Modular perverse sheaves on flag varieties III: positivity conditions
aa r X i v : . [ m a t h . R T ] A ug MODULAR PERVERSE SHEAVES ON FLAG VARIETIES III:POSITIVITY CONDITIONS
PRAMOD N. ACHAR AND SIMON RICHE
Abstract.
We further develop the general theory of the “mixed modularderived category” introduced by the authors in a previous paper in this series.We then use it to study positivity and Q -Koszulity phenomena on flag varieties. Introduction
The category P ( B ) ( B , C ) of Bruhat-constructible perverse C -sheaves on theflag variety B of a complex connected reductive algebraic group G has been exten-sively studied for decades, with much of the motivation coming from applications tothe representation theory of complex semisimple Lie algebras. Two salient featuresof this category are as follows :(1) C The stalks and costalks of the simple perverse sheaves IC w ( C ) enjoy aparity-vanishing property (see [KL]).(2) C The category P ( B ) ( B , C ) admits a Koszul grading (see [BGS]).It was long expected that the obvious analogues of statements (1) C and (2) C wouldalso hold for modular perverse sheaves (i.e. for perverse sheaves with coefficients in afinite field F of characteristic ℓ >
0) under mild restrictions on ℓ , with consequencesfor the representation theory of algebraic groups; see e.g. [So]. But Williamson’swork [Wi] implies that both of these statements fail in a large class of examples.The next question one may want to consider is then: what could take the placeof (1) C and (2) C in the setting of modular perverse sheaves? Fix a finite extension K of Q ℓ whose ring of integers O has F as residue field. In this paper, we considerthe following statements as possible substitutes for those above:(1) F The stalks of the O -perverse sheaves IC w ( O ) are torsion-free. Equivalently,the stalks of the F -perverse sheaves F ⊗ L IC w ( O ) enjoy a parity-vanishingproperty.(2) F The category P ( B ) ( B , F ) admits a standard Q -Koszul grading.The definition of a standard Q -Koszul category—a generalization of the ordinaryKoszul property, due to Parshall–Scott [PS1]—will be recalled in § § F and (2) F are nearlyequivalent to each other. Statement (1) F may be compared to (and was inspiredby) the Mirkovi´c–Vilonen conjecture [MV] (now a theorem in most cases [ARd]),which asserts that spherical IC -sheaves on the affine Grassmannian have torsion-free stalks. Statement (2) F is closely related to certain conjectures of Cline, Parshall,and Scott [CPS, PS1] on representations of algebraic groups. P.A. was supported by NSF Grant No. DMS-1001594. S.R. was supported by ANR GrantsNo. ANR-09-JCJC-0102-01, ANR-2010-BLAN-110-02 and ANR-13-BS01-0001-01.
In the characteristic zero case, state-ments (1) C and (2) C are best understood in the framework of mixed Q ℓ -sheaves.In [AR3] we defined and studied a replacement for these objects in the modularcontext (when ℓ is good for G ). More precisely, for E = K , O , or F we defined atriangulated category D mix( B ) ( B , E ), endowed with a “Tate twist” h i and a “perverset-structure” whose heart we denote by P mix( B ) ( B , E ). This category is also endowedwith a t-exact “forgetful” functor D mix( B ) ( B , E ) → D b( B ) ( B , E ), where the usualBruhat-constructible derived category D b( B ) ( B , E ) is endowed with the usual per-verse t-structure. The main tool in this construction is the category Parity ( B ) ( B , E )of parity complexes on B in the sense of Juteau–Mautner–Williamson [JMW]. Theindecomposable objects in the latter category are naturally parametrized by W × Z ;we denote as usual by E w the object associated with ( w, P mix( B ) ( B , F ) is a graded quasihereditary category, and can be consid-ered a “graded version” of the category P ( B ) ( B , F ). The analogue of this categorywhen F is replaced by K can be identified with the category studied in [BGS, § standard Koszul ). One might wonder if thecategory P ( B ) ( B , F ) enjoys a similar property, or some weaker analogues. The maintheme of this paper is to relate these properties to properties of the usual perversesheaves on B or the flag variety ˇ B of the Langlands dual reductive group. Moreprecisely, we consider the following four properties:(1) The category P mix( B ) ( B , F ) is positively graded.(2) The category P mix( B ) ( B , F ) is standard Q -Koszul.(3) The category P mix( B ) ( B , F ) is metameric.(4) The category P mix( B ) ( B , F ) is standard Koszul.Here, condition (1) is a natural condition defined and studied in § § § E w , resp. ˇ IC w , is the paritysheaf, resp. IC -sheaf on ˇ B naturally associated with w . This statement combinesparts of Theorems 5.1, 5.2, and 5.5.) Theorem.
Assume that ℓ is good for G . (1) The following conditions are equivalent: (a)
The category P mix( B ) ( B , F ) is positively graded. (b) For all w ∈ W , the parity sheaf ˇ E w ( F ) on ˇ B is perverse. (2) The following conditions are equivalent: (a)
The category P mix( B ) ( B , F ) is metameric. (b) The category P mix( ˇ B ) ( ˇ B , F ) is standard Q -Koszul. (c) For all w ∈ W , the parity sheaf E w ( F ) is perverse, and the O -perversesheaf ˇ IC w ( O ) on ˇ B has torsion-free stalks. (3) The following conditions are equivalent: (a)
The category P mix( B ) ( B , F ) is standard Koszul. (b) For all w ∈ W we have E w ( F ) ∼ = IC w ( F ) . ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 3 (c)
For all w ∈ W , the O -perverse sheaf IC w ( O ) on B has torsion-freestalks and costalks.Moreover, these statements hold if and only if the analogous statements forthe Langlands dual group hold. In this theorem, part (1) is an immediate consequence of the results of [AR3].Part (3) is also not difficult to prove. However, as noted earlier, Williamson [Wi]has exhibited counterexamples to condition (3c) in groups of arbitrarily large rank.Part (2) of the theorem is the most interesting and delicate case, and its proof re-quires the introduction of new tools. Williamson has informed us that condition (2c)holds for G = GL( n ) with n ≤ IC w ( O ), not in their stalks. Thus, asof this writing, there are no known counterexamples to the conditions in part (2). To prove part (2) of the theorem above, we introduce a formal-ism which plays a role similar to Deligne’s theory of weights for mixed Q ℓ -perversesheaves. (However, it is much less powerful than Deligne’s theory: in particu-lar, the existence of a “weight filtration” on mixed modular perverse sheaves isnot automatic.) More precisely, in § D mix( B ) ( B , E ) to have weights ≤ n or ≥ n , and we prove that the !- and ∗ -pullback andpushforward functors associated with locally closed inclusions of unions of Bruhatcells enjoy the same stability properties for this formalism as in the case of mixed Q ℓ -sheaves (cf. [BBD, Stabilit´es 5.1.14]).Next, in § P ◦ ( B ) ( B , E ) ⊂ D mix( B ) ( B , E ). This is not the heart of a t-structure on D mix( B ) ( B , E ); for instance, when E = Q ℓ , it is the category consisting of semisimplepure perverse sheaves of weight 0. The category P ◦ ( B ) ( B , F ) need not be semisim-ple, but it is always quasihereditary, so one may speak of standard and costandardobjects in P ◦ ( B ) ( B , F ). These objects are parametrized by W , and the standard,resp. costandard, object associated with w is denoted ∆ ◦ w ( F ), resp. ∇ ◦ w ( F ). A care-ful study of the structure of the ∆ ◦ w ( F ), carried out in § ∆ ◦ w ( F ) . In the course of the proof, we will see that if P ( B ) ( B , F ) is positively graded, then ∆ ◦ w ( F ) ∼ = F ⊗ L IC mix w ( O ). This property isanalogous to the fact [MV, §
8] that in the category of spherical perverse sheaveson the affine Grassmannian, standard objects are of the form F ⊗ L IC λ ( O ). Ofcourse, in the setting of [MV], there is a representation-theoretic interpretation forthese objects as well: they correspond to Weyl modules under the geometric Satakeequivalence.If one hopes to prove that the conditions in part (2) of the theorem are actuallytrue, it will likely be useful to find a representation-theoretic interpretation of the∆ ◦ w ( F ). One candidate is the class of reduced standard modules introduced by Cline–Parshall–Scott [CPS]. These are certain representations of an algebraic group, ob-tained by modular reduction of irreducible quantum group representations. It islikely that under the equivalence of [AR2, Theorem 2.4], reduced standard modulescorrespond to objects of the form F ⊗ L IC w ( O ). With this in mind, condition (2a) PRAMOD N. ACHAR AND SIMON RICHE should be compared to [CPS, Conjecture 6.5], and condition (2c) to [CPS, Con-jecture 6.2]. (See [PS1, PS2] for other results about standard Q -Koszulity in thecontext of representations of algebraic groups.)There are further parallels between P ◦ ( B ) ( B , F ) and the affine Grassmannian thatmay lead to future insights. We have already noted that condition (2c) resemblesthe Mirkovi´c–Vilonen conjecture. In fact, a version of the metameric property(see [BK, Corollary 5.1.13]) plays a role in the proof of that conjecture. Separately,the conditions in part (2) imply that the ˇ E w ( F ) are precisely the tilting objects in P ◦ ( ˇ B ) ( ˇ B , F ). This is similar to the main result of [JMW2], which relates sphericalparity sheaves to tilting modules via the geometric Satake equivalence. We thank Geordie Williamson for stimulating discus-sions.
Section 2 contains general results on positively graded quasihered-itary categories, including metameric and standard Q -Koszul categories. In Sec-tions 3 and 4, we work in the general setting of a stratified variety satisfying theassumptions of [AR3, §§ D mix S ( X, F ), and contain the definition of P ◦ S ( X, F ). Finally, in Section 5 we con-centrate on the case of flag varieties, and prove our main theorems. Positivity conditions for graded quasihereditary categories
Throughout this section, k will be a field, and A will be a finite-length k -linearabelian category. We begin by recalling the definitionof graded quasihereditary categories. We refer to [AR3, Appendix A] for reminderson the main properties of these categories.Assume A is equipped with an automorphism h i : A → A . Let Irr( A ) be theset of isomorphism classes of irreducible objects of A , and let S = Irr( A ) / Z , where n ∈ Z acts on Irr( A ) by h n i . Assume that S is equipped with a partial order ≤ , andthat for each s ∈ S , we have a fixed representative simple object L gr s . Assume alsowe are given, for any s ∈ S , objects ∆ gr s and ∇ gr s , and morphisms ∆ gr s → L gr s and L gr s → ∇ gr s . For T ⊂ S , we denote by A T the Serre subcategory of A generatedby the objects L gr t h n i for t ∈ T and n ∈ Z . We write A ≤ s for A { t ∈ S | t ≤ s } , andsimilarly for A
The category A (with the data above) is said to be graded quasi-hereditary if the following conditions hold:(1) The set S is finite.(2) For each s ∈ S , we haveHom( L gr s , L gr s h n i ) = ( k if n = 0;0 otherwise.(3) The kernel of ∆ gr s → L gr s and the cokernel of L gr s → ∇ gr s belong to A
Recall (see [AR3, Theorem A.3]) that if A is graded quasihereditary then ithas enough projective objects, and that each projective object admits a standardfiltration, i.e. a filtration with subquotients of the form ∆ gr t h n i ( t ∈ S , n ∈ Z ).Moreover, if we denote by P gr s the projective cover of L gr s , then a graded form ofthe reciprocity formula holds:(2.1) ( P gr s : ∆ gr t h n i ) = [ ∇ gr t h n i : L gr s ] , where the left-hand side denotes the multiplicity of ∆ gr t h n i in any standard filtrationof P gr s , and the right-hand side denotes the usual multiplicity as a compositionfactor. Similar claims hold for injective objects.Below we will also consider some (ungraded) quasihereditary categories: theseare categories satisfying obvious analogues of the conditions in Definition 2.1.Later we will need the following properties. Lemma 2.2.
Let T ⊂ S be a closed subset. (1) The subcategory A T ⊂ A is a graded quasihereditary category, with stan-dard (resp. costandard) objects ∆ gr t (resp. ∇ gr t ) for t ∈ T . Moreover, thefunctor ι T : D b A T → D b A induced by the inclusion A T ⊂ A is fullyfaithful. (2) The Serre quotient A / A T is a graded quasihereditary category for the orderon S r T obtained by restriction from the order on S . The standard(resp. costandard) objects are the images in the quotient of the objects ∆ gr s (resp. ∇ gr s ) for s ∈ S r T . (3) The natural functor D b ( A ) /D b ( A T ) → D b ( A / A T ) (where the left-handside is the Verdier quotient) is an equivalence. Moreover, the functors Π T : D b ( A ) → D b ( A / A T ) and ι T admit left and right adjoints, denoted Π R T , Π L T , ι R T , ι L T , which satisfy (2.2) Π L T ◦ Π T (∆ gr s ) ∼ = ∆ gr s , Π R T ◦ Π T ( ∇ gr s ) ∼ = ∇ gr s and such that, for any M in D b ( A ) , the adjunction morphisms inducefunctorial triangles ι T ι R T M → M → Π R T Π T M [1] −→ , Π L T Π T M → M → ι T ι L T M [1] −→ . Proof. (1) It is clear that A T satisfies the first four conditions in Definition 2.1.To check that it satisfies the fifth condition, one simply observes that the naturalmorphism Ext A T (∆ gr s , ∇ gr t h n i ) → Ext A (∆ gr s , ∇ gr t h n i ) is injective for s, t ∈ T , n ∈ Z , see e.g. [BGS, Lemma 3.2.3]. Since the second space is trivial by assumption,the first one is trivial also.Now it follows from the definitions that the category D b A T is generated (as atriangulated category) by the objects ∆ gr t h n i for t ∈ T and n ∈ Z , as well as bythe objects ∇ gr t h n i for t ∈ T and n ∈ Z . Hence, by a standard argument, to provethat ι T is fully faithful, it is enough to prove that for s, t ∈ T and k, n ∈ Z thenatural morphism Ext k A T (∆ gr s , ∇ gr t h n i ) → Ext k A (∆ gr s , ∇ gr t h n i )is an isomorphism. However in both categories A and A T we haveExt k (∆ gr s , ∇ gr t h n i ) = ( k if s = t , k = n = 0;0 otherwise, PRAMOD N. ACHAR AND SIMON RICHE see e.g. [AR3, Equation (A.1)]. Hence this claim is clear.(2) It is clear that the quotient A / A T satisfies conditions (1), (2), and (3) ofDefinition 2.1. To check that it satisfies condition (4), we denote by π T : A → A T the quotient morphism. Then one can easily check that if s ∈ S r T , for any M in A the morphismsHom A (∆ gr s , M ) → Hom A / A T ( π T (∆ gr s ) , π T ( M )) , Hom A ( M, ∇ gr s ) → Hom A / A T ( π T ( M ) , π T ( ∇ gr s ))induced by π T are isomorphisms. Using [Ga, Corollaire 3 on p. 369], one easilydeduces that condition (4) holds.To prove condition (5), we observe that, by [Ga, Corollaire 1 on p. 375], thesubcategory A T is localizing; by [Ga, Corollaire 2 on p. 375] we deduce that A / A T has enough injectives, and that every injective object is of the form π T ( I ) for some I injective in A . In particular, since π T ( ∇ gr s ) is either 0 or a costandard objectof A / A T , we deduce that injective objects in A / A T admit costandard filtrations.By a standard argument (see e.g. [Rin, Corollary 3]), this implies condition (5).(3) Observe that the objects { ∆ gr s , s ∈ S } form a graded exceptional set in D b ( A ) in the sense of [Be2, § ι T and the quotient functor ′ Π T : D b ( A ) → D b ( A ) /D b ( A T ) admit left and right adjoints, which induce functorialtriangles as in the lemma. If we denote by ′ Π L T (resp. ′ Π R T ) the left (resp. right)adjoint to ′ Π T , it is easily checked that we have( ′ Π L T ) ◦ (Π T )(∆ gr s ) ∼ = ∆ gr s and ( ′ Π R T ) ◦ (Π T )( ∇ gr s ) ∼ = ∇ gr s for any s ∈ S r T (see e.g. [Be1, Lemma 4(d)] for a similar claim). Using thisproperty and an argument similar to the one used to prove that ι T is fully faithful,one can deduce that the natural functor D b ( A ) /D b ( A T ) → D b ( A / A T ) is anequivalence, which finishes the proof. (cid:3) In this section we willmainly consider graded quasihereditary categories which exhibit some positivityproperties. The precise definition is as follows.
Definition 2.3.
Let A be a graded quasihereditary category. We say that A is positively graded if for all s, t ∈ S , we have [ P gr s : L gr t h n i ] = 0 whenever n > Remark . The condition in Definition 2.3 is equivalent to requiring that we haveHom( P gr t , P gr s h n i ) = 0 whenever n <
0. In other words, if we let P gr = L s ∈ S P gr s ,then A is positively graded if and only if the graded ring R := M n ∈ Z Hom( P gr , P gr h n i )is concentrated in nonnegative degrees. Note that R is a finite dimensional k -algebra, and that the functor M L n Hom A ( P gr , M h n i ) induces an equivalenceof categories between A and the category of finite dimensional graded right R -modules. Proposition 2.5.
Let A be a graded quasihereditary category. The following con-ditions are equivalent: (1) A is positively graded. (2) We have [∆ gr s : L gr t h n i ] = ( P gr s : ∆ gr t h n i ) = 0 whenever n > . ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 7 (3)
We have [∆ gr s : L gr t h n i ] = [ ∇ gr s h n i : L gr t ] = 0 whenever n > . (4) We have
Ext ( L gr s , L gr t h n i ) = 0 for n > . (5) Every object M ∈ A admits a canonical filtration W • M with the propertythat every composition factor of Gr Wi M is of the form L gr s h i i , and everymorphism in A is strictly compatible with this filtration.Proof. (1) = ⇒ (2). Since ∆ gr s is a quotient of P gr s , we clearly have [∆ gr s : L gr t h n i ] = 0for n >
0. If we had ( P gr s : ∆ gr t h n i ) = 0 for some s, t and some n >
0, then wewould also have ( P gr s : L gr t h n i ) = 0, contradicting the assumption.(2) = ⇒ (1). This is obvious.The equivalence (2) ⇐⇒ (3) follows from the reciprocity formula (2.1).(1) = ⇒ (4). Let K be the kernel of P gr s → L gr s . Note that if n >
0, then[ K : L gr t h n i ] = 0, and hence Hom( K, L gr t h n i ) = 0. We deduce the desired resultfrom the exact sequence · · · → Hom(
K, L gr t h n i ) → Ext ( L gr s , L gr t h n i ) → Ext ( P gr s , L gr t h n i ) → · · · . (4) = ⇒ (5). This follows from [BBD, Lemme 5.3.6] (see also the proof of [BBD,Th´eor`eme 5.3.5]).(5) = ⇒ (1). Consider the weight filtration W • P gr s of P gr s . Let n be the largestinteger such that Gr Wn P gr s = 0. Then Gr Wn P gr s is a quotient of P gr s , and in particular, P gr s has a quotient of the form L gr t h n i . But L gr s is the unique simple quotient of P gr s , so we must have n = 0, and the result follows. (cid:3) Let us note the following consequence of Proposition 2.5, which is immediatefrom condition (3) of the proposition.
Corollary 2.6. If A is a positively graded quasihereditary category and if T ⊂ S is closed, then the graded quasihereditary category A / A T is positively graded. It is easy to see that in a positively graded quasihereditary category, any L gr s admits a projective resolution whose terms are direct sums of various P gr t h n i with n ≤
0. As a consequence, for all k ≥ k ( L gr s , L gr t h n i ) = 0 for n > Proposition 2.7 (cf. [PS1, Proposition 3.1(a)]) . Let A be a positively graded quasi-hereditary category, and let A ◦ be the Serre subcategory generated by the simpleobjects { L gr s | s ∈ S } (i.e., without Tate twists). Then A ◦ is a quasihereditary cat-egory (with weight poset S ), with standard and costandard objects given respectivelyby ∆ ◦ s := Gr W ∆ gr s and ∇ ◦ s := Gr W ∇ gr s . Proof.
It is clear that A is a finite length category, and that its simple objectsare parametrized by S . It is also clear from the definitions that ∆ ◦ s is a quotientof ∆ gr s , and that the surjection ∆ gr s → L gr s factors through a surjection ∆ ◦ s → L gr s .Similarly, ∇ ◦ s is a subobject of ∇ gr s , and the injection L gr s → ∇ gr s factors throughan injection L gr s → ∇ ◦ s . The ungraded analogues of axioms (1), (2) and (3) ofDefinition 2.1 are clear.We now turn to axiom (4). Since ∆ ◦ s is a quotient of ∆ gr s , it has a unique simplequotient, isomorphic to L gr s . Next, let T ⊂ S be closed, with s maximal in T .For t ∈ T , consider the exact sequence · · · → Hom A T ( W − ∆ gr s , L gr t ) → Ext A T (∆ ◦ s , L gr t ) → Ext A T (∆ gr s , L gr t ) → · · · . PRAMOD N. ACHAR AND SIMON RICHE
The first term vanishes because W − ∆ gr s has only composition factors of the form L gr t h n i with n <
0, and the last term vanishes by axiom (4) for A . So the middleterm does as well. It clear that Ext A ◦ T (∆ ◦ s , L gr t ) = Ext A T (∆ ◦ s , L gr t ), so we haveshown that ∆ ◦ s is a projective cover of L gr s in A ◦ T .A similar argument shows that ∇ ◦ s is an injective envelope of L gr s in A ◦ T ; we omitfurther details.Finally, we consider the analogue of axiom (5). Consider the exact sequence · · · → Ext A ( W − ∆ gr s , ∇ gr t ) → Ext A (∆ ◦ s , ∇ gr t ) → Ext A (∆ gr s , ∇ gr t ) → · · · . The first term vanishes by Proposition 2.5(4), and the last by axiom (5) for A , sothe middle term does as well. That term is also the last term in the exact sequence · · · → Ext A (∆ ◦ s , ∇ gr t /W ∇ gr t ) → Ext A (∆ ◦ s , ∇ ◦ t ) → Ext A (∆ ◦ s , ∇ gr t ) → · · · , whose first term again vanishes by Proposition 2.5(4). We have now shown thatExt A (∆ ◦ s , ∇ ◦ t ) = 0. By a standard argument (see e.g. [BGS, Lemma 3.2.3]), thenatural map Ext A ◦ (∆ ◦ s , ∇ ◦ t ) → Ext A (∆ ◦ s , ∇ ◦ t ) is injective, so the former vanishesas well, as desired. (cid:3) Remark . With the notation of Remark 2.4, if A is a positively graded quasi-hereditary category, then the category A ◦ identifies with the subcategory of thecategory of finite-dimensional graded right R -modules consisting of modules con-centrated in degree 0; in other words, with the category of finite-dimensional rightmodules over the 0-th part R of R .The determination of Ext A (∆ ◦ s , ∇ ◦ t ) at the end of the preceding proof can easilybe adapted to higher Ext-groups: by using (2.3) in place of Proposition 2.5(4),and [AR3, Eq. (A.1)] in place of axiom (5) for A , we find that(2.4) Ext k A (∆ ◦ s , ∇ ◦ t ) = 0 for all k ≥ Lemma 2.9.
Let A be a positively graded quasihereditary category. The naturalfunctor D b A ◦ → D b A is fully faithful. We have seen above that any positively gradedquasihereditary category contains two classes of objects worthy of being called“standard”: the usual ∆ gr s , and the new ∆ ◦ s . In this subsection, we study cate-gories in which these two classes are closely related. Definition 2.10.
Let A be a positively graded quasihereditary category. Wesay that A is a metameric category if for all s ∈ S and all i ∈ Z , the object (cid:0) Gr Wi ∆ gr s (cid:1) h− i i ∈ A ◦ admits a standard filtration, and (cid:0) Gr Wi ∇ gr s (cid:1) h− i i ∈ A ◦ admitsa costandard filtration.This term is borrowed from biology, where metamerism refers to a body plancontaining repeated copies of some smaller structure. The analogy is that in oursetting, each ∆ s is made up of copies of the smaller objects ∆ ◦ u . Theorem 2.11.
Let A be a metameric category. For any s ∈ S , there exists aunique object e ∆ gr s ∈ A which satisfies the following properties. (1) e ∆ gr s has a unique simple quotient, isomorphic to L gr s . (2) For all r ∈ S and k ∈ Z > , we have Ext k ( e ∆ gr s , L gr r ) = 0 if r ≤ s . ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 9 (3)
For all r ∈ S and k ∈ Z > , we have Ext k ( e ∆ gr s , L gr r h n i ) = 0 if n = 0 . (4) There is a surjective map e ∆ gr s → ∆ gr s whose kernel admits a filtration whosesubquotients are various ∆ gr u h n i with u > s and n < .Dually, for any s ∈ S , there exists a unique object e ∇ gr s ∈ A which satisfies thefollowing properties. (1 ′ ) e ∇ gr s has a unique simple subobject, isomorphic to L gr s . (2 ′ ) For all r ∈ S and k ∈ Z > , we have Ext k ( L gr r , e ∇ gr s ) = 0 if r ≤ s . (3 ′ ) For all r ∈ S and k ∈ Z > , we have Ext k ( L gr r h n i , e ∇ gr s ) = 0 if n = 0 . (4 ′ ) There is an injective map ∇ gr s → e ∇ gr s whose cokernel admits a filtrationwhose subquotients are various ∇ gr u h n i with u > s and n > .Conversely, if A is a positively graded quasihereditary category which contains ob-jects e ∆ gr s and e ∇ gr s satisfying the above properties for all s ∈ S , then A is metameric.Proof. Suppose that A is metameric. We first remark that, if e ∆ gr s exists, then it isthe projective cover of L gr s in the Serre subcategory of A generated by the objects L gr r with r ≤ s and the objects L gr t h n i for all t ∈ S and n = 0. Hence uniquenessis clear. It also follows from this remark that the map in (4) is the unique (up toscalar) nonzero morphism e ∆ gr s → ∆ gr s .To prove existence, we can assume without loss of generality that S is theset { , , . . . , N } (with its natural order). We proceed by induction on N . Let A ′ := A ≤ N − , and assume the theorem is known to hold for A ′ . For each i ≤ N − e ∆ gr ′ i be the object in A ′ satisfying the properties of Theorem 2.11 for A ′ .We begin by constructing the objects e ∆ gr i . For i = N , we simply set e ∆ gr N := ∆ gr N . This object clearly has properties (1)–(4). Now suppose i < N . For n < E n := Ext ( e ∆ gr ′ i , ∆ gr N h n i ). Let ǫ n be the canonical element of E ∗ n ⊗ E n ∼ =Ext ( e ∆ gr ′ i , E ∗ n ⊗ ∆ gr N h n i ), and let ǫ := M n< ǫ n ∈ Ext e ∆ gr ′ i , M n< E ∗ n ⊗ ∆ gr N h n i ! . (Note that only finitely many of the spaces E n are nonzero, so these direct sums arefinite.) Define e ∆ gr i to be the middle term of the corresponding short exact sequence:(2.5) 0 → M n< E ∗ n ⊗ ∆ gr N h n i → e ∆ gr i → e ∆ gr ′ i → . Then, for any m <
0, the natural map(2.6) Hom (cid:0)M n< E ∗ n ⊗ ∆ gr N h n i , ∆ gr N h m i (cid:1) → Ext ( e ∆ gr ′ i , ∆ gr N h m i )is an isomorphism. For brevity, we henceforth write C := L n< E ∗ n ⊗ ∆ gr N h n i .Suppose j ≤ N −
1. Then Ext k ( C, L gr j h m i ) = 0 for all k ≥ m ∈ Z , so(2.7) Ext k ( e ∆ gr i , L gr j h m i ) ∼ = Ext k ( e ∆ gr ′ i , L gr j h m i ) for all k ≥
0, if j ≤ N − e ∆ gr i , L gr j h m i ) = ( j = i and m = 0,0 in all other cases with j ≤ N − and, using induction and Lemma 2.2,(2.9) Ext k ( e ∆ gr i , L gr j h m i ) = 0 for k ≥
1, if ( j ≤ N − m = 0, or j ≤ i and m = 0.Next, let K be the kernel of the map ∆ gr N → L gr N . Note that if m <
0, thenevery composition factor of K h m i is isomorphic to some L gr j h n i with n < j ≤ N −
1. Assume m <
0, and consider the following long exact sequences:
Hom( e ∆ gr ′ i , K h m i ) / / (cid:15) (cid:15) Hom( e ∆ gr ′ i , ∆ gr N h m i ) / / (cid:15) (cid:15) Hom( e ∆ gr ′ i , L gr N h m i ) / / (cid:15) (cid:15) Ext ( e ∆ gr ′ i , K h m i ) (cid:15) (cid:15) Hom( e ∆ gr i , K h m i ) / / Hom( e ∆ gr i , ∆ gr N h m i ) / / Hom( e ∆ gr i , L gr N h m i ) / / Ext ( e ∆ gr i , K h m i ) . Since Hom(
C, K h m i ) = 0, the first vertical map is an isomorphism. By (2.7)and (2.9), both groups in the last column vanish. It follows from (2.6) that thesecond vertical map is an isomorphism. Therefore, by the five lemma, the third oneis also an isomorphism, and we have Hom( e ∆ gr i , L gr N h m i ) = 0 for m <
0. In fact, wehave(2.10) Hom( e ∆ gr i , L gr N h m i ) = 0 for all m ∈ Z .For m ≥
0, this follows from (2.5), since Hom(
C, L gr N h m i ) = 0 for m ≥ m <
0, consider the exact sequence · · · →
Ext ( e ∆ gr i , ∆ gr N h m i ) → Ext ( e ∆ gr i , L gr N h m i ) → Ext ( e ∆ gr i , K h m i ) → · · · . The isomorphism (2.6) implies that the first term vanishes. On the other hand, thelast term vanishes by (2.9). We conclude that(2.11) Ext ( e ∆ gr i , L gr N h m i ) = 0 for m < M be the cokernel of L gr N ֒ → ∇ gr N . We will study Ext-groups involving M h m i with m <
0. Let M ′ = W − m − M and M ′′ = M/W − m − M . In other words, M ′ h m i = W − ( M h m i ) and M ′′ h m i = ( M h m i ) /W − ( M h m i ). All compositionfactors of M ′ h m i are of the form L gr j h n i with n < j ≤ N −
1, so by (2.9), wehave(2.12) Ext k ( e ∆ gr i , M ′ h m i ) = 0 for all k ≥ → Gr W − m M → M ′′ → M/W − m M →
0. It follows from (2.3) that Ext k ( e ∆ gr i , ( M/W − m M ) h m i ) = 0 for all k ≥
0, soExt k ( e ∆ gr i , M ′′ h m i ) ∼ = Ext k ( e ∆ gr i , Gr W − m M h m i ). Another invocation of (2.3) showsthat Ext k ( e ∆ gr i , Gr W − m M h m i ) ∼ = Ext k (Gr W e ∆ gr i , Gr W − m M h m i ). Now, by construc-tion, Gr W e ∆ gr i ∼ = Gr W ∆ gr i = ∆ ◦ i . On the other hand, since − m >
0, Gr W − m M h m i ∼ =Gr W − m ∇ gr N h m i . The latter object has a costandard filtration as an object of A ◦ , since A is metameric by assumption. By (2.4), we have that Ext k (∆ ◦ i , Gr W − m M h m i ) = 0for k ≥
1. Unwinding the last few sentences, we find that Ext k ( e ∆ gr i , M ′′ h m i ) = 0for all k ≥
1. Combining this with (2.12) yieldsExt k ( e ∆ gr i , M h m i ) = 0 for all k ≥ k ( e ∆ gr i , L gr N ) → Ext k ( e ∆ gr i , ∇ gr N ) is an isomor-phism for k ≥
2. The latter group vanishes because e ∆ gr i has a standard filtration. ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 11
We conclude that(2.13) Ext k ( e ∆ gr i , L gr N h m i ) = 0 for m < k ≥ m <
0. But they both hold for m > e ∆ gr i is, by construction, of the form L gr u h n i with n ≤ e ∆ gr i . To summarize, property (4) in the theoremholds by construction, and property (1) holds by (2.8) and (2.10). Property (2) iscovered by (2.9), and property (3) is obtained by combining (2.9), (2.11), and (2.13).The construction of e ∇ gr i is similar and will be omitted.We now turn to the last assertion in the theorem. Assume that A contains afamily of objects { e ∆ gr s , s ∈ S } satisfying properties (1)–(4). Let s, t ∈ S , and let m <
0. A routine argument with weight filtrations, using (2.3) and property (3)(similar to the discussion following (2.12)) shows thatExt ( e ∆ gr s , ∇ gr t h m i ) ∼ = Ext (cid:0) Gr W e ∆ gr s , (Gr W − m ∇ gr t ) h m i (cid:1) . The left-hand side vanishes because e ∆ gr s has a standard filtration. On the otherhand, it follows from property (4) that Gr W e ∆ gr s ∼ = Gr W ∆ gr s ∼ = ∆ ◦ s . Therefore,Ext (cid:0) ∆ ◦ s , (Gr W − m ∇ gr t ) h m i (cid:1) = 0. We have computed this Ext -group in A , but itsvanishing implies thatExt A ◦ (cid:0) ∆ ◦ s , (Gr W − m ∇ gr t ) h m i (cid:1) = 0 for all s ∈ S as well. By a standard argument (see e.g. [Do, Proposition A2.2(iii)]), we concludethat (Gr W − m ∇ gr t ) h m i has a costandard filtration for all m <
0. A dual argumentshows that each (Gr Wm ∆ gr t ) h− m i has a standard filtration, so A is metameric. (cid:3) Remark . In a metameric category, the description of projectives from [BGS,Theorem 3.2.1] or [AR3, Theorem A.3] can be refined somewhat, as follows. Let A be a metameric category, and let P gr s and P ◦ s be projective covers of L gr s in A andin A ◦ , respectively. Then P gr s admits a filtration whose subquotients are various e ∆ gr t with t ≥ s . Moreover, we have( P gr s : e ∆ gr t ) = ( P ◦ s : ∆ ◦ t ) . The proof is a straightforward generalization of that of [BGS, Theorem 3.2.1]. Asthis result will not be needed in this paper, we omit the details.
Recall that a graded quasi-hereditary category is said to be
Koszul if it satisfies(2.14) Ext k ( L gr s , L gr t h n i ) = 0 unless n = − k .(A Koszul category is automatically positively graded by Proposition 2.5.) It issaid to be standard Koszul if it satisfies(2.15) Ext k ( L gr s , ∇ gr t h n i ) = Ext k (∆ gr s , L gr t h n i ) = 0 unless n = − k .(See [ADL, Maz] for this notion; see also [Ir] for an earlier study of this condition.)The following well-known result follows from [ADL]. Since the latter paper usesa vocabulary which is is quite different from ours, we include a proof. Proposition 2.13.
Let A be a graded quasihereditary category. If A is standardKoszul, then it is Koszul. Proof.
We prove the result by induction on the cardinality of S . The claim isobvious if S consists of only one element, since in this case A is semisimple.Now assume that S has at least two elements, and that A is standard Koszul.Let s ∈ S be minimal, and set A ′ := A { s } , A ′′ := A / A { s } , ι := ι { s } , Π := Π { s } .By Lemma 2.2, these categories are graded quasihereditary. We claim that A ′′ isstandard Koszul. Indeed for t, u ∈ S r { s } we haveExt k A ′′ (Π( L gr t ) , Π( ∇ gr u ) h n i ) ∼ = Ext k A ( L gr t , Π R ◦ Π( ∇ gr u ) h n i ) ∼ = Ext k A ( L gr t , ∇ gr u h n i )by (2.2), and the right-hand side vanishes unless n = − k by assumption. Similarlywe have Ext k A ′′ (Π(∆ gr t ) , Π( L gr u ) h n i ) ∼ = Ext k A (∆ gr t , L gr u h n i ) , and again the right-hand side vanishes unless n = − k .By induction, we deduce that A ′′ is Koszul. Now let t ∈ S , and consider thedistinguished triangle ι ◦ ι R ( L gr t ) → L gr t → Π R ◦ Π( L gr t ) [1] −→ of Lemma 2.2. Applying the functor Hom( L gr u , −h n i ) (for some u ∈ S and n ∈ Z )we obtain a long exact sequence · · · → Ext k A ′ ( ι L ( L gr u ) , ι R ( L gr t ) h n i ) → Ext k A ( L gr u , L gr t h n i ) → Ext k A ′′ (Π( L gr u ) , Π( L gr t ) h n i ) → · · · Since A ′′ is Koszul, the third term vanishes unless n = − k . Hence to conclude itsuffices to prove that the first term also vanishes unless n = − k .We claim that ι L ( L gr u ) is a direct sum of objects of the form L gr s h m i [ − m ] forsome m ∈ Z . Indeed, since A ′ is semisimple, this object is a direct sum of objectsof the form L gr s h a i [ b ]. But if such an object appears as a direct summand thenHom D b ( A ′ ) ( ι L ( L gr u ) , L gr s h a i [ b ]) ∼ = Ext b A ( L gr u , L gr s h a i ) = 0 , which implies that a = − b since L gr s = ∇ gr s ; this finishes the proof of the claim.Similar arguments show that ι R ( L gr t ) is also a direct sum of objects of the form L gr s h m ′ i [ − m ′ ] for m ′ ∈ Z . One deduces that indeed Ext k A ′ ( ι L ( L gr u ) , ι R ( L gr t ) h n i ) = 0unless n = − k , which finishes the proof. (cid:3) Remark . The proof shows that the condition that Ext k ( L gr s , ∇ gr t h n i ) = 0 un-less k = − n already implies that A is Koszul. Similar arguments using the func-tors ι L , Π L instead of ι R , Π R show that if A satisfies the dual condition thatExt k (∆ gr s , L gr t h n i ) = 0 unless k = − n , then A is Koszul. Q -Koszul and standard Q -Koszul categories. In this subsection we studya generalization of the notions considered in § § Definition 2.15.
Let A be a positively graded quasihereditary category. It is saidto be Q -Koszul if Ext k A (∆ ◦ s , ∇ ◦ t h n i ) = 0 unless n = − k .It is said to be standard Q -Koszul ifExt k A (∆ ◦ s , ∇ gr t h n i ) = Ext k A (∆ gr s , ∇ ◦ t h n i ) = 0 unless n = − k . ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 13
The following result is an analogue of Proposition 2.13 in this context. The sameresult appears in [PS2, Corollary 3.2], but in a somewhat different language, so aswith Proposition 2.13, we include a proof.
Proposition 2.16.
Let A be a positively graded quasihereditary category. If A isstandard Q -Koszul, then it is Q -Koszul.Proof. The proof is very similar to that of Proposition 2.13. We proceed by in-duction on the cardinality of S , the base case being obvious. We choose s ∈ S minimal, and set A ′ := A { s } , A ′′ := A / A ′ , ι := ι { s } , Π := Π { s } . By Corollary 2.6,the category A ′′ is positively graded. It is also clear that Gr W (Π(∆ gr t )) ∼ = Π(∆ ◦ t )and Gr W (Π( ∇ gr t )) ∼ = Π( ∇ ◦ t ) for t = s . Then using (2.2) as in the proof of Propo-sition 2.13, one obtains that A ′′ is standard Q -Koszul; hence by induction it is Q -Koszul.Now consider, for t ∈ S , the distinguished triangle ι ◦ ι R ( ∇ ◦ t ) → ∇ ◦ t → Π R ◦ Π( ∇ ◦ t ) [1] −→ . Applying the functor Hom(∆ ◦ u , −h n i ) (for some u ∈ S and n ∈ Z ) we obtain along exact sequence · · · → Ext k A ′ ( ι L (∆ ◦ u ) , ι R ( ∇ ◦ t ) h n i ) → Ext k A (∆ ◦ u , ∇ ◦ t h n i ) → Ext k A ′′ (Π(∆ ◦ u ) , Π( ∇ ◦ t ) h n i ) → · · · By induction, the third term vanishes unless n = − k . Now one can easily checkthat both ι L (∆ ◦ u ) and ι R ( ∇ ◦ t ) are direct sums of objects of the form L gr s h m i [ − m ]for m ∈ Z , and we deduce that the first term also vanishes unless n = − k , whichfinishes the proof. (cid:3) Remark . It is natural to ask whether there is a notion of “Koszul duality” for Q -Koszul categories.Recall that classical Koszul duality is a kind of derived equivalence that sendssimple objects in one category to projective objects in the other. There is a gen-eralization of this notion due to Madsen [Mad]. Suppose A is a finite-length (butnot necessarily quasihereditary) category satisfying conditions (4) or (5) of Propo-sition 2.5. Then it still makes sense to define a Serre subcategory A ◦ as in Proposi-tion 2.7. Assume that A ◦ has the structure of a quasihereditary category, and thatfor any two tilting objects T ◦ s , T ◦ t ∈ A ◦ , we haveExt k A ( T ◦ s , T ◦ t h n i ) = 0 unless n = − k .Such a category A is said to be T -Koszul . Madsen’s theory leads to a new T -Koszulabelian category B and a derived equivalence D b ( A ) ∼ −→ D b ( B ) such that tiltingobjects of A ◦ correspond to projective objects in B . If A ◦ happens to be semisimple,then Madsen’s notion reduces to ordinary Koszul duality.Clearly, every Q -Koszul category is T -Koszul. But it is not known whether the T -Koszul dual of a Q -Koszul category must be Q -Koszul, see [PS2, Questions 4.2]. Weights
In this section (and the next one) we work in the setting of [AR3, §§ ℓ and a finite extension K of Q ℓ . We denote by O the ring of integers of K and by F the residue field of O . We usethe letter E to denote any member of ( K , O , F ).We fix a complex algebraic variety X endowed with a finite stratification X = F s ∈ S X s where each X s is isomorphic to an affine space. We denote by D b S ( X, E )the derived S -constructible category of sheaves on X , with coefficients in E . Thecohomological shift in this category will be denoted { } . We assume that the as-sumptions (A1) (“existence of enough parity complexes”) and (A2) (“standardand costandard objects are perverse”) of [AR3] are satsified. Then one can con-sider the additive category Parity S ( X, E ) of parity complexes on X (in the senseof [JMW]; see [AR3, § D mix S ( X, E ) := K b Parity S ( X, E ). This categorypossesses two important autoequivalences: the cohomological shift [1], and the “in-ternal” shift { } induced by the shift functor on parity complexes. We also set h i := {− } [1]. If h : Y → X is a locally closed inclusion of a union of strata, thenwe have well-defined functors h ∗ , h ! : D mix S ( Y, E ) → D mix S ( X, E ) , h ∗ , h ! : D mix S ( Y, E ) → D mix S ( X, E )which satisfy all the usual properties; see [AR3, § S the restriction of the stratification to Y .) We also have “extension ofscalars” functors K : D mix S ( X, O ) → D mix S ( X, K ) , F : D mix S ( X, O ) → D mix S ( X, F )and a “Verdier duality” antiequivalence D : D mix S ( X, E ) ∼ −→ D mix S ( X, E ) . The triangulated category D mix S ( X, E ) can be endowed with a “perverse t-struc-ture”; see [AR3, Definition 3.3]. We denote by P mix S ( X, E ) the heart of this t-structure. Objects of P mix S ( X, E ) will be called “mixed perverse sheaves.” If E = O or K , this category is a graded quasihereditary category, with shift func-tor h i , simple objects IC mix s := i s ! ∗ E X s , standard objects ∆ mix s := i s ! E X s , andcostandard objects ∇ mix s := i s ∗ E X s . (Here, i s : X s → X is the inclusion, and E X s := E X s { dim( X s ) } , where E X s is the constant sheaf on X s , an object of Parity S ( X s , E ).) We denote by P mix s the projective cover of IC mix s , and by T mix s the indecomposable tilting object associated with s . When necessary, we add amention of the coefficients “ E ” we consider. Note in particular that we have K ( IC mix s ( O )) ∼ = IC mix s ( K ) , F ( P mix s ( O )) ∼ = P mix s ( F ) , F ( T mix s ( O )) ∼ = T mix s ( F ) . (For all of this, see [AR3, §§ E s the unique indecomposable parity complex whichis supported on X s and whose restriction to X s is E X s . We denote this same objectby E mix s when it is regarded as an object of D mix S ( X, E ).We do not know whether P mix S ( X, F ) is positively graded under these assump-tions. The main result of this section, Proposition 3.15, gives a number of conditionsthat are equivalent to P mix S ( X, F ) being positively graded. Along the way to thatresult, we construct a candidate abelian category P ◦ S ( X, F ) that “should” be thecategory A ◦ of Proposition 2.7 in this case. However, P ◦ S ( X, F ) is defined evenwhen P mix S ( X, F ) is not positively graded. ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 15
We begin by introducing a notion that will “morally” play the samerole in D mix S ( X, E ) that is played by Deligne’s theory of weights (see [BBD, § ℓ -adic ´etale sheaves. Definition 3.1.
An object M ∈ D mix S ( X, E ) is said to have weights ≤ n (resp. ≥ n )if it is isomorphic to a complex · · · → M − → M → M → · · · of objects in Parity S ( X, E ) in which M i = 0 for all i < − n (resp. i > − n ). It is said to be pureof weight n if has weights ≤ n and ≥ n .The full subcategory of D mix S ( X, E ) consisting of objects with weights ≤ n (resp. ≥ n ) is denoted D mix S ( X, E ) ≤ n (resp. D mix S ( X, E ) ≥ n ). Using standard ar-guments in triangulated categories one can check that these categories admit thefollowing alternative characterizations: D mix S ( X, E ) ≤ n = { M | Hom( M, E mix s { m } [ k ]) = 0 for all m ∈ Z and all k > n } ,D mix S ( X, E ) ≥ n = { M | Hom( E mix s { m } [ k ] , M ) = 0 for all m ∈ Z and all k < n } , and moreover that an object in D mix S ( X, E ) is pure of weight n if and only if it is adirect sum of objects of the form E mix s { m } [ n ].Note that weights are stable under extensions. That is, if the first and thirdterms of a distinguished triangle have weights ≤ n (resp. ≥ n ), then the sameholds for the middle term. Example . Consider a single stratum X s . For a finitely-generated E -module N ,let N denote the corresponding constant sheaf on X s , and let N = N { dim X s } .(Here we use the same convention as in [AR3, § E = O and N is notfree.) Every object M ∈ D mix S ( X s , E ) is isomorphic (canonically if E = F or K , andnoncanonically if E = O ) to a finite direct sum(3.1) M ∼ = M i,j ∈ Z M ij { j } [ − i ]where the M ij are various finitely generated E -modules. With this in mind:(1) If E = F or K , then M has weights ≤ M ij = 0 for all i < E = O , then M has weights ≤ M ij = 0 for all i <
0, andall M j are torsion-free. Lemma 3.3.
For any s ∈ S , ∆ mix s has weights ≤ , and ∇ mix s has weights ≥ .Proof. It is clear by adjunction (and using [AR3, Remark 2.7]) that we haveHom(∆ mix s , E mix t { m } [ k ]) = 0 if k = 0, and similarly for ∇ mix s . (cid:3) Lemma 3.4.
Let j : U ֒ → X be the inclusion of an open union of strata, and let i : Z ֒ → X be the complementary closed inclusion. (1) j ∗ and i ∗ preserve weights. (2) j ! sends D mix S ( U, E ) ≤ n to D mix S ( X, E ) ≤ n , and j ∗ sends D mix S ( U, E ) ≥ n to D mix S ( X, E ) ≥ n . (3) i ∗ sends D mix S ( X, E ) ≤ n to D mix S ( Z, E ) ≤ n , and i ! sends D mix S ( X, E ) ≥ n to D mix S ( Z, E ) ≥ n . (4) If Z consists of a single stratum, then i ∗ and i ! preserve weights. (5) D exchanges D mix S ( X, E ) ≤ n and D mix S ( X, E ) ≥− n . Proof.
Parts (1), (4), and (5) are clear, because in those cases, the functors takeparity complexes to parity complexes. Parts (2) and (3) then follow from part (1)by adjunction. (cid:3)
Lemma 3.5.
Let
F ∈ D mix S ( X, E ) . We have: (1) F has weights ≤ n if and only if i ∗ s F has weights ≤ n for all s ∈ S . (2) F has weights ≥ n if and only if i ! s F has weights ≥ n for all s ∈ S .Proof. We will only treat the first assertion. The “only if” direction is part ofLemma 3.4, so we need only prove the “if” direction. In that case, we proceed byinduction on the number of strata in X . If X consists of a single stratum, thereis nothing to prove. Otherwise, suppose i ∗ s F has weights ≤ n for all s . Choose aclosed stratum X s ⊂ X . Let U = X r X s , and let j : U ֒ → X be the inclusionmap. Then j ∗ F has weights ≤ n by induction. The first and last terms of thedistinguished triangle j ! j ∗ F → F → i s ∗ i ∗ s F → have weights ≤ n by Lemma 3.4, sothe middle term does as well. (cid:3) For n ∈ Z , we define two full triangulatedsubcategories of D mix S ( X, E ) as follows:(3.2) D mix S ( X, E ) E n := the subcategory generated by the ∆ mix s h m i for m ≤ nD mix S ( X, E ) D n := the subcategory generated by the ∇ mix s h m i for m ≥ n We also put D mix S ( X, E ) ◦ := D mix S ( X, E ) E ∩ D mix S ( X, E ) D . Example . With the notation of Example 3.2, the object M in (3.1) lies in D mix S ( X s , E ) E if and only if M ij = 0 for all j < Lemma 3.7. (1)
For any A ∈ D mix S ( X, E ) E n and B ∈ D mix S ( X, E ) D n +1 , wehave Hom(
A, B ) = 0 . (2) The inclusion D mix S ( X, E ) E n ֒ → D mix S ( X, E ) admits a right adjoint β E n : D mix S ( X, E ) → D mix S ( X, E ) E n , which is a triangulated functor. Similarly,the inclusion D mix S ( X, E ) D n ֒ → D mix S ( X, E ) admits a left adjoint β D n : D mix S ( X, E ) → D mix S ( X, E ) D n , which is a triangulated functor. (3) For every object M ∈ D mix S ( X, E ) and every n ∈ Z , there is a functorialdistinguished triangle β E n M → M → β D n +1 M [1] −→ . Moreover, if M ′ ∈ D mix S ( X, E ) E n and M ′′ ∈ D mix S ( X, E ) D n +1 , for any dis-tinguished triangle M ′ → M → M ′′ [1] −→ there exist canonical isomorphisms ϕ : M ′ ∼ −→ β E n M and ψ : M ′′ ∼ −→ β D n +1 M such that ( ϕ, id M , ψ ) is anisomorphism of distinguished triangles. (4) All the β E n and β D m commute with one another.Proof. Part (1) is immediate from the definitions and [AR3, Lemma 3.2]. Next, weprove a weak version of part (3). It is clear that the collection of objects C = { ∆ mix s h m i | m ≤ } ∪ {∇ mix s h m i | m ≥ } generates D mix S ( X, E ) as a triangulated category. Let us express this another way,using the “ ∗ ” notation of [BBD, § M ∈ D mix S ( X, E ), there are ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 17 objects C , . . . , C n ∈ C and integers k , . . . , k n such that(3.3) M ∈ C [ k ] ∗ · · · ∗ C n [ k n ] . Now, observe that if a ≤ b ≥
1, then ∇ mix s h b i [ p ] ∗ ∆ mix t h a i [ q ] contains onlythe object ∇ mix s h b i [ p ] ⊕ ∆ mix t h a i [ q ], because Hom(∆ mix t h a i [ q ] , ∇ mix s h b i [ p + 1]) = 0.Thus, ∇ mix s h b i [ p ] ∗ ∆ mix t h a i [ q ] ⊂ ∆ mix t h a i [ q ] ∗ ∇ mix s h b i [ p ] . Using this fact, we can rearrange the expression (3.3) so that the following holds:there is some n ′ ≤ n such that C , . . . , C n ′ are all of the form ∆ mix s h m i with m ≤ C n ′ +1 , . . . , C n are of the form ∇ mix s h m i with m ≥
1. Then (3.3) says thatthere is a distinguished triangle A → M → B → where A ∈ C ∗ · · ·∗ C n ′ ⊂ D mix S ( X, E ) E , and B ∈ C n ′ +1 ∗ · · ·∗ C n ⊂ D mix S ( X, E ) D .We have not yet proved that this triangle is functorial. However, we have shownthat the collection of categories ( { D mix S ( X, E ) E n } , { D mix S ( X, E ) D n } ) n ∈ Z satisfies theaxioms of a so-called baric structure [AT, Definition 2.1]. The remaining statementsin the lemma are general properties of baric structures from [AT, Propositions 2.2& 2.3]. (cid:3) Remark . If M is an object of D mix S ( X, E ), then M is in D mix S ( X, E ) E iffHom( M, ∇ mix s h m i [ k ]) = 0 for all s ∈ S , k ∈ Z and m ∈ Z > . Indeed, the “onlyif” part follows from Lemma 3.7(1). To prove the “if” part, consider the barictruncation triangle β E M → M → β D M [1] −→ of Lemma 3.7(3). Our assumptionimplies that the second arrow in this triangle is trivial, hence we deduce an iso-morphism β E M ∼ = M ⊕ β D M [ − β D M were non zero, then the projection β E M → β D M [ −
1] would be non zero, contradicting Lemma 3.7(1).
Lemma 3.9.
Let j : U ֒ → X be the inclusion of an open union of strata, and let i : Z ֒ → X be the complementary closed inclusion. (1) j ∗ and i ∗ commute with all β E n and β D n . (2) j ! sends D mix S ( U, E ) E n to D mix S ( X, E ) E n , and j ∗ sends D mix S ( U, E ) D n to D mix S ( X, E ) D n . (3) i ∗ sends D mix S ( X, E ) E n to D mix S ( Z, E ) E n , and i ! sends D mix S ( X, E ) D n to D mix S ( Z, E ) D n . (4) D exchanges D mix S ( X, E ) E n and D mix S ( X, E ) D − n .Proof. For the first three parts, it suffices to observe that j ∗ , i ∗ , j ! , and i ∗ sendstandard objects to standard objects (or to zero), while j ∗ , i ∗ , j ∗ , and i ! sendcostandard objects to costandard objects (or to zero). Similarly, the last partfollows from the fact that D exchanges standard and costandard objects. (cid:3) Lemma 3.10.
The functors β E n and β D n commute with K ( − ) and F ( − ) .Proof. Since extension of scalars sends standard objects to standard objects andcostandard objects to costandard objects, it is clear that F ( − ) sends D mix S ( X, O ) E n to D mix S ( X, F ) E n and D mix S ( X, O ) D n to D mix S ( X, F ) D n , and similarly for K ( − ). Thenthe result follows from Lemma 3.7(3). (cid:3) Lemma 3.11.
Suppose X = X s consists of a single stratum. Then the functors β E n and β D n are t-exact for the perverse t-structure on D mix S ( X s , E ) . In fact, for M ∈ D mix S ( X s , E ) , there exists a canonical isomorphism M ∼ = β E n M ⊕ β D n +1 M . Proof.
Given M ∈ D mix S ( X s , E ), write a decomposition as in (3.1), and form thedistinguished triangle M i ∈ Z j ≥− n M ij { j } [ − i ] → M → M i ∈ Z j ≤− n − M ij { j } [ − i ] [1] −→ . Referring to Example 3.6, we see that the first term belongs to D mix S ( X s , E ) E n ,and the third one to D mix S ( X s , E ) D n +1 . By Lemma 3.7(3), this triangle must becanonically isomorphic to β E n M → M → β D n +1 M [1] −→ . This triangle is clearlysplit. Since Hom( β D n +1 M, β E n M ) vanishes, the splitting is canonical. Finally,since any direct summand of a perverse sheaf is a perverse sheaf, the functors β E n and β D n are t-exact. (cid:3) D mix S ( X, E ) ◦ . In the following statement we use the notionof recollement from [BBD, § Proposition 3.12.
Let j : U ֒ → X be the inclusion of an open union of strata,and let i : Z ֒ → X be the complementary closed inclusion. We have a recollementdiagram D mix S ( Z, E ) ◦ i ∗ / / D mix S ( X, E ) ◦ j ∗ / / β D i ∗ s s β E i ! k k D mix S ( U, E ) ◦ . β D j ! s s β E j ∗ k k Proof.
The required adjunction properties for these functors, and the fact that j ∗ i ∗ = 0, follow from the corresponding result for the mixed derived category;see [AR3, Proposition 2.3]. Next, for M ∈ D mix S ( Z, E ) ◦ , consider the natural maps i ∗ i ∗ M → ( β D i ∗ ) i ∗ M → M. It is easily checked that the composition is the morphism induced by adjunction,and so is an isomorphism. In particular, i ∗ i ∗ M lies in D mix S ( Z, E ) D , so the map i ∗ i ∗ M → β D i ∗ i ∗ M is an isomorphism. We conclude that the adjunction map( β D i ∗ ) i ∗ M → M is an isomorphism as well. Similar arguments show that theadjunction morphisms id → j ∗ ( β D j ! ), id → ( β E i ! ) i ∗ , and j ∗ ( β E j ∗ ) → id areisomorphisms.Finally, given M ∈ D mix S ( X, E ) ◦ , form the triangle j ! j ∗ M → M → i ∗ i ∗ M [1] −→ ,and then apply β D . Using Lemma 3.9, we obtain a distinguished triangle( β D j ! ) j ∗ M → M → i ∗ ( β D i ∗ ) M [1] −→ . Similar reasoning leads to the triangle i ∗ ( β E i ! ) M → M → ( β E j ∗ ) j ∗ M [1] −→ . (cid:3) Proposition 3.13.
The following two full subcategories of D mix S ( X, E ) ◦ constitutea t-structure: D mix S ( X, E ) ◦ , ≤ = { M | β D i ∗ s M ∈ p D mix S ( X s , E ) ≤ for all s ∈ S } ,D mix S ( X, E ) ◦ , ≥ = { M | β E i ! s M ∈ p D mix S ( X s , E ) ≥ for all s ∈ S } . Moreover, if E = K or F , this t-structure is preserved by D . ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 19
Proof.
Let us first treat the special case where X consists of a single stratum X s . Inthis case, the definition reduces to D mix S ( X, E ) ◦ , ≤ = D mix S ( X, E ) ◦ ∩ p D mix S ( X s , E ) ≤ and D mix S ( X, E ) ◦ , ≥ = D mix S ( X, E ) ◦ ∩ p D mix S ( X s , E ) ≥ . Because β E and β D aret-exact here (see Lemma 3.11), these categories do indeed constitute a t-structureon D mix S ( X, E ) ◦ .The proposition now follows by induction on the number of strata in X usinggeneral properties of recollement; see [BBD, Th´eor`eme 1.4.10]. (cid:3) We denote the heart of this t-structure by P ◦ S ( X, E ) := D mix S ( X, E ) ◦ , ≤ ∩ D mix S ( X, E ) ◦ , ≥ . We saw in the course of the proof that on a single stratum, we have P ◦ S ( X s , E ) = P mix S ( X s , E ) ∩ D mix S ( X s , E ) ◦ , but this does not necessarily hold for larger varieties.For another description of this t-structure, we introduce the objects∆ ◦ s := β D j ! E X s and ∇ ◦ s := β E j ∗ E X s . By adjunction, we have(3.4) D mix S ( X, E ) ◦ , ≤ = { M | for all s ∈ S and k <
0, Hom( M, ∇ ◦ s [ k ]) = 0 } ,D mix S ( X, E ) ◦ , ≥ = { M | for all s ∈ S and k >
0, Hom(∆ ◦ s [ k ] , M ) = 0 } . Note that by definition we have ∆ ◦ s ∈ D mix S ( X, E ) ◦ , ≤ and ∇ ◦ s ∈ D mix S ( X, E ) ◦ , ≥ ,but it is not clear in general whether ∆ ◦ s and ∇ ◦ s belong to P ◦ S ( X, E ).Let ◦ H i : D mix S ( X, E ) ◦ → P ◦ S ( X, E ) denote the i -th cohomology functor withrespect to this t-structure. For s ∈ S , we put IC ◦ s := im (cid:0) ◦ H (∆ ◦ s ) → ◦ H ( ∇ ◦ s ) (cid:1) , where the map is induced by the natural map ∆ mix s → ∇ mix s . If E is a field, then P ◦ S ( X, E ) is a finite-length category, and its simple objects are precisely the objects IC ◦ s . Moreover, in this case, these objects are preserved by D . The description in (3.4) matches the frame-work of [Be2, Proposition 2(c)]. That statement (see also [Be2, Remark 1]) tellsus that when E is a field, P ◦ S ( X, E ) always satisfies ungraded analogues of axioms(1)–(3) of Definition 2.1 (with respect to the objects ∆ ◦ s and ∇ ◦ s ). Under additionalassumptions, we can obtain finer information about this category. Lemma 3.14.
Assume that E = F or K , and that for all s ∈ S , ∆ ◦ s and ∇ ◦ s lie in P mix S ( X, E ) . (1) The category P ◦ S ( X, E ) is quasihereditary, the ∆ ◦ s and the ∇ ◦ s being, respec-tively, the standard and costandard objects. Moreover, if T ◦ S ( X, E ) denotesthe category of tilting objects in P ◦ S ( X, E ) , the natural functors K b T ◦ S ( X, E ) → D b P ◦ S ( X, E ) → D mix S ( X, E ) ◦ are equivalences of categories. (2) We have P ◦ S ( X, E ) ⊂ P mix S ( X, E ) , and the inclusion functor D mix S ( X, E ) ◦ → D mix S ( X, E ) is t-exact. (3) If the objects E mix s are perverse, then they lie in P ◦ S ( X, E ) , and they areprecisely the indecomposable tilting objects therein. Proof.
If all the objects ∆ ◦ s and ∇ ◦ s lie in P mix S ( X, E ), then there are no nonvanishingnegative-degree Ext-groups among them, so we see from (3.4) that these objects liein P ◦ S ( X, E ). Next, the proof of [AR3, Lemma 3.2] is easily adapted to show thatfor any s, t ∈ S , we haveHom D mix S ( X, E ) ◦ (∆ ◦ s , ∇ ◦ t [ i ]) = 0 if i = 0.With these observations in hand, the rest of the proof of part (1) is essentiallyidentical to that of [AR3, Proposition 3.10 and Lemma 3.14].We prove part (2) by induction on the number of strata in X . If X consists ofa single stratum, the statement holds trivially.Otherwise, choose an open stratum X s ⊂ X . It suffices to prove that every simpleobject of P ◦ S ( X, E ) lies in P mix S ( X, E ). For t = s , the object IC ◦ t is supported onthe smaller variety X r X s , so we know by induction that it lies in P mix S ( X, E ). Itremains to consider IC ◦ s . Let K be the kernel of the natural map ∆ ◦ s → IC ◦ s . Since K is also supported on X r X s , we know that K ∈ P mix S ( X, E ). By assumption,∆ ◦ s ∈ P mix S ( X, E ), so by considering the distinguished triangle K → ∆ ◦ s → IC ◦ s [1] −→ ,we see that IC ◦ s ∈ p D mix S ( X, E ) ≤ . Since D ( IC ◦ s ) ∼ = IC ◦ s , this object also lies in p D mix S ( X, E ) ≥ , and hence in P mix S ( X, E ), as desired.Finally, we consider part (3). We claim that Hom( E mix s , ∇ mix t { n } [ k ]) = 0 for all n <
0. When k = 0, this follows from the assumption that E mix s is perverse, andwhen k = 0, it follows from the same arguments as for Lemma 3.3. Thus, E mix s lies in D mix S ( X, E ) E . Since D ( E mix s ) ∼ = E mix s , this object also lies in D mix S ( X, E ) D ,hence in D mix S ( X, E ) ◦ . Similar arguments show thatHom( E mix s , ∇ mix t [ k ]) ∼ = Hom( E mix s , ∇ ◦ t [ k ])vanishes for k >
0. That condition and its dual together imply that E mix s belongs to P ◦ S ( X, E ) and is a tilting object therein, by, say, the criterion in [Be2, Lemma 4].The E mix s are indecomposable and parametrized by S , so they must coincide withthe indecomposable tilting objects of P ◦ S ( X, E ). (cid:3) We conclude this section with a result col-lecting a number of conditions equivalent to P mix S ( X, E ) being positively graded.The proof makes use of Verdier duality, but no other tools coming from geometry.Indeed, if A is any graded quasihereditary category equipped with an antiautoe-quivalence satisfying similar formal properties to D , one can formulate an analogueof the following proposition for D b ( A ). The argument below will go through essen-tially verbatim. Proposition 3.15.
Assume that E = F or K . The following are equivalent: (1) The category P mix S ( X, E ) is positively graded. (2) We have [∆ mix s : IC mix t h n i ] = 0 if n > . (3) We have ( P mix s : ∆ mix t h n i ) = 0 if n > . (4) We have IC mix s ∈ D mix S ( X, E ) ◦ for all s ∈ S . (5) For all n ∈ Z , the functors β E n and β D n are t-exact for the perverse t-structure on D mix S ( X, E ) . (6) We have IC ◦ s ∼ = IC mix s for all s ∈ S .Moreover, if these conditions hold, then P ◦ S ( X, E ) can be identified with the Serresubcategory of P mix S ( X, E ) generated by all the IC mix s (without Tate twists). ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 21
Remark . The last assertion says that when the above conditions hold, weare in the setting of Proposition 2.7; in this case the two definitions of ∆ ◦ s and of ∇ ◦ s coincide. Moreover, under this assumption all the objects ∆ ◦ s and ∇ ◦ s lie in P mix S ( X, E ), so the conclusions of Lemma 3.14 hold as well. Proof. (1) ⇐⇒ (2) ⇐⇒ (3). We saw in Proposition 2.5 that (1) holds if andonly if both (2) and (3) hold. But by Verdier duality, (2) holds if and only if[ ∇ mix s h n i : IC mix t ] = 0 for all n >
0. By the reciprocity formula, the latter isequivalent to (3).(1) = ⇒ (4). As observed in the proof of (2.3), IC mix s admits a finite resolution · · · → P − → P such that every term P i is a direct sum of objects of the form P mix t h n i with n ≤
0. Using (3), we see that every term of this projective resolutionlies in D mix S ( X, E ) E , so IC mix s ∈ D mix S ( X, E ) E as well. Since IC mix s is stable underVerdier duality D , we also have IC mix s ∈ D mix S ( X, E ) D .(4) = ⇒ (5). The assumption implies that(3.5) β E n ( IC mix s h m i ) ∼ = ( IC mix s h m i if m ≤ n ,0 if m > n ,along with a similar formula for β D n . Since β E n and β D n send every simple objectof P mix S ( X, E ) to an object of P mix S ( X, E ), they are both t-exact.(5) = ⇒ (6). First we note that, if (5) holds, then the assumptions of Lemma 3.14are satisfied. Consider the distinguished triangle β E − IC mix s → IC mix s → β D IC mix s [1] −→ . Since β E − and β D are exact, this is actually a short exact sequence in P mix S ( X, E ).The middle term is simple, so either the first or last term must vanish. The nonzeromorphism IC mix s → ∇ mix s shows that β D IC mix s = 0. Thus, we have β E − IC mix s = 0,and IC mix s ∼ = β D IC mix s . A dual argument shows that we actually have IC mix s ∼ = β E β D IC mix s . Moreover, applying the exact functor β E β D to the canonical morphism ∆ mix s →∇ mix s tells us that IC mix s is the image (in P mix S ( X, E )) of the map ∆ ◦ s → ∇ ◦ s . On theother hand, Lemma 3.14 tells us that this map is also a morphism in P ◦ S ( X, E ),where its image is IC ◦ s . Since the inclusion functor P ◦ S ( X, E ) → P mix S ( X, E ) is exact(again by Lemma 3.14), the image of ∆ ◦ s → ∇ ◦ s is the same in both categories.(6) = ⇒ (1). The assumption implies that IC mix s ∈ D mix S ( X, E ) E , and that if n >
0, then IC mix t h n i [1] ∈ D mix S ( X, E ) D . Therefore,Ext ( IC mix s , IC mix t h n i ) = Hom D mix S ( X, E ) ( IC mix s , IC mix t h n i [1]) = 0by Lemma 3.7(1). By Proposition 2.5, it follows that P mix S ( X, E ) is positivelygraded.The last assertion in the proposition is immediate from part (6). (cid:3) For later use, we conclude this section with a description of themost favorable situation. (See [RSW, Proposition 5.7.2] and [We, Theorem 5.3] forrelated results.)
Corollary 3.17.
Assume that E = K or F , and that for all s ∈ S we have IC mix s ∼ = E mix s . Then the category P mix S ( X, E ) is Koszul (and hence in particularpositively graded). Proof.
Under our assumptions we haveExt k P mix S ( X, E ) ( IC mix s , IC mix t h n i ) ∼ = Hom D mix S ( X, E ) ( E mix s , E mix t {− n } [ k + n ]) , which clearly vanishes unless k + n = 0. (cid:3) Remark . One can easily show that, under these assumptions, P mix S ( X, E ) iseven standard Koszul. Further study of mixed perverse O -sheaves We continue in the setting of Section 3, with the goal of furthering our under-standing of positivity. The arguments in the previous section were mostly based ongeneral principles of homological algebra, and in some cases were restricted to fieldcoefficients. To make further progress, we need to bring in concrete geometric factsabout our variety. In this section, we will focus on O -sheaves as an intermediarybetween F - and K -sheaves, and the main results will involve the assumption that IC mix s ( K ) ∼ = E mix s ( K ). This holds, of course, on flag varieties, by [KL]. We begin with a brief review ofa convenient language for describing objects in D mix S ( X, E ) with a specified restric-tion to some open subset of X (see e.g. [JMW, Lemma 2.18] for a similar statementin the classical setting). The descriptions below are valid for arbitrary coefficients,although they will be used in this paper mainly in the case where E = O .Let X t ⊂ X be a closed stratum, and let j : U ֒ → X be the inclusion of thecomplementary open subset. Let M U ∈ D mix S ( U, E ). Then there is a bijectionbetween the set of isomorphism classes of pairs ( M, α ) where M ∈ D mix S ( X, E ) and α : j ∗ M ∼ −→ M U is an isomorphism in D mix S ( U, E ), and the set of isomorphismclasses of distinguished triangles(4.1) A → i ! t j ! M U → B [1] −→ in D mix S ( X t , E ). Specifically, given such a triangle, one can recover M as the coneof the composite morphism i t ∗ A → i t ∗ i ! t j ! M U → j ! M U . On the other hand, to M we associate the natural triangle with A = i ∗ t M [ −
1] and B = i ! t M. Here are some specific examples: If M U is perverse, the extension M = j ! ∗ M U corresponds to A = τ ≤ i ! t j ! M U , B = τ ≥ i ! t j ! M U (see [BBD, Proposition 1.4.23]). The extension M = j ! M U corresponds to A = 0, B = i ! t j ! M U . The extension M = j ∗ M U corresponds to A = i ! t j ! M U , B = 0. If M U ∈ D mix S ( U, E ) ◦ , then β D j ! M U corresponds to A = β E − i ! t j ! M U , B = β D i ! t j ! M U . (Indeed we have A = i ∗ t β D j ! M U [ − β D A = ( β D i ∗ t )( β D j ! ) M U [ −
1] = 0,which implies that A is in D mix S ( X t , E ) E − . On the other hand, B = i ! t β D j ! M U is in D mix S ( X t , E ) D by Lemma 3.9. Hence triangle (4.1) must be the truncationtriangle for the baric structure.) ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 23 If M U ∈ Parity S ( U, E ), then i ! t j ! M U ∈ D mix S ( X t , E ) has weights in the inter-val [ − , F • in K b Parity S ( X t , E )in which the only nonzero terms are F and F . If E = K or F , then the “parityextension” of M U constructed in [JMW, Lemma 2.27] (considered as an object in D mix S ( X, E )) corresponds to A = F [ − , B = F . ∆ ◦ s ( O ) . If M is in D mix S ( X, E ), we will say that the stalks of M are pure of weight s ∈ S , the object i ∗ s M ∈ D mix S ( X s , E ) is pureof weight 0, i.e. a direct sum of objects of the form E X s { i } for i ∈ Z . Typicalobjects that satisfy this condition are the parity sheaves E mix s . Note that if M isin D mix S ( X, O ), then the stalks of M are pure of weight 0 iff the stalks of F ( M ) arepure of weight 0.In the proofs below we will use the following notation. Recall from Lemma 3.11that on a single stratum X s , the functors β E n and β D n are t-exact. For objects in D mix S ( X s , E ), we set p H kr := p H k ◦ β E r ◦ β D r ∼ = β E r ◦ β D r ◦ p H k . The following result relates “pointwise purity” to a “torsion-free” condition.
Lemma 4.1.
For each s ∈ S , the following conditions are equivalent: (1) The stalks of ∆ ◦ s ( F ) are pure of weight . (2) The stalks of ∆ ◦ s ( O ) are pure of weight .Moreover, if IC mix s ( K ) ∼ = E mix s ( K ) , then these statements are also equivalent to thefollowing one: (3) We have ∆ ◦ s ( O ) ∼ = IC mix s ( O ) , and the stalks of IC mix s ( O ) are free.Proof. Conditions (1) and (2) are equivalent because F (∆ ◦ s ( O )) ∼ = ∆ ◦ s ( F ) (seeLemma 3.10).Assume now that IC mix s ( K ) ∼ = E mix s ( K ). If condition (3) holds, then the stalksof IC mix s ( O ) are pure of weight 0, since those of K ( IC mix s ( O )) ∼ = IC mix s ( K ) are,implying (2).Conversely, suppose that condition (2) holds. We will prove condition (3) byinduction on the number of strata in X . If X consists of a single stratum, thestatement is trivial. Otherwise, let X t ⊂ X be a closed stratum, and let j : U ֒ → X be the inclusion of the complementary open subset. Let X s be a stratum in U . Let M U := ∆ ◦ U,s ( O ) ∼ = IC mix U,s ( O ), and let L = i ! t j ! M U .We begin by showing that p H ( L ) is a torsion O -module. Observe that K ( L ) ∼ = i ! t j ! ( K ( IC mix U,s ( O ))) ∼ = i ! t j ! IC mix U,s ( K ). According to § τ ≤ ( K ( L )) ∼ = i ∗ t IC mix s ( K )[ − ∼ = i ∗ t E mix s ( K )[ − k ≤ p H kr ( K ( L )) vanishesunless r = k −
1. In particular, p H ( K ( L )) ∼ = K ( p H ( L )) = 0. This implies that p H ( L ) is torsion.Next, we carry out a similar line of reasoning using the fact that β E − L ∼ = i ∗ t ∆ ◦ X,s ( O )[ −
1] (see § − r ≤ − p H kr ( L ) vanishes unless k = r + 1. In particular, p H kr ( L ) vanishes for all k > r ≤ −
1. In other words, β E − L ∈ p D mix S ( X t , O ) ≤ .Finally, assumption (2) implies that ∆ ◦ U,s ( O ) has weights ≤ L has weights ≤ p H kr ( L ) = 0 for k < r , and it must be free when k = r . But we previously saw that p H ( L ) is torsion, so infact, it must vanish. For r ≥
1, we have that p H kr ( L ) = 0 for all k ≤
0. Combiningthese, we find that β D L ∈ p D mix S ( X t , O ) ≥ . This fact, together with the previousparagraph, tells us that the two distinguished triangles β E − L → L → β D L → and τ ≤ L → L → τ ≥ L → coincide. From the discussion in § § ◦ X,s ( O ) ∼ = IC mix X,s ( O ). The stalks of IC mix X,s ( O ) are torsion-free because those of ∆ ◦ X,s ( O ) are byassumption. (cid:3) The main result of this section is the follow-ing.
Theorem 4.2.
Assume that IC mix s ( K ) ∼ = E mix s ( K ) for all s ∈ S . Then the followingare equivalent: (1) P mix S ( X, F ) is positively graded. (2) For all s, t ∈ S , we have [ F ( IC mix s ( O )) : IC mix t ( F ) h n i ] = 0 unless n = 0 . (3) For all s ∈ S , K ( P mix s ( O )) is a direct sum of objects of the form P mix t ( K ) (i.e., without Tate twists). (4) For all s ∈ S , we have ∆ ◦ s ( O ) ∼ = IC mix s ( O ) .Proof of the equivalence of parts (1) – (3) . We begin by proving the equivalence ofparts (2) and (3). By the same arguments as in the proof of [AR2, Lemma 5.2] (seealso [AR3, Lemma 2.10]), the O -module Hom( P mix s ( O ) , IC mix t ( O ) h n i ) is free, andwe have natural isomorphisms F ⊗ O Hom( P mix s ( O ) , IC mix t ( O ) h n i ) ∼ = Hom( P mix s ( F ) , F ( IC mix t ( O )) h n i ) , K ⊗ O Hom( P mix s ( O ) , IC mix t ( O ) h n i ) ∼ = Hom( K ( P mix s ( O )) , IC mix t ( K ) h n i ) . Condition (2) expresses the property that the first vector space can be nonzero onlyif n = 0, and condition (3) expresses the property that the second vector space canbe nonzero only if n = 0. Hence these conditions are indeed equivalent.To prove the other equivalences we need to introduce Grothendieck groups. For E = K , O or F , consider the Grothendieck group K mix S ( X, E ) of the abelian category P mix S ( X, E ). This abelian group naturally has the structure of a Z [ v, v − ]-module,where v acts via the shift h i . The classes of the objects IC mix s ( E ) form a basisof this Z [ v, v − ]-module, and similarly for the objects ∆ mix s ( E ). (When E = O ,this assertion relies on the fact that E has finite global dimension.) Moreover, thefunctors K ( − ) and F ( − ) induce morphisms of Z [ v, v − ]-modules e K : K mix S ( X, O ) → K mix S ( X, K ) , r F : K mix S ( X, O ) → K mix S ( X, F ) . For any s ∈ S , write [∆ mix s ( O )] = X t ∈ S d s,t [ IC mix t ( O )]where d s,t ∈ Z [ v, v − ].Now we can prove that (2) implies (1). First, it follows from our assumptionthat P mix S ( X, K ) is positively graded (see Corollary 3.17). Therefore, applying e K , ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 25 we see that we must have d s,t ∈ Z [ v − ] for any s, t . Now assumption (2) ensuresthat r F ( IC mix t ( O )) ∈ X u ∈ S Z · [ IC mix u ( F )] . It follows that [∆ mix s ( F )] = r F ([∆ mix s ( O )]) is a Z [ v − ]-linear combination of the[ IC mix u ( F )]. In other words, Proposition 3.15(2) holds, so P mix S ( X, F ) is positivelygraded.For the converse, suppose that (1) holds. Write[ P mix s ( O )] = X t ∈ S p s,t [∆ mix t ( O )]where p s,t ∈ Z [ v, v − ]. Applying r F , we obtain that p s,t ∈ Z [ v − ]. Since P mix S ( X, K )is also positively graded, we deduce that the indecomposable direct summands of K ( P mix s ( O )) are of the form P mix t ( K ) h n i with n ≤
0. Assume that P mix t ( K ) h n i ap-pears for some n <
0. By the remarks in the equivalence of (2) and (3), this impliesthat IC mix s h− n i is a composition factor of the mixed perverse sheaf F ( IC mix t ( O )).Then IC mix s h− n i is also a composition factor of F (∆ mix t ( O )) = ∆ mix t ( F ), whichcontradicts Proposition 3.15(2). (cid:3) Remark . Since IC mix s ( O ) and D ( IC mix s ( O )) differ only by torsion, the mixedperverse sheaves F ( IC mix s ( O )) and D (cid:0) F ( IC mix s ( O ) (cid:1) have the same composition fac-tors. Hence condition (2) is equivalent to the property that all composition factorsof all F ( IC mix s ( O )) are of the form IC mix t ( F ) h n i with n ≤
0, or all of the form IC mix t ( F ) h n i with n ≥
0. A similar remark applies to (3).
Lemma 4.4.
Assume that IC mix s ( K ) ∼ = E mix s ( K ) for all s ∈ S . In addition, assumethat conditions (1) – (4) of Theorem hold for every locally closed union of strata Y ( X . Then, for all s ∈ S , the objects β D IC mix s ( F ) , β E IC mix s ( F ) , ∆ ◦ s ( F ) , and ∇ ◦ s ( F ) are all perverse.Proof. If X s is not an open stratum, then the objects in question are all supportedon a proper closed subvariety of X , and so are perverse by assumption and Proposi-tion 3.15. Assume henceforth that X s is an open stratum, and let Y = X r X s . Wewill treat β D IC mix s ( F ) and ∆ ◦ s ( F ); the statement follows for the other two objectsby Verdier duality.Let Q denote the cokernel of the map IC mix s ( F ) → ∇ mix s ( F ). Since Q is supportedon Y , Proposition 3.15(5) tells us that the triangle β E − Q → Q h → β D Q [1] −→ is actually a short exact sequence in P mix S ( X, F ). In particular, the map h is sur-jective. Now consider the commutative diagram IC mix s ( F ) / / (cid:15) (cid:15) ∇ mix s ( F ) p / / Q h (cid:15) (cid:15) β D IC mix s ( F ) / / β D ∇ mix s ( F ) q / / β D Q Since h and p are both surjective maps in P mix S ( X, F ), q is as well. It follows thatthe cocone of q (i.e. β D IC mix s ( F )) lies in P mix S ( X, F ). Next, let K denote the kernel of the map ∆ mix s ( F ) → IC mix s ( F ), and form thedistinguished triangle β D K → β D ∆ mix s ( F ) → β D IC mix s ( F ) [1] −→ . Since K is supported on Y , Proposition 3.15(5) again tells us that the first term liesin P mix S ( X, F ). We have just seen above that the last term also lies in P mix S ( X, F ),so the middle term (which is ∆ ◦ s ( F ) by definition) does as well. (cid:3) End of the proof of Theorem . We will show that condition (4) is equivalent tocondition (6) of Proposition 3.15, by induction on the number of strata in X . If X consists of a single stratum, it is clear that both statements are true.Otherwise, let X s ⊂ X be an open stratum, and let X t ⊂ X be a closed stratum.Let U = X r X t and Y = X r X s . Note that if either (4) or condition (6) ofProposition 3.15 holds on X , the same statement holds on both U and Y , and hence,by induction, all parts of Theorem 4.2 hold on both U and Y . For the remainder ofthe proof, we assume that this is the case. We must show that ∆ ◦ s ( O ) ∼ = IC mix s ( O )if and only if IC ◦ s ( F ) ∼ = IC mix s ( F ). By Lemma 4.4, β D IC mix s ( F ) and ∆ ◦ s ( F ) areperverse.For E = K , O or F , let M U ( E ) := ∆ ◦ U,s ( E ). Note that F ( M U ( O )) ∼ = M U ( F ) and K ( M U ( O )) ∼ = M U ( K ) (see Lemma 3.10), and that M U ( E ) ∼ = IC mix U,s ( E ) if E = K or O . Let j : U ֒ → X be the inclusion map, and let L ( E ) = i ! t j ! M U ( E ). Since F ( L ( O )) = L ( F ), there is a natural short exact sequence of F -vector spaces(4.2) 0 → F ⊗ O p H k ( L ( O )) → p H k ( L ( F )) → Tor O ( F , p H k +1 ( L ( O ))) → . On the other hand, we have M U ( K ) ∼ = IC mix U,s ( K ) ∼ = E mix U,s ( K ). By assumption, j ! ∗ M U ( K ) coincides with the parity extension E mix s ( K ) of M U ( K ). Comparing theconstructions in § § τ ≤ L ( K )[1] and τ ≥ L ( K ) are paritysheaves. In other words,(4.3) p H kr ( L ( K )) = 0 unless ( k ≤ r = k −
1, or k ≥ r = k .We now proceed in several steps. Step 1. If k > , then p H k ( β E − L ( O )) = 0 . If k < , then p H k ( β D L ( O )) = 0 . Recall that ∆ ◦ s ( O ) ∼ = β D j ! M U . From § β E − L ( O ) ∼ = i ∗ t ∆ ◦ s ( O )[ −
1] and β D L ( O ) ∼ = i ! t ∆ ◦ s ( O ) . Since ∆ ◦ s ( F ) ∼ = F (∆ ◦ s ( O )) is perverse, we have by [AR3, Lemma 3.5] that ∆ ◦ s ( O )lies in P mix S ( X, O ). This implies that i ∗ t ∆ ◦ s ( O )[ − ∈ p D mix S ( X t , O ) ≤ , or in otherwords, p H k ( β E − L ( O )) = 0 for k > i ! t ∆ ◦ s ( O ) ∈ p D mix S ( X t , O ) ≥ . We claim, furthermore, that p H ( i ! t ∆ ◦ s ( O )) is torsion-free: otherwise, F ( i ! t ∆ ◦ s ( O )) ∼ = i ! t ∆ ◦ s ( F ) would fail to liein p D mix S ( X t , F ) ≥ , contradicting the fact that ∆ ◦ s ( F ) is perverse. To reiterate, p H k ( β D L ( O )) vanishes for k < k = 0. But it fol-lows from (4.3) that p H ( β D L ( K )) ∼ = K ⊗ O p H ( β D L ( O )) vanishes. Therefore, p H ( β D L ( O )) = 0 as well, finishing the proof of Step 1. Step 2. We have p H ( β E − i ∗ t ∆ ◦ s ( F )) ∼ = F ⊗ O p H ( β E − L ( O )) . From Step 1,we know that p H ( β E − L ( O )) = 0, so (4.2) tells us that p H ( β E − L ( F )) ∼ = F ⊗ O p H ( β E − L ( O )). On the other hand, as in Step 1, β E − L ( F ) ∼ = i ∗ s ∆ ◦ s ( F )[ − ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 27
Step 3. We have ∆ ◦ s ( O ) ∼ = IC mix s ( O ) if and only if p H ( β E − L ( O )) = 0 . Fromthe descriptions in § § ◦ s ( O ) ∼ = IC mix s ( O ) if and only if τ ≤ L ( O ) ∼ = β E − L ( O ) and τ ≥ L ( O ) ∼ = β D L ( O ) . According to Step 1, we always have β E − L ( O ) ∈ p D mix S ( X t , O ) ≤ and β D L ( O ) ∈ p D mix S ( X t , O ) ≥ . Thus, the conditions above hold if and only if p H ( β E − L ( O )) = 0. Step 4. We have IC ◦ s ( F ) ∼ = IC mix s ( F ) if and only if p H k ( β E − i ∗ t IC ◦ s ( F )) = 0 forall k ≥ . We already know that the restrictions of IC ◦ s ( F ) and IC mix s ( F ) to U agree. Recall that IC mix s ( F ) is characterized (among all objects whose restrictionto U is IC mix U,s ( F )) by the following two properties:(4.4) i ∗ t IC mix s ( F ) ∈ p D mix S ( X t , F ) ≤− and i ! t IC mix s ( F ) ∈ p D mix S ( X t , F ) ≥ . Since IC ◦ s ( F ) is self-Verdier-dual, if it satisfies one of these properties then it mustsatisfy both. Thus, IC ◦ s ( F ) ∼ = IC mix s ( F ) if and only if p H k ( i ∗ t IC ◦ s ( F )) = 0 for k ≥ IC ◦ s ( F ) is itself characterized by similar properties to those in (4.4), com-ing from the recollement structure in Proposition 3.12. In particular, we have p H k ( β D i ∗ t IC ◦ s ( F )) = 0 for k ≥
0. By Lemma 3.11, we deduce that for k ≥ p H k ( i ∗ t IC ◦ s ( F )) ∼ = p H k ( β E − i ∗ t IC ◦ s ( F )), which finishes the proof of Step 4. Step 5. We have p H k ( β E − i ∗ t ∆ ◦ s ( F )) ∼ = p H k ( β E − i ∗ t IC ◦ s ( F )) for k ≥ . Let K bethe kernel of the map ∆ ◦ s ( F ) → IC ◦ s ( F ). This kernel is to be taken in P ◦ S ( X, F ):we do not know at the moment whether IC ◦ s ( F ) lies in P mix S ( X, F ). However, we doknow that K lies in P mix S ( X, F ), because K is supported on Y , where the conclusionsof Lemma 3.14 hold. In fact, for each composition factor IC ◦ u ( F ) ∼ = IC mix u ( F ) of K ,we have p H k ( β E − i ∗ t IC ◦ u ( F )) = 0 for k ≥ u = t , this holds because i ∗ t IC ◦ u ( F ) ∈ p D mix S ( X t , F ) ≤− ; if u = t , we clearly have β E − i ∗ t IC ◦ u ( F ) = 0.) Therefore, p H k ( β E − K ) = 0 for k ≥
0. The result followsfrom the long exact sequence in perverse cohomology associated with β E − i ∗ t K → β E − i ∗ t ∆ ◦ s ( F ) → β E − i ∗ t IC ◦ s ( F ) [1] −→ . Conclusion of the proof.
Since ∆ ◦ s ( F ) is perverse, we know that p H k ( i ∗ t ∆ ◦ s ( F )) = 0for k >
0, and so p H k ( β E − i ∗ t ∆ ◦ s ( F )) = 0 for k > p H k ( β E − i ∗ t IC ◦ s ( F )) = 0 for k >
0, so we can rephrase Step 4 as follows: IC ◦ s ( F ) ∼ = IC mix s ( F ) if and only if p H ( β E − i ∗ t IC ◦ s ( F )) = 0. Using Step 5 again together withStep 2, we have that IC ◦ s ( F ) ∼ = IC mix s ( F ) if and only if F ⊗ O p H ( β E − L ( O )) = 0. Thelatter holds if and only if p H ( β E − L ( O )) = 0, and then Step 3 lets us conclude. (cid:3) Corollary 4.5.
Assume that IC mix s ( K ) ∼ = E mix s ( K ) for all s ∈ S . Then the follow-ing conditions are equivalent: (1) The category P mix S ( X, F ) is standard Q -Koszul. (2) For all s ∈ S , we have ∆ ◦ s ( O ) ∼ = IC mix s ( O ) , and IC mix s ( O ) has torsion-freestalks.Proof. Each of these conditions independently implies that all parts of Theorem 4.2and of Proposition 3.15 hold for X . In particular, both conditions imply at least that P mix S ( X, F ) is positively graded, and that the perverse-sheaf meaning of the notation ∆ ◦ s is compatible its usage in Definition 2.15. By Verdier duality, standard Q -Koszulity can be checked by a one-sided condition: P mix S ( X, F ) is standard Q -Koszulif and only if Ext k (∆ ◦ s ( F ) , ∇ mix t ( F ) h n i ) = 0 whenever n = − k . By adjunction, thelatter holds if and only if the stalks of ∆ ◦ s ( F ) are pure of weight 0 for all s . Thatcondition is equivalent to (2) by Lemma 4.1, as desired. (cid:3) Positivity and Q -Koszulity for flag varieties In this section we choose a connected reductivealgebraic group G , a Borel subgroup B ⊂ G and a maximal torus T ⊂ B , andfocus on the case where X = B := G/B is the flag variety of G , endowed withthe stratification by Bruhat cells (i.e. by orbits of B ). We use the symbol “( B )”to denote this stratification. The strata are parametrized by the Weyl group W := N G ( T ) /T of G ; the dimension of B w is the length ℓ ( w ) of w (for the naturalCoxeter group structure on W determined by our choice of B ). By [AR3, § ℓ is good for G . Note also that the assumption of Lemma 4.1,Theorem 4.2, and Corollary 4.5 is satisfied in this case, by [KL].We will also consider a connected reductive group ˇ G , a Borel subgroup ˇ B ⊂ ˇ G ,and a maximal torus ˇ T ⊂ G , such that the based root datum of ˇ G determined byˇ T and ˇ B is dual to the based root datum of G determined by T and B . As abovewe have a flag variety ˇ B := ˇ G/ ˇ B , endowed with the Bruhat stratification. Thestrata are also parametrized by W (since the Weyl groups of ( G, T ) and ( ˇ G, ˇ T ) canbe canonically identified). We will use h´aˇcek accents to denote objects attached toˇ G rather than to G . For instance, ˇ∆ w ( E ) is a standard object in P ( ˇ B ) ( ˇ B , E ), andˇ T mix w ( E ) is a tilting object in P mix( ˇ B ) ( ˇ B , E ).Recall that by [AR3, Theorem 5.4] there exists an equivalence of triangulatedcategories κ : D mix( B ) ( B , E ) ∼ −→ D mix( ˇ B ) ( ˇ B , E )which satisfies in particular κ ◦ h n i ∼ = h− n i [ n ] ◦ κ and κ ( ∇ mix w ) ∼ = ˇ ∇ mix w − , κ ( T mix w ) ∼ = ˇ E mix w − , κ ( E mix w ) ∼ = ˇ T mix w − . Below we will also use the Radon transform R mix : D mix( B ) ( B , E ) ∼ −→ D mix( B ) ( B , E ) . This equivalence of triangulated categories satisfies R mix ( ∇ mix w h n i ) ∼ = ∆ mix ww h n i , R mix ( T mix w h n i ) ∼ = P mix ww h n i . (See [AR3, Proposition 4.11].) We also set σ := κ ◦ ( R mix ) − : D mix( B ) ( B , E ) ∼ −→ D mix( ˇ B ) ( ˇ B , E ) . This functor has the property that σ (∆ mix w h n i ) ∼ = ˇ ∇ mix w w − h− n i [ n ] and σ ( P mix w h n i ) ∼ = ˇ E mix w w − h− n i [ n ] . In [AR3, Proposition 5.5] we have also constructed a t-exact “forgetful” functor µ : D mix( B ) ( B , E ) → D b( B ) ( B , E ) ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 29 (where the right-hand side is endowed with the usual perverse t-structure) and anisomorphism µ ◦ h i such that for all F , G ∈ D mix( B ) ( B , E ) the morphism(5.1) M n ∈ Z Hom( F , Gh n i ) → Hom( µ F , µ G )induced by µ is an isomorphism, and such that µ (∆ mix w ) ∼ = ∆ w , µ ( ∇ mix w ) ∼ = ∇ w , µ ( IC mix w ) ∼ = IC w ,µ ( T mix w ) ∼ = T w , µ ( E mix w ) ∼ = E w . (Here ∆ w , ∇ w , IC w , T w are the obvious “non-mixed” analogues of ∆ mix w , ∇ mix w , IC mix w , T mix w , which are objects of the usual category P ( B ) ( B , E ) of Bruhat-constru-ctible perverse sheaves on B .) There is also a functor ˇ µ : D mix( ˇ B ) ( ˇ B , E ) → D b( ˇ B ) ( ˇ B , E )with similar properties. The next two theorems are the main results of the paper.
Theorem 5.1 (Positivity) . The following are equivalent: (1)
The category P mix( B ) ( B , F ) is positively graded. (2) For all w ∈ W , we have ∆ ◦ w ( O ) ∼ = IC mix w ( O ) . (3) For all w ∈ W , the object ˇ E w ( O ) ∈ D b( ˇ B ) ( ˇ B , O ) is perverse. (4) For all w ∈ W , the object ˇ E w ( F ) ∈ D b( ˇ B ) ( ˇ B , O ) is perverse.Proof. The equivalence of the first two statements follows from Theorem 4.2. Theequivalence of the last two statements follows from the fact that the objects ˇ E w ( O )have free stalks and costalks by definition.By [AR3, Corollary 5.6], the last statement is equivalent to the condition that( T mix v : ∇ mix u h n i ) = 0 for all n > u, v ∈ W . Using the equivalence R mix ,the latter is equivalent to requiring that ( P mix v : ∆ mix u h n i ) = 0 for all n > u, v ∈ W . By Proposition 3.15, we conclude that the first and third statements areequivalent. (cid:3) Theorem 5.2 ( Q -Koszulity) . The following are equivalent: (1)
The category P mix( B ) ( B , F ) is metameric. (2) The category P mix( ˇ B ) ( ˇ B , F ) is standard Q -Koszul. (3) For all w ∈ W , we have ˇ∆ ◦ w ( O ) ∼ = ˇ IC mix w ( O ) , and ˇ IC mix w ( O ) has torsion-freestalks. (4) For all w ∈ W , we have ˇ∆ ◦ w ( O ) ∼ = ˇ IC mix w ( O ) , and ˇ IC w ( O ) has torsion-freestalks.Proof. The equivalence (2) ⇐⇒ (3) follows from Corollary 4.5. The equivalence(3) ⇐⇒ (4) follows from (5.1) (or rather its analogue for ˇ B ), using the fact thatˇ µ ( ˇ IC mix w ( O )) ∼ = ˇ IC w ( O ) and ˇ µ ( ˇ ∇ mix v ( O )) ∼ = ˇ ∇ v ( O ), and the observation that an ob-ject M of the derived category of finitely generated O -modules has free cohomologyobjects iff Hom k ( M, O ) is a free O -module for all k ∈ Z .(1) = ⇒ (3). Assuming that P mix( B ) ( B , F ) is metameric, Theorem 2.11 gives us aclass of objects { e ∆ mix w } w ∈ W in P mix( B ) ( B , F ). Form the short exact sequence K w ֒ → e ∆ mix w ։ ∆ mix w ( F ). Recall that K w has a filtration by various ∆ mix u h n i with n < Therefore, σ ( K w ) is an iterated extension of various ˇ ∇ mix u h− n i [ n ] with n <
0. Inparticular, σ ( K w ) ∈ D mix( ˇ B ) ( ˇ B , F ) D .On the other hand, the e ∆ mix w have the property that Ext k ( e ∆ mix w , ∆ mix u h n i ) = 0for all k and all n <
0. Applying σ , we obtain thatExt k ( σ ( e ∆ mix w ) , ˇ ∇ mix w u − h− n i [ n ]) = 0 for all k and all n < σ ( e ∆ mix w ) ∈ D mix( ˇ B ) ( ˇ B , F ) E (see Remark 3.8). Thus, the followingtwo distinguished triangles must be isomorphic: σ ( e ∆ mix w ) → σ (∆ mix w ) → σ ( K w [1]) [1] −→ , ˇ ∇ ◦ w w − → ˇ ∇ mix w w − → β D ˇ ∇ mix w w − [1] −→ (see Lemma 3.7(3)); in particular we obtain an isomorphism σ ( e ∆ mix w ) ∼ = ˇ ∇ ◦ w w − . Now, e ∆ mix w is an iterated extension of various ∆ mix u h n i , so ˇ ∇ ◦ w w − is an iteratedextension of various ˇ ∇ mix u { n } . In particular, the costalks of ˇ ∇ ◦ w w − are extensionsof the costalks of the ˇ ∇ mix u { n } . The latter are pure of weight 0, so the same holds forˇ ∇ ◦ w w − . By Verdier duality, the stalks of the objects ˇ∆ ◦ w w − are pure of weight 0.By Lemma 4.1, we find that condition (3) holds.(3) = ⇒ (1). When (3) holds, by Theorem 5.1 the category P mix( ˇ B ) ( ˇ B , F ) ispositively graded. First, let us prove that the category P mix( B ) ( B , F ) also is posi-tively graded. By Lemma 3.14, P ◦ ( ˇ B ) ( ˇ B , F ) is a quasihereditary category (see Re-mark 3.16). We claim that the indecomposable tilting objects in this category arethe parity sheaves ˇ E mix w . Indeed, let ˇ T ◦ w be the unique indecomposable tilting objectof P ◦ ( ˇ B ) ( ˇ B , F ) whose support is ˇ B w . Then ˇ T ◦ w has a filtration by various ˇ∆ ◦ u , whosestalks are pure of weight 0 by Lemma 4.1. Therefore, the stalks of ˇ T ◦ w are alsopure of weight 0. By Lemma 3.5, it follows that ˇ T ◦ w has weights ≤
0. Since ˇ T ◦ w isVerdier-self-dual, it also has weights ≥
0, so it must be pure of weight 0, and hencea direct sum of various ˇ E mix u { m } by the remarks in § T ◦ w ∼ = ˇ E mix w { m } for some m ∈ Z .Considering the restriction to ˇ B w , we obtain that m = 0, i.e. that ˇ T ◦ w ∼ = ˇ E mix w , asclaimed. Now the objects ˇ∆ ◦ u are perverse sheaves (see Remark 3.16), so the objectsˇ T ◦ w are also perverse; we deduce that ˇ E mix w is a perverse sheaf for any w ∈ W . UsingTheorem 5.1 again, this finishes the proof of the fact that P mix( B ) ( B , F ) is positivelygraded.To conclude, we will essentially reverse the argument used in the proof of theimplication (1) = ⇒ (3). Let us define e ∆ mix w ∈ D mix( B ) ( B , F ) to be σ − ( ˇ ∇ ◦ w w − ).Since (3) holds, using Lemma 4.1 and Verdier duality, we know that the costalks ofˇ ∇ ◦ w are pure of weight 0 for all w ∈ W . In other words, for any u ∈ W , the objectˇ ı u ∗ ˇ ı ! u ˇ ∇ ◦ w is a direct sum of various ˇ ∇ mix u { n } with n ∈ Z . In fact, since ˇ ∇ ◦ w is perverse(see Remark 3.16) we must have n ≤
0. We even have n < u = w , andin that case, we have ˇ ∇ mix w = ˇ ı w ∗ ˇ ı ! w ˇ ∇ ◦ w . (The first claim follows from the followingcomputation for u < w : Hom( F , ˇ ı ! u ˇ ∇ ◦ w ) ∼ = Hom( ˇ∆ mix u , ˇ ∇ ◦ w ) ∼ = Hom( ˇ∆ ◦ u , ˇ ∇ ◦ w ) = 0,where the second isomorphism follows from (2.3). The second claim is obvious fromthe construction of ˇ ∇ ◦ w in Proposition 2.7.) A routine recollement argument showsthat ˇ ∇ ◦ w is an iterated extension of the various ˇ ı u ∗ ˇ ı ! u ˇ ∇ ◦ w , and hence of ˇ ∇ mix w together ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 31 D mix( B ) ( B , F ) σ + + D mix( B ) ( B , F ) R mix ∼ o o κ ∼ / / D mix( B ) ( B , F )? ∆ mix w ✤ o o ✤ / / ˇ∆ mix w − ∆ mix ww ∇ mix w ✤ o o ✤ / / ˇ ∇ mix w − ? E mix w ∼ = T ◦ w ✤ o o ✤ / / ˇ T mix w − P mix ww T mix w ✤ o o ✤ / / ˇ E mix w − ∼ = ˇ T ◦ w − e ∆ mix ww ? ✤ o o ✤ / / ˇ ∇ ◦ w − Figure 1.
Behavior of various objects under the equivalence σ .with various ˇ ∇ mix u { n } with n < u < w . Applying σ − to this description,we find that e ∆ mix w is an iterated extension of ∆ mix w and various ∆ mix u h n i with n < u < w . In particular, e ∆ mix w is a perverse sheaf with a standard filtration.Next, we claim that Ext k ( e ∆ mix w , IC mix v h n i ) = 0 for all k ≥ n = 0, or else if n = 0 and v ≤ w . For n > v that the conditions for n ≤ k ( e ∆ mix w , ∆ mix v h n i ) = 0 for all k ≥ n <
0, or else if n = 0 and v ≤ w .Applying σ , this is equivalent to a similar vanishing claim aboutHom( ˇ ∇ ◦ w w − , ˇ ∇ mix w v − h− n i [ k + n ]) . If n <
0, this claim follows from (2.3). If n = 0, it holds for reasons of support.Referring to Theorem 2.11, we see that we have already shown that the objects e ∆ mix w enjoy properties (2), (3), and (4). We will now show that they satisfy prop-erty (1) as well. The Ext -case of the vanishing proved above shows that e ∆ mix w is projective as an object of the Serre subcategory of P mix( B ) ( B , F ) generated by all IC mix v h n i with n <
0, together with the IC mix v with v ≤ w . It is indecomposablebecause ˇ ∇ ◦ w is, so it is the projective cover of some simple object. Its unique simplequotient must be the head of one of the standard objects in its standard filtration.By weight filtration considerations, that unique simple quotient must be IC mix w .We have shown that the objects e ∆ mix w satisfy the properties listed in Theo-rem 2.11. It is clear that the objects e ∇ mix w := D ( e ∆ mix w ) will satisfy the dual condi-tions, so that Theorem 2.11 implies that P mix( B ) ( B , F ) is metameric. (cid:3) Remark . When the conditions of Theorem 5.2 are satisfied, one can completethe description [AR3, Figure 1] of the behavior of the various special objects underthe equivalence κ , as shown in Figure 1. (Here the isomorphism on the third linefollows from Lemma 3.14, and question marks indicate objects for which we don’thave an explicit description.) Remark . Suppose that the conditions in Theorem 5.2 hold. By Lemma 3.14,the ˇ E mix w ( F ) are precisely the indecomposable tilting objects in P ◦ ( ˇ B ) ( ˇ B , F ). Sincethe equivalence σ − : D mix( ˇ B ) ( ˇ B , F ) → D mix( B ) ( B , F ) takes these to projective objects in P mix( B ) ( B , F ), the category P mix( B ) ( B , F ) is the “ T -Koszul dual” to P mix( ˇ B ) ( ˇ B , F ) in thesense of Madsen [Mad]. (See Remark 2.17.) We conclude this paper with a proof of the converse to Corol-lary 3.17, in the case of flag varieties.
Theorem 5.5.
The following are equivalent: (1)
For all w ∈ W we have E w ( F ) ∼ = IC w ( F ) . (2) The category P mix( B ) ( B , F ) is Koszul. (3) The category P mix( B ) ( B , F ) is positively graded, and P mix( B ) ( B , F ) ◦ is a semisim-ple category.Moreover, these statements hold if and only if their analogues for ˇ B hold.Proof. In this proof, we will write (1) ∨ to refer to the analogue of statement (1) forˇ B , and likewise for the other assertions in the theorem.The implications (1) = ⇒ (2) and (1) ∨ = ⇒ (2) ∨ follow from Corollary 3.17.The implications (2) = ⇒ (3) and (2) ∨ = ⇒ (3) ∨ are obvious.(3) = ⇒ (1) ∨ . Since P mix( B ) ( B , F ) is positively graded, by Theorem 5.1, ˇ E w ( F ) isperverse. Now, the fact that P mix( B ) ( B , F ) ◦ is semisimple implies that the ringHom P mix( B ) ( B , F ) M v ∈ W P mix v ( F ) , M v ∈ W P mix v ( F ) ! is isomorphic to L v k (where 1 in the copy of k parametrized by v corresponds tothe identity morphism of P mix v ( F )). Using equivalence σ , we deduce a similar claimfor the objects ˇ E mix v ( F ), v ∈ W . It follows that(5.2) Hom D b( B ) ( B , F ) (cid:0) ˇ E v ( F ) , ˇ E w ( F ) (cid:1) = 0 unless v = w .Now assume that there exists w ∈ W such that the perverse sheaf ˇ E w ( F ) is notsimple, and choose w ∈ W minimal (for the Bruhat order) with this property. Sinceˇ E w ( F ) is supported on the closure of ˇ B w , and since its restriction to ˇ B w is F , eitherthe top or the socle of ˇ E w ( F ) contains a simple object ˇ IC v ( F ) with v < w . Thenthere exists either a non zero morphism ˇ E w ( F ) → ˇ IC v ( F ), or a nonzero morphismˇ IC v ( F ) → ˇ E w ( F ). Since ˇ IC v ( F ) ∼ = ˇ E v ( F ) by minimality, this contradicts (5.2) andfinishes the proof of the implication.By symmetry we also obtain the implication (3) ∨ = ⇒ (1), which finishes theproof. (cid:3) References [AR1] P. Achar and S. Riche,
Koszul duality and semisimplicity of Frobenius , Ann. Inst. Fourier (2013), 1511–1612.[AR2] P. Achar and S. Riche, Modular perverse sheaves on flag varieties I: tilting and paritysheaves , preprint arXiv:1401.7245 (with a joint appendix with G. Williamson).[AR3] P. Achar and S. Riche,
Modular perverse sheaves on flag varieties II: Koszul duality andformality , preprint arXiv:1401.7256.[ARd] P. Achar, L. Rider,
Parity sheaves on the affine Grassmannian and the Mirkovi´c–Vilonenconjecture , preprint arXiv:1305.1684.[AT] P. Achar, D. Treumann,
Baric structures on triangulated categories and coherent sheaves ,Int. Math. Res. Not. (2011), 3688–3743.[ADL] I. ´Agoston, V. Dlab, E. Luk´acs,
Quasi-hereditary extension algebras , Algebr. Repre-sent. Theory (2003), 97–117. ODULAR PERVERSE SHEAVES ON FLAG VARIETIES III 33 [ABG] S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg,
Quantum groups, the loop Grassman-nian, and the Springer resolution , J. Amer. Math. Soc. (2004), 595–678.[BBD] A. Be˘ılinson, J. Bernstein, and P. Deligne, Faisceaux pervers , in
Analysis and topologyon singular spaces, I (Luminy, 1981) , Ast´erisque (1982), 5–171.[BGS] A. Be˘ılinson, V. Ginzburg, and W. Soergel,
Koszul duality patterns in representationtheory , J. Amer. Math. Soc. (1996), 473–527.[Be1] R. Bezrukavnikov, Quasi-exceptional sets and equivariant coherent sheaves on the nilpo-tent cone , Represent. Theory (2003), 1–18.[Be2] R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and t -structureson derived categories of coherent sheaves , Invent. Math. (2006), 327–357.[BK] M. Brion and S. Kumar, Frobenius splitting methods in geometry and representationtheory , Progr. Math., vol. 231, Birkh¨auser Boston, Boston, MA, 2005.[CPS] E. Cline, B. Parshall, and L. Scott,
Reduced standard modules and cohomology , Trans.Amer. Math. Soc. (2009), 5223–5261.[Do] S. Donkin,
The q -Schur algebra , London Mathematical Society Lecture Note Series, 253,Cambridge University Press, Cambridge, 1998.[Ga] P. Gabriel, Des cat´egories ab´eliennes , Bulletin de la S.M.F. (1962), 323–448.[Ir] R. Irving, Graded BGG algebras , Abelian groups and noncommutative rings, 181–200,Contemp. Math. 130, Amer. Math. Soc., Providence, RI, 1992.[J] D. Juteau,
Modular Springer correspondence and decomposition matrices , Ph.D. thesis,Universit´e Paris 7, 2007.[JMW] D. Juteau, C. Mautner, G. Williamson,
Parity sheaves , preprint arXiv:0906.2994, J.Amer. Math. Soc., to appear.[JMW2] D. Juteau, C. Mautner, and G. Williamson,
Parity sheaves and tilting modules , preprintarXiv:1403.1647.[KL] D. Kazhdan and G. Lusztig,
Schubert varieties and Poincar´e duality , Geometry of theLaplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979),Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, RI, 1980, pp. 185–203.[Mad] D. Madsen,
On a common generalization of Koszul duality and tilting equivalence , Adv.Math. (2011), 2327–2348.[Maz] V. Mazorchuk,
Some homological properties of the category O , Pacific J. Math. (2007), 313–341.[MV] I. Mirkovi´c and K. Vilonen, Geometric Langlands duality and representations of algebraicgroups over commutative rings , Ann. of Math. (2) (2007), 95–143.[PS1] B. Parshall and L. Scott,
New graded methods in the homological algebra of semisimplealgebraic groups , arXiv:1304.1461.[PS2] B. Parshall and L. Scott, Q -Koszul algebras and three conjectures , arXiv:1405.4419.[RSW] S. Riche, W. Soergel, G. Williamson, Modular Koszul duality , Compos. Math. (2014), 273–332.[Rin] C. M. Ringel,
The category of modules with good filtrations over a quasi-hereditaryalgebra has almost split sequences , Math. Z. (1991), 209–223.[So] W. Soergel,
On the relation between intersection cohomology and representation theoryin positive characteristic , Commutative algebra, homological algebra and representationtheory (Catania/Genoa/Rome, 1998), J. Pure Appl. Algebra (2000), 311–335.[We] J. Weidner,
Grassmannians and Koszul duality , preprint arXiv:1311.1975.[Wi] G. Williamson,
Schubert calculus and torsion , preprint arXiv:1309.5055.
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,U.S.A.
E-mail address : [email protected] Universit´e Blaise Pascal - Clermont-Ferrand II, Laboratoire de Math´ematiques,CNRS, UMR 6620, Campus universitaire des C´ezeaux, F-63177 Aubi`ere Cedex, France
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