EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES. 1. Introduction

Transcription

1 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES RINA ANNO, VINOTH NANDAKUMAR Abstract. We study the exotc t-structure o D, the derved category of coheret sheaves o two-block Sprger fbre (.e. for a lpotet matrx of type (m +, ) type A). The exotc t-structure has bee defed by Bezrukavkov ad Mrkovc for Sprger theoretc varetes order to study represetatos of Le algebras postve characterstc. Usg work of Cauts ad Kamtzer, we costruct fuctors dexed by affe tagles, betwee categores of coheret sheaves o dfferet two-block Sprger fbres (.e. for dfferet values of ). After checkg some exactess propertes of these fuctors, we descrbe the rreducble objects the heart of the exotc t-structure o D ad eumerate them by crossgless (m, m + 2) matchgs. We compute the Ext s betwee the rreducble objects, ad show that the resultg algebras are a aular varat of Khovaov s arc algebras. I subsequet work we wll make a lk wth aular Khovaov homology, ad use these results to gve a characterstc p aalogue of some categorfcato results usg two-block parabolc category O (by Berste-Frekel-Khovaov, Bruda, Stroppel, et al). 1. Itroducto Let G be a sem-smple Le group, wth Le algebra g, flag varety B ad lpotet coe N. It s well-kow that there s a atural map π : T B N whch s a resoluto of sgulartes (kow as the Sprger resoluto). Gve e N, let B e = π 1 (e); these varetes are kow as Sprger fbers, ad are of specal terest represetato theory. For stace, type A, Sprger showed that the top cohomology of a Sprger fber ca be equpped wth a represetato of the Weyl group, ad further realzes a rreducble represetato. Ths specal case whe G = SL(m + 2), ad the lpotet e has Jorda type (m +, ), s easer to uderstad, ad has bee studed extesvely. I [18], Stroppel ad Webster study the geometry ad combatorcs of these two-block Sprger fbers ad vestgate coectos wth Khovaov s arc algebras. I [16], Russell studes the topology of these varetes, ad descrbes a certa bass the Sprger represetato. I [8], Bezrukavkov ad Mrkovc troduce exotc t-structures o derved categores of coheret sheaves o Sprger theoretc varetes, order to study the modular represetato theory of g. These exotc t-structures are defed usg a certa acto of the affe brad group B aff o these categores, whch was defed by Bezrukavkov ad Rche (see [10]). More precsely, let k be a algebracally closed feld of characterstc p wth p > h (here h s the Coxeter umber), ad let g be a arbtrary reductve group defed over k. Let λ h k be tegral ad regular; ad let e N (k) be a lpotet. Let Mod fg,λ e (Ug) be the category of modules wth geeralzed cetral character (λ, e). Theorem from [9] (see also Secto from [8]) states that there s a equvalece: (1) D b (Coh Be,k ( g k )) D b (Mod fg,λ e (Ug k )) 1

2 2 RINA ANNO, VINOTH NANDAKUMAR Further, t s prove that the tautologcal t-structure o the derved category of modules, correspods to the exotc t-structure o the derved category of coheret sheaves. Here we wll study exotc t-structures for the case of two-block Sprger fbers type A (e. for a lpotet of Jorda type (m +, )), gve a descrpto of the rreducble objects the heart of the t-structure, ad the Ext spaces betwee these rreducbles. I [11], Bruda ad Stroppel show that the prcpal block of parabolc category O p, for the parabolc p wth Lev gl m gl sde gl m+, s govered by a dagram algebra that s closely related to Khovaov s arc algebra. Further, work by Berste-Frekel-Khovaov (see [4]) ad Stroppel (see [17]) shows that Reshetkh-Turaev varats for sl 2, dexed by lear tagles, may be categorfed by certa fuctors betwee these categores. Usg the machery developed ths paper, future work we wll gve a characterstc p aalogue of ths story; (Ug) ca be thought of as a characterstc p aalogue of O p. Whle the former costructo ( [4]) aturally gves rse to Khovaov homology, the characterstc p settg oe wll obta aular Khovaov homology (whch was developed by Grgsby, Lcata ad Wehrl [6]). See secto 6.2 ad 6.3 for more detals. the category Mod fg,λ e Now let us descrbe the cotets of ths paper more detal Two-block Sprger fbers. I Secto 1, we recall the defto ad some propertes of two-block Sprger fbers, ad defe the categores that we wll be studyg. Let m 0 be fxed; ad let Z 0 vary. Cosder the Le algebra g = sl m+2, ad deote the lpotet coe of sl m+2 (the varety cosstg of lpotet matrces of sze m + 2) by N. Deote by z the stadard lpotet of type (m +, ): z = Let B be the flag varety for GL m+2. The Sprger resoluto s T B : B = {(0 V 1 V m+2 = C m+2 ) dm V = } T B = {(0 V 1 V m+2 = C m+2, x) x sl m+2, xv V 1 } The atural projecto π : T B N s a resoluto of sgulartes. The two-block Sprger fber s the varety B z = π 1 (z ) = {(0 V 1 V m+2 ) B z V V 1 }

3 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 3 The Mrkovc-Vyborov trasverse slces S g s a varat of the Slodowy slce. The followg varety s of terest, sce t s a resoluto of S N. U = π 1 (S ) T B = {(0 V 1 V m+2 = C m+2, x) x S, xv V 1 } Let D = D b (Coh Bz (U )) be the bouded derved category of coheret sheaves o U, whch are supported o B z. These are the categores that we wll be studyg Affe tagles. I Secto 2, we recall the defto, ad some propertes, of affe tagles. Defto. Let p, q be postve tegers of the same party. A (p, q) affe tagle s a embeddg of p+q arcs ad a fte umber of crcles to the rego {(x, y) C R 1 x 2}, such 2 that the ed-pots of the arcs are (1, 0), (ζ p, 0),, (ζp p 1, 0), (2, 0), (2ζ q, 0),, (2ζq q 1, 0) some order (where ζ k = e 2π k ). Defto. Let ATa be the category wth objects {k} for k Z 0, ad the morphsms betwee p ad q cosst of all affe (p, q) tagles (up to sotopy). The above defto s cosstet, sce a (p, q) affe tagle α, ad a (q, r) affe tagle β, the β α s a (p, r) affe tagle. We recall the well-kow presetato of ths category usg geerators ad relatos. The geerators cosst of cups, g, whch are ( 2, ) tagles; caps, f, whch are (, 2) tagles, crossgs, t (1), t (2) ad rotatos r, r, whch are (, ) tagles. The relatos are lsted Defto 3.7. I ths paper we work wth the category AFTa of affe framed tagles that has addtoal geerators w (1) ad w (2) that twst the framg of the th strad Fuctors assocated to affe tagles. Defto. Let AFTa m be the full subcategory of AFTa, cotag the objects {m + 2} for Z 0. A weak represetato of the category AFTa m s a assgmet of a tragulated category C for each Z 0, ad a fuctor Ψ(α) : D p D q for each affe framed (m + 2p, m + 2q) tagle α, such that the relatos betwee tagles hold for these fuctors:.e. f β s a (m + 2q, m + 2r)-tagle, the there s a somorphsm Ψ(β) Ψ(α) Ψ(β α). The ma result of ths secto s a costructo of a weak represetato of AFTa m usg the categores D above. To do ths, we mmc the strategy used by Cauts ad Kamtzer [5], where they costruct a weak represetato of the category OTa of oreted (o-affe) tagles, usg slghtly larger categores The exotc t-structure o D. I Secto 5, we recall the defto of exotc t- structures (troduced by Bezrukavkov ad Mrkovc [8]), ad descrbe how they are related to the acto of affe tagles costructed above. Let B aff be the affe brad group. As a specal case of the costructo Secto 1 of [8] (see also Bezrukavkov-Rche, [10]), we have a acto of B aff o D (e. for every b B aff, there exsts a fuctor Ψ(b) : D D, ad a somorphsm Ψ(b 1 b 2 ) Ψ(b 1 ) Ψ(b 2 ) for b 1, b 2 B aff ). It turs out that B aff ca be detfed as a subgroup of the mood of

4 4 RINA ANNO, VINOTH NANDAKUMAR (m + 2, m + 2)-tagles; ad uder ths detfcato, the acto of B aff cocdes wth the acto costructed above. Let B + aff B aff be the semgroup geerated by the lfts of the smple reflectos s α the Coxeter group Waff Cox. Bezrukavkov-Mrkovc s costructo [8] specalzes to gve a exotc t-structure o D, whch s defed as follows: D 0 = {F RΓ(Ψ(b 1 )F) D 0 (Vect) b B + aff } D 0 = {F RΓ(Ψ(b)F) D 0 (Vect) b B + aff } We also prove that the cup fuctors Ψ(g ) are exact wth the exotc t-structures, ad sed rreducble objects to rreducble objects (Theorem 5.6) Irreducble objects the heart of the exotc t-structure o D. I Secto 6, we gve a descrpto of the rreducble objects the exotc t-structure o D, ad compute the Ext spaces betwee them. Let Cross(m, ) be the set of affe (m, m + 2) tagles, where the m er pots are ot labelled, the m+2 outer pots are labelled, ad whose vertcal projectos to C do ot have crossgs. For every α Cross() we have a fuctor Ψ(α) : D 0 D ; let Ψ α = Ψ(α)(C) (here C D b (Vect) D 0 ). We show that that {Ψ α α Cross(m, )} costtute the rreducble objects D 0 (Proposto 5.10). We also prove that for β Cross(m, ), Ext (Ψ α, Ψ β ) s gve by the below formula. Here Λ deotes a complex D b (Vect) cocetrated degrees 1 ad 1; ad ˇα s the (m + 2, m) affe tagle obtaed by vertg α. A a (m, m) affe tagle γ wth o crossgs s sad to be good f t has o cups or caps, ad ω(γ) deote the umber of crcles preset. I Theorem 5.17, we prove that Ext (Ψ α, Ψ β ) = { Λ ω(ˇα β) [ ] f ˇα β s good 0 otherwse We also gve a cojectural descrpto of the multplcato the algebra Ext ( Ψ α ) α Cross(m,) 1.6. Further drectos. I the equvalece (1), the heart of the exotc t-structure s detfed wth a abela category of modules over Ug havg a fxed cetral character. Thus the smple objects that we have classfed the heart of the exotc t-structure wll correspod to rreducble represetatos wth that fxed cetral character. I future work, we pla to study these modules (e.g. compute dmesos, ad gve character formulaes) by usg our descrpto of these exotc sheaves. Usg techques developed by Cauts ad Kamtzer, we ca show the Grothedeck group of the category D ca be aturally detfed wth V m+2 [m], the m-weght space V m+2

5 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 5 (here V = C 2, cosdered as a sl 2 represetato). By lookg at the mages of the fuctors Ψ(α) the Grothedeck group, we obta a map ˆψ : {(m + 2k, m + 2l)-affe tagles} Hom(V m+2k [m], V m+2l [m] ) We expect that ths map wll cocde wth a well-kow varat for affe tagles, ad that the mages of the rreducble objects Ψ α the Grothedeck group wll be the caocal bass (or perhaps the dual caocal bass). Ispred by Khovaov s costructo [15] ad [12], we also expect that t wll be possble to gve a alterate categorfcato of ˆψ, usg categores of modules over the Ext algebras cotrollg D (whch closely resemble Khovaov s arc algebras) Ackowledgemets. We would lke to thak Roma Bezrukavkov, for suggestg ths project to us, ad for umerous helpful dscussos ad sghts. We would also lke to thak Paul Sedel for suggestg the study of m = 0 case to the frst author a whle ago. We are also grateful to Joel Kamtzer, Mkhal Khovaov, Cathera Stroppel, Be Webster ad Davd Yag for may helpful dscussos; ad to Athoy Hederso for help wth the proof of Lemma 2.8. The frst author would lke to thak the Uversty of Pttsburgh ad the secod author would lke to thak the Uversty of Sydey, where part of ths work was completed. 2. Two-block Sprger fbres 2.1. Trasverse slces for two-block lpotets. Fx m 0. For Z 0, let z be the stadard lpotet of Jorda type (m +, ). Let S sl m+2 deote the Mrkovc-Vyborov trasverse slce to the lpotet z (see secto [7]): S = {z + a e m+, + b e m+2, } 1 m+2 {1,,,m++1,,m+2} Defto 2.1. Deote by N the lpotet coe for sl m+2. Let B deote the complete flag varety for GL m+2 (C), ad for 0 < k < m + 2 defe the varetes P k, as follows: P k, = {(0 V 1 V k V m+2 = C m+2 )}. The the varetes T B, T P k, ca be descrbed as follows: T B = {(0 V 1 V m+2 = C m+2, x) x sl m+2, xv V 1 }; T P k, = {(0 V 1 V k V m+2 = C m+2 ), x x sl m+2, xv k+1 V k 1, xv V 1 for k, k + 1}. Pck a bass e 1,..., e m++1, f 1,..., f +1 of C m+2+2 so that z +1 e = e 1, z +1 f j = f j 1 (where we set e 0 = f 0 = 0). Lemma 2.2. For ay x S +1 such that dm(ker x) = 2, we have Ker x = Ce 1 Cf 1, ad there s a atural somorphsm φ x : xv m+2+2 C m+2.

6 6 RINA ANNO, VINOTH NANDAKUMAR Proof. By the costructo [7, secto 3.3.1] we ca assume that xe = e 1 + a e m++1 + c f m+1 f m + 1, xe = e 1 + a e m++1 f > m + 1, ad xf j = f j 1 + b j e m++1 + d j f m+1. The we have: ( x λ e + ) ν j f j = ( λ +1 e + a λ + ) b j ν j e m++1 1 m++1 1 j m m+ 1 j m ν j+1 f j + ( 1 m+1 1 m++1 a λ + 1 j m+1 1 j m+1 d j ν j ) f m+1 So xv = x ( 1 m++1 λ e + 1 j m+1 ν ) jf j = 0 mples that λ = ν j = 0 for, j > 1,.e. that v Ce 1 Cf 1. If xv = 0 t follows that a 1 = b 1 = c 1 = d 1 = 0. So: { ( xv m+2+2 = λ e + a +1 λ + ) b j+1 ν j e m m+ 1 m+ 1 j 1 j µ j f j + ( 1 m c +1 λ + 1 j d j+1 ν j ) f +1 } Let γ m, : C m+2+2 = ( 1 m+ Ce 1 j Cf j) (Ce m++1 Cf +1 ) ( 1 m+ Ce 1 j Cf j) deote the atural projecto map. Now φ x := γ m, xvm+2+2 : xv m m+ Ce 1 j Cf j s a somorphsm. Proposto 2.3. For every 0 < k < m we have a somorphsm of varetes S +1 slm+2+2 T P k,+1 S slm+2 T B. Proof. By defto: S +1 slm+2+2 T P k,+1 ={(0 V 1 V k V m+2+2, x) x S +1, xv k+1 V k 1, xv V 1 for k, k + 1}; S slm+2 T B = {(0 W 1 W m+2 = C m+2, y) y S, yw W 1 }. Sce x S +1, the Jorda type of x s a two-block partto, ad dm(ker(x)) 2; but xv k+1 V k 1 so we must have xv k+1 = V k 1. Cosder the flag (0 V 1 V k 1 = xv k+1 xv k+2 xv m+2+2 ). Recall the somorphsm φ x : xv m+2+2 C m+2 from Lemma 2.2 ad deote by Φ(x) Ed(C m+2 ) the edomorphsm duced o C m+2 by the acto of x o xv m+2+2. Costruct a map α : S +1 slm+2+2 T P k,+1 T B as follows: α(0 V 1 V m+2, x) = = ((0 φ x (V 1 ) φ x (V k 1 ) = φ x (xv k+1 ) φ x (xv k+2 ) C m+2 ), Φ(x)) We clam that α gves the requred somorphsm S +1 slm+2+2 T P k,+1 S slm+2 T B. Frst we check that Φ(x) S. From the argumet Lemma 2.2, Φ(x)e = e 1 + a +1 e m+ + c +1 f f, Φ(x)e = e 1 +a +1 e m+ f >, ad Φ(x)f j = f j 1 +c j+1 e m+ +d j+1 f. Thus Φ gves a bjecto betwee {x S +1 N +1 dm(ker x) = 2} ad S N. It follows that α has mage S slm+2 T B ad that α s a somorphsm oto ts mage, as requred.

7 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 7 Let us defe the varetes ad categores that we are gog to use throughout the paper. Defto 2.4. Uder the Sprger resoluto map π : T B N, let B z = π 1 (z ). Let U = S slm+2 T B ={(0 V 1 V m+2, x) x S, xv j V j 1 j}. X, = S slm+2 T P, P, B ={(0 V 1 V m+2, x) x S, xv +1 V 1, xv j V j 1 j}. Defe D = D b (Coh Bz (U )) to be the bouded derved category of coheret sheaves o U supported o B z. Note that X, s a closed subvarety of U of codmeso 1. O the other had, the projecto of X, oto S slm+2 T P,, whch s by Proposto 2.3 somorphc to U 1, s a P 1 -budle. Ideed, the fber over each pot (0 V 1 V k V m+2+2, x) s somorphc to P(V +1 /V 1 ) Descrpto of the geeral setup. Our geometrc setup s gog to be be smlar to that of [5], so we wll descrbe both alogsde ad pot out the depedeces ad the dffereces. Cosder a 2(m+2)-dmesoal vector space V m, wth bass e 1,..., e m+2, f 1,..., f m+2 ad a lpotet z such that ze = e 1, zf = f 1. Let W m, V m, deote the vector subspace wth bass e 1,..., e m+, f 1,..., f, so that z Wm, has Jorda type (m +, ); we wll detfy W m, wth V m+2. Let P : V m, W m, deote the projecto defed by P e = e f m +, P e = 0 f > m + ; P f = f f, P f = f f >. I Secto 2 of [5], the followg four seres of varetes are defed (for m = 0): Y m+2 = {(L 1 L m+2 V m, ) dm L =, zl L 1 }; Q m+2 = {(L 1 L m+2 ) Y m+2 P (L m+2 ) = W m, }; X m+2 = {(L 1 L 2 L m+2 ) L +1 = z 1 (L 1 )}; Z m+2 = {(L, L ) Y m+2 Y m+2 L j = L j j }. I the otato of [5], the varety Q m+2 should be deoted by U m+2, but we chose to call t Q m+2 here to avod the cofuso wth our U. The relatoshps betwee these varetes are as follows: Q m+2 Y m+2 s a ope subset, ad X m+2 Y m+2 s a closed subset. Moreover, X m+2 s fbered over Y m+2 2 wth fber P 1, ad thus ca be cosdered a closed subset Y m+2 Y m+2 2. Cauts ad Kamtzer use the categores D(Y m+2 ) for ther categorfcato, ad utlze the varetes X m+2 Y m+2 Y m+2 2 ad Z m+2 Y m+2 Y m+2 to costruct Fourer-Muka fuctors that geerate the tagle category acto. We are gog to use the varetes U ad X, U U 1 from Defto 2.4 that have smlar propertes, amely X, U s a closed embeddg, ad X, U 1 s a P 1 budle. We are gog to use the categores D = D b (Coh Bz (U )) for the categorfcato, ad the varetes X, wll provde the cup ad cap tagle geerators (see Secto 3 for the descrpto of the tagle category). Whle t s true that there s a embeddg U Y m+2 (see secto 2.3 below) ad uder ths embeddg we ca detfy X, U X m+2, certa geometrc facts such as Lemma?? below do ot follow drectly from ther couterparts [5]. We wll ot eed the aalogue of Z m+2 to descrbe

8 8 RINA ANNO, VINOTH NANDAKUMAR the crossg geerators sce our Theorem 4.15 allows us to defe the crossg geerators ad prove most tagle relatos wthout drect computatos wth Fourer-Muka kerels The varetes U. We are gog to show that U s a somorphc to a closed subvarety of Q m+2 ad thus a locally closed subvarety of Y m+2. To do ths, we recall that U S slm+2 T B ad preset Q m+2 a smlar form. Defto 2.5. S = { a 1 a 2 a m+ b 1 b 2 b }. c 1 c 2 c m+ d 1 d 2 d Note that we have S S, ad U S slm+2 T B. Now we ca prove the followg lemma: Lemma 2.6. Gve x S N, there exsts a uque subspace L m+2 V m,, wth P L m+2 = W m,, such that zl m+2 L m+2 ad P zp 1 = x. Proof. Sce P L m+2 = W m,, to specfy the subspace L m+2 t suffces to specfy ẽ := P 1 (e ) = e + a (k) e m++k + c (l) f +l 1 k f j := P 1 (f j ) = f j + 1 k 1 l m+ b (k) j e m++k + 1 l m+ d (l) j f +l Suppose for 1 m +, 1 j, xe = e 1 + a e m+ + c f, xf j = f j 1 + b j e m+ + d j f ; the the detty P zp 1 = x s equvalet to a (1) = a, c (1) = c, b (1) j = b j ad d (1) j = d j. The statemet zl m+2 L m+2,.e. zẽ, z f j L m+2, s equvalet to sayg that: 1 k zẽ = ẽ 1 + a ẽ m+ + c f z f j = f j 1 + b j ẽ m+ + d j f Expadg the above two equatos: e 1 + a (k) e m++k 1 + c (l) f +l 1 = e 1 + +a e m+ + f j k +b j e m+ + 1 k 1 l m+ a (k) m+e m++k + b (k) j e m++k k c (l) m+f +l 1 l m+ 1 l m+ a (k) m+e m++k + 1 k +c f + d (l) j f +l 1 = f j 1 + c (l) m+f +l 1 l m+ 1 k +d j f + a (k) 1 e m++k + 1 k 1 l m+ b (k) e m++k + b (k) j 1 e m++k + 1 k 1 l m+ b (k) e m++k + c (l) 1 f +l+ d (l) f +l 1 l m+ d (l) j 1 f +l+ d (l) f +l 1 l m+ ;.

9 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 9 Extractg coeffcets of e m++k ad f +l the above two equatos gves: a (k+1) c (l+1) = a (k) 1 + a a (k) m+ + c b (k), b (k+1) j = b (k) j 1 + b ja (k) m+ + d j b (k) = c (l) + a c (l) m+ + c d (l), d (l+1) j = d (l) j 1 + b jc (l) m+ + d j d (l) Cosder the matrx coeffcets (x k ) p,q for 1 p, q m + 2. It follows by ducto that we have a (k) = (x k ) m+,, b (k) j = (x k ) m+,m++j, c (l) = (x l ) m+2,, d (l) j = (x l ) m+2,m++j. Ideed, the case where k = l = 1 s clear; ad the ducto step follows from expadg the equato (x r+1 ) uv = 1 w m+2 (xr ) uw (x) wv for u = m + ad u = m + 2. Usg the above recursve defto of a (k) a (+1) = b (+1) j = 0 ad c (m++1), b (k) j, c (l), ad d (l) j, t remas to prove that = d (m++1) j = 0. Thus we must show that (x +1 ) m+,p = (x m++1 ) m+2,p = 0 gve 1 p m+2. Usg the equato (x r+1 ) uv = 1 w m+2 (x) uw(x r ) wv, we compute that: (x +1 ) m+,p = (x +2 ) m+ 1,p = = (x m+2 ) 1,p = 0 (x m++1 ) m+2,p = (x m++2 ) m+2 1,p = = (x m+2 ) m++1,p = 0 Ths completes the proof of the exstece ad uqueess of a z-stable subspace L m+2 V m, wth P L m+2 = W m, ad P zp 1 = x. Now we ca prove the followg geeralzato of Proposto 2.4 [5]: Lemma 2.7. There s a somorphsm Q m+2 S slm+2 T B. Proof. Gve (L 1 L m+2 ) Q m+2, sce P : L m+2 W m, s a somorphsm, we have a lpotet edomorphsm x = P zp 1 Ed(V m+2 ) (here we detfy W m, ad V m+2 ). If P 1 e = e + v, where v les the spa of e m++1,, e m+2, f +1,, f m+2, the zp 1 e = e 1 + v where v s the spa of e m+,, e m+2 1, f,, f m+2 1. Hece P zp 1 e = xe spa(e 1, e m+, f ), ad smlarly xf spa(f 1, e m+, f ); so x S. Thus we have a map α : Q m+2 S slm+2 T B gve by α(l 1,, L m+2 ) = (P zp 1, (P (L 1 ), P (L 2 ),, P (L m+2 ))). For the coverse drecto, from the below Lemma 2.6 we kow that gve x S N there exsts a uque z-stable subspace L m+2 V m, such that P L m+2 = W m, ad P zp 1 = x; call ths subspace L m+2 = Θ(x). We have a somorphsm P : Θ(x) W m,. Thus gve a elemet ((0 V 1 V m+2 ), x) S slm+2 T B, let β(x) = (0 P 1 V 1 P 1 V 2 Θ x ). It s clear that α ad β are verse to oe aother The varetes X,. We have a P 1 -budle π, : X, S slm+2 T P, S 1 slm+2 2 T B 1 = U 1, ad the embeddg of the dvsor j, : X, S slm+2 T B = U. Thus we ca vew X, as a subvarety of U 1 U. Lemma 2.8. For j, the varetes X, ad X,j tersect trasversely sde U.

10 10 RINA ANNO, VINOTH NANDAKUMAR Proof. We wll vew U (ad also X, ad X,j ) as a subvarety of G B, ad compute taget spaces to X, ad X,j at pots X, X,j to show trasversalty. Gve (g, x) G B ; frst we wll calculate the taget space T (g,x) (G B ). Gve X 1 g, X 2, a curve through (g, x) G wth taget drecto (g X 1, X 2 ) s (g exp(ɛx 1 ), x+ ɛx 2 ). Iftesmally, (g exp(ɛx 1 ), x + ɛx 2 ) = (g, x) G B provded that X 1 b (e. exp(ɛx 1 ) B), ad exp(ɛx 1 )(x + ɛx 2 )exp( ɛx 1 ) x Dscardg o-lear powers of ɛ, the latter traslates to x + ɛ(x 2 + [X 1, x]) = x,.e. X 2 = [X 1, x]. Thus the kerel of the map g = T (g,x) (G ) T (g,x) (G B ) s the subspace {(X, [X, x]) X b}, so: T (g,x) (G B g ) {(X, [X, x]) X b}. Suppose (g, x) G B les U ; or equvaletly, that x := gxg 1 S. Now gve (X 1, X 2 ) T (g,x) (G B ), we have that (X 1, X 2 ) T (g,x) (U ) whe the curve (g exp(ɛx 1 ), x+ ɛx 2 ) les U. Ths happes precsely whe g exp(ɛx 1 )(x + ɛx 2 )exp( ɛx 1 ) g 1 S (ftesmally). Dscardg o-lear powers of ɛ, ths s equvalet to sayg that g (x + ɛ(x 2 + [X 1, x]) g 1 S. Sce gxg 1 S, ths s equvalet to X 2 + [X 1, x] g 1 C g (recall that S = z + C where C s a vector subspace). Thus: T (g,x) (U ) {(X 1, X 2 ) g X 2 + [X 1, x] g 1 C g} {(X, [X, x]) X b} {(X, Y ) g C [X, x] + Y g } g b 0 For the last somorphsm, use the substtuto X = gx 1 g 1, Y = g(x 2 + [X 1, x])g 1. Recall from the dscusso Secto 1.4 of [7] that the map π : g C g, π(x, Y ) = [X, x] + Y s surjectve. Hece: dm(t g,x (U )) = dm() + dm(c ) dm(b) I partcular, ths shows that U s smooth. Now suppose that (g, x) X, X,j. It s clear that X, = U (G B ), where s the lradcal of the mmal parabolc correspodg to. The above argumet s vald after replacg wth, ad we obta: T (g,x) (X, ) {(X, Y ) g C [X, x] + Y g } g b 0 T (g,x) (X,j ) {(X, Y ) g C [X, x] + Y g j } g b 0 Usg the surjectvty of π, t s clear that T (g,x) (X, ) ad T (g,x) (X,j ) are dstct co-dmeso 1 subspaces T (g,x) (U ). Hece T (g,x) (X, ) + T (g,x) (X,j ) = T (g,x) (U ), ad X, ad X,j tersect trasversely U. Corollary 2.9. The followg tersectos are trasverse: (1) π12 1 (X, ) π23 1 (X,j ) sde U 1 U U 1 for j. (2) π12 1 (X, ) π23 1 (X +1,j ) sde U 1 U U +1.

11 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 11 Proof. Both statemets follow usg Lemma 5.3 from [5]; for the frst, we also eed Lemma Affe tagles. 3. Tagles Defto 3.1. If p q (mod 2), a (p, q) affe tagle s a embeddg of p+q arcs ad a 2 fte umber of crcles to the rego {(x, y) C R 1 x 2}, such that the edpots of the arcs are (1, 0), (ζ p, 0),, (ζp p 1, 0), (2, 0), (2ζ q, 0),, (2ζq q 1, 0) some order; here ζ k = e 2π k. Remark 3.2. Gve a (p, q) affe tagle α, ad a (q, r) affe tagle β, we ca compose them usg scalg ad cocateato. Ths composto s assocatve up to sotopy. The composto β α s a (p, r) affe tagle. Defto 3.3. Gve 1, defe the followg affe tagles: Let g deote the ( 2, ) tagle wth a arc coectg (2ζ, 0) to (2ζ +1, 0). Let other strads coect (ζ 2, k 0) to (2ζ, k 0) for 1 k < ad (ζ 2, k 0) to (2ζ k+2, 0) for + 1 < k 2. Let f deote the (, 2) tagle wth a arc coectg (ζ, 0) ad (ζ +1, 0). Let other strads coect (ζ, k 0) to (2ζ 2, k 0) for 1 k < ad (ζ, k 0) to (2ζ 2, k 2 0) for + 1 < k 2. Let t (1) (respectvely, t (2)) deote the (, ) tagle whch a strad coectg (ζ, 0) to (2ζ +1, 0) passes above (respectvely, beeath) a strad coectg (ζ +1, 0) to (2ζ, 0). Let other strads coect (ζ, k 0) to (2ζ, k 0) for k, + 1. Let r deote the (, ) tagle coectg (ζ, j 0) to (2ζ j 1, 0) for each 1 j (clockwse rotato of all strads), ad let r deote the (, ) tagle coectg (ζ, j 0) to (2ζ j+1, 0) for each 1 j (couterclockwse rotato). The fgure below has dagrams depctg some of these elemetary tagles; see s 4 4 s defed below Defto 3.9.

12 12 RINA ANNO, VINOTH NANDAKUMAR Defto 3.4. Defe a lear tagle to be a affe tagle that s sotopc to a product of the geerators g, f, t (1) ad t (2) for. Remark 3.5. Lear tagles ca be moved away from the half-le e ɛ R 0 where ɛ s a small postve umber. If we cut the aulus 1 z 2 by that le ad apply the logarthm map, lear tagles tur to the usual tagles that lve betwee two parallel les. Lemma 3.6. Ay affe tagle s sotopc to a composto of the above geerators. Proof. For a curve C, defe ts affe crtcal pot as a pot where ths curve s taget to a crcle wth ceter at 0. We ca adjust a tagle wth ts sotopy class so that ts projecto oto C has a fte umber of trasversal crossgs ad affe crtcal pots. We ca also assume that o two of these pots le o the same crcle wth ceter at 0. Cut the projecto of the tagle by crcles wth ceter at 0 to aul so that each aulus cotas oly oe crossg or affe crtcal pot. We ca further adjust the tagle so that we have a tagle sde each aulus, ad by costructo these tagles have to be g, f, or t (p), possbly composed wth a power of r. Defto 3.7. Let ATa (resp. Ta) deote the category wth objects k for k Z 0, ad the set of morphsms betwee p ad q cosst of all affe (resp. lear) (p, q) tagles. I the category ATa we record the followg relatos betwee the above geerators; here let 1 1, 1 p, q 2, k 2: (1) (Redemester 0) f g +1 = f +1 g = d (2) (Redemester 1) f t ±1 (2) g = f t ±1 (1) g = d (3) (Redemester 2) t (1) t (2) = t (2) t (1) = d (4) (Redemester 3) t (1) t +1 (1) t (1) = t +1 (1) t (1) t +1 (5) (Cup-cup sotopy) g +k +2 g = g +2 g +k 2 (6) (Cap-cap sotopy) f +k 2 (7) (Cup-cap sotopy) g +k 2 (8) (Cup-crossg sotopy) g t +k 2 2 (9) (Cap-crossg sotopy) f t +k f+2 = f f+2 +k f = f+2 g +k (q) = t +k (q) = t +k 2 2 (10) (Crossg-crossg sotopy) t (p) t +k +2, g f +k 2 (q) g, g +k (q) f, f +k (q) = t +k (q) t (p) (1). (11) (Ptchfork move) t (1) g +1 = t +1 (2) g, t (2) g +1 = t +1 (12) (Rotato) r r = r r = d (13) (Cap rotato) r 2 f r = f +1, f 1 (14) (Cup rotato) r g r 2 = g +1, r 2 g 1 (15) (Crossg rotato) r t (q) r = t +1 (q), r 2 = f +k +2 g +2 t 2(q) = t (q) g +k t (q) = t 2(q) f +k (1) g. r 2 = f 1 = g 1 t 1 (q) r 2 = t 1 (q). By Lemma 4.1 from [5], ay relato betwee lear tagles ca be expressed as a composto of the relatos (1)-(11) above. We ca geeralze that to affe tagles: Proposto 3.8. Ay relato betwee affe tagles ca be expressed as a composto of the relatos (1)-(15) above. Proof. Frst, let us reduce ay relato to a composto of relatos (1)-(11) volvg g, f, t (p) for 1 (for the defto of g, f, t (p) see the proof of Lemma 3.6). The, we

13 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 13 ca express the relatos (1)-(11) volvg g, f, t (p) usg relatos (1)-(15), by a drect computato. Let us call a sotopy lear f t fxes a segmet of the form [(ζ, 0), (2ζ, 0)] for some ζ. Note that a lear sotopy s a composto of elemetary sotopes (1)-(11) (possbly volvg g, f, t (p)) sce the pots where the tagle tersects [(ζ, 0), (2ζ, 0)] stay fxed. Now, f two affe tagles are sotopc, the they are also sotopc through a composto of two lear sotopes, whch completes the proof. For our purposes, t wll be more coveet to replace the relatos (13)-(15) by the equvalet set of defg relatos below. Defto 3.9. Let s deote the (, )-tagle wth a strad coectg (ζ j, 0) to (2ζ j, 0) for each j, ad a strad coectg (ζ, 0) to (2ζ, 0) passg clockwse aroud the crcle, beeath all the other strads. Lemma The followg relatos are equvalet to the relatos (13)-(15) above. s g = g s 2 2, s 2 2 f = f s, s t (p) = t (p) s ; f 1 s t 1 (2) s t 1 (2) = f 1 ; s t 1 (2) s t 1 (2) g 1 = g 1 ; t 1 (2) s t 1 (2) s t 1 (2) = s t 1 (2) s t 1 (2) t 1 (2). Proof. It s straghtforward to verfy the that we have the relato r = s t 1 (2) t 1 (2). It remas to see that the relatos (13)-(15) the follow from the those lsted the statemet of ths Lemma, ad the relatos (1)-(11). Ths ca be doe by drect computato; as a example, see the below calculato for relato (13). r 2 f r = t 1 2(1) t 3 2(1) (s 2 2) 1 f s t 1 (2) t 1 (2) = t 1 2(1) t 3 2(1) (s 2) 1 s 2 2 f t 1 (2) t 1 (2) = t 1 2(1) t 3 2(1) f t 1 (2) t 1 (2) = f Framed tagles. All precedg costructos may be carred out for framed tagles. Defe the geerators ĝ (resp. ˆf, resp. ˆt (l), resp. ˆr ) as tagles g (resp. f, resp. t (l), resp. r ) wth blackboard framg. Itroduce ew geerators ŵ (1) ad ŵ (2), whch correspod to postve ad egatve twsts of framg of the th strad of a (, ) detty tagle. Defto Defe a framed lear tagle to be a framed affe tagle that sotopc to a product of the geerators ĝ, ˆf, ˆt (1), ˆt (2) for, ad ŵ (1), ŵ (2). Defto Cosder the category AFTa (resp. FTa), wth objects k for k Z 0, ad the set of morphsms betwee p ad q cosst of all framed affe (resp. framed lear) (p, q) tagles. The relatos for framed tagles are trasformed as follows: (1) ˆf ĝ +1 = d = ˆf +1 ĝ

14 14 RINA ANNO, VINOTH NANDAKUMAR (2) (Redemester 1) ˆf ˆt ±1 (l) ĝ = ŵ(l) (3) ˆt (2) ˆt (1) = d = ˆt (1) ˆt (2) (4) ˆt (l) ˆt +1 (l) ˆt (l) = ˆt +1 (l) ˆt (l) ˆt +1 (l) (5) ĝ+2 +k ĝ = ĝ+2 ĝ +k 2 +k 2 (6) ˆf ˆf +2 = ˆf +k ˆf +2 (7) ĝ +k 2 ˆf = ˆf +2 ĝ+2, +k ĝ +k 2 +k ˆf = ˆf +2 ĝ+2 (8) ĝ ˆt +k 2 2 (l) = ˆt +k (l) ĝ, ĝ +k ˆt 2(l) = ˆt (l) ĝ +k (9) ˆf ˆt +k (l) = ˆt +k 2 2 (l) ˆf, +k ˆf ˆt (l) = ˆt +k 2(l) ˆf (10) ˆt (l) ˆt +k (m) = ˆt +k (m) ˆt (l) (11) ˆt (1) ĝ +1 = ˆt +1 (2) ĝ, ˆt (2) ĝ +1 = ˆt +1 (1) ĝ (12) ˆr ˆr = d = ˆr ˆr (13) ˆr 2 ˆf +1 1 ˆr = ˆf, = 1,..., 2; ˆf (ˆr ) 2 = ˆf 1 (14) ˆr ĝ ˆr 2 = ĝ +1, = 1,..., 2; (ˆr ) 2 ĝ 1 = ĝ 1 (15) ˆr ˆt (l) ˆr = ˆt +1 (l); (ˆr ) 2 ˆt 1 (l) (ˆr ) 2 = ˆt 1 (l) We have the followg addtoal relatos for twsts: (16) ŵ(1) ŵ(2) = d, ŵ(l) ŵ(k) j = ŵ(k) j ŵ(l), j (17) ŵ(k) ĝ = ŵ +1 (k) ĝ, ŵ(k) ĝ j = ĝ j ŵ +1±1 (k), j, j + 1 (18) ˆf ŵ(k) = ˆf ŵ +1 (k), ŵ(k) ˆf j = ˆf j ŵ 1±1 (k), j, j + 1 (19) ŵ(k) ˆt = ŵ +1 (k) ˆt, ŵ(k) ˆt j = ˆt j ŵ(k), j, j + 1 (20) ˆt ŵ(k) = ˆf ŵ +1 (k), ŵ(k) ˆf j = ˆt j ŵ(k), j, j + 1 (21) ŵ(k) ˆr = ˆr ŵ 1 (k), ŵ(k) ˆr = ˆr ŵ +1 (k) Note how the Redemester 1 move (2) s the oly relato betwee the o-twst geerators that dffers from the relatos ATa. Proposto Ay sotopy of affe framed tagles s equvalet to a composto of elemetary sotopes (1)-(21). Proof. There s a forgetful fuctor from the 2-category of framed tagles ad ther sotopes to the 2-category of o-framed tagles ad ther sotopes, whch forgets the framg. Thus, for every sotopy there s a composto of relatos (1)-(15) ( ATa) whch dffers oly framg, ad that ca be ruled out by the commutato laws (16)-(21) ( AFTa) of twsts wth all other geerators. Lemma 3.10 stll holds ths cotext, after replacg s by t s framed verso ŝ. 4. Fuctors assocated to affe tagles Defto 4.1. Recall that AFTa (resp Ta, FTa) has objects {k} for k Z 0, ad the set of morphsms betwee {p} ad {q} cossts of all framed affe (resp. framed lear) (p, q) tagles. Defe the category AFTa m (resp. Ta m, FTa m ) to be the full subcategory of AFTa (resp. Ta, FTa) wth objects {m + 2k} for k Z 0. Defto 4.2. A weak represetato of the category AFTa m s a assgmet of a tragulated category C k for each k Z 0, ad a fuctor Ψ(α) : C p C q for each framed affe (m + 2p, m + 2q)-tagle, so that the relatos betwee tagles hold for these fuctors:.e. f β s a (m + 2q, m + 2r) tagle, the there s a somorphsm Ψ(β) Ψ(α) Ψ(β α).

15 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 15 Smlarly oe ca defe the oto of a weak represetato of the categores Ta m, FTa m. The goal of ths secto s to costruct a weak represetato of AFTa m usg the categores D k. I [5] Cauts ad Kamtzer costruct a weak represetato of the category of oreted tagles. We are gog to adapt ther costructo to our settg of framed tagles, ad the geeralze t to the category AFTa m of affe framed tagles. The relatos betwee the geerators for oreted tagles are mostly the same as the relatos we use here, wth a otable excepto of Redemester I move Cauts ad Kamtzer s represetato of the oreted tagle calculus. Let D = D b (Coh(Y m+2 )). I secto 4 of [5], Cauts ad Kamtzer costruct a weak represetato of the category OTa m of oreted tagles usg the categores D. I fact, Cauts ad Kamtzer costruct a weak represetato of the full category OTa (whch gves a weak represetato of the subcategory OTa m ). Also, Cauts ad Kamtzer deal wth the C -equvarat derved categores; but we wll omt ths C -equvarace as we do ot eed t. I ths subsecto we are gog to recall ther costructo, altered so that t becomes a weak represetato of FTa. Recall the defto of Fourer-Muka trasforms (see [13] for a exteded treatmet). Here all pullbacks, pushforwards, Homs ad tesor products of sheaves wll deote the correspodg derved fuctors. Defto 4.3. ([13]) Let X, Y be two complex algebrac varetes, ad let π 1 : X Y X, π 2 : X Y Y deote the two projectos. For a object T D b (Coh(X Y )), defe the Fourer-Muka trasform Ψ T : D b (Coh(X)) D b (Coh(Y )) by Ψ T (F) = π 2 (π 1F T ). The object T s the called the Fourer-Muka kerel of Ψ T. Let Ṽk deote the tautologcal vector budle o Y m+2 correspodg to V k, ad let Ẽk be the quotet le budle Ẽk = Ṽk/Ṽk 1. The followg two deftos are based o [5], but ot detcal to the deftos there: Defto 4.4. Defe the followg Fourer-Muka kerels: G m+2 = O X m+2 π 2Ẽ D b (Coh(Y m+2 2 Y m+2 )), F m+2 = O X m+2 π 1Ẽ 1 +1 Db (Coh(Y m+2 Y m+2 2 )) T m+2(1) = O Z m+2 D b (Coh(Y m+2 Y m+2 )) T m+2(2) = O Z m+2 π 1Ẽ 1 +1 π 2Ẽ D b (Coh(Y m+2 Y m+2 )) Defto 4.5. Defe the fuctors G m+2 = Ψ(g m+2) = Ψ G m+2 : D 1 D F m+2 = Ψ(f m+2) = Ψ F m+2 : D D 1 T m+2(1) = Ψ(t m+2(1)) = Ψ T m+2 (1) : D D T m+2(2) = Ψ(t m+2(2)) = Ψ T m+2 (2) : D D

16 16 RINA ANNO, VINOTH NANDAKUMAR W m+2(1) = Ψ(w m+2(1)) = [ 1] : D D W m+2(2) = Ψ(w m+2(2)) = [1] : D D Note that the dfferece wth the defto [5] s that we oly use two kds of twsts T (1) ad T (2) where they use four, ad our twsts dffer from ther twsts by a shft. The reasos for ths chage are, frst, that there are oly two dfferet crossg geerators the category FTa whle there are four OTa; secod, ths s the chage that turs the oreted tagle relatos to the framed tagle relatos (see Proposto 4.7 below); ad thrd, t gves us the ske relato a ce form of a exact tragle Id Ψ(t (2)) Ψ(g f ) the sprt of Khovaov s homology costructo as descrbed [14] (see Lemma 4.6 below). The fuctors G m+2 : D 1 D admt the followg alterate descrpto: G m+2 (F) = j (p F Ẽ) for F D 1. Smlarly, the fuctor F m+2 : D D 1 admts the followg descrpto: F m+2 (G) = p (j G Ẽ 1 +1 ) for G D. The followg calculato of the left ad rght adjots to G m+2, ad a alteratve descrpto of the fuctors T m+2(1), T m+2(2), from [5] wll be of use to us. Lemma 4.6. We have ( G m+2) R = F m+2[ 1] ad ( G m+2) L = F m+2[1]. Also, for F D, there are dstgushed tragles G m+2( G m+2) R F F T m+2(2)f ad T m+2(1)f F G m+2( G m+2) L. Proof. Ths follows from Lemma 4.4, ad Theorem 4.6 [5]. Recall that ay framed lear tagle ca be expressed as a composto of the above geerators, ad that ay relato betwee lear tagles ca be expressed va the relatos (1)-(11), (16)- (20) Defto Hece defg fuctors Ψ(α) for each (m + 2p, m + 2q)-tagle α, whch are compatble uder composto, s equvalet to defg fuctors for each of the geerators, satsfyg the relatos (1)-(11), (16)-(20) (up to somorphsm). Proposto 4.7. The fuctors Ψ(f m+2), Ψ(g m+2), Ψ(t m+2(l)), Ψ(w m+2(l)) satsfy the relatos (1)-(11), (16)-(20). Thus, gve a lear (m+2p, m+2q) tagle, α, wrtte as a product of geerators, we ca defe Ψ(α) by composto (ad up to somorphsm, the result does ot deped o the choce of decomposto as a product of geerators). Ths gves a weak represetato of FTa m usg the categores D. Proof. By Theorem 4.2 [5], the fuctors G m+2, F m+2, T m+2 (1)[1], ad T m+2(2)[ 1] satsfy the relatos the category OTa that dffer slghtly from the relatos (1)-(11). The relatos (1), (3)-(11) are detcal for OTa ad FTa, ad they hold for the fuctors G m+2, F m+2, T m+2(1), T m+2(2) as well sce every relato has the same umber of each type of crossgs o both sdes, so after shftg every type 1 crossg by [1] ad every type 2 crossg by [ 1] the relatos stll hold. The oreted Redemester move I relato F m+2 T ±1 m+2(1)[1] G m+2 Id F m+2 T ±1 m+2(2)[ 1] G m+2

17 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 17 s exactly the relato (2) for G m+2, F m+2, T m+2(1), T m+2(2), ad W m+2(l): F m+2 F m+2 The relatos (16)-(20) are straghtforward. ±1 T m+2(1) G m+2 [ 1] = W m+2(1) ±1 T m+2(2) G m+2 [1] = W m+2(2) 4.2. Costructg fuctors Ψ(α) : D p D q dexed by lear tagles: cups ad caps. I the prevous secto we costructed a weak represetato of the category FTa of framed tagles usg the tragulated categores D = D b (Coh(Y m+2 )). Our ext goal s to costruct a weak represetato of the category AFTa of affe framed tagles usg the categores D = D b (Coh Bz (U )). The embeddg : U Y m+2 duces a fuctor : D D for each, thus oe may hope to lft the fuctor Ψ(α) : Dp D q to a fuctor Ψ(α) : D p D q. I more precse terms, we am to costruct a fuctor Ψ(α) such that q Ψ(α) = Ψ(α) p. Note that ths somorphsm together wth the somorphsm Ψ(β α) Ψ(β) Ψ(α) does ot yet mply the somorphsm Ψ(β α) Ψ(β) Ψ(α), so we wll eed to prove the latter separately alog wth our costructo of Ψ(α), employg a argumet smlar to oe [5]. Let V k deote the tautologcal vector budle o S slm+2 T B correspodg to V k, ad let E k be the quotet le budle E k = V k /V k 1. Defto 4.8. Defe the followg Fourer-Muka kerels: Defto 4.9. Defe the fuctors: G m+2 = O X, π 2E D b (Coh(U 1 U )), F m+2 = O X, π 1E 1 +1 Db (Coh(U U 1 )) G m+2 = Ψ(g m+2) = Ψ G m+2 : D 1 D F m+2 = Ψ(f m+2) = Ψ F m+2 : D D 1 Remark A pror, the fuctor G m+2 maps D b (Coh(U 1 )) to D b (Coh(U )). However, t s easy to see that G m+2 maps the subcategory D 1 = D b (Coh Bz 1 (U 1 )) D b (Coh(U 1 )) to the subcategory D = D b (Coh Bz (U )) D b (Coh(U 1 )); smlarly F m+2 maps D to D 1. The fuctors G m+2 : D 1 D admt the followg alterate descrpto: G m+2(f) = j, (π,f E k ) for F D 1. Smlarly, the fuctor Fm+2 : D D 1 ca be expressed as follows: Fm+2(G) = π, (j,g E 1 k+1 ) for G D. We wll defe the fuctors Ψ(t m+2(1)) ad Ψ(t m+2(2)) the ext secto, by provg a aalogue of Lemma 4.6 above Costructg fuctors Ψ(α) : D p D q dexed by lear tagles: crossgs ad the framg. Recall the deftos of sphercal twsts ad sphercal fuctors from [3]: Defto Suppose we have two tragulated categores C ad D, ad a fuctor S : C D, wth a left adjot L : D C ad a rght adjot R : D C. Assume that the categores C ad D admt DG-ehacemets, ad the fuctors S, R, ad L desced from DGfuctors betwee those (ths holds for Fourer-Muka trasforms betwee derved categores

18 18 RINA ANNO, VINOTH NANDAKUMAR of coheret sheaves, see [3] Example 4.3). The the four adjucto maps for (L, S, R) have caocal coes, ad we ca defe these coes to be the twst T S (1), the dual twst T S (2), the cotwst F S (1), ad the dual co-twst F S (2): SR d T S (1); T S (2) d SL; F S (1) d RS; LS d F S (2). Defto The fuctor S s called sphercal f the followg four codtos hold: (1) T S (1) ad T S (2) are quas-verse autoequvaleces of D; (2) F S (1) ad F S (2) are quas-verse autoequvaleces of C; (3) The composto LT S (1)[ 1] LSR R of caocal maps s a somorphsm of fuctors; (4) The composto R RSL F S (1)L[1] of caocal maps s a somorphsm of fuctors. Theorem ([3]) Ay two codtos Defto 4.12 mply all four. The usual way to prove that a fuctor s sphercal s to use codto (2) ad oe of the codtos (3) ad (4). We are gog to focus o fuctors for whch a stroger verso of (2) holds: Defto A sphercal fuctor S : C D s called strogly sphercal f F S (1) = [ 3]. It turs out that f we use strogly sphercal fuctors ad ther adjots ad twsts to costruct weak represetatos of FTa m, the oly relatos we eed to check are the Redemester 0 move ad the commutato relatos betwee o-adjacet cups ad caps; all relatos volvg crossgs follow automatcally. Theorem Suppose we have a tragulated category C m+2k for each k Z 0 ; ad for each k 1, 1 < m+2k, a strogly sphercal fuctor Sm+2k : C m+2k 2 C m+2k. Let L m+2k be t s left adjot; Rm+2k be t s rght adjot; T m+2k (1) ts twst, ad T m+2k (2) ts dual twst. If the followg codtos hold: (1) Sm+2k L±1 m+2k [ 1] d (2) S +l m+2k+2 S m+2k S m+2k+2 S+l 2 m+2k for l 2 (3) S +l 2 m+2k L m+2k L m+2k+2 S+l m+2k+2, S m+2k L+l 2 m+2k L+l m+2k+2 S m+2k+2 the assg: Ψ(gm+2k ) S m+2k, Ψ(f m+2k ) = L m+2k [ 1] R m+2k [1] Ψ(t m+2k (1)) = T m+2k (1), Ψ(t m+2k (2)) = T m+2k (2) Ψ(wm+2k (1)) = [ 1], Ψ(w m+2k ( 1)) = [1] These fuctors wll gve a weak represetato of FTa m. for l 2. Proof. Let us check that the relatos (1)-(11), (16)-(20) from Defto 3.12 hold for the above choce of fuctors. The Redemester move 0, cup-cup sotopy ad cup-cap sotopy relatos hold by the assumptos of the theorem, ad the cap-cap sotopy relato follows mmedately from the

19 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 19 cup-cup sotopy relato ad the fact that caps are adjot to cups up to a shft. The cap-crossg sotopy, cup-crossg sotopy ad crossg-crossg sotopy relatos follow the from the above relatos ad the defto of a twst. The Redemester move II relato Tm+2k (1)T m+2k (2) d T m+2k (2)T m+2k (1) follows from the fact that S m+2k are sphercal fuctors, hece Tm+2k (l) are equvaleces of categores. The commutato relatos wth twsts (16)-(20) hold because all exact fuctors commute wth shfts. The remag less trval relatos are Redemester move I (2), Redemester move III (4) ad the ptchfork move (8). For smplcty of otato assume that k = 3 ad deote Υ m+6 by Υ, where Υ stads for L, R, T (1) or T (2). Redemester move I: L 2 T 1 (1)S 2 [ 1] [1]. We have a exact tragle L 2 S 1 R 1 S 2 L 2 S 2 L 2 T 1 (1)S 2 by the defto of T 1 (1) ad aother exact tragle d[2] L 2 S 2 d sce S 2 s a strog sphercal fuctor. Note that the composto of maps L 2 S 1 R 1 S 2 L 2 S 2 d from these two exact tragles s fact the adjucto cout for the par of L 2 S 1 ad ts rght adjot R 1 S 2. By the assumptos of the theorem, L 2 S 1 s a equvalece, so ths composto s a somorphsm. Therefore by the octahedral axom we have L 2 T 1 (1)S 2 d[2], qed. Ptchfork move: T 1 (1)S 2 T 2 (2)S 1. Cosder the followg dagram: S 1 R 1 S 2 S 2 T 1 (1)S 2 S 1 [ 1] S 2 L 2 S 1 [ 1] T 2 (2)S 1 where the rows are exact tragles ad the two vertcal morphsms are duced by the somorphsms R 1 S 2 [1] d ad ts dual d L 2 S 1 [ 1]. The dagram commutes (aga because the adjucto maps for (L 2 S 1, R 1 S 2 ) are compostos of adjucto maps for (L 1, S 1, R 1 ) ad (L 2, S 2, R 2 )), therefore there s a somorphsm T 1 (1)S 2 T 2 (2)S 1, qed. Redemester move III: T 1 (1)T 2 (1)T 1 (1) T 2 (1)T 1 (1)T 2 (1). Ths follows from [3], Theorem 1.2, sce L S are equvaleces of categores, so the maps L S j R j S d have zero coes Checkg the tagle relatos. To apply Theorem 4.15 wth C m+2k = D k, ad S m+2k = G m+2k, we wll eed to prove that G m+2 : D 1 D are strogly sphercal fuctors, ad check the three relatos from Theorem Recall that we have the cluso of the dvsor X, U, as well as the P 1 -budle X, U 1. Deote these maps by j, ad π, respectvely. By abuse of otato we wll deote j,(e k ) smply by E k. The tautologcal sheaves V k exst o X, as well as o U, ad so do ther quotets. Lemma The followg sheaves are somorphc: (1) O U (X, ) E 1 +1 E ; (2) ω X, /U E 1 +1 E ω X, /U 1.

20 20 RINA ANNO, VINOTH NANDAKUMAR Proof. The proof of the frst part s detcal to the proof of Lemma 4.3 () [5]. Note that we have prove that X, ad X,+1 tersect trasversally sde U Lemma 2.8 here. The frst somorphsm the secod part, as part () of the same Lemma [5], follows mmedately from the frst part ad the fact that X, s a smooth dvsor U, so ω X, /U = j,o! U [1] j,o U (X, ). The secod somorphsm follows from the caocal somorphsm ω P(V ) = E (V/E) 1, where V s a two-dmesoal space, E s the tautologcal le budle o P(V ), ad V s a costat vector budle wth fber V. Lemma We have (G m+2) R F m+2[ 1] ad (G m+2) L F m+2[1]. Proof. As the proof of Lemma 4.4 [5], ths follows from a drect computato of the Fourer-Muka kerels, usg the secod part of Lemma 4.16 here. Lemma We have F m+2 G m+2 d[ 1] d[1]. Proof. Ths s aga a drect computato. The fuctor G m+2 ca be expressed as (j, ) (E π,( )). The for ts rght adjot (G m+2) R, whch by Lemma 4.17 s somorphc to Fm+2[ 1], we have (G m+2) R (π, ) (E 1 j,( )).! Sce j, : X, U s a embeddg of a smooth dvsor, we have j,(j!, ) d ( ) O X, (X, )[ 1] d ( ) E E 1 +1 [ 1]. The F m+2 G m+2 (π, ) (E 1 j!,((j, ) (E π,( ))))[1] (π, ) π,( )[1] (π, ) (E E 1 +1 π,( )) d[1] d[ 1] sce π, s a Fao fbrato, so (π, ) π, d (π, ) π,,! ad E E 1 +1 ω X, /U 1 whle X, has dmeso 1 over U 1, so (π, ) (E E 1 +1 π,) (π, ) (π,)[ 1]! d[ 1]. Now we ca show that the fuctors G m+2 satsfy the codtos of Theorem Proposto The fuctors G m+2 : D 1 D are sphercal. Proof. By defto, the fuctor G m+2 dffers from the Fourer-Muka fuctor wth kerel O X, D b (Coh(U 1 U )) by tesorg wth a le budle, so the two fuctors are sphercal smultaeously. By [2], Theorem 4.2 the latter fuctor s sphercal f for ay p U 1 two codtos hold: frst, H (Λ j N lp ) = 0 uless = j = 0 or = j = 1; secod, (ω X, /U ) lp ω lp. Here l p P 1 s the fber over p, ad N s the ormal budle of X, U. Sce X, U s a dvsor, we have N O U (X, ), whch by Lemma 4.16 s somorphc to E 1 +1 E, ad sce E lp O( 1) ad E +1 lp O(1), we have N lp O( 2) ad the frst codto holds. The, by Lemma 4.16 we have ω X, /U E 1 +1 E ad aga, sce E 1 +1 E lp O( 2), the secod codto holds as well. Corollary The fuctors G m+2 : D 1 D are strogly sphercal. Proof. By Lemmas 4.17 ad 4.18, (G m+2) R G m+2 F m+2g m+2[ 1] d[ 2] d. The kerel of ths Fourer-Muka trasform s somorphc to O [ 2] O, where U U s the dagoal. Sce O s a sheaf o a smooth algebrac varety, we have Hom(O, O [ 2]) = 0, so the adjucto ut d (G m+2) R G m+2 must be a multple of the embeddg d 0 d d[ 2] d. Ths map s o-zero, sce we ca multply t by

21 EXOTIC t-structures FOR TWO-BLOCK SPRINGER FIBRES 21 G m+2 to get the map G m+2 G m+2(g m+2) R G m+2 that composes to detty wth the map G m+2(g m+2) R G m+2 G m+2 duced by the adjucto cout. Therefore, the coe of the adjucto ut s somorphc to d[ 2], whch proves the asserto. The followg two propostos are duplcates of Propostos 5.6 ad 5.16 [5], ad the proofs from [5], whch are drect computatos wth Fourer-Muka kerels, work our case verbatm, except that we eed to use our Corollary 2.9 stead of Corollary 5.4 from [5] for a certa trasversalty statemet. Proposto F m+2 G +1 m+2 d F +1 m+2 G m+2. Proposto The followg relatos hold: (1) G +l m+2k+2 G m+2k G m+2k+2 G+l 2 m+2k for l 2; (2) G +l 2 m+2k F m+2k F m+2k+2 G+l m+2k+2, G m+2k F +l 2 m+2k F +l m+2k+2 G m+2k+2 for l 2. Now we have verfed the codtos of Theorem 4.15, so we troduce the twsts T m+2(l) ad costruct a weak represetato of FTa m. Defto Defe the fuctors T m+2(1) ad T m+2(2) va the dstgushed tragles: G m+2(g m+2) R d T m+2(1), T m+2(2) d G m+2(g m+2) L Theorem The assgmets Ψ(gm+2) = G m+2, Ψ(fm+2) = Fm+2 Ψ(t m+2(1)) = Tm+2(1), Ψ(t m+2(2)) = Tm+2(2) Ψ(wm+2(1)) = [ 1], Ψ(wm+2( 1)) = [1] gve rse to a weak represetato of FTa m usg the categores D k Fuctors Ψ(α) : D p D q dexed by affe tagles. At ths pot, we have costructed a fuctor Ψ(α) : D p D q for each framed lear (m + 2p, m + 2q)-tagle α. To exted ths costructo to framed affe tagles, t suffces to costruct a fuctor Ψ(s m+2 m+2) : D D satsfyg the relatos Lemma Defe Sm+2(F) = F Em+2, 1 ad let Ψ(s m+2 m+2) := Sm+2. The relatos that we must check are the followg: Proposto The followg dettes hold, where 1 m + 2 2, 1 p 2: (1) S m+2 2 m+2 2 F m+2 F m+2 S m+2 m+2; (2) S m+2 m+2 G m+2 G m+2 S m+2 2 m+2 2; (3) Sm+2 Tm+2(p) Tm+2(p) Sm+2; (4) Fm+2 m+2 1 Sm+2 Tm+2 m+2 1 (2) S m+2 (5) Sm+2 Tm+2 m+2 1 (2) Sm+2 Tm+2 m+2 1 m+2 Tm+2 m+2 1 (2) F m+2 1 (2) G m+2 1 m+2 G m+2 1 (2) Sm+2 Tm+2 m+2 1 (2) Sm+2 Tm+2 m+2 1 (2) Sm+2 Tm+2 m+2 1 (2) Sm+2 Tm+2 m+2 1 (2) Tm+2 m+2 1 (2). (6) T m+2 1 m+2 m+2 ; m+2 ;

22 22 RINA ANNO, VINOTH NANDAKUMAR Proof. The sheaf E m+2 s costat o the fbers of π, for < m + 2 1, thus tesorg wth Em+2 1 commutes wth all parts of G m+2 ad Fm+2, so the frst three statemets follow mmedately. We wll prove the fourth statemet by drect computato, ad the ffth statemet ca be prove smlarly. The sxth statemet follows from the fourth, the ffth, ad the exact tragle Tm+2 m+2 1 (2) d G m+2 1 m+2 Fm+2 m+2 1 [1]. To save space, let us skp the dces whe there s o ambguty wth the curret proof: deote G m+2 1 m+2 by G, Fm+2 m+2 1 by F, Tm+2 m+2 1 (l) by T (l), Sm+2 by S, X,m+2 1 by X, j,m+2 1 by j, π,m+2 1 by π. Recall that we use the same otato for E ad j E, so we ca say that tesor multplcato by E commutes wth the fuctors j ad j. By defto, T (2) = {d GG L }[ 1], ad by Lemma 4.17 we have G L F [1], so we ca wrte F ST (2)S {π (E 3 m+2 j ( )) π (E 2 m+2 E m+2 1 j j π π (E 2 m+2 j ( )))[1]}[ 1]. The adjucto map the coe flters through the map π (Em+2 3 j ( )) π (Em+2 3 j j j ( )). Recall that j j d Em+2 1 E m+2 1 ( )[1], ad the adjucto morphsm j j j j j E m+2 Em j has detty for the frst compoet j j. Observe that π (Em+2 1 π ( )) 0 sce the restrcto of Em+2 1 o a fber of π s somorphc to O( 1). Therefore, we ca further evaluate F ST (2)S as the coe {π (E 3 m+2 j ( )) π (E 2 m+2 E m+2 1 π π (E 2 m+2 j ( )))[1]}[ 1] where the map s duced by the adjucto map d π! π E 1 m+2 E m+2 1 π π [1]. By projecto formula, that turs to (2) {π (E 3 m+2 j ( )) π (E 2 m+2 E m+2 1 ) π (E 2 m+2 j ( )))[1]}[ 1]. By Grothedeck-Serre dualty, sce ω X/U 1 Em+2 E 1 m+2 1, we have π (Em+2 E 2 m+2 1 ) (π E m+2 ) [ 1]. Observe that π s by costructo the projectvzato of V m+2 /V m+2 2, whereas E m+2 = V m+2 /V m+2 1 s the fberwse O(1), so π E m+2 V m+2 /V m+2 2. We ca ow rewrte (2) as follows, pullg π ad j out of the coe: π { E 3 m+2 (V m+2 /V m+2 2 ) E 2 m+2} j ( )[ 1]. The map wth the coe s gve by ι d, where ι : (E m+2 ) (V m+2 /V m+2 2 ) s dual to the projecto V m+2 /V m+2 2 E m+2, ad d : Em+2 2 Em+2. 2 From the short exact sequece 0 (E m+2 ) (V m+2 /V m+2 2 ) (E m+2 1 ) 0 we see that the coe s somorphc to π (Em+2 2 Em+2 1 j ( ))[ 1]. Now, the sheaf E m+2 E m+2 1 = Λ 2 (V m+2 /V m+2 2 ) ad hece by Lemma 4.26 below t s trval o X, so we have prove F ST (2)S π (E 1 m+2 j ( ))[ 1] F [ 1]. Sce G s sphercal, we have G L T (1)[ 1] G R, ad sce by Lemma 4.17 we kow that G L F [1] ad G R F [ 1], t follows that F T (1) F [ 1]. The F ST (2)S F [ 1] mples F ST (2)S F T (1), whch cocludes the proof. It remas to prove the followg techcal lemma: