth varetes and Φ: D qcoh (X) D qcoh (Y) a functor gven by a kernel n D(X Y). If the restrcton of Φ to D(X) s conservatve then for any F D qcoh (X), Φ(F) D(Y) f and only f F D(X).

10 10 SABIN CAUTIS AND CLEMENS KOPPENSTEINER Proof. It suffces to show the statement for the nfnty-categorcal enhancements of the correspondng categores (the orgnal statement then follows by takng homotopy categores). If X s smooth, then D(X) = Perf(X) can be dentfed wth the subcategory of compact obects of D qcoh (X), and D qcoh (X) s the nd-completon of D(X) as stable -categores (see for nstance [BFN]). Let us frst show that Φ s conservatve on all of D qcoh (X). Equvalently we show that Φ L (D qcoh (Y)) weakly generates D qcoh (X). Suppose F D qcoh (X) and Hom(Φ L (G),F) = 0 for all G D qcoh (Y). We know that Φ L (D(Y)) weakly generates D(X). Thus the nd-completon of Φ L (D(Y)) contans D(X) and hence all of D qcoh (X). Thus Φ L (D(Y)) weakly generates D qcoh (X). Suppose now F D qcoh (X) wth Φ(F) D(Y). We can wrte F as a fltered (homotopy) colmt of compact obects, F = colm α F α. If Φ(F) = colm α Φ(F α ) s compact then there exsts an ndex β such that Φ(F β ) Φ(F) s an somorphsm. Snce Φ s conservatve ths mples that F β F s an somorphsm and hence that F s compact,.e. s an element of D(X). Remark 2.4. The proof of Lemma 2.3 makes essental use of the fact that f X s smooth then D(X) conssts exactly of the compact obects on D qcoh (X). If the underlyng spaces are not smooth, then ths s no longer true. On the other hand, D(X) s always the compact obects n the category of nd-coherent sheaves. Thus for general X and Y one can smply replace quas-coherent sheaves by nd-coherent sheaves n the above arguments. We fnsh ths secton wth some techncal results that we need about t-structures. Lemma 2.5 ([Po, Lemma 1.1.1]). Let Φ: C 1 C 2 be a conservatve t-exact functor between trangulated categores. Then, C 0 1 = { X C 1 : Φ(X) C 0 } 2, C 0 1 = { X C 1 : Φ(X) C 0 } 2. We recall that f (Φ,Ψ) s a par of adont functors, then Φ s rght t-exact f and only f Ψ s left t-exact. In partcular an equvalence of categores s t-exact f and only f ts nverse s t-exact. Lemma 2.6. Let (C 0,C 0 ) be a t-structure on C and suppose X,Y C. If X Y C 0 then already X,Y C 0 and lkewse f X Y C 0 then X,Y C 0. Proof. We prove the frst clam (the second s smlar). Consder the dstngushed trangle X X Y Y 0. Applyng τ >0 we get the dstngushed trangle τ >0 (X) 0 τ >0 (Y) 0 whch mples that τ >0 (X) = τ >0 (Y) = 0. Lemma 2.7. Let (C 0,C 0 ) be a t-structure on C and suppose we have a dstngushed trangle X Y Y[k]

11 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 11 for some k > 0 such that X C 0 and Hom (Y,Y) = 0 for 0. Then Y C 0. Proof. Let φ: Y Y[k] be the second map n the trangle and consder the composton Y φ Y[k] φ[k] Y[2k]. From the octahedral axom we have a trangle Cone(φ) Cone(φ 2 ) Cone(φ[k]). As Cone(φ) = X[1] and Cone(φ[k]) = X[k + 1] are n C 1, so s Cone(φ 2 ). Repeatng ths argument we fnd that Cone(φ 2n ) s n C 1 for all n 1. But φ 2n = 0 for n 0 whch means that Cone(Y 0 Y[2 n k]) C 1. But the left sde s ust Y[1] Y[2 n k]. It follows that Y[1] C 1 and hence Y C 0. Lemma 2.8. Suppose we have two t-structures (C 10, C 10 ) and (C 20, C 20 ) on C such that C 10 C 20 and C 10 C 20. Then the two t-structures are dentcal. Proof. In any t-structurec 0 s the rght orthogonal ofc 1. Hence, C 1 1 C 2 1 mples C 10 C 20. Smlarly one shows that C 10 C Induced brad groups actons. One of the applcatons of categorcal g actons s to the constructon of brad group actons. We recall these results followng [CK2, C2]. Gven an sl 2 acton generated by E,F one consders the Rckard complexes [F (λ) EF (λ+1) 1 E (2) F (λ+2) 2...] f λ 0, [E ( λ) FE ( λ+1) 1 F (2) E ( λ+2) 2...][λ] λ f λ 0, where the left hand terms are n cohomologcal degree zero and where the dfferentals can be defned usng the fact that E and F are adont to each other. These complexes lve n Kom(K) and we denote them [T]. On the other hand, as s explaned n [CKL2], f K s trangulated then [T] has a unque convoluton whch we denote T. Ths allows us to avod havng to pass from K to Kom(K) f K s already trangulated. Remark 2.9. For techncal reasons we use the opposte conventon from [CKL2, CK2, CK4]. Namely, the complexes for T go to the rght rather comng from the left. Ths means that T n ths paper corresponds to T 1 n the old notaton. In partcular, ths means that one needs to take the nverse of a number of results that we use from [CK4]. More generally, gven a (Lgl n,θ) KM acton we get such a [T ] for each Î. Followng [CK2] these satsfy the standard bradng relatons [T ][T ] = [T ][T ] f, = 0 [T ][T ][T ] = [T ][T ][T ] f, = 1.

12 12 SABIN CAUTIS AND CLEMENS KOPPENSTEINER The bradng property s a consequence of the followng more fundamental relatons (2) [T ][T ]E = E [T ][T ] and [T ][T ]F = F [T ][T ] for, = 1 [C2, Corollary 7.3]. The same relatons hold for T,T. Note that f we denote by s ( Î) the generators of the affne Weyl group Ŵn then T 1 k = 1 s kt. It s also useful to consder the followng shfted brad group acton. We set T 1 k := T 1 k [k ] k. Ths notaton agrees wth that from [C2, CK4] (wth the same caveat as before that we have swtched T wth ts nverse). We wll mostly use T n the rest of the paper. These T have several nce propertes. Frst, the correspondng complexes always le n non-postve cohomologcal degrees. Second, they stll satsfy the relatons descrbed n (2). Thrd, we have (3) F (p) T 1 k = T E (p) 1 k p( k,α +p). Ths relaton follows from [C2, Cor. 4.6] by keepng n mnd that our T s a shfted verson of T 1 from that paper. Notce that there s no cohomologcal shft on the rght hand sde of (3). Fnally, the followng s an mportant observaton. Lemma The fnte brad group acton generated by T for I canoncally dentfes any two obects K(k),K(k ) f k can be obtaned from k by rearrangng the poston of some zeros. Proof. The key (and farly clear) observaton s that (T )2 1 k = 1 k f k = 0 or k +1 = 0. Ths means that the brad group acton dscussed above descends to a symmetrc group acton. The result now follows Symmetrc/skew sdes and some termnology. For the applcatons we have planned, the obects K(k) wll be Z-graded trangulated categores, where we denote the nternal gradng by {1}. Our goal s to construct t-structures on these trangulated categores. The man example to keep n mnd s where K(k) s the derved category of coherent sheaves D(Y(k)) on some varety Y(k) and where Hom(k,k ) s D(Y(k) Y(k )) (the space of kernels). The space of 2-morphsms s then ust the space of maps between two kernels. See Secton 7.1 for a bref revew of kernels. We wll always assume that K(k) s zero f k < 0 for some. Moreover, we wll fx N and assume that K(k) s zero f k N. For convenence we wll also take n > N. One can show ths s always possble to do by extendng any Lgl n acton to one of Lgl n+k for any k 0. We do not explan ths n detal snce n all our examples t wll be obvous how to ensure that n > N. We wll consder two cases dependng on the evaluaton of the gradng shft 1 from the (Lgl n,θ) acton. On the symmetrc sde we wll have 1 = {1} whle on the skew sde 1 = [1]{ 1}. To more easly dstngush whch sde we are workng on we wll use K(k) to denote the symmetrc sde and K(k) the skew. The reason

13 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 13 for these names s that n our man applcaton we have K( K(k)) = K C (Y(k)) = Sym k 1 (V) Sym kn (V) K(K(k)) = K C (Y(k)) = Λ k 1 (V) Λ kn (V) where K C (X) denotes the complexfed Grothendeck group of coherent sheaves on X tensored over Z[q,q 1 ] wth C(q) whle Y(k) and Y(k) are certan flag-lke varetes and V s the standard representaton of SL m (for some fxed m). Suppose now we have a fxed a t-structure on k W n kk(k ) where W n s the fnte Weyl group. We wll say that the t-structure s brad postve at k f the endofunctor T actng on k W n kk(k ) s rght t-exact for every Î. Smlarly, we say that a t-structure on k K(k) s brad postve f T actng on t s rght t-exact for every Î. Snce n > N each nonzero weght space K(k) has at least one k = 0. By Lemma 2.10 let us assume that = n. We can then defne R 1 k := T n 2...T 1 T 0 1 k. One can readly check that R 1 k = 1 r k R where r (k 1,k 2,...,k n ) = (k 2,...,k n,k 1 ). Moreover, E R = R E +1 and F R = R F +1 for all Î. Fnally, we denote by η the hghest weght (0,...,0,N) and by µ the mddle weght (0 n N,1 N ). 3. Structure results In ths secton we collect some general results regardng an acton wth a target 2-category K as n Secton 2.5. In partcular, the obects n ths category are ndexed by sequences k Z n. Recall that 1 k = 0 f k < 0 for some or f k N. The man result, Proposton 3.1, dentfes the possble 1-endomorphsms of K(η) where η = (0,...,0,N) s the hghest weght A remark about trees. In [CKM] one used skew Howe dualty to gve a descrpton of the category of SL m -representatons (more precsely, the full subcategory generated by tensor products of fundamental representatons) n terms of webs. For our purposes, the man pont s that webs on an annulus descrbe precsely the U q (Lgl n ) KM relatons n an acton on a 2-category K as above (4) 2 = 2 3 For example, the planar webs above descrbe equvalent 1-morphsms from (3) to (1,1,1) (the web s read from bottom to top). We wll not recall the defnton 3

14 14 SABIN CAUTIS AND CLEMENS KOPPENSTEINER of webs here because we do not use them n a substantal way. However, the relatonshp s conceptually helpful. For example, any planar web between (N) and (1,...,1) can be decomposed as a drect sum of trees lke the one n dagram (4). Moreover, any two such trees (1-morphsms) are equvalent because of relatons lke those llustrated n the dagram. Translatng ths nto our language, ths means that any 1-morphsm between η = (0,...,0,N) and µ = (0 n N,1 N ) that does not nvolve any F 0 or E 0 (.e. the correspondng web s planar) can be decomposed as a drect sum of 1-morphsms equvalent to F (N 3) n 3 F (N 2) n 2 F (N 1) n 1. We call such a morphsm a tree (and lkewse for maps from µ and η). One can also consder a collecton of sde-by-sde trees. For example, fgure (5) depcts a 1-morphsm from (0,0,3,0,2) to (1,1,1,1,1). As above, the space of planar webs from (k) to (1,...,1) s spanned by any choce of a collecton of trees (5) 2 3 Fnally, one can consder such trees embedded on the annulus rather than the plane. We call these annular trees rather than planar trees. These descrbe 1- morphsms that also nvolve E 0 and F 0 (n contrast to planar trees) Endomorphsms of the hghest weght. For N l N we set { A (l) F (l) 0 = F(l) 1 F(l) n 1 f l 0 E ( l) n 1 E( l) n 2 E( l) 0 f l 0 Recall that η denotes the hghest weght (0,...,0,N). Note that A (l) 1 η begns and ends at η. In terms of webs A (±N) corresponds to twstng the strand labeled N around the annulus (clockwse or counterclockwse dependng on the sgn). In partcular, ths means that A (N) and A ( N) are nverses of each other. The followng s the man result n ths secton. Proposton 3.1. Any 1-endomorphsm of K(η) s a drect summand of an element of the algebra generated by A (l) 1 η for N l N and ±1. Remark 3.2. It actually suffces to take A (l) only for 1 l N and l = N. In the remander of ths secton we wll prove ths result. For any two 1-morphsms X, Y: K(k) K(k ) we wrte X Y f X s a drect summand of f Y for some f N[q,q 1 ]. For example, we have F (a) F (b) F (a+b). More generally, f {Y α : α S} s any fnte collecton of 1-morphsms K(k) K(k ) we wrte X {Y α : α S} 2

15 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 15 f X s a drect summand of α f α Y α for some f α N[q,q 1 ]. Note that s a transtve relaton. Lemma 3.3. Let k be any weght and a,b N. Then E (a) F (b) 1 k { F (b l) F (b) E (a) where F (c) = E (c) = 0 for c < 0. 1 k { E (a l) E (a l) 1 k : l 0 } F (b l) 1 k : l 0 } Proof. Let us prove the frst clam (the second s smlar). Snce s a transtve relaton and F (a) F (b) F (a+b) F (a) F (b) (and lkewse for Es) t suffces to prove the case a = b = 1. If k,α 0 then the result s clear sncee F 1 k = F E 1 k [ k,α ] 1 k. Otherwse, f k,α 0 then E F 1 k s a drect summand of F E 1 k and we are done agan. Lemma 3.4. Let k be any weght,, = 1 and a,b,c N. Then F (a) F (b) F (c) 1 k {F (l) F (a+c) F (b l) 1 k : 0 l b}. Proof. We use nested nductons on b and then on a. The case b = 1 s [CK2, Corollary 4.8]. So assume that b 2. If a = 1, then F F (b) F (c) F (b 1) F F F (c) by [CK2, Corollary 4.8]. By the base case b = 1 ths s then a drect summand of a drect sum of shfts of F (b 1) and F (b 1) F F (c+1) F (b) Now let a > 1. Then F (c+1). F (a) F (b) F (c) F (a) F F (b 1) F (c) F (a 1) F F F (b 1) F (c). F (c+1) F By nducton on b appled to F F (b 1) F (c) ths s a drect summand of a sum of shfts of F (a 1) F F (l) F (c+1) F (b 1 l) F (a 1) F (1+l) F (c+1) F (b 1 l), 0 l b 1. If l = b 1, ths s of the requred form by nducton on a. Otherwse t s of the requred form by nducton on b. By anf-strng we wll mean a sequence of the formf (lm) m...f (l 1) 1 for0 n 1 and l N. We call m the length of the F-strng. Lemma 3.5. Let X1 η be any 1-endomorphsm K(η). Then X1 η {(A ( N) ) a YY1 η } for a fnte collecton of a Y 0 and F-strngs Y. Proof. We use nducton on the number of E s n X. If there are none, we are done. Otherwse put all Es to the left of the sequence usng Lemma 3.3. Ths does not ncrease the number of Es. Thus we can assume X = E (l) n 1 X where X has fewer Es. But then A (N) X = F (N) 0 F (N) n 2 F(N) n 1 E(l) n 1 X (N) = F 0 F (N) n 2 F(N l) n 1 X and the result follows by nducton.

16 16 SABIN CAUTIS AND CLEMENS KOPPENSTEINER We say an F-strng X = F (lm) m...f (l 1) 1 s strctly ordered f +1 = 1 mod n and l 0 for all. In ths case we wrte # 0 X for the number of wth = 0 (the number of F (l) 0 n the strng). For l 0,..., l n 1 N let F (l 0,...,l n 1 ) denote F (l 0) 0...F (l n 1) n 1. For smplcty n the rest of ths secton we wll take the all ndces of F modulo n. Lemma 3.6. Suppose X s a strctly ordered strng. Then 1 η XF (a) {1 η Y} where Y belongs to a fnte collecton of strctly ordered strngs. Moreover, # 0 Y # 0 X unless = 0 and X = X F (b) n 1 n whch case # 0Y = # 0 X+1. Proof. We can assume 1 η X = 1 η F ( ) F( ) F( ) F( ) F( ) 0 1 n 1 0 where the ( ) represent nonzero exponents. Note that 1 η X has to begn on the left wth a F ( ) 0 because otherwse t s zero. Note also that there mght be many appearances of F ( ) 0 n the strng for X. Our argument s by nducton on the length of X. If = or = + 1 we are done. Otherwse we start movng the F (a) n 1 η XF (a) to the left untl t hts an F +1. Ths way we end up wth (6) 1 η F (b) F (c) +1 F(a) F ( ). Now, by Lemma 3.4, we get (7) 1 η XF (a) {1 η F (l) +1 [F(a+b) F (c l) F( ) +1 ] : 0 l c}. If c = l then we are done by nducton on the length. Otherwse the expresson n the brackets s strctly ordered and we contnue movng the F (l) +1 to the left. Ths procedure must end snce the strng s fnte. Fnally we need to explan why the number off ( ) 0 does not ncrease (unless = 0). In the argument above the only tme an extra F ( ) 0 appears s when, n equaton (6), = n 1. In ths case the rght sde of equaton (7) reads {1 η F (l) 0 [F(a+b) n 1 F(c l) 0 F ( ) ] : 0 l c}. We now contnue movng the F (l) 0 to the left. It wll eventually ht an F ( ) 1 at whch pont we wll have somethng of the form 1 η F ( ) F( ) 0 1 F(l) 0 whch, upon smplfyng agan, decreases the number of F ( ) 0. So overall the number of F( ) 0 does not ncrease. The only excepton s f = 0 and = n 1 n whch case the algorthm mmedately stops and we get an extra F ( ) 0. Corollary 3.7. For any F-strng X we have 1 η X {1 η Y} where Y vares over a fnte collecton of strctly ordered F-strngs. Proof. Ths follows by applyng Lemma 3.6 repeatedly startng on the left. Lemma 3.8. Let X be an F-strng. Then 1 η X1 η { A (l 1) A (lr) 1 η } where r 0 and l w X for some fxed w X whch depends on X.

17 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 17 Proof. By Corollary 3.7 we can assume that X s strctly ordered. If # 0 X 1 then 1 η X1 η = 0 s ether zero or equal to 1 η X1 η = A (l) 1 η for some l. So assume that # 0 X 2 and proceed by nducton on # 0 X. Wrte X1 η = X F (b 0,...,b n 1 ) F (a 0,...,a n 1 ) 1 η for some strctly ordered F-strng X wth a 0 and b 0 for all. Further we can assume that a 0 a 1 a n 1 as otherwse the functor vanshes. We wll now nduct on the sum a 0 + +a n 1. If a 0 = a 1 = = a n 1, then F (a 0,...,a n 1 ) = A (a 0) so that X F (b 0,...,b n 1 ) A (a 0) and we are done by nducton on # 0 X. Otherwse let be the largest ndex such that a a 1. We have to dstngush whether = n 1 or not. Frst assume that 0 < < n 1. Let k = (a 1 a 0,...,a n 1 a n 2,N+a 0 a n 1 ),.e. such that 1 k F (a 0,...,a n 1 ) 1 η. Then we have k = a a 1 > 0 and k +1 = a +1 a = 0 so that 1 k F E : X F (b 0,...,b n 1 ) F (a 0,...,a n 1 ) 1 η X F (b 0,...,b n 1 ) F E F (a 0,...,a n 1 ) 1 η = [ X F (b 0,...,b n 1 ) ] F F (a 0,...,a 1,a 1,a +1,...,a n 1 ) 1 η { YF (a 0,...,a 1,a 1,a +1,...,a n 1 ) 1 η }, where by Lemma 3.6 Y belongs to a fnte set of strctly ordered F-strngs wth # 0 Y # 0 X + 1 = # 0 X 1. Thus ether we are done by nducton on # 0 X, or otherwse by nducton on a. If = n 1, then, usng Lemma 3.4, we have X F (b 0,...,b n 1 ) F (a 0,...,a n 1 ) 1 η = X F (b 0) 0 F (b n 1) n 1 F (a 0) 0 F (a n 1) n 1 1 η = X F (b 0) 0 [F (b n 1) n 1 F (a 0) 0 F (a n 1 a n 2 ) n 1 { X F (b 0) 0 F (b n 2) n 2 F (l) 0 F(b n 1+a n 1 a n 2 ) n 1 ] (a F 1 ) 1 F (a n 2) n 2 F (a n 2) n 1 1 η F (a 0 l) 0 F (a 1) 1 F (a n 2) n 2 F (a } n 2) n 1 1 η wth 0 l a 0. If l 0, we use Lemma 3.6 appled to ( X F (b 0) 0 F (b ) n 2) (l) n 2 F 0, to reduce each of the functors n the set to functors YF (b n 1+a n 1 a n 2 ) n 1 F (a 0 l,a 1,...,a n 2,a n 2 ) 1 η where Y s strctly ordered wth # 0 Y # 0 X +1 = # 0 X 1 (f l = 0 ths s form s obtaned trvally). If l = a 0, we can use Lemma 3.6 to reduce ths further to strngs wth less F ( ) 0 than X and apply nducton on # 0 X. Otherwse we can apply nducton on a. Lemma 3.8 above proves Proposton 3.1 n the case of F-strngs. The general case follows by applyng Lemma t-structures on the symmetrc sde In ths secton we work on the symmetrc sde. Ths means that we have a (Lgl n,θ) KM acton on a trangulated 2-category K where the obects K(k) are

18 18 SABIN CAUTIS AND CLEMENS KOPPENSTEINER graded trangulated categores ndexed by k Z n and where 1 = {1} s the nternal gradng. Snce [k] k = [k]{ k} the complexes T now become T 1 k = T 1 k [k ]{ k }. Recall that µ denotes the weght (0 n N,1 N ) and η the weght (0,...,0,N) and that 1 k = 0 f k < 0 for some or k N. Theorem 4.1. Suppose that K(µ) s weakly generated by the mage of K(η) under the Lgl n acton. Moreover, suppose that K(η) s equpped wth a t-structure such that for all N l N the 1-endomorphsms A (l) of K(η) are t-exact. Then one can unquely extend ths t-structure to all other categores K(k) so that E and F are t-exact for all Î. Moreover, ths t-structure s brad postve. Proof. Frst note that every K(k) s weakly generated by the mage of K(η). Ths s because for each K(k) we have a 1-morphsm ψ k : K(k) K(µ) correspondng to a planar tree (see Secton 3.1). Ths 1-morphsm has the property that ψ L k ψ k contans at least one copy of the dentty 1-morphsm of K(k). So f X K(µ) weakly generates then ψ L k (X) K(k) also weakly generates snce Hom(ψ L k (X)[],Y ) = 0 Hom(X[],ψ k(y)) = 0 ψ k (Y) = 0 Y = 0. Next we defne the t-structure on K(k). Choose a 1-morphsm ψ: K(η) K(k) whose mage weakly generates K(k) and such that ψ R ψ contans at least one copy of the dentty 1-morphsm (ths s possble snce we can ust add nto ψ a copy of the planar tree morphsm from K(η) to K(k)). Now ψ R ψ s an endomorphsm of K(η) and hence, by Proposton 3.1, ψ R ψ {A (l 1) A (l k) } for some fnte collecton of l 1,...,l k Z. By assumpton A (l 1) A (l k) s t-exact. Now, 1 = {1} s t-exact and so by Lemma 2.6 t follows that ψ R ψ s t-exact. We can now apply Theorem 2.2 to ψ R : K(k) K(η) to obtan an nduced t-structure on K(k). Snce the mage of ψ weakly generates K(k), ψ R s conservatve. Thus the t-structure on K(k) s gven by (8) K(k) 0 = { A K(k) : ψ R (A) K(η) 0}, K(k) 0 = { A K(k) : ψ R (A) K(η) 0}. The t-structure above does not depend on the choce of ψ as long as t weakly generates. Ths s because gven two choces ψ, ψ we can consder the t-structures defned by the 1-morphsms ψ and ψ ψ respectvely. Then, usng (8), Lemma 2.6 mples that K(k) 0 and K(k) 0 for the latter are contaned n the ones for the former. Hence by Lemma 2.8 the two t-structures are equal. Lkewse, ψ ψ and ψ defne the same t-structure. Hence ψ and ψ also nduce the same t-structure.

19 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 19 From ths t follows that E and F are exact. More precsely, to show E 1 k s t- exact t suffces to show that ψ R E 1 k s t-exact. But ths s t-exact by the dscusson above. Moreover, F s exact snce t s badont to E (up to some { } shft whch s exact). To see unqueness of the t-structure suppose that there exst another t-structure on all K(k) extendng the t-structure on K(η) such that E and F are t-exact for all Î. Then the functors ψr defned above wll also be t-exact. Thus (8) shows that the two t-structures agree. Fnally, we have to show that for ths t-structure the functors T are rght t-exact. We assume that k k +1 (the case k k +1 s smlar). Set A = E (k ) F (k +1 ) for 0. Then T 1 k s the convoluton of the complex A k A 1 A 0 where A 0 s n cohomologcal degree zero. Snce each A s t-exact t follows that T s rght t-exact. 5. t-structures on the skew sde In ths secton we work on the skew sde. Ths means that we have an (Lgl n,θ) KM acton on a trangulated 2-category K where the obects K(k) are graded trangulated categores ndexed by k Z n and where 1 = [1]{ 1} where [1] s the cohomologcal gradng and {1} s the nternal gradng. Snce [k] k = {k} we now have T 1 k = T 1 k {k }. Recall that µ denotes the weght (0 n N,1 N ) and that 1 k = 0 f k < 0 for some or k N. Theorem 5.1. Suppose we have a brad postve t-structure on K(µ ). µ W n µ Then there exsts a unque t-structure on kk(k) determned by exactness of the 1- morphsm ψ: K(k) K(µ) correspondng to a collecton of planar trees. Moreover, ths t-structure s brad postve. Remark 5.2. The 1-morphsm ψ: K(k) K(µ) above s conservatve because the composton ψ R ψ contans at least one summand 1 k. Hence Lemma 2.5 mples that the t-structure n Theorem 5.1 s gven by K(k) 0 = { A K(k) : ψ(a) K(µ) 0} K(k) 0 = { A K(k) : ψ(a) K(µ) 0}. In the rest of ths secton we wll prove ths theorem Some prelmnares. Let us denote by k the number of k 0. The dea for provng Theorem 5.1 s to use the categorcal acton to defne a t-structure on all K(k) by usng the gven t-structure on K(µ). We wll do ths by (decreasng) nducton on k

20 20 SABIN CAUTIS AND CLEMENS KOPPENSTEINER Lemma 5.3. Consder k wth k > 0 and k +1 = 1. Then we have dstngushed trangles for some X End(K(k)). T T 1 k 1 k {2} X X E F 1 k [k ]{ k +2} E F 1 k [k +2]{ k }. Proof. Ths follows from Proposton 6.8 of [C1] where t s shown that [T ][T ]1 k s homotopc to a complex of the form 1 k {2} E F 1 k [k ]{ k +2} E F 1 k [k +2]{ k }. Note that the extra shft of {2} here s ust a result of our conventons for T. Corollary 5.4. Suppose agan that we have k wth k > 0 and k +1 = 1. If (T )2 1 k s rght t-exact then E E L 1 k s rght exact. Proof. Snce F 1 k = E L 1 k [k ]{ k } we get dstngushed trangles T T 1 k 1 k {2} X X E E L 1 k{2} E E L 1 k[2]. From the frst trangle t follows that X s rght exact. From the second trangle t then follows by Lemma 2.7 that E E L 1 k s also rght exact Defnng the t-structure on K(k). We wll now defne the t-structure on every K(k). Ths wll be done nductvely and usng only the gl n part of the Lgl n acton (the entre acton wll be dealt wth subsequently). By assumpton we have a t-structure on K(k) where k = N (ths s the base case). Suppose we have nductvely constructed t-structures on all K(k) wth k > r and that T for = 1,...,n 1 are rght t-exact for these t-structures. Now consder some k wth k = r. By Lemma 2.10 we can assume that there exsts an ndex wth k > 1 and k +1 = 0. Consder the 1-morphsm E 1 k. By nducton we have a t-structure on K(k+α ) so that T s rght t-exact for = 1,...,n 1. Corollary 5.4 mples that E E L 1 k+α s rght t-exact. Subsequently, we get an nduced t-structure on K(k) usng Theorem 2.2. Moreover, snce E 1 k s conservatve n ths case (snce F E 1 k contans several copes of the dentty) we fnd that the t-structure s gven by K(k) 0 = { A K(k) : E (A) K(k +α ) 0} K(k) 0 = { A K(k) : E (A) K(k +α ) 0}. Lemma 5.5. The t-structure on K(k) defned above s ndependent of any choces made along the way. Proof. By constructon, the t-structure on K(k) s the unque t-structure wth K(k) 0 = { A K(k) : ψ(a) K(µ) 0} where ψ: K(k) K(µ) s the functor assocated to some planar tree from k to µ. As remarked n Secton 3.1 even though ths tree s not unque all such trees gve

21 EXOTIC T-STRUCTURES AND ACTIONS OF QUANTUM AFFINE ALGEBRAS 21 the same 1-morphsms. In ths sense, the resultng t-structure on K(k) does not depend on any choces made n defnng the other K(k ) wth k > r. Corollary 5.6. If k = 0 or k +1 = 0 then T 1 k s t-exact. Proof. Ths s ust because for 1-morphsm ψ: K(s k) K(µ) correspondng to a planar tree, the composton ψt 1 k corresponds to another planar tree (snce k = 0 or k +1 = 0). So now we can assume that we have t-structures for all K(k) wth k r. The next step s to show thatt for = 1,...,n 1 are rght t-exact for these t-structures. Proposton 5.7. Usng the t-structure on K(k) defned above T s rght t-exact for = 1,...,n 1. Proof. Consder some T 1 k. Usng Corollary 5.6 we can assume that (say) k +2 = 0. To show T 1 k s rght exact t suffces to show that E +1 T 1 kt +1 s, snce 1 kt +1 s t-exact by Corollary 5.6. But E +1 T 1 kt +1 = T T +1 1 s +1 k+α E 1 s+1 k and T T +1 1 s +1 k+α s rght exact by nducton. Moreover, E 1 s+1 k s t-exact by constructon. The result follows Extendng to the affne case. Recall that R 1 k s defned ast n 2...T 1 T 0 1 k f k n = 0 (f k n 0 then we use the fnte brad group acton to move a zero nto that spot). Lemma 5.8. The functor R 1 k : K(k) K(r k) s rght t-exact. Proof. We prove ths by decreasng nducton on k. The base case follows snce R 1 (k1,...,k n,0) = T n 2...T 1T 01 (k1,...,k n,0) and each term on the rght hand sde s rght exact by our basc hypothess. To prove the nducton step consder now the composton K(0,k 1,...,k n,0) R R K(k 2,...,k n,0,0,k 1 ) F n+1 K(k 2,...,k n,0,1,k 1 1). The frst R 1 (0,k1,...,k n,0) s t-exact snce T 0 1 (0,k 1,...,k n,0) s the dentty (and hence exact). Thus the composton s rght exact f and only f R 1 (k1,...,k n,0,0) s rght exact. On the other hand, the composton also equals R R 1 (1,k1 1,...,k n,0)f 1 whch s rght exact by decreasng nducton on k. Corollary 5.9. The functor R 1 k : K(k) K(r k) s t-exact. Proof. In Lemma 5.8 we proved that R 1 k s rght exact. Note that n the descrpton of R 1 k as T n 2...T 1 T 0 1 k each T equals ts nverse because t always exchanges a zero wth somethng. Thus a smlar argument to that used n Lemma 5.8 also shows that R 1 k s left exact. Remark Notce that the proofs used n Lemma 5.8 and Corollary 5.9 also work to show that R 1 k : K(k) K(r k) s t-exact on the symmetrc sde.