REPRESENTATION THEORYAn Electronic Journal of the American Mathematical SocietyVolume 16, Pages 212ā269 (April 30, 2012)S 1088-4165(2012)00416-2
GRADED DECOMPOSITION MATRICES OF v-SCHUR
ALGEBRAS VIA JANTZEN FILTRATION
PENG SHAN
Abstract. We prove that certain parabolic Kazhdan-Lusztig polynomials cal-culate the graded decomposition matrices of v-Schur algebras given by theJantzen filtration of Weyl modules, confirming a conjecture of Leclerc andThibon.
Contents
Introduction 2121. Jantzen filtration of standard modules 2132. Affine parabolic category O and v-Schur algebras 2183. Generalities on D-modules on ind-schemes 2284. Localization theorem for affine Lie algebras of negative level 2355. The geometric construction of the Jantzen filtration 2406. Proof of the main theorem 245Appendix A. Kashiwara-Tanisakiās construction, translation functors and
Proof of Proposition 4.2 250Appendix B. Proof of Proposition 5.4 260Index of notation 266Acknowledgement 267References 267
Introduction
Let v be a r-th root of unity in C. The v-Schur algebra Sv(n) over C is a finitedimensional quasi-hereditary algebra. Its standard modules are the Weyl modulesWv(Ī») indexed by partitions Ī» of n. The module Wv(Ī») has a simple quotientLv(Ī»). See Section 2.9 for more details.
The decomposition matrix of Sv(n) is given by the following algorithm. LetFq be the Fock space of level one. It is a Q(q)-vector space with a basis {|Ī»ć}indexed by the set of partitions. Moreover, it carries an action of the quantum
enveloping algebra Uq(slr). Let L+ (resp. Lā) be the Z[q]-submodule (resp. Z[qā1]-
submodule) in Fq spanned by {|Ī»ć}. Following Leclerc and Thibon [LT1, Theorem4.1], the Fock space Fq admits two particular bases {G+
Ī» }, {GāĪ» } with the properties
that
G+Ī» ā” |Ī»ć mod qL+, Gā
Ī» ā” |Ī»ć mod qā1Lā.
Received by the editors March 27, 2011.2010 Mathematics Subject Classification. Primary 20G43.
cĀ©2012 American Mathematical Society
212
JANTZEN FILTRATION 213
Let dĪ»Ī¼(q), eĪ»Ī¼(q) be the elements in Z[q] such that
G+Ī¼ =
āĪ»
dĪ»Ī¼(q)|Ī»ć, GāĪ» =
āĪ¼
eĪ»Ī¼(āqā1)|Ī¼ć.
For any partition Ī» write Ī»ā² for the transposed partition. Then the Jordan-Holdermultiplicity of Lv(Ī¼) in Wv(Ī») is equal to the value of dĪ»ā²Ī¼ā²(q) at q = 1. This resultwas conjectured by Leclerc and Thibon [LT1, Conjecture 5.2] and has been provedby Varagnolo and Vasserot [VV1].
We are interested in the Jantzen filtration of Wv(Ī») [JM]:
Wv(Ī») = J0Wv(Ī») ā J1Wv(Ī») ā . . . .
It is a filtration by Sv(n)-submodules. The graded decomposition matrix of Sv(n)counts the multiplicities of Lv(Ī¼) in the associated graded module of Wv(Ī»). Thegraded version of the above algorithm was also conjectured by Leclerc and Thibon[LT1, Conjecture 5.3]. Our main result is a proof of this conjecture under a mildrestriction on v.
Theorem 0.1. Suppose that v = exp(2Ļi/Īŗ) with Īŗ ā Z and Īŗ ļæ½ ā3. Let Ī», Ī¼ bepartitions of n. Then
(0.1) dĪ»ā²Ī¼ā²(q) =āiļæ½0
[J iWv(Ī»)/Ji+1Wv(Ī») : Lv(Ī¼)]q
i.
Let us outline the idea of the proof. We first show that certain equivalenceof highest weight categories preserves the Jantzen filtrations of standard modules(Proposition 1.8). By constructing such an equivalence between the module cate-gory of the v-Schur algebra and a subcategory of the affine parabolic category Oof negative level, we then transfer the problem of computing the Jantzen filtrationof Weyl modules into the same problem for parabolic Verma modules (Corollary2.14). The latter is solved using Beilinson-Bernsteinās techniques (Sections 4, 5, 6).
1. Jantzen filtration of standard modules
1.1. Notation. We will denote by A -mod the category of finitely generated mod-ules over an algebra A, and by A -proj its subcategory consisting of projectiveobjects. Let R be a commutative noetherian C-algebra. By a finite projectiveR-algebra we mean an R-algebra that belongs to R -proj.
A R-category C is a category whose Hom sets are R-modules. All the functorsbetween R-categories will be assumed to be R-linear, i.e., they induce morphismsof R-modules on the Hom sets. Unless otherwise specified, all the functors will beassumed to be covariant. If C is abelian, we will write C -proj for the full subcategoryconsisting of projective objects. If there exists a finite projective algebra A togetherwith an equivalence of R-categories F : C ā¼= A -mod, then we define C ā© R -proj tobe the full subcategory of C consisting of objects M such that F (M) belongs toR -proj. By Morita theory, the definition of C ā© R -proj is independent of A or F .Further, for any C-algebra homomorphism R ā Rā² we will abbreviate Rā²C for thecategory (Rā² āR A) -mod. The definition of Rā²C is independent of the choice of Aup to equivalence of categories.
For any abelian category C we will write [C] for the Grothendieck group of C. Anyexact functor F from C to another abelian category Cā² yields a group homomorphism[C] ā [Cā²], which we will again denote by F .
214 PENG SHAN
A C-category C is called artinian if the Hom sets are finite dimensional C-vectorspaces and every object has a finite length. The Jordan-Holder multiplicity of asimple object L in an object M of C will be denoted by [M : L].
We abbreviate ā = āC and Hom = HomC.
1.2. Highest weight categories. Let C be a R-category that is equivalent tothe category A -mod for some finite projective R-algebra A. Let Ī be a finite setof objects of C together with a partial order <. Let CĪ be the full subcategoryof C consisting of objects which admit a finite filtration such that the successivequotients are isomorphic to objects in
{D ā U |D ā Ī, U ā R -proj}.
We have the following definition; see [R, Definition 4.11].
Definition 1.1. The pair (C,Ī) is called a highest weight R-category if the follow-ing conditions hold:
ā¢ the objects of Ī are projective over R,ā¢ we have EndC(D) = R for all D ā Ī,ā¢ given D1, D2 ā Ī, if HomC(D1, D2) = 0, then D1 ļæ½ D2,ā¢ if M ā C satisfies HomC(D,M) = 0 for all D ā Ī, then M = 0,ā¢ given D ā Ī, there exists P ā C -proj and a surjective morphism f : P ā Dsuch that ker f belongs to CĪ. Moreover, in the filtration of ker f onlyDā² ā U with Dā² > D appears.
The objects in Ī are called standard. We say that an object has a standardfiltration if it belongs to CĪ. There is another setā of objects in C, called costandardobjects, given by the following proposition.
Proposition 1.2. Let (C,Ī) be a highest weight R-category. Then there is a setā = {DāØ |D ā Ī} of objects of C, unique up to isomorphism, with the followingproperties:
(a) the pair (Cop,ā) is a highest weight R-category, where ā is equipped withthe same partial order as Ī,
(b) for D1, D2 ā Ī we have ExtiC(D1, DāØ2 )
ā¼={R if i = 0 and D1 = D2,
0 else.
See [R, Proposition 4.19].
1.3. Base change for highest weight categories. From now on, unless other-wise specified we will fix R = C[[s]], the ring of formal power series in the variables. Let ā be its maximal ideal and let K be its fraction field. For any R-module M ,any morphism f of R-modules and any i ā N we will write
M(āi) = M āR (R/āiR), MK = M āR K,
f(āi) = f āR (R/āiR), fK = f āR K.
We will abbreviate
C(ā) = R(ā)C, CK = KC.Let us first recall the following basic facts.
JANTZEN FILTRATION 215
Lemma 1.3. Let A be a finite projective R-algebra. Let P ā A -mod.(a) The A-module P is projective if and only if P is a projective R-module and
P (ā) belongs to A(ā) -proj.(b) If P belongs to A -proj, then we have a canonical isomorphism
HomA(P,M)(ā)ā¼ā HomA(ā)(P (ā),M(ā)), ā M ā A -mod .
Further, if M belongs to R -proj, then HomA(P,M) also belongs to R -proj.
We will also need the following theorem of Rouquier [R, Theorem 4.15].
Proposition 1.4. Let C be a R-category that is equivalent to A -mod for somefinite projective R-algebra A. Let Ī be a finite poset of objects of C ā©R -proj. Thenthe category (C,Ī) is a highest weight R-category if and only if (C(ā),Ī(ā)) is ahighest weight C-category.
Finally, the costandard objects can also be characterized in the following way.
Lemma 1.5. Let (C,Ī) be a highest weight R-category. Assume that āā²={āØD |DāĪ} is a set of objects of C ā©R -proj such that for any D ā Ī we have
(āØD)(ā) ā¼= D(ā)āØ, (āØD)K ā¼= DK .
Then we have āØD ā¼= DāØ ā ā.
Proof. We prove the lemma by showing that āā² has the properties (a), (b) inProposition 1.2 with āØD playing the role ofDāØ. This will imply that āØD ā¼= DāØ ā ā.To check (a) note thatāā²(ā) is the set of costandard modules of C(ā) by assumption.So (C(ā)op,āā²(ā)) is a highest weight C-category. Therefore (Cop,āā²) is a highestweight R-category by Proposition 1.4. Now, let us concentrate on (b). Given D1,D2 ā Ī, let Pā¢ = 0 ā Pn ā Ā· Ā· Ā· ā P0 be a projective resolution of D1 in C. ThenExtiC(D1,
āØD2) is the cohomology of the complex
Cā¢ = HomC(Pā¢,āØD2).
Since D1 and all the Pi belong to R -proj and R is a discrete valuation ring, by theUniversal Coefficient Theorem the complex
Pā¢(ā) = 0 ā Pn(ā) ā Ā· Ā· Ā· ā P0(ā)
is a resolution of D1(ā) in C(ā). Further, each Pi(ā) is a projective object in C(ā)by Lemma 1.3(a). So ExtiC(ā)(D1(ā),
āØD2(ā)) is given by the cohomology of thecomplex
Cā¢(ā) = HomC(ā)(Pā¢(ā),āØD2(ā)).
Again, by the Universal Coefficient Theorem, the canonical map
Hi(Cā¢)(ā) āā Hi(C(ā)ā¢)
is injective. In other words, we have a canonical injective map
(1.1) ExtiC(D1,āØD2)(ā) āā ExtiC(ā)(D1(ā),
āØD2(ā)).
Note that each R-module Ci is finitely generated. Therefore ExtiC(D1,āØD2) is also
finitely generated over R. Note that if i > 0, or i = 0 and D1 = D2, then theright hand side of (1.1) is zero by assumption. So ExtiC(D1,
āØD2)(ā) = 0, and
216 PENG SHAN
hence ExtiC(D1,āØD2) = 0 by Nakayamaās lemma. Now, let us concentrate on the
R-module HomC(D, āØD) for D ā Ī. First, we have
HomC(D, āØD)āR K = HomCK(DK , (āØD)K)
= EndCK(DK)
= EndC(D)āR K
= K.(1.2)
Here the second equality is given by the isomorphism DKā¼= (āØD)K and the last
equality follows from EndC(D) = R. Next, note that HomC(D, āØD)(ā) is includedinto the vector space HomC(ā)(D(ā), āØD(ā)) = C by (1.1). So its dimension overC is less than one. Together with (1.2) this yields an isomorphism of R-modulesHomC(D, āØD) ā¼= R, because R is a discrete valuation ring. So we have verified thatāā² satisfies both properties (a) and (b) in Proposition 1.2. Therefore it coincideswith ā and āØD is isomorphic to DāØ. ļæ½
1.4. The Jantzen filtration of standard modules. Let (CC,ĪC) be a highestweight C-category and (C,Ī) be a highest weight R-category such that (CC,ĪC) ā¼=(C(ā),Ī(ā)). Then any standard module in ĪC admits a Jantzen type filtrationassociated with (C,Ī). It is given as follows.
Definition 1.6. For any D ā Ī let Ļ : D ā DāØ be a morphism in C such thatĻ(ā) = 0. For any positive integer i let
(1.3) Ļi : DāØ āā DāØ/āiDāØ
be the canonical quotient map. Set
Di = ker(Ļi ā¦ Ļ) ā D, J iD(ā) = (Di + āD)/āD.
The Jantzen filtration of D(ā) is the filtration
D(ā) = J0D(ā) ā J1D(ā) ā Ā· Ā· Ā· .
To see that the Jantzen filtration is well defined, one notices first that the mor-phism Ļ always exists because HomC(D,DāØ)(ā) ā¼= R(ā). Further, the filtration isindependent of the choice of Ļ. Because if Ļā² : D ā DāØ is another morphism suchthat Ļā²(ā) = 0, the fact that HomC(D,DāØ) ā¼= R and Ļ(ā) = 0 implies that thereexists an element a in R such that Ļā² = aĻ. Moreover, Ļā²(ā) = 0 implies that a isinvertible in R. So Ļ and Ļā² define the same filtration.
Remark 1.7. If the category CK is semi-simple, then the Jantzen filtration of anystandard module D(ā) is finite. In fact, since EndC(D) = R we have EndCK
(DK) =K. Therefore DK is an indecomposable object in CK . So the semi-simplicity ofCK implies that DK is simple. Similarly, DāØ
K is also simple. So the morphismĻK : DK ā DāØ
K is an isomorphism. In particular, Ļ is injective. Now, consider theintersection ā
i
J iD(ā) =āi
(Di + āD)/āD.
Since we have Di ā Di+1, the intersection on the right hand side is equal to((ā
i Di)+āD)/āD. The injectivity of Ļ implies that
āiD
i = kerĻ is zero. Henceāi J
iD(ā) = 0. Since D(ā) ā C(ā) has a finite length, we have J iD(ā) = 0 for ilarge enough.
JANTZEN FILTRATION 217
1.5. Equivalences of highest weight categories and Jantzen filtrations.Let (C1,Ī1), (C2,Ī2) be highest weight R-categories (resp. C-categories or K-categories). A functor F : C1 ā C2 is an equivalence of highest weight categories ifit is an equivalence of categories and if for any D1 ā Ī1 there exists D2 ā Ī2 suchthat F (D1) ā¼= D2. Note that for such an equivalence F we also have
(1.4) F (DāØ1 )
ā¼= DāØ2 ,
because the two properties in Proposition 1.2 which characterize the costandardobjects are preserved by F .
Let F : C1 ā C2 be an exact functor. Since C1 is equivalent to A -mod for somefinite projective R-algebra A, the functor F is represented by a projective object Pin C1, i.e., we have F ā¼= HomC1
(P,ā). Set
F (ā) = HomC1(ā)(P (ā),ā) : C1(ā) ā C2(ā).
Note that the functor F (ā) is unique up to equivalence of categories. It is an exactfunctor, and it is isomorphic to the functor HomC1
(P,ā)(ā); see Lemma 1.3. Inparticular, for D ā Ī1 there are canonical isomorphisms
(1.5) F (D)(ā) ā¼= F (ā)(D(ā)), F (DāØ)(ā) ā¼= F (ā)(DāØ(ā)).
Proposition 1.8. Let (C1,Ī1), (C2,Ī2) be two equivalent highest weight R-categor-ies. Fix an equivalence F : C1 ā C2. Then the following holds.
(a) The functor F (ā) is an equivalence of highest weight categories.(b) The functor F (ā) preserves the Jantzen filtration of standard modules, i.e.,
for any D1 ā Ī1 let D2 = F (D1) ā Ī2, then
F (ā)(J iD1(ā)) = J iD2(ā), ā i ā N.
Proof. (a) If G : C2 ā C1 is a quasi-inverse of F , then G(ā) is a quasi-inverse ofF (ā). So F (ā) is an equivalence of categories. It maps a standard object to astandard one because of the first isomorphism in (1.5).
(b) The functor F yields an isomorphism of R-modules
HomC1(D1, D
āØ1 )
ā¼ā HomC2(F (D1), F (DāØ
1 )),
where the right hand side identifies with HomC2(D2, D
āØ2 ) via the isomorphism (1.4).
Let Ļ1 be an element in HomC1(D1, D
āØ1 ) such that Ļ1(ā) = 0. Let
Ļ2 = F (Ļ1) : D2 ā DāØ2 .
Then we also have Ļ2(ā) = 0.For a = 1, 2 and i ā N let Ļa,i : D
āØa ā DāØ
a (āi) be the canonical quotient map.
Since F is R-linear and exact, the isomorphism F (DāØ1 )
ā¼= DāØ2 maps F (āiDāØ
1 ) toāiDāØ
2 and induces an isomorphism
F (DāØ1 (ā
i)) ā¼= DāØ2 (ā
i).
Under these isomorphisms the morphism F (Ļ1,i) is identified with Ļ2,i. So we have
F (Di1) = F (ker(Ļ1,i ā¦ Ļ1))
= ker(F (Ļ1,i) ā¦ F (Ļ1))ā¼= ker(Ļ2,i ā¦ Ļ2)
= Di2.
218 PENG SHAN
Now, apply F to the short exact sequence
(1.6) 0 ā āD1 ā Di1 + āD1 ā J iD1(ā) ā 0,
we get
F (J iD1(ā)) ā¼= (F (Di1) + āF (D1))/āF (D1)
ā¼= J iD2(ā).
Since F (J iD1(ā)) = F (ā)(J iD1(ā)), the proposition is proved. ļæ½
2. Affine parabolic category O and v-Schur algebras
2.1. The affine Lie algebra. Fix an integer m > 1. Let G0 ā B0 ā T0 be respec-tively the linear algebraic group GLm(C), the Borel subgroup of upper triangularmatrices and the maximal torus of diagonal matrices. Let g0 ā b0 ā t0 be their Liealgebras. Let
g = g0 ā C[t, tā1]ā C1ā Cā
be the affine Lie algebra of g0. Its Lie bracket is given by
[Ī¾āta+x1+yā, Ī¾ā²ātb+xā²1+yā²ā] = [Ī¾, Ī¾ā²]āta+b+aĪ“a,āb tr(Ī¾Ī¾ā²)1+byĪ¾ā²ātbāayā²Ī¾āta,
where tr : g0 ā C is the trace map. Set t = t0 ā C1ā Cā.For any Lie algebra a over C, let U(a) be its enveloping algebra. For any C-
algebra R, we will abbreviate aR = aāR and U(aR) = U(a)ā R.In the rest of the paper, we will fix once for all an integer c such that
(2.1) Īŗ = c+m ā Z<0.
Let UĪŗ be the quotient of U(g) by the two-sided ideal generated by 1 ā c. TheUĪŗ-modules are precisely the g-modules of level c.
Given a C-linear map Ī» : t ā R and a gR-module M we set
(2.2) MĪ» = {v ā M |hv = Ī»(h)v, ā h ā t}.Whenever MĪ» is nonzero, we call Ī» a weight of M .
We equip tā = HomC(t,C) with the basis Īµ1, . . . , Īµm, Ļ0, Ī“ such that Īµ1, . . . , Īµm ātā0 is dual to the canonical basis of t0,
Ī“(ā) = Ļ0(1) = 1, Ļ0(t0 ā Cā) = Ī“(t0 ā C1) = 0.
Let ćā¢ : ā¢ć be the symmetric bilinear form on tā such that
ćĪµi : Īµjć = Ī“ij , ćĻ0 : Ī“ć = 1, ćtā0 ā CĪ“ : Ī“ć = ćtā0 ā CĻ0 : Ļ0ć = 0.
For h ā tā we will write ||h||2 = ćh : hć. The weights of a UĪŗ-module belong to
Īŗtā = {Ī» ā tā | ćĪ» : Ī“ć = c}.
Let a denote the projection from tā to tā0. Consider the map
(2.3) z : tā ā C
such that Ī» ļæ½ā z(Ī»)Ī“ is the projection tā ā CĪ“.Let Ī be the root system of g with simple roots Ī±i = Īµi ā Īµi+1 for 1 ļæ½ i ļæ½ mā 1
and Ī±0 = Ī“ āāmā1
i=1 Ī±i. The root system Ī 0 of g0 is the root subsystem of Ī
generated by Ī±1, . . . , Ī±mā1. We will write Ī +, Ī +0 for the sets of positive roots in
Ī , Ī 0, respectively.The affine Weyl group S is a Coxeter group with simple reflections si for 0 ļæ½
i ļæ½ mā 1. It is isomorphic to the semi-direct product of the symmetric group S0
JANTZEN FILTRATION 219
with the lattice ZĪ 0. There is a linear action of S on tā such that S0 fixes Ļ0, Ī“,and acts on tā0 by permuting Īµiās, and an element Ļ ā ZĪ 0 acts by
(2.4) Ļ (Ī“) = Ī“, Ļ (Ļ0) = Ļ + Ļ0 ā ćĻ : ĻćĪ“/2, Ļ (Ī») = Ī»ā ćĻ : Ī»ćĪ“, ā Ī» ā tā0.
Let Ļ0 be the half sum of positive roots in Ī 0 and Ļ = Ļ0 +mĻ0. The dot actionof S on tā is given by w Ā· Ī» = w(Ī»+ Ļ)ā Ļ. For Ī» ā tā we will denote by S(Ī») thestabilizer of Ī» in S under the dot action. Let l : S ā N be the length function.
2.2. The parabolic Verma modules and their deformations. The subset Ī 0
of Ī defines a standard parabolic Lie subalgebra of g, which is given by
q = g0 ā C[t]ā C1ā Cā.
It has a Levi subalgebra
l = g0 ā C1ā Cā.
The parabolic Verma modules of UĪŗ associated with q are given as follows. Let Ī»be an element in
Ī+ = {Ī» ā Īŗtā | ćĪ» : Ī±ć ā N, ā Ī± ā Ī +
0 }.Then there is a unique finite dimensional simple g0-module V (Ī») of highest weighta(Ī»). It can be regarded as a l-module by letting h ā C1 ā Cā act by the scalarĪ»(h). It is, furthermore, a q-module if we let the nilpotent radical of q act trivially.The parabolic Verma module of highest weight Ī» is given by
MĪŗ(Ī») = U(g)āU(q) V (Ī»).
It has a unique simple quotient, which we denote by LĪŗ(Ī»).Recall that R = C[[s]] and ā is its maximal ideal. Set
c = c+ s and k = Īŗ+ s.
They are elements in R. Write Uk for the quotient of U(gR) by the two-sided idealgenerated by 1āc. So if M is a Uk-module, then M(ā) is a UĪŗ-module. Now, notethat R admits a qR-action such that g0āC[t] acts trivially and t acts by the weightsĻ0. Denote this qR-module by RsĻ0
. For Ī» ā Ī+ the deformed parabolic Vermamodule Mk(Ī») is the gR-module induced from the qR-module V (Ī»)āRsĻ0
. It is aUk-module of highest weight Ī»+ sĻ0, and we have a canonical isomorphism
Mk(Ī»)(ā) ā¼= MĪŗ(Ī»).
We will abbreviate Ī»s = Ī»+ sĻ0 and will write
ktā = {Ī»s |Ī» ā Īŗt
ā}.
Lemma 2.1. The gK-module Mk(Ī»)K = Mk(Ī»)āR K is simple.
Proof. Assume that Mk(Ī»)K is not simple. Then it contains a nontrivial submod-ule. This submodule must have a highest weight vector of weight Ī¼s for someĪ¼ ā Ī+, Ī¼ = Ī». By the linkage principle, there exists w ā S such that Ī¼s = w Ā· Ī»s.Therefore w fixes Ļ0, so it belongs to S0. But then we must have w = 1, becauseĪ», Ī¼ ā Ī+. So Ī» = Ī¼. This is a contradiction. ļæ½
220 PENG SHAN
2.3. The Jantzen filtration of parabolic Verma modules. For Ī» ā Ī+ theJantzen filtration of MĪŗ(Ī») is given as follows. Let Ļ be the R-linear anti-involutionon gR such that
Ļ(Ī¾ ā tn) = tĪ¾ ā tān, Ļ(1) = 1, Ļ(ā) = ā.
Here Ī¾ ā g0 and tĪ¾ is the transposed matrix. Let gR act on HomR(Mk(Ī»), R) via(xf)(v) = f(Ļ(x)v) for x ā gR, v ā Mk(Ī»). Then
(2.5) DMk(Ī») =āĪ¼ākt
ā
HomR
(Mk(Ī»)Ī¼, R
)is a gR-submodule of HomR(Mk(Ī»), R). It is the deformed dual parabolic Vermamodule with highest weight Ī»s. The Ī»s-weight spaces of Mk(Ī») and DMk(Ī») areboth free R-modules of rank one. Any isomorphism between them yields, by theuniversal property of Verma modules, a gR-module morphism
Ļ : Mk(Ī») ā DMk(Ī»)
such that Ļ(ā) = 0. The Jantzen filtration(J iMĪŗ(Ī»)
)of MĪŗ(Ī») defined by [J] is
the filtration given by Definition 1.6 using the morphism Ļ above.
2.4. The deformed parabolic category O. The deformed parabolic category O,denoted by Ok, is the category of Uk-modules M such that
ā¢ M =ā
Ī»āktā MĪ» with MĪ» ā R -mod,
ā¢ for any m ā M the R-module U(qR)m is finitely generated.
It is an abelian category and contains deformed parabolic Verma modules. Replac-ing k by Īŗ and R by C we get the usual parabolic category O, denoted OĪŗ.
Recall the map z in (2.3). For any integer r setrĪŗtā = {Ī¼ ā Īŗt
ā | r ā z(Ī¼) ā Zļæ½0}.Define r
ktā in the same manner. Let rOĪŗ (resp. rOk) be the Serre subcategory of
OĪŗ (resp. Ok) consisting of objects M such that MĪ¼ = 0 implies that Ī¼ belongs torĪŗtā (resp. r
ktā). Write rĪ+ = Ī+ ā© r
Īŗtā. We have the following lemma.
Lemma 2.2. (a) For any finitely generated projective object P in rOk and anyM ā rOk the R-module HomrOk
(P,M) is finitely generated and the canonical map
HomrOk(P,M)(ā) ā HomrOĪŗ
(P (ā),M(ā))
is an isomorphism. Moreover, if M is free over R, then HomrOk(P,M) is also free
over R.(b) The assignment M ļæ½ā M(ā) yields a functor
rOk ā rOĪŗ.
This functor gives a bijection between the isomorphism classes of simple objects anda bijection between the isomorphism classes of indecomposable projective objects.
For any Ī» ā rĪ+ there is a unique finitely generated projective cover rPĪŗ(Ī») ofLĪŗ(Ī») in
rOĪŗ; see [RW, Lemma 4.12]. Let Lk(Ī»),rPk(Ī») be respectively the simple
object and the indecomposable projective object in rOk that map respectively toLĪŗ(Ī»), PĪŗ(Ī») by the bijections in Lemma 2.2(b).
Lemma 2.3. The object rPk(Ī») is, up to isomorphism, the unique finitely generatedprojective cover of Lk(Ī») in rOk. It has a filtration by deformed parabolic Vermamodules. In particular, it is a free R-module.
JANTZEN FILTRATION 221
The proof of Lemmas 2.2 and 2.3 can be given by imitating [F, Section 2].There, Fiebig proved the analogue of these results for the (nonparabolic) deformedcategory O by adapting arguments of [RW]. The proof here goes in the same way,because the parabolic case is also treated in [RW]. We left the details to the reader.Note that the deformed parabolic category O for reductive Lie algebras has alsobeen studied in [St].
2.5. The highest weight category Ek. Fix a positive integer n ļæ½ m. Let Pn
denote the set of partitions of n. Recall that a partition Ī» of n is a sequence ofintegers Ī»1 ļæ½ Ā· Ā· Ā· ļæ½ Ī»m ļæ½ 0 such that
āmi=1 Ī»i = n. We associate Ī» with the
elementām
i=1 Ī»iĪµi ā tā0, which we denote again by Ī». We will identify Pn with asubset of Ī+ by the following inclusion:
(2.6) Pn ā Ī+, Ī» ļæ½ā Ī»+ cĻ0 āćĪ» : Ī»+ 2Ļ0ć
2ĪŗĪ“.
We will also fix an integer r large enough such that Pn is contained in rĪ+. EquipĪ+ with the partial order ļæ½ given by Ī» ļæ½ Ī¼ if and only if there exists w ā S suchthat Ī¼ = w Ā· Ī» and a(Ī¼)ā a(Ī») ā NĪ +
0 . Let ļæ½ denote the dominance order on Pn
given by
Ī»ļæ½ Ī¼ āāiā
j=1
Ī»j ļæ½iā
j=1
Ī¼j , ā 1 ļæ½ i ļæ½ m.
Note that for Ī», Ī¼ ā Pn we have
(2.7) Ī» ļæ½ Ī¼ =ā Ī»ļæ½ Ī¼,
because Ī» ļæ½ Ī¼ implies that Ī¼ā Ī» ā NĪ +0 , which implies that
iāj=1
Ī¼j āiā
j=1
Ī»j = ćĪ¼ā Ī», Īµ1 + Ā· Ā· Ā·+ Īµić ļæ½ 0, ā 1 ļæ½ i ļæ½ m.
Now consider the following subset of rĪ+:
E = {Ī¼ ā rĪ+ |Ī¼ = w Ā· Ī» for some w ā S, Ī» ā Pn}.
Lemma 2.4. The set E is finite.
Proof. Since Pn is finite, it is enough to show that for each Ī» ā Pn the set SĀ·Ī»ā©rĪ+
is finite. Note that for w ā S0 and Ļ ā ZĪ 0 we have z(wĻ Ā· Ī») = z(Ļ Ā· Ī»). By (2.4)we have
z(Ļ Ā· Ī») = z(Ī»)ā Īŗ
2(||Ļ +
Ī»+ Ļ
Īŗ||2 ā ||Ī»+ Ļ
Īŗ||2).
If z(Ļ Ā· Ī») ļæ½ r, then
||Ļ +Ī»+ Ļ
Īŗ||2 ļæ½ 2
āĪŗ(r ā z(Ī»)) + ||Ī»+ Ļ
Īŗ||2.
There exists only finitely many Ļ ā ZĪ 0 which satisfies this condition, hence theset E is finite. ļæ½
Let EĪŗ be the full subcategory of rOĪŗ consisting of objects M such that
Ī¼ ā rĪ+, Ī¼ /ā E =ā HomrOĪŗ(rPĪŗ(Ī¼),M) = 0.
Note that since rPĪŗ(Ī¼) is projective in rOĪŗ, an object M ā rOĪŗ is in EĪŗ if and onlyif each simple subquotient of M is isomorphic to LĪŗ(Ī¼) for Ī¼ ā E. In particular EĪŗis abelian and it is a Serre subcategory of rOĪŗ. Furthermore, EĪŗ is also an artinian
222 PENG SHAN
category. In fact, each object M ā EĪŗ has a finite length because E is finite andfor each Ī¼ ā E the multiplicity of LĪŗ(Ī¼) in M is finite because dimCMĪ¼ < ā. Letgā² denote the Lie subalgebra of g given by
gā² = g0 ā C[t, tā1]ā C1.
Forgetting the ā-action yields an equivalence of categories from EĪŗ to a category ofgā²-modules; see [So, Proposition 8.1] for details. Since Īŗ is negative, this categoryof gā²-modules is equal to the category studied in [KL2].
Lemma 2.5. (a) For Ī» ā E, Ī¼ ā rĪ+ such that [MĪŗ(Ī») : LĪŗ(Ī¼)] = 0 we have Ī¼ ā Eand Ī¼ ļæ½ Ī».
(b) The module rPĪŗ(Ī») admits a filtration by UĪŗ-modulesrPĪŗ(Ī») = P0 ā P1 ā Ā· Ā· Ā· ā Pl = 0
such that P0/P1 is isomorphic to MĪŗ(Ī») and Pi/Pi+1ā¼= MĪŗ(Ī¼i) for some Ī¼i ļæ½ Ī».
(c) The category EĪŗ is a highest weight C-category with standard objects MĪŗ(Ī»),Ī» ā E. The indecomposable projective objects in EĪŗ are the modules rPĪŗ(Ī») withĪ» ā E.
Proof. Let Uv be the quantized enveloping algebra of g0 with parameter v =exp(2Ļi/Īŗ). Then the Kazhdan-Lusztig tensor equivalence [KL2, Theorem IV.38.1]identifies EĪŗ with a full subcategory of the category of finite dimensional Uv-modules. It maps the module MĪŗ(Ī») to the Weyl module of Uv with highest weighta(Ī»). Since v is a root of unity, part (a) follows from the strong linkage principlefor Uv; see [An, Theorem 3.1]. Part (b) follows from (a) and [KL2, PropositionI.3.9]. Finally, part (c) follows directly from parts (a), (b). ļæ½
Now, let us consider the deformed version. Let Ek be the full subcategory of rOk
consisting of objects M such that
Ī¼ ā rĪ+, Ī¼ /ā E =ā HomrOk(rPk(Ī¼),M) = 0.
Lemma 2.6. An object M ā rOk belongs to Ek if and only if M(ā) belongs to EĪŗ.In particular Mk(Ī») and
rPk(Ī») belong to Ek for Ī» ā E.
Proof. By Lemma 2.2: (a) for any Ī¼ ā rĪ+ the R-module HomrOk(rPk(Ī¼),M) is
finitely generated and we have
HomrOk(rPk(Ī¼),M)(ā) = HomrOĪŗ
(rPĪŗ(Ī¼),M(ā)).
Therefore HomrOk(rPk(Ī¼),M) is nonzero if and only if HomrOĪŗ
(rPĪŗ(Ī¼),M(ā)) isnonzero by Nakayamaās lemma. So the first statement follows from the definitionof Ek and EĪŗ. The rest follows from Lemma 2.5(c). ļæ½
LetPk(E) =
āĪ»āE
rPk(Ī»), PĪŗ(E) =āĪ»āE
rPĪŗ(Ī»).
We have the following corollary.
Corollary 2.7. (a) The category Ek is abelian.(b) For M ā Ek there exists a positive integer d and a surjective map
Pk(E)ād āā M.
(c) The functor HomrOk(Pk(E),ā) yields an equivalence of R-categories
Ek ā¼= EndrOk(Pk(E))op -mod .
JANTZEN FILTRATION 223
Proof. Let M ā Ek, N ā rOk. First assume that N ā M . For Ī¼ ā rĪ+ ifHomrOk
(rPk(Ī¼), N) = 0, then HomrOk(rPk(Ī¼),M) = 0, so Ī¼ belongs to E. Hence
N belongs to Ek. Now, if N is a quotient of M , then N(ā) is a quotient of M(ā).Since M(ā) belongs to EĪŗ, we also have N(ā) ā EĪŗ. Hence N belongs to Ek byLemma 2.6. This proves part (a). Let us concentrate on (b). Since M ā Ek wehave M(ā) ā EĪŗ. The category EĪŗ is artinian with PĪŗ(E) a projective generator.Hence there exists a positive integer d and a surjective map
f : PĪŗ(E)ād āā M(ā).
Since Pk(E)ād is projective in rOk, this map lifts to a map of Uk-modules f :Pk(E)ād ā M such that the following diagram commutes
Pk(E)ād f ļæ½ļæ½
ļæ½ļæ½ļæ½ļæ½
M
ļæ½ļæ½ļæ½ļæ½PĪŗ(E)ād f ļæ½ļæ½ ļæ½ļæ½ M(ā).
Now, since the map f preserves weight spaces and all the weight spaces of Pk(E)ār
and M are finitely generated R-modules, by Nakayamaās lemma, the surjectivityof f implies that f is surjective. This proves (b). Finally part (c) is a directconsequence of parts (a) and (b) by Morita theory. ļæ½
Proposition 2.8. The category Ek is a highest weight R-category with standardmodules Mk(Ī¼), Ī¼ ā E.
Proof. Note that EndrOk(Pk(E))op is a finite projective R-algebra by Lemmas 2.2
and 2.3. Since EĪŗ is a highest weight C-category by Lemma 2.5(c), the result followsfrom Proposition 1.4. ļæ½
2.6. The highest weight category Ak. By definition Pn is a subset of E. LetAk be the full subcategory of Ek consisting of the objects M such that
HomrOk(Mk(Ī»),M) = 0, ā Ī» ā E, Ī» /ā Pn.
We define the subcategory AĪŗ of EĪŗ in the same way. Let
Īk = {Mk(Ī») |Ī» ā Pn}, ĪĪŗ = {MĪŗ(Ī») |Ī» ā Pn}.Recall that E ā rĪ+ is equipped with the partial order ļæ½, and that Pn ā E. Wehave the following lemma.
Lemma 2.9. The set Pn is an ideal in E, i.e., for Ī» ā E, Ī¼ ā Pn, if Ī» ļæ½ Ī¼, thenwe have Ī» ā Pn.
Proof. Let Ī» ā E and Ī¼ ā Pn and assume that Ī» ļæ½ Ī¼. Recall that E ā Īŗtā, so we
can write a(Ī») =ām
i=1 Ī»iĪµi. Since E ā rĪ+ we have Ī»i ā Z and Ī»i ļæ½ Ī»i+1. Weneed to show that Ī»m ā N. Since Ī» ļæ½ Ī¼ there exist ri ā N such that a(Ī¼)ā a(Ī») =āmā1
i=1 riĪ±i. Therefore we have Ī»m = Ī¼m + rmā1 ļæ½ 0. ļæ½
Now, we can prove the following proposition.
Proposition 2.10. The category (Ak,Īk) is a highest weight R-category withrespect to the partial order ļæ½ on Pn. The highest weight category (Ak(ā),Īk(ā))given by base change is equivalent to (AĪŗ,ĪĪŗ).
224 PENG SHAN
Proof. Since Ek is a highest weight R-category and Pn is an ideal of E, [R, Proposi-tion 4.14] implies that (Ak,Īk) is a highest weight R-category with respect to thepartial order ļæ½ on Pn. By (2.7) this implies that (Ak,Īk) is also a highest weightR-category with respect to ļæ½. Finally, the equivalence Ak(ā) ā¼= AĪŗ follows fromthe equivalence Ek(ā) ā¼= EĪŗ and loc. cit. ļæ½
2.7. Costandard objects of Ak. Consider the (contravariant) duality functor Don Ok given by
(2.8) DM =āĪ¼ākt
ā
HomR(MĪ¼, R),
where the action of Uk on DM is given as in Section 2.3, with the module Mk(Ī»)there replaced by M . Similarly, we define the (contravariant) duality functor D onOĪŗ by
(2.9) DM =āĪ¼ākt
ā
Hom(MĪ¼,C),
with the UĪŗ-action given in the same way. This functor fixes the simple modules inOĪŗ. Hence it restricts to a duality functor on AĪŗ, because AĪŗ is a Serre subcategoryof OĪŗ. Therefore (AĪŗ,ĪĪŗ) is a highest weight category with duality in the sense of[CPS]. It follows from [CPS, Proposition 1.2] that the costandard module MĪŗ(Ī»)
āØ
in AĪŗ is isomorphic to DMĪŗ(Ī»).
Lemma 2.11. The costandard module Mk(Ī»)āØ in Ak is isomorphic to DMk(Ī»)
for any Ī» ā Pn.
Proof. By definition we have a canonical isomorphism
(DMk(Ī»))(ā) ā¼= D(MĪŗ(Ī»)) ā¼= MĪŗ(Ī»)āØ.
Recall from Lemma 2.1 that Mk(Ī»)K is a simple Uk,K -module. Therefore we have(DMk(Ī»))K ā¼= Mk(Ī»)K . So the lemma follows from Lemma 1.5 applied to thehighest weight category (Ak,Īk) and the set {DMk(Ī») |Ī» ā Pn}. ļæ½
2.8. Comparison of the Jantzen filtrations. By Definition 1.6 for any Ī» ā Pn
there is a Jantzen filtration of MĪŗ(Ī») associated with the highest weight category(Ak,Īk). Lemma 2.11 implies that this Jantzen filtration coincides with the onegiven in Section 2.3.
2.9. The v-Schur algebra. In this section let R denote an arbitrary integral do-main. Let v be an invertible element in R. The Hecke algebra Hv over R is anR-algebra, which is free as a R-module with basis {Tw |w ā S0}, the multiplicationis given by
Tw1Tw2
= Tw1w2, if l(w1w2) = l(w1) + l(w2),
(Tsi + 1)(Tsi ā v) = 0, 1 ļæ½ i ļæ½ mā 1.
Next, recall that a composition of n is a sequence Ī¼ = (Ī¼1, . . . , Ī¼d) of positive
integers such thatād
i=1 Ī¼i = n. Let Xn be the set of compositions of n. ForĪ¼ ā Xn let SĪ¼ be the subgroup of S0 generated by si for all 1 ļæ½ i ļæ½ d ā 1 suchthat i = Ī¼1 + Ā· Ā· Ā·+ Ī¼j for any j. Write
xĪ¼ =ā
wāSĪ¼
Tw and yĪ¼ =ā
wāSĪ¼
(āv)āl(w)Tw.
JANTZEN FILTRATION 225
The v-Schur algebra Sv of parameter v is the endomorphism algebra of the rightHv-module
āĪ¼āXn
xĪ¼Hv. We will abbreviate
Av = Sv -mod .
Consider the composition ļæ½ of n such that ļæ½i = 1 for 1 ļæ½ i ļæ½ n. Then xļæ½Hv =Hv. So the Hecke algebra Hv identifies with a subalgebra of Sv via the canonicalisomorphism Hv
ā¼= EndHv(Hv).
For Ī» ā Pn let Ī»ā² be the transposed partition of Ī». Let ĻĪ» be the element inSv given by ĻĪ»(h) = xĪ»h for h ā xļæ½Hv and ĻĪ»(xĪ¼Hv) = 0 for any compositionĪ¼ = ļæ½. Then there is a particular element wĪ» ā S0 associated with Ī» such thatthe Weyl module Wv(Ī») is the left ideal in Sv generated by the element
zĪ» = ĻĪ»TwĪ»yĪ»ā² ā Sv.
See [JM] for details. We will write
Īv = {Wv(Ī») |Ī» ā Pn}.
2.10. The Jantzen filtration of Weyl modules. Now, set again R = C[[s]]. Fix
v = exp(2Ļi/Īŗ) ā C and v = exp(2Ļi/k) ā R.
Below we will consider the v-Schur algebra over C with the parameter v, and thev-Schur algebra over R with the parameter v. The category (Av,Īv) is a highestweight C-category. Write Lv(Ī») for the simple quotient of Wv(Ī»). The canonicalalgebra isomorphism Sv(ā) ā¼= Sv implies that (Av,Īv) is a highest weight R-category and there is a canonical equivalence
(Av(ā),Īv(ā)) ā¼= (Av,Īv).
We define the Jantzen filtration (J iWv(Ī»)) of Wv(Ī») by applying Definition 1.6to (Av,Īv). This filtration coincides with the one defined in [JM], because thecontravariant form on Wv(Ī») used in [JM]ās definition is equivalent to a morphismfrom Wv(Ī») to the dual standard module Wv(Ī»)
āØ = HomR(Wv(Ī»), R).
2.11. Equivalence of Ak and Av. In this section we will show that the highestweight R-categories Ak and Av are equivalent. The proof uses rational double affineHecke algebras and Suzukiās functor. Let us first give some reminders. Let h = Cn,let y1, . . . , yn be its standard basis and x1, . . . , xn ā hā be the dual basis. Let H1/Īŗ
be the rational double affine Hecke algebra associated with Sn with parameter 1/Īŗ.It is the quotient of the smash product of the tensor algebra T (hā hā) with CSn
by the relations
[yi, xi] = 1 +1
Īŗ
āj ļæ½=i
sij , [yi, xj ] =ā1
Īŗsij , 1 ļæ½ i, j ļæ½ n, i = j.
Here sij denotes the element of Sn that permutes i and j. Denote by BĪŗ thecategory O of H1/Īŗ as defined in [GGOR]. It is a highest weight C-category. Let{BĪŗ(Ī») |Ī» ā Pn} be the set of standard modules.
Now, let V = Cm be the dual of the vectorial representation of g0. For anyobject M in AĪŗ consider the action of the Lie algebra g0āC[z] on the vector space
T (M) = V ān āM ā C[h]
226 PENG SHAN
given by
(Ī¾ ā za)(v āmā f) =nā
i=1
Ī¾(i)(v)āmā xai f + v ā (ā1)a(Ī¾ ā tāa)mā f
for Ī¾ ā g0, a ā N, v ā V ān, m ā M , f ā C[h]. Here Ī¾(i) is the operator on V ān
that acts on the i-th copy of V by Ī¾ and acts on the other copies of V by identity.Suzuki defined a natural action of H1/Īŗ on the space of coinvariants
EĪŗ(M) = H0(g0 ā C[z], T (M)).
The assignment M ļæ½ā EĪŗ(M) gives a right exact functor
EĪŗ : AĪŗ ā BĪŗ.
See [Su] or [VV2, Section 2] for details. We have
EĪŗ(MĪŗ(Ī»)) = BĪŗ(Ī»),
and EĪŗ is an equivalence of highest weight categories [VV2, Theorem A.5.1].Next, we consider the rational double affine Hecke algebra H1/k over R with
parameter 1/k. The category O of H1/k is defined in the obvious way. It is ahighest weight R-category. We will denote it by Bk. The standard modules will bedenoted by Bk(Ī»). The Suzuki functor over R,
Ek : Ak ā Bk, M ļæ½ā H0(g0 ā C[z], T (M)).
is defined in the same way. It has the following properties.
Lemma 2.12. (a) We have Ek(Mk(Ī»)) = Bk(Ī») for Ī» ā Pn.(b) The functor Ek restricts to an exact functor AĪ
k ā BĪk .
(c) The functor Ek maps a projective generator of Ak to a projective generatorof Bk.
Proof. The proof of part (a) is the same as in the nondeformed case. For part(b), since Ek is right exact over Ak, it is enough to prove that for any injectivemorphism f : M ā N with M , N ā AĪ
k the map
Ek(f) : Ek(M) ā Ek(N)
is injective. Recall from Lemma 2.1 that the Uk,K -module Mk(Ī»)K is simple forany Ī». So the functor
Ek,K : Ak,K ā Bk,K
is an equivalence. Hence the map
Ek(f)āR K : Ek,K(MK) ā Ek,K(NK)
is injective. Since both Ek(M) and Ek(N) are free R-modules, this implies thatEk(f) is also injective. Now, let us concentrate on (c). Let P be a projectivegenerator of Ak. Then P (ā) is a projective generator of AĪŗ. Since EĪŗ is anequivalence of categories, we have EĪŗ(P (ā)) is a projective generator of BĪŗ. By (b)the object Ek(P ) belongs to BĪ
k , so it is free over R. Therefore by the UniversalCoefficient Theorem we have
(Ek(P ))(ā) ā¼= EĪŗ(P (ā)).
Hence Ek(P ) is a projective object of Bk. Note that for any Ī» ā Pn there is asurjective map P ā Mk(Ī»). The right exact functor Ek sends it to a surjectivemap Ek(P ) ā Bk(Ī»). This proves that Ek(P ) is a projective generator of Bk. ļæ½
JANTZEN FILTRATION 227
Proposition 2.13. Assume that Īŗ ļæ½ ā3. Then there exists an equivalence ofhighest weight R-categories
Akā¼ā Av,
which maps Mk(Ī») to Wv(Ī») for any Ī» ā Pn.
Proof. We first give an equivalence of highest weight categories
Ī¦ : Ak ā Bk
as follows. Let P be a projective generator of Ak. Then Q = Ek(P ) is a projectivegenerator of Bk by Lemma 2.12(c). By Morita theory we have equivalences ofcategories
HomAk(P,ā) : Ak
ā¼ā EndAk(P )op -mod,
HomBk(Q,ā) : Bk
ā¼ā EndBk(Q)op -mod .
We claim that the algebra homomorphism
(2.10) EndAk(P ) ā EndBk
(Q), f ļæ½ā Ek(f),
is an isomorphism. To see this, note that we have
Q(ā) = EĪŗ(P (ā)), (EndAk(P ))(ā) = EndAĪŗ
(P (ā)),
(EndBk(Q))(ā) = EndBĪŗ
(Q(ā)).
Since EĪŗ is an equivalence, it yields an isomorphism
EndAĪŗ(P (ā))
ā¼ā EndBĪŗ(Q(ā)), f ļæ½ā EĪŗ(f).
Since both EndAk(P ) and EndBk
(Q) are finitely generated free R-modules, byNakayamaās lemma the morphism (2.10) is an isomorphism. In particular, it yieldsan equivalence of categories
EndAk(P )op -mod ā¼= EndBk
(Q)op -mod .
Combined with the other two equivalences above, we get an equivalence of categories
Ī¦ : Ak ā Bk.
It remains to show that
Ī¦(Mk(Ī»)) ā¼= Bk(Ī»), Ī» ā Pn.
Note that the functor Ek yields a morphism of finitely generated R-modules
HomBk(Q,Ī¦(Mk(Ī»))) = EndBk
(Q)op āEndAk(P )op HomAk
(P,Mk(Ī»))
ā HomBk(Q,Ek(Mk(Ī»)))
= HomBk(Q,Bk(Ī»)).
Let us denote it by Ļ. Note also that we have isomorphisms
HomAk(P,Mk(Ī»))(ā) = HomAĪŗ
(P (ā),MĪŗ(Ī»)),
HomBk(Q,Bk(Ī»))(ā) = HomBĪŗ
(Q(ā), BĪŗ(Ī»)),
and note that EĪŗ is an equivalence of categories. So the map Ļ(ā) is an isomorphism.Furthermore, HomBk
(Q,Bk(Ī»)) is free over R, so Nakayamaās lemma implies thatĻ is also an isomorphism. The preimage of Ļ under the equivalence HomBk
(Q,ā)yields an isomorphism
Ī¦(Mk(Ī»)) ļæ½ Bk(Ī»).
228 PENG SHAN
Finally, if v = ā1, i.e., Īŗ ļæ½ ā3, then by [R, Theorem 6.8] the categories Bk
and Av are equivalent highest weight R-categories with Bk(Ī») corresponding toWv(Ī»). This equivalence composed with Ī¦ gives the desired equivalence in theproposition. ļæ½
Corollary 2.14. Assume that Īŗ ļæ½ ā3. Then for any Ī», Ī¼ ā Pn and i ā N we have
(2.11) [J iMĪŗ(Ī»)/Ji+1MĪŗ(Ī») : LĪŗ(Ī¼)] = [J iWv(Ī»)/J
i+1Wv(Ī») : Lv(Ī¼)].
Proof. This follows from the proposition above and Proposition 1.8. ļæ½
To prove the main theorem, it remains to compute the left hand side of (2.11).This will be done by generalizing the approach of [BB] to the affine parabolic case.To this end, we first give some reminders on D-modules on affine flag varieties.
3. Generalities on D-modules on ind-schemes
In this section, we first recall basic notion for D-modules on (possibly singular)schemes. We will also discuss twisted D-modules and holonomic D-modules. Thenwe introduce the notion of D-modules on ind-schemes following [BD] and [KV].
3.1. Reminders on D-modules. Unless specified otherwise, all the schemes willbe assumed to be of finite type over C, quasi-separated and quasi-projective. Al-though a large number of statements are true in a larger generality, we will only usethem for quasi-projective schemes. For any scheme Z, let OZ be the structure sheafover Z. We write O(Z) for the category of quasi-coherent OZ-modules on Z. Notethat we abbreviate OZ-module for sheaf of OZ-modules over Z. For f : Z ā Ya morphism of schemes, we write fā, f
ā for the functors of direct and inverse im-ages on O(Z), O(Y ). If f is a closed embedding and M ā O(Y ), we consider thequasi-coherent OZ-module
f !M = fā1 HomOY(fāOZ ,M ).
It is the restriction to Z of the subsheaf of M consisting of sections supportedscheme-theoretically on f(Z) ā Y .
Let Z be a smooth scheme. Let DZ be the ring of differential operators onZ. We denote by M(Z) the category of right DZ-modules that are quasi-coherentas OZ-modules. It is an abelian category. Let Ī©Z denote the sheaf of differentialforms of highest degree on Z. The category of right DZ-modules is equivalent to thecategory of left DZ-modules via M ļæ½ā Ī©Z āOZ
M . Let i : Y ā Z be a morphismof smooth schemes. We consider the (DY , i
ā1DZ)-bimodule
DYāZ = iāDZ = OY āiā1OZiā1DZ .
We define the following functors:
iā : M(Z) ā M(Y ), M ļæ½ā Ī©Y āOY
(DYāZ āDZ
(Ī©Z āOZM )
),
iā¢ : M(Y ) ā M(Z), M ļæ½ā iā(M āDYDYāZ).
For any M ā M(Y ) let M O denote the underlying OY -module of M . Then wehave
iā(M O) = iā(M )O .
If i is a locally closed affine embedding, then the functor iā¢ is exact. For anyclosed subscheme Z ā² of Z, we denote by M(Z,Z ā²) the full subcategory of M(Z)consisting of DZ-modules supported set-theoretically on Z ā². If i : Y ā Z is a
JANTZEN FILTRATION 229
closed embedding of smooth varieties, then by a theorem of Kashiwara, the functoriā¢ yields an equivalence of categories
(3.1) M(Y ) ā¼= M(Z, Y ).
We refer to [HTT] for more details about D-modules on smooth schemesNow, let Z be a possibly singular scheme. We consider the abelian category
M(Z) of right D-modules on Z with a faithful forgetful functor
M(Z) ā O(Z), M ļæ½ā M O
as in [BD, 7.10.3]. If Z is smooth, it is equivalent to the category M(Z) above; see[BD, 7.10.12]. For any closed embedding i : Z ā X there is a left exact functor
i! : M(X) ā M(Z)
such that (i!(M ))O = i!(M O) for all M . It admits an exact left adjoint functor
iā¢ : M(Z) ā M(X).
In the smooth case these functors coincide with the one before. If X is smooth,then iā¢ and i! yield mutually inverse equivalences of categories
(3.2) M(Z) ā¼= M(X,Z)
such that i! ā¦ iā¢ = Id; see [BD, 7.10.11]. Note that when Z is smooth, this isKashiwaraās equivalence (3.1). We will always consider D-modules on a (possiblysingular) scheme Z which is given an embedding into a smooth scheme. Finally, ifj : Y ā Z is a locally closed affine embedding and Y is smooth, then we have thefollowing exact functor
(3.3) jā¢ = i! ā¦ (i ā¦ j)ā¢ : M(Y ) ā M(Z).
Its definition is independent of the choice of i.
3.2. Holonomic D-modules. Let Z be a scheme. If Z is smooth, we denote byMh(Z) the category of holonomic DZ-modules; see e.g., [HTT, Definition 2.3.6].Otherwise, let i : Z ā X be a closed embedding into a smooth scheme X. Wedefine Mh(Z) to be the full subcategory of M(Z) consisting of objects M suchthat iā¢M is holonomic. The category Mh(Z) is abelian. There is a (contravariant)duality functor on Mh(Z) given by
D : Mh(Z) ā Mh(Z), M ļæ½ā i!(Ī©X āOX
ExtdimXDX
(iā¢M ,DX)).
For a locally closed affine embedding i : Y ā Z with Y a smooth scheme, thefunctor iā¢ given by (3.3) maps Mh(Y ) to Mh(Z). We put
i! = D ā¦ iā¢ ā¦ D : Mh(Y ) ā Mh(Z).
There is a canonical morphism of functors
Ļ : i! ā iā¢.
The intermediate extension functor is given by
i!ā¢ : Mh(Y ) ā Mh(Z), M ļæ½ā Im(Ļ(M ) : i!M ā iā¢M ).
Note that the functors iā¢, i! are exact. Moreover, if the embedding i is closed, thenĻ is an isomorphism of functors i! ā¼= iā¢.
230 PENG SHAN
3.3. Weakly equivariant D-modules. Let T be a linear group. For any T -scheme Z there is an abelian category MT (Z) of weakly T -equivariant right D-modules on Z with a faithful forgetful functor
(3.4) MT (Z) ā M(Z).
If Z is smooth, an object M of MT (Z) is an object M of M(Z) equipped with astructure of T -equivariant OZ-module such that the action map M āOZ
DZ ā Mis T -equivariant. For any T -scheme Z with a T -equivariant closed embedding i :Z ā X into a smooth T -scheme X, the functor iā¢ yields an equivalence MT (Z) ā¼=MT (X,Z), where MT (X,Z) is the subcategory of MT (X) consisting of objectssupported set-theoretically on Z.
3.4. Twisted D-modules. Let T be a torus, and let t be its Lie algebra. LetĻ : Zā ā Z be a right T -torsor over the scheme Z. For any object M ā MT (Zā )the OZ-module Ļā(M O) carries a T -action. Let
M ā = Ļā(MO)T
be the OZ-submodule of Ļā(MO) consisting of the T -invariant local sections. We
have
Ī(Z,M ā ) = Ī(Zā ,M )T .
For any weight Ī» ā tā we define the categories MĪ»(Z), MĪ»(Z) as follows.First, assume that Z is a smooth scheme. Then Zā is also smooth. So we have
a sheaf of algebras on Z given by
Dā Z = (DZā )ā ,
and M ā is a right Dā Z-module for any M ā MT (Zā ). For any open subscheme
U ā Z the T -action on Ļā1(U) yields an algebra homomorphism
(3.5) Ī“r : U(t) ā Ī(U,Dā Z),
whose image lies in the center of the right hand side. Thus there is also an action
of U(t) on M ā commuting with the Dā Z-action. For Ī» ā tā let mĪ» ā U(t) be the
ideal generated by
{h+ Ī»(h) | h ā t}.
We define MĪ»(Z) (resp. MĪ»(Z)) to be the full subcategory of MT (Zā ) consistingof the objects M such that the action of mĪ» on M ā is zero (resp. nilpotent). In
particular MĪ»(Z) is a full subcategory of MĪ»(Z) and both categories are abelian.We will write
Ī(Z,M ) = Ī(Z,M ā ), ā M ā MĪ»(Z).(3.6)
Now, let Z be any scheme. We say that a T -torsor Ļ : Zā ā Z is admissible ifthere exists a T -torsor Xā ā X with X smooth and a closed embedding i : Z ā Xsuch that Zā ā¼= Xā ĆX Z as a T -scheme over Z. We will only use admissible
T -torsors. Let MĪ»(X,Z), MĪ»(X,Z) be respectively the subcategories of MĪ»(X),
MĪ»(X) consisting of objects supported on Zā . We define MĪ»(Z), MĪ»(Z) to be thefull subcategories of MT (Zā ) consisting of objects M such that iā¢(M ) belongs to
MĪ»(X,Z), MĪ»(X,Z) respectively. Their definition only depends on Ļ.
JANTZEN FILTRATION 231
Remark 3.1. Let Z be a smooth scheme. Let M(Dā Z) be the category of right
Dā Z-modules on Z that are quasi-coherent as OZ-modules. The functor
(3.7) MT (Zā )ā¼ā M(Dā
Z), M ļæ½ā M ā
is an equivalence of categories. A quasi-inverse is given by Ļā; see e.g., [BB, Lemma1.8.10].
Remark 3.2. We record the following fact for a further use. For any smooth T -torsorĻ : Zā ā Z, the exact sequence of relative differential 1-forms
Ļā(Ī©1Z) āā Ī©1
Zā āā Ī©1Zā /Z āā 0
yields an isomorphismĪ©Zā = Ļā(Ī©Z)āO
Zā Ī©Zā /Z .
Since Ļ is a T -torsor we have indeed
Ī©Zā /Z = OZā
as a line bundle. Therefore we have an isomorphism of OZā -modules
Ī©Zā = Ļā(Ī©Z).
Below, we will identify them whenever needed.
3.5. Twisted holonomic D-modules and duality functors. Let Ļ : Zā ā Zbe an admissible T -torsor. We define MT
h (Zā ) to be the full subcategory of MT (Zā )
consisting of objects M whose image via the functor (3.4) belongs to Mh(Zā ). We
define the categories MĪ»h(Z), MĪ»
h(Z) in the same manner.Assume that Z is smooth. Then the category MT (Zā ) has enough injective
objects; see e.g., [Kas, Proposition 3.3.5] and the references there. We define a(contravariant) duality functor on MT
h (Zā ) by
Dā² : MTh (Z
ā ) ā MTh (Z
ā ), M ļæ½ā Ī©Zā āOZā ExtdimZā
MT (Zā )(M ,DZā ).
We may write Dā² = Dā²Z . Note that by Remark 3.2 and the equivalence (3.7) we
have
(3.8) (Dā²ZM )ā = Ī©Z āOZ
ExtdimZā
Dā Z
(M ā ,Dā Z), ā M ā MT (Zā ).
For any Ī» ā tā the functor Dā² restricts to (contravariant) equivalences of categories
(3.9) Dā² : MĪ»h(Z) ā M
ĖāĪ»h (Z), Dā² : MĪ»
h(Z) ā MāĪ»h (Z);
see e.g., [BB, Remark 2.5.5(iv)]. In particular, if Ī» = 0, then Dā² yields a duality onM0
h(Zā ). It is compatible with the duality functor D on Mh(Z) defined in Section
3.2 via the equivalence
Ī¦ : M0h(Z) ā Mh(Z), M ļæ½ā M ā
given by (3.7). More precisely, we have the following lemma.
Lemma 3.3. We have Ī¦ ā¦ Dā² = D ā¦ Ī¦.
The proof is standard, and is left to the reader.Similarly, for an admissible T -torsor Ļ : Zā ā Z with an embedding i into a
smooth T -torsor Xā ā X, we define the functor
Dā² : MTh (Z
ā ) ā MTh (Z
ā ), M ļæ½ā i!Dā²X(iā¢(M )).
Its definition only depends on Ļ. The equivalence (3.9) and Lemma 3.3 hold again.
232 PENG SHAN
A weight Ī» ā tā is integral if it is given by the differential of a character eĪ» :T ā Cā. For such a Ī» we consider the invertible sheaf L Ī»
Z ā O(Z) defined by
Ī(U,L Ī»Z ) = {Ī³ ā Ī(Ļā1(U),OZā ) | Ī³(xhā1) = eĪ»(h)Ī³(x), (x, h) ā Ļā1(U)Ć T}
for any open set U ā Z. We define the following translation functor :
ĪĪ» : MTh (Z
ā ) ā MTh (Z
ā ), M ļæ½ā M āOZā Ļā(L Ī»
Z ).
It is an equivalence of categories. A quasi-inverse is given by ĪāĪ». For any Ī¼ ā tā
the restriction of ĪĪ» yields equivalences of categories
(3.10) ĪĪ» : MĪ¼h(Z) ā MĪ¼+Ī»
h (Z), ĪĪ» : MĪ¼h(Z) ā MĪ¼+Ī»
h (Z).
We define the duality functor on MĪ»h(Z) to be
D : MĪ»h(Z) ā MĪ»
h(Z), M ļæ½ā Ī2Ī» ā¦ Dā²(M ).
It restricts to a duality functor on MĪ»h(Z), which we denote again by D. To avoid
any confusion, we may write D = DĪ». The equivalence ĪĪ» intertwines the dualityfunctors, i.e., we have
(3.11) DĪ»+Ī¼ ā¦ĪĪ» = ĪĪ» ā¦ DĪ¼.
For any locally closed affine embedding of T -torsors i : Z ā Y with Z smooth,we define the functor
i! = D ā¦ iā¢ ā¦ D : MĪ»h(Z) ā MĪ»
h(Y ).
As in Section 3.2, we have a morphism of exact functors Ļ : i! ā iā¢ which is anisomorphism if i is a closed embedding. The intermediate extension functor i!ā¢ isdefined in the same way.
Remark 3.4. Assume that Z is smooth. Let M ā MĪ»h(Z). Put Ī¼ = 0 in (3.10).
Using the equivalence Ī¦ we see that M ā is a right module over the sheaf of algebras
DĪ»Z = L āĪ»
Z āOZDZ āOZ
L Ī»Z .
Furthermore, we have
D(M )ā = Ī©Z āOZāOZ
L 2Ī»Z āOZ
ExtdimZDĪ»
Z(M ā ,DĪ»
Z)
by Lemma 3.3 and (3.11), compare [KT1, (2.1.2)].
3.6. Injective and projective limit of categories. Let us introduce the follow-ing notation. Let A be a filtering poset. For any inductive system of categories(CĪ±)Ī±āA with functors iĪ±Ī² : CĪ± ā CĪ² , Ī± ļæ½ Ī², we denote by 2limāāCĪ± its inductive
limit, i.e., the category whose objects are pairs (Ī±,MĪ±) with Ī± ā A, MĪ± ā CĪ± and
Hom2limāā CĪ±((Ī±,MĪ±), (Ī²,NĪ²)) = limāā
Ī³ļæ½Ī±,Ī²
HomCĪ³(iĪ±Ī³(MĪ±), iĪ²Ī³(NĪ²)).
For any projective system of categories (CĪ±)Ī±āA with functors jĪ±Ī² : CĪ² ā CĪ±,Ī± ļæ½ Ī², we denote by 2limāāCĪ± its projective limit, i.e., the category whose objectsare systems consisting of objects MĪ± ā CĪ± given for all Ī± ā A and isomorphismsjĪ±Ī²(MĪ²) ā MĪ± for each Ī± ļæ½ Ī² and satisfying the compatibility condition for eachĪ± ļæ½ Ī² ļæ½ Ī³. Morphisms are defined in the obvious way. See e.g., [KV, 3.2, 3.3].
JANTZEN FILTRATION 233
3.7. The O-modules on ind-schemes. An ind-scheme X is a filtering inductivesystem of schemes (XĪ±)Ī±āA with closed embeddings iĪ±Ī² : XĪ± ā XĪ² for Ī± ļæ½ Ī²such that X represents the ind-object ā limāā āXĪ±. See [KS, 1.11] for details on ind-objects. Below we will simply write limāā for ā limāā ā, hoping this does not create
any confusion. The categories O(XĪ±) form a projective system via the functorsi!Ī±Ī² : O(XĪ²) ā O(XĪ±). Following [BD, 7.11.4] and [KV, 3.3] we define the categoryof O-modules on X as
O(X) = 2limāāO(XĪ±).
It is an abelian category. An object M of O(X) is represented by
M = (MĪ±, ĻĪ±Ī² : i!Ī±Ī²MĪ² ā MĪ±)
where MĪ± is an object of O(XĪ±) and ĻĪ±Ī² , Ī± ļæ½ Ī², is an isomorphism in O(XĪ±).Note that any object M of O(X) is an inductive limit of objects from O(XĪ±).
More precisely, we first identify O(XĪ±) as a full subcategory of O(X) in the fol-lowing way: since the poset A is filtering, to any MĪ± ā O(XĪ±) we may associate acanonical object (NĪ²) in O(X) such that NĪ² = iĪ±Ī²ā(MĪ±) for Ī± ļæ½ Ī² and the struc-ture isomorphisms ĻĪ²Ī³ , Ī² ļæ½ Ī³, are the obvious ones. Let us denote this object inO(X) again by MĪ±. Given any object M ā O(X) represented by M = (MĪ±, ĻĪ±Ī²),these MĪ± ā O(X), Ī± ā A, form an inductive system via the canonical morphismsMĪ± ā MĪ² . Then, the ind-object limāāMĪ± of O(X) is represented by M . So, wedefine the space of global sections of M to be the inductive limit of vector spaces
(3.12) Ī(X,M ) = limāāĪ(XĪ±,MĪ±).
We will also use the category O(X) defined as the limit of the projective systemof categories (O(XĪ±), i
āĪ±Ī²); see [BD, 7.11.3] or [KV, 3.3]. Note that the canonical
isomorphisms iāĪ±Ī²OXĪ²= OXĪ±
yield an object (OXĪ±)Ī±āA in O(X). We denote
this object by OX . An object F ā O(X) is said to be flat if each FĪ± is a flatOXĪ±
-module. Such a F yields an exact functor
(3.13) O(X) ā O(X), M ļæ½ā M āOXF = (MĪ± āOXĪ±
FĪ±)Ī±āA.
For F ā O(X) the vector spaces Ī(XĪ±,FĪ±) form a projective system with thestructure maps induced by the functors iāĪ±Ī² . We set
(3.14) Ī(X,F ) = limāāĪ(XĪ±,FĪ±).
3.8. The D-modules on ind-schemes. The category of D-modules on the ind-schemeX is defined as the limit of the inductive system of categories (M(XĪ±), iĪ±Ī²ā¢);see e.g., [KV, 3.3]. We will denote it by M(X). Since M(XĪ±) are abelian categoriesand iĪ±Ī²ā¢ are exact functors, the category M(X) is abelian. Recall that an objectof M(X) is represented by a pair (Ī±,MĪ±) with Ī± ā A, MĪ± ā M(XĪ±). There is anexact and faithful forgetful functor
M(X) ā O(X), M = (Ī±,MĪ±) ā M O = (iĪ±Ī²ā¢(MĪ±)O)Ī²ļæ½Ī±.
The global sections functor on M(X) is defined by
Ī(X,M ) = Ī(X,M O).
Next, we say that X is a T -ind-scheme if X = limāāXĪ± with each XĪ± being
a T -scheme and iĪ±Ī² : XĪ± ā XĪ² being T -equivariant. We define MT (X) to
234 PENG SHAN
be the abelian category given by the limit of the inductive system of categories(MT (XĪ±), iĪ±Ī²ā¢). The functors (3.4) for each XĪ± yield an exact and faithful functor
(3.15) MT (X) ā M(X).
The functor Ī on MT (X) is given by the functor Ī on M(X).Finally, given a T -ind-scheme X = limāāXĪ± let Ļ : Xā ā X be a T -torsor over
X, i.e., Ļ is the limit of an inductive system of T -torsors ĻĪ± : Xā Ī± ā XĪ±. We
say that Ļ is admissible if each of the ĻĪ± is admissible. Assume this is the case.
Then the categories MĪ»(XĪ±), MĪ»(XĪ±) form, respectively, two inductive systems
of categories via iĪ±Ī²ā¢. Let
MĪ»(X) = 2limāāMĪ»(XĪ±), MĪ»(X) = 2limāāMĪ»(XĪ±).
They are abelian subcategories of MT (Xā ). For any object M = (Ī±,MĪ±) ofMT (Xā ), the OXĪ²
-modules (iĪ±Ī²ā¢MĪ±)ā with Ī² ļæ½ Ī± give an object of O(X). We
will denote it by M ā . The functor
MT (Xā ) ā O(X), M ļæ½ā M ā
is exact and faithful. For M ā MĪ»(X) we will write
(3.16) Ī(M ) = Ī(X,M ā ).
Note that it is also equal to Ī(Xā ,M )T . We will consider the following categories
MĪ»h(X) = 2limāāMĪ»
h(XĪ±), MĪ»h(X) = 2limāāMĪ»
h(XĪ±).
Let Y be a smooth scheme. A locally closed affine embedding i : Y ā X is thecomposition of an affine open embedding i1 : Y ā Y with a closed embedding
i2 : Y ā X. For such a morphism the functor iā¢ : MĪ»h(Y ) ā MĪ»
h(X) is defined by
iā¢ = i2ā¢ ā¦ i1ā¢, and the functor i! : MĪ»h(Y ) ā MĪ»
h(X) is defined by i! = i2ā¢ ā¦ i1!.3.9. The sheaf of differential operators on a formally smooth ind-scheme.Let X be a formally smooth1 ind-scheme. Fix Ī² ļæ½ Ī± in A. Let Diff Ī²,Ī± be theOXĪ²ĆXĪ±
-submodule of HomC(OXĪ², iĪ±Ī²āOXĪ±
) consisting of local sections supportedset-theoretically on the diagonal XĪ± ā XĪ² Ć XĪ±. Here HomC(OXĪ²
, iĪ±Ī²āOXĪ±)
denotes the sheaf of morphisms between the sheaves of C-vector spaces associatedwith OXĪ²
and iĪ±Ī²āOXĪ±. As a left OXĪ±
-module Diff Ī²,Ī± is quasi-coherent; see e.g.,[BB, Section 1.1]. So it is an object in O(XĪ±). For Ī² ļæ½ Ī³ the functor iĪ²Ī³ā and thecanonical map OXĪ³
ā iĪ²Ī³āOXĪ²yield a morphism of OXĪ±
-modules
HomC(OXĪ², iĪ±Ī²āOXĪ±
) ā HomC(OXĪ³, iĪ±Ī³āOXĪ±
).
It induces a morphism Diff Ī²,Ī± ā Diff Ī³,Ī± in O(XĪ±). The OXĪ±-modules Diff Ī²,Ī±,
Ī² ļæ½ Ī±, together with these maps form an inductive system. Let
Diff Ī± = limāāĪ²ļæ½Ī±
Diff Ī²,Ī± ā O(XĪ±).
The system consisting of the Diff Ī±ās and the canonical isomorphisms iāĪ±Ī² Diff Ī² āDiff Ī± is a flat object in O(X); see [BD, 7.11.11]. We will call it the sheaf ofdifferential operators on X and denote it by DX . It carries canonically a structureof OX-bimodules, and a structure of algebra given by
Diff Ī³,Ī² āOXĪ²Diff Ī²,Ī± ā Diff Ī³,Ī±, (g, f) ļæ½ā g ā¦ f, Ī± ļæ½ Ī² ļæ½ Ī³.
1See [BD, 7.11.1] and the references there for the definition of formally smooth.
JANTZEN FILTRATION 235
Any object M ā M(X) admits a canonical right DX -action given by a morphism
(3.17) M āOXDX ā M
in O(X) which is compatible with the multiplication in DX .
4. Localization theorem for affine Lie algebras of negative level
In this section we first consider the affine localization theorem which relates rightD-modules on the affine flag variety (an ind-scheme) to a category of modules overthe affine Lie algebra with integral weights and a negative level. Then we computethe D-modules corresponding to Verma and parabolic Verma modules. All theconstructions here hold for a general simple linear group. We will only use the caseof SLm, since the multiplicities on the left hand side of (2.11) that we want tocompute are the same for slm and glm. We will use, for slm, the same notation asin Section 2 for glm. In particular g0 = slm and tā0 is now given the basis consistingof the weights Īµi ā Īµi+1 with 1 ļæ½ i ļæ½ m ā 1. We identify Pn as a subset of tā0 viathe map
Pn ā tā0, Ī» = (Ī»1, . . . , Ī»m) ļæ½ā
māi=1
(Ī»i ā n/m)Īµi.
Finally, we will modify slightly the definition of g by extending C[t, tā1] to C((t)),i.e., from now on we set
g = g0 ā C((t))ā C1ā Cā.
The bracket is given in the same way as before. We will again denote by b, n, q,etc., the corresponding Lie subalgebras of g.
4.1. The affine Kac-Moody group. Consider the group ind-scheme LG0 =G0(C((t))) and the group scheme L+G0 = G0(C[[t]]). Let I ā L+G0 be the Iwahorisubgroup. It is the preimage of B0 via the canonical map L+G0 ā G0. For z ā Cā
the loop rotation t ļæ½ā zt yields a Cā-action on LG0. Write
LG0 = Cā ļæ½ LG0.
Let G be the Kac-Moody group associated with g. It is a group ind-scheme whichis a central extension
1 ā Cā ā G ā LG0 ā 1;
see e.g., [Ku2, Section 13.2]. There is an obvious projection pr : G ā LG0. We set
B = prā1(I), Q = prā1(L+G0), T = prā1(T0).
Finally, let N be the prounipotent radical of B. We have
g = Lie(G), b = Lie(B), q = Lie(Q), t = Lie(T ), n = Lie(N).
4.2. The affine flag variety. Let X = G/B be the affine flag variety. It is aformally smooth ind-scheme. The enhanced affine flag variety Xā = G/N is aT -torsor over X via the canonical projection
(4.1) Ļ : Xā ā X.
The T -action on Xā is given by gN ļæ½ā ghā1N for h ā T , g ā G. The T -torsor Ļ isadmissible; see the end of Section A.1. The ind-scheme Xā is also formally smooth.
236 PENG SHAN
For any subscheme Z of X we will write Zā = Ļā1(Z). The B-orbit decompositionof X is
X =āwāS
Xw, Xw = BwB/B,
where w is a representative of w in the normalizer of T in G. Each Xw is an affinespace of dimension l(w). Its closure Xw is an irreducible projective variety. Wehave
Xw =ā
wā²ļæ½w
Xwā² , X = limāāw
Xw.
4.3. Localization theorem. Recall the sheaf of differential operators DXā āO(Xā ). The space of sections of DXā is defined as in (3.14). The left action ofG on Xā yields an algebra homomorphism
(4.2) Ī“l : U(g) ā Ī(Xā ,DXā ).
Since the G-action on Xā commutes with the right T -action, the image of the mapabove lies in the T -invariant part of Ī(Xā ,DXā ). So for M ā MT (Xā ) the DXā -action on M given by (3.17) induces a g-action2 on M ā via Ī“l. In particular thevector space Ī(M ) as defined in (3.16) is a g-module. Let M(g) be the category ofg-modules. We say that a weight Ī» ā tā is antidominant (resp. dominant, regular)if for any Ī± ā Ī + we have ćĪ» : Ī±ć ļæ½ 0 (resp. ćĪ» : Ī±ć ļæ½ 0, ćĪ» : Ī±ć = 0).
Proposition 4.1. (a) The functor
Ī : MĪ»(X) ā M(g), M ļæ½ā Ī(M )
is exact if Ī»+ Ļ is antidominant.(b) The functor
Ī : MĪ»(X) ā M(g), M ļæ½ā Ī(M )
is exact if Ī»+ Ļ is antidominant.
Proof. A proof of part (a) is sketched in [BD, Theorem 7.15.6]. A detailed proofcan be given using similar techniques as in the proof of the Proposition A.2 below.This is left to the reader. See also [FG, Theorem 2.2] for another proof of this result.
Now, let us concentrate on part (b). Let M = (Ī±,MĪ±) be an object in MĪ»(X). Bydefinition the action of mĪ» on M ā is nilpotent. Let Mn be the maximal subobjectof M such that the ideal (mĪ»)
n acts on M ā n by zero. We have Mnā1 ā Mn and
M = limāāMn. Write RkĪ(X,ā) for the k-th derived functor of the global sections
functor Ī(X,ā). Given n ļæ½ 1, suppose that
RkĪ(X,M ā n) = 0, ā k > 0.
Since Mn+1/Mn is an object of MĪ»(X), by part (a) we have
RkĪ(X, (Mn+1/Mn)ā ) = 0, ā k > 0.
The long exact sequence for RĪ(X,ā) applied to the short exact sequence
0 āā M ā n āā M ā
n+1 āā (Mn+1/Mn)ā āā 0
2More precisely, here by g-action we mean the g-action on the associated sheaf of vector spaces(M ā )C; see Step 1 of the proof of Proposition A.2 for details.
JANTZEN FILTRATION 237
implies that RkĪ(X,M ā n+1) = 0 for any k > 0. Therefore by induction the vector
space RkĪ(X,M ā n) vanishes for any n ļæ½ 1 and k > 0. Finally, since the functor
RkĪ(X,ā) commutes with direct limits (see e.g., [TT, Lemma B.6]), we have
RkĪ(X,M ā ) = limāāRkĪ(X,M ā n) = 0, ā k > 0. ļæ½
4.4. The category OĪŗ and Verma modules. For a t-module M and Ī» ā tā let
(4.3) MĪ» = {m ā M | (hā Ī»(h))Nm = 0, ā h ā t, N ļæ½ 0}.We call a t-module M a generalized weight module if it satisfies the conditions
M =āĪ»āĪŗt
ā
MĪ»,
dimCMĪ» < ā, ā Ī» ā tā.
Its character ch(M) is defined as the formal sum
(4.4) ch(M) =āĪ»ātā
dimC(MĪ»)eĪ».
Let O be the category consisting of the U(g)-modules M such that
ā¢ as a t-module M is a generalized weight module,ā¢ there exists a finite subset Ī ā tā such that MĪ» = 0 implies that Ī» āĪ +
āmā1i=0 Zļæ½0Ī±i.
The category O is abelian. We define the duality functor D on O by
(4.5) DM =āĪ»ātā
Hom(MĪ»,C),
with the action of g given by the involution Ļ; see Section 2.3. Let OĪŗ be the fullsubcategory of O consisting of the g-modules on which 1 ā (Īŗ ā m) acts locally
nilpotently. The category OĪŗ is also abelian. It is stable under the duality functor,because Ļ(1) = 1. The category OĪŗ is a Serre subcategory of OĪŗ. For Ī» ā Īŗt
ā weconsider the Verma module
NĪŗ(Ī») = U(g)āU(b) CĪ».
Here CĪ» is the one-dimensional b-module such that n acts trivially and t acts byĪ». It is an object of OĪŗ. Let LĪŗ(Ī») be the unique simple quotient of NĪŗ(Ī»). We
have DLĪŗ(Ī») = LĪŗ(Ī») for any Ī». A simple subquotient of a module M ā OĪŗ isisomorphic to LĪŗ(Ī») for some Ī» ā Īŗt
ā. The classes [LĪŗ(Ī»)] form a basis of the vector
space [OĪŗ], because the characters of the LĪŗ(Ī»)ās are linearly independent.Denote by Ī the set of integral weights in Īŗt
ā. Let Ī» ā Ī and w ā S. Recall theline bundle L Ī»
Xwfrom Section 3.5. Let
(4.6) A Ī»w = Ī©Xā
wāO
Xā w
Ļā(L Ī»Xw
).
It is an object of MĪ»h(Xw) and
D(A Ī»w ) = A Ī»
w .
Let iw : Xā w ā Xā be the canonical embedding. It is locally closed and affine. We
have the following objects in MĪ»h(X),
A Ī»w! = iw!(A
Ī»w ), A Ī»
w!ā¢ = iw!ā¢(AĪ»w ), A Ī»
wā¢ = iwā¢(AĪ»w )
238 PENG SHAN
We will consider the Serre subcategory MĪ»0 (X) of MĪ»
h(X) generated by the simpleobjects A Ī»
w!ā¢ for w ā S. It is an artinian category. Since D(A Ī»w!ā¢) = A Ī»
w!ā¢, thecategory MĪ»
0 (X) is stable under the duality. We have the following proposition.
Proposition 4.2. Let Ī» ā Ī such that Ī»+ Ļ is antidominant. Then(a) Ī(A Ī»
w!) = NĪŗ(w Ā· Ī»),(b) Ī(A Ī»
wā¢) = DNĪŗ(w Ā· Ī»),
(c) Ī(A Ī»w!ā¢) =
{LĪŗ(w Ā· Ī») if w is the shortest element in wS(Ī»),
0 else.
The proof of the proposition will be given in Appendix A. It relies on results ofKashiwara-Tanisaki [KT1] and uses translation functors for the affine category O.
4.5. The parabolic Verma modules. Let QS be the set of the longest repre-sentatives of the cosets S0\S. Let w0 be the longest element in S0. Recall thefollowing basic facts.
Lemma 4.3. For w ā S if w Ā· Ī» ā Ī+ for some Ī» ā Ī with Ī» + Ļ antidominant,then w ā QS. Further, if w ā QS then we have
(a) the element w is the unique element v in S0w such that Ī +0 ā āv(Ī +),
(b) for any v ā S0 we have l(vw) = l(w)ā l(v),(c) the element w0w is the shortest element in S0w.
The Q-orbit decomposition of X is given by
X =ā
wāQS
Yw, Yw = QwB/B.
Each Yw is a smooth subscheme of X, and Xw is open and dense in Yw. The closureof Yw in X is an irreducible projective variety of dimension l(w) given by
Y w =ā
wā²āQS, wā²ļæ½w
Ywā² .
Recall that Y ā w = Ļā1(Yw). The canonical embedding jw : Y ā
w ā Xā is locallyclosed and affine; see Remark 5.2(b). For Ī» ā Ī and w ā QS let
(4.7) BĪ»w = Ī©Y ā
wāO
Yā w
Ļā(L Ī»Yw
).
We have the following objects in MĪ»h(X)
BĪ»w! = jw!(B
Ī»w), BĪ»
w!ā¢ = jw!ā¢(BĪ»w), BĪ»
wā¢ = jwā¢(BĪ»w).
Now, consider the canonical embedding r : Xā w ā Y ā
w. Since r is open and affine,we have rā = r! and the functors
(r!, r! = rā, rā¢)
form a triple of adjoint functors between the categories MĪ»h(Yw) and MĪ»
h(Xw).Note that rā(BĪ»
w)ā¼= A Ī»
w .
Lemma 4.4. For Ī» ā Ī and w ā QS the following holds.(a) The adjunction map r!r
ā ā Id yields a surjective morphism in MĪ»h(Yw),
(4.8) r!(AĪ»w ) ā BĪ»
w.
(b) The adjunction map Id ā rā¢rā yields an injective morphism in MĪ»
h(Yw),
(4.9) BĪ»w ā rā¢(A
Ī»w ).
JANTZEN FILTRATION 239
Proof. We will only prove part (b). Part (a) follows from (b) by applying the dualityfunctor D. To prove (b), it is enough to show that the OY ā
w-module morphism
(4.10) (BĪ»w)
O ā(rā¢r
ā(BĪ»w)
)O
is injective. Since r is an open embedding, the right hand side is equal to rārā((BĪ»
w)O).
Now, consider the closed embedding
i : Y ā w āXā
w āā Y ā w.
The morphism (4.10) can be completed into the following exact sequence in O(Y ā w),
0 ā iāi!((BĪ»
w)O)ā (BĪ»
w)O ā rār
ā((BĪ»w)
O);
see e.g., [HTT, Proposition 1.7.1]. The OY ā w-module (BĪ»
w)O is locally free. So it
has no subsheaf supported on the closed subscheme Y ā w ā Xā
w. We deduce thatiāi
!((BĪ»
w)O)= 0. Hence the morphism (4.10) is injective. ļæ½
Lemma 4.5. For Ī» ā Ī and w ā QS we have
A Ī»w!ā¢
ā¼= BĪ»w!ā¢.
Proof. By applying the exact functor jwā¢ to the map (4.9) we see that BĪ»wā¢ is a
subobject of A Ī»wā¢ in MĪ»
h(X). In particular BĪ»w!ā¢ is a simple subobject of A Ī»
wā¢; soit is isomorphic to A Ī»
w!ā¢. ļæ½
Proposition 4.6. Let Ī» ā Ī such that Ī»+ Ļ is antidominant, and let w ā QS.(a) If there exists Ī± ā Ī +
0 such that ćw(Ī»+ Ļ) : Ī±ć = 0, then
Ī(BĪ»w!) = 0.
(b) We have
ćw(Ī»+ Ļ) : Ī±ć = 0, ā Ī± ā Ī +0 āā w Ā· Ī» ā Ī+.
In this case, we have
Ī(BĪ»w!) = MĪŗ(w Ā· Ī»), Ī(BĪ»
wā¢) = DMĪŗ(w Ā· Ī»).(c) We have
Ī(BĪ»w!ā¢) =
{LĪŗ(w Ā· Ī») if w is the shortest element in wS(Ī»),
0 else.
Proof. The proof is inspired by the proof in the finite type case; see e.g., [M,Theorem G.2.10]. First, by Kazhdan-Lusztigās algorithm (see Remark 6.3), thefollowing equality holds in [MĪ»
0 (X)]:
(4.11) [BĪ»w!] =
āyāS0
(ā1)l(y)[A Ī»yw!].
Since Ī»+Ļ is antidominant, the functor Ī is exact on MĪ»0 (X) by Proposition 4.1(a).
Therefore we have the following equalities in [OĪŗ]:
[Ī(BĪ»w!)] =
āyāS0
(ā1)l(y)[Ī(A Ī»yw!)]
=āyāS0
(ā1)l(y)[NĪŗ(yw Ā· Ī»)].(4.12)
240 PENG SHAN
Here the second equality is given by Proposition 4.2. Now, suppose that there existsĪ± ā Ī +
0 such that ćw(Ī»+ Ļ) : Ī±ć = 0. Let sĪ± be the corresponding reflection in S0.Then we have
sĪ±w Ā· Ī» = w Ā· Ī» .
By Lemma 4.3(b) we have l(w) = l(sĪ±w) + 1. So the right hand side of (4.12)vanishes. Therefore we have Ī(BĪ»
w!) = 0. This proves part (a). Now, let usconcentrate on part (b). Note that Ī»+ Ļ is antidominant. Thus by Lemma 4.3(a)we have ćw(Ī»+ Ļ) : Ī±ć ā N for any Ī± ā Ī +
0 . Hence
ćw(Ī»+ Ļ) : Ī±ć = 0 āā ćw(Ī»+ Ļ) : Ī±ć ļæ½ 1 āā ćw Ā· Ī» : Ī±ć ļæ½ 0.
By consequence ćw(Ī»+ Ļ) : Ī±ć = 0 for all Ī± ā Ī +0 if and only if w Ā· Ī» belongs
to Ī+. In this case, the right hand side of (4.12) is equal to [MĪŗ(w Ā· Ī»)] by theBGG-resolution. We deduce that
(4.13) [Ī(BĪ»w!)] = [MĪŗ(w Ā· Ī»)].
Now, applying the exact functor j! to the surjective morphism in (4.8) yields aquotient map BĪ»
w! ā A Ī»w! in MĪ»(X). The exactness of Ī implies that Ī(BĪ»
w!) isa quotient of NĪŗ(w Ā· Ī») = Ī(A Ī»
w!). Since MĪŗ(w Ā· Ī») is the maximal q-locally-finitequotient of NĪŗ(w Ā· Ī») and Ī(BĪ»
w!) is q-locally finite, we deduce that Ī(BĪ»w!) is a
quotient of MĪŗ(w Ā· Ī»). So the first equality in part (b) follows from (4.13). Theproof of the second one is similar. Finally, part (c) follows from Lemma 4.5 andProposition 4.2. ļæ½
Remark 4.7. Note that if w ā QS is a shortest element in wS(Ī»), then we havećw(Ī»+ Ļ) : Ī±ć = 0 for all Ī± ā Ī +
0 . Indeed, if there exists Ī± ā Ī +0 such that
ćw(Ī»+ Ļ) : Ī±ć = 0. Let sā² = wā1sĪ±w. Then sā² belongs to S(Ī»). Therefore wehave l(wsā²) > l(w). But wsā² = sĪ±w and sĪ± ā S0, by Lemma 4.3 we have l(wsā²) =l(sĪ±w) < l(w). This is a contradiction.
5. The geometric construction of the Jantzen filtration
In this part, we give the geometric construction of the Jantzen filtration in theaffine parabolic case by generalizing the result of [BB].
5.1. Notation. Let R be any noetherian C-algebra. To any abelian category C weassociate a category CR whose objects are the pairs (M,Ī¼M ) with M an object of Cand Ī¼M : R ā EndC(M) a ring homomorphism. A morphism (M,Ī¼M ) ā (N,Ī¼N )is a morphism f : M ā N in C such that Ī¼N (r) ā¦ f = f ā¦ Ī¼M (r) for r ā R. Thecategory CR is also abelian. We have a faithful forgetful functor
(5.1) for : CR ā C, (M,Ī¼M ) ā M.
Any functor F : C ā Cā² gives rise to a functor
FR : CR ā Cā²R, (M,Ī¼M ) ļæ½ā (F (M), Ī¼F (M))
such that Ī¼F (M)(r) = F (Ī¼M (r)) for r ā R. The functor FR is R-linear. If F isexact, then FR is also exact. We have for ā¦ FR = F ā¦ for. Given an inductivesystem of categories (CĪ±, iĪ±Ī²), it yields an inductive system ((CĪ±)R, (iĪ±Ī²)R), and wehave a canonical equivalence
(2limāāCĪ±)R = 2limāā((CĪ±)R).
JANTZEN FILTRATION 241
5.2. The function fw. Let Qā² = (Q,Q) be the commutator subgroup of Q. Itacts transitively on Y ā
w for w ā QS. We have the following lemma.
Lemma 5.1. For any w ā QS there exists a regular function fw : Yā w ā C such
that fā1w (0) = Y
ā w ā Y ā
w and
fw(qxhā1) = ew
ā1Ļ0(h)fw(x), q ā Qā², x ā Y ā w, h ā T.
Proof. Let V denote the simple g-module of highest weight Ļ0. It is integrable,hence it admits an action of G. Let v0 ā V be a nonzero vector in the weight spaceVĻ0
. It is fixed under the action of Qā². So the map
Ļ : G ā V, g ļæ½ā gā1v0
maps QwB to Bwā1v0 for any w ā QS. Let V (wā1) be the U(b)-submodule of Vgenerated by the weight space Vwā1Ļ0
. We have Bwā1v0 ā V (wā1). Recall thatfor wā² ā QS we have
wā² < w āā (wā²)ā1 < wā1
āā (wā²)ā1v0 ā nwā1v0;(5.2)
see e.g., [Ku2, Proposition 7.1.20]. Thus, if wā² ļæ½ w, then Ļ(Qwā²B) ā V (wā1).The C-vector space V (wā1) is finite dimensional. We choose a linear form lw :V (wā1) ā C such that
lw(wā1v0) = 0 and lw(nw
ā1v0) = 0.
Set fw = lw ā¦ Ļ. Then for q ā Qā², h ā T , u ā N we have
fw(qwhā1u) = lw(u
ā1hwā1v0)
= ewā1Ļ0(h)lw(w
ā1v0)
= ewā1Ļ0(h)fw(w
ā1).
A similar calculation together with (5.2) yields that fw(Qwā²B) = 0 for wā² < w.
Hence fw defines a regular function onā
wā²ļæ½w Qwā²B which is invariant under the
right action of N . By consequence it induces a regular function fw on Yā w which
has the required properties. ļæ½
Remark 5.2. (a) The function fw above is completely determined by its value onthe point wN/N , hence is unique up to scalar.
(b) The lemma implies that the embedding jw : Y ā w ā Xā is affine.
(c) The function fw is an analogue of the function defined in [BB, Lemma 3.5.1]in the finite type case. Below we will use it to define the Jantzen filtration on BĪ»
w!.Note that [BB]ās function is defined on the whole enhanced flag variety (which is
a smooth scheme). Although our fw is only defined on the singular scheme Yā w,
this does not create any problem, because the definition of the Jantzen filtration is
local (see Section 5.5), and each point of Yā w admits a neighborhood V which can
be embedded into a smooth scheme U such that fw extends to U . The choice ofsuch an extension will not affect the filtration; see [BB, Remark 4.2.2(iii)].
242 PENG SHAN
5.3. The D-module B(n). Fix Ī» ā Ī and w ā QS. In the rest of Section 5, wewill abbreviate
j = jw, f = fw, B = BĪ»w, B! = BĪ»
w!, etc.
Following [BB] we introduce the deformed version of B. Recall that R = C[[s]] andā is the maximal ideal. Let x denote a coordinate on C. For each integer n > 0 setR(n) = R(ān). Consider the left DCā-module
I (n) = (OCā āR(n))xs.
It is a rank one OCā ā R(n)-module generated by a global section xs such thatthe action of DCā is given by xāx(x
s) = s(xs). The restriction of f yields a mapY ā w ā Cā. Thus fāI (n) is a left DY ā
wāR(n)-module. So we get a right DY ā
wāR(n)-
module
B(n) = B āOY
ā w
fāI (n).
Lemma 5.3. The right DY ā wāR(n)-module B(n) is an object of MĪ»
h(Yw).
Proof. Since R(n) is a C-algebra of dimension n and B is locally free of rank oneover OY ā
w, the OY ā
w-module B(n) is locally free of rank n. Hence it is a holonomic
DY ā w-module. Note that the DCā-module I (n) is weakly T -equivariant such that xs
is a T -invariant global section. Since the map f is T -equivariant, the DY ā w-module
fāI (n) is also weakly T -equivariant. Let fs be the global section of fāI (n) givenby the image of xs under the inclusion
Ī(Cā,I (n)) ā Ī(Y ā w, f
āI (n)).
Then fs is T -invariant. It is nowhere vanishing on Y ā w. Thus it yields an isomor-
phism of OY ā wāR(n)-modules
fāI (n) ā¼= OY ā wāR(n).
By consequence we have the following isomorphism
(B(n))ā = Ļā(Ļā(Ī©Yw
āOYwL Ī»
Yw)āO
Yā w
fāI (n))T
= Ī©YwāOYw
L Ī»Yw
āOYwĻā(f
āI (n))T
ā¼= Ī©YwāOYw
L Ī»Yw
āOYw(OYw
āR(n)).
See Remark 3.2 for the first equality. Next, recall from (3.5) that the right T -actionon Y ā
w yields a morphism of Lie algebras
Ī“r : t ā Ī(Y ā w,DY ā
w).
The right Dā Yw
-module structure of fā(I (n)) is such that
(fs Ā· Ī“r(h))(m) = swā1Ļ0(āh)fs(m), ā m ā Y ā w.
So the action of the element
h+ Ī»(h) + swā1Ļ0(h) ā U(t)āR(n)
on (B(n))ā via the map Ī“r vanishes. Since the multiplication by s on (B(n))ā isnilpotent, the action of the ideal mĪ» is also nilpotent. Therefore B(n) belongs to
the category MĪ»h(Yw). ļæ½
JANTZEN FILTRATION 243
It follows from the lemma that we have the following objects in MĪ»h(X)
B(n)! = j!(B
(n)), B(n)!ā¢ = j!ā¢(B
(n)), B(n)ā¢ = jā¢(B
(n)).
5.4. Deformed parabolic Verma modules. Fix Ī» ā Ī and w ā QS as before.Let n > 0. If w Ā· Ī» ā Ī+ we will abbreviate
MĪŗ = MĪŗ(w Ā· Ī»), Mk = Mk(w Ā· Ī»), M(n)k = Mk(ā
n), DM(n)k = (DMk)(ā
n).
Note that the condition is satisfied when w is a shortest element in wS(Ī»); seeRemark 4.7.
Proposition 5.4. Assume that Ī» + Ļ is antidominant and that w is a shortestelement in wS(Ī»). Then there are isomorphisms of gR(n)-modules
Ī(B(n)! ) = M
(n)k , Ī(B
(n)ā¢ ) = DM
(n)k .
The proof will be given in Appendix B.
5.5. The geometric Jantzen filtration. Now, we define the Jantzen filtration
on B! following [BB, Sections 4.1,4.2]. Recall that B(n) is an object of MĪ»h(Yw).
Consider the map
Ī¼ : R(n) ā EndMĪ»
h(Yw)(B(n)), Ī¼(r)(m) = rm,
where m denotes a local section of B(n). Then the pair (B(n), Ī¼) is an object of the
category MĪ»h(Yw)R(n) defined in Section 5.1. We will abbreviate B(n) = (B(n), Ī¼).
Fix an integer a ļæ½ 0. Recall the morphism of functors Ļ : j! ā jā¢. We consider themorphism
Ļ(a, n) : B(n)! ā B
(n)ā¢
in the category MĪ»h(X)R(n) given by the composition of the chain of maps
(5.3) B(n)!
j!(Ī¼(sa)) ļæ½ļæ½ B(n)
!
Ļ(B(n)) ļæ½ļæ½ B(n)ā¢ .
The category MĪ»h(X)R(n) is abelian. The obvious projection R(n) ā R(nā1) yields
a canonical map
Coker(Ļ(a, n)) ā Coker(Ļ(a, nā 1)).
By [B, Lemma 2.1] this map is an isomorphism when n is sufficiently large. Wedefine
(5.4) Ļa(B) = Coker(Ļ(a, n)), n ļæ½ 0.
This is an object of MĪ»h(X)R(n) . We view it as an object of MĪ»
h(X) via the forgetfulfunctor (5.1). Now, let us consider the maps
Ī± : B! ā Ļ1(B), Ī² : Ļ1(B) ā Ļ0(B)
in MĪ»h(X) given as follows. First, since
Ļ0(B) = Coker(Ļ(B(n))) and Ļ1(B) = Coker(Ļ(B(n)) ā¦ j!(Ī¼(s))),there is a canonical projection Ļ1(B) ā Ļ0(B). We define Ī² to be this map. Next,the morphism Ļ(B(n)) maps j!(s(B(n))) to Im(Ļ(1, n)). Hence it induces a map
j!(B(n)/s(B(n))) ā Ļ1(B), n ļæ½ 0.
244 PENG SHAN
Composing it with the isomorphism B ā¼= B(n)/s(B(n)) we get the map Ī±. Let Ī¼1
denote the R(n)-action on Ļ1(B). Then by [B] the sequence
(5.5) 0 āā B!Ī±āā Ļ1(B)
Ī²āā Ļ0(B) āā 0,
is exact and Ī± induces an isomorphism
B! ā Ker(Ī¼1(s) : Ļ1(B) ā Ļ1(B)).
The Jantzen filtration of B! is defined by
(5.6) J i(B!) = Ker(Ī¼1(s)) ā© Im(Ī¼1(s)i), ā i ļæ½ 0.
5.6. Comparison of the Jantzen filtrations. Fix Ī» ā Ī and w ā QS. Considerthe Jantzen filtration (J iMĪŗ) on MĪŗ as defined in Section 2.3. The followingproposition compares it with the geometric Jantzen filtration on B!.
Proposition 5.5. Assume that Ī» + Ļ is antidominant and that w is a shortestelement in wS(Ī»). Then we have
J iMĪŗ = Ī(J iB!), ā i ļæ½ 0.
Proof. By Proposition 4.6(b) and Proposition 5.4 we have
Ī(B!) = MĪŗ, Ī(B(n)! ) = M
(n)k , Ī(B
(n)ā¢ ) = DM
(n)k .
So the map
Ļ(n) = Ī(Ļ(B(n))) : Ī(B(n)! ) ā Ī(B
(n)ā¢ ).
identifies with a gR(n)-module homomorphism
Ļ(n) : M(n)k ā DM
(n)k .
Consider the projective systems (M(n)k ), (DM
(n)k ), n > 0, induced by the quotient
map R(n) ā R(nā1). Their limits are respectively Mk and DMk. The morphismsĻ(n), n > 0, yield a morphism of gR-modules
Ļ = limāāĻ(n) : Mk ā DMk
such thatĻ(ā) = Ļ(1) = Ī(Ļ(B)).
The functor Ī is exact by Proposition 4.1. So the image of Ļ(ā) is Ī(B!ā¢). It isnonzero by Proposition 4.6(c). Hence Ļ satisfies the condition of Definition 1.6 andwe have
J iMĪŗ =({x ā Mk |Ļ(x) ā siDMk}+ sMk
)/sMk.
By Lemma 2.1 and Remark 1.7 the map Ļ is injective. So the equality above canbe rewritten as
J iMĪŗ =(Ļ(Mk) ā© siDMk + sĻ(Mk)
)/sĻ(Mk).
Now, for a ļæ½ 0 let
Ļ(a, n) : M(n)k ā DM
(n)k
be the gR(n)-module homomorphism given by the composition
(5.7) M(n)k
Ī¼(sa) ļæ½ļæ½ M (n)k
Ļ(n)
ļæ½ļæ½ DM(n)k .
Then we have Ī(Ļ(a, n)) = Ļ(a, n). Since Ī is exact, we have
Coker(Ļ(a, n)) = Ī(Coker(Ļ(a, n))
).
JANTZEN FILTRATION 245
So the discussion in Section 5.5 and the exactness of Ī yields that the canonicalmap
Coker(Ļ(a, n)) ā Coker(Ļ(a, nā 1))
is an isomorphism if n is large enough. We deduce that
DMk/saMk = Coker(Ļ(a, n)) = Ī(Ļa(B)), n ļæ½ 0;
see (5.4). The action of Ī¼(s) on DMk/saĻ(Mk) is nilpotent, because Ī¼(s) is nilpo-
tent on DM(n)k . Furthermore, Ī maps the exact sequence (5.5) to an exact sequence
(5.8) 0 āā MĪŗ āā DMk/sĻ(Mk) āā DMk/Ļ(Mk) āā 0,
and the first map yields an isomorphism
MĪŗ = Ker(Ī¼(s) : DMk/sĻ(Mk) ā DMk/sĻ(Mk)
).
Note that since DMk is a free R-module, for x ā DMk if sx ā sĻ(Mk), thenx ā Ļ(Mk). So by (5.6) and the exactness of Ī, we have for i ļæ½ 0,
Ī(J iB!) = Ker(Ī¼(s)) ā© Im(Ī¼(s)i)
=(Ļ(Mk) ā© siDMk + sĻ(Mk)
)/sĻ(Mk)
= J iMĪŗ.
The proposition is proved. ļæ½
6. Proof of the main theorem
6.1. Mixed Hodge modules. Let Z be a smooth scheme. Let MHM(Z) be thecategory of mixed Hodge modules on Z [Sa]. It is an abelian category. Each objectM of MHM(Z) carries a canonical filtration
W ā¢M = Ā· Ā· Ā·W kM ā W kā1M Ā· Ā· Ā· ,
called the weight filtration. For each k ā Z the Tate twist is an auto-equivalence
(k) : MHM(Z) ā MHM(Z), M ļæ½ā M (k)
such that W ā¢(M (k)) = W ā¢+2k(M ). Let Perv(Z) be the category of perversesheaves on Z with coefficient in C. There is an exact forgetful functor
ļæ½ : MHM(Z) ā Perv(Z).
For any locally closed affine embedding i : Z ā Y we have exact functors
i!, iā¢ : MHM(Z) ā MHM(Y )
which correspond via ļæ½ to the same named functors on the categories of perversesheaves.
If Z is not smooth we embed it into a smooth variety Y and we define MHM(Z)as the full subcategory of MHM(Y ) consisting of the objects supported on Z. It isindependent of the choice of the embedding for the same reason as for D-modules.
246 PENG SHAN
6.2. The graded multiplicities of BĪ»x!ā¢ in BĪ»
w!. Now, let us calculate the multi-plicities of a simple object BĪ»
x!ā¢ in the successive quotients of the Jantzen filtrationof BĪ»
w! for x, w ā QS with x ļæ½ w.We fix once for all an element v ā S, and we consider the Serre subcategory
MĪ»0 (Xv) of MĪ»
h(Xv) generated by the objects A Ī»w!ā¢ with w ļæ½ v, w ā S. The
de Rham functor yields an exact fully faithful functor
DR : MĪ»0 (Xv) āā Perv(Xv).
See e.g., [KT1, Section 4]. Let MHM0(Xv) be the full subcategory of MHM(Xv)consisting of objects whose image by ļæ½ belong to the image of the functor DR.There exists a unique exact functor
Ī· : MHM0(Xv) ā MĪ»0 (Xv)
such that DR ā¦Ī· = ļæ½. An object M in MHM0(Xv) is pure of weight i if we haveW kM /W kā1M = 0 for any k = i. For any w ā S, w ļæ½ v, there is a unique
simple object A Ī»w in MHM(Xw) pure of weight l(w) such that Ī·(A Ī»
w ) = A Ī»w ; see
e.g. [KT2]. Let
A Ī»w! = (iw)!(A
Ī»w ), A Ī»
w!ā¢ = (iw)!ā¢(AĪ»w ).
They are objects of MHM0(Xv) such that
Ī·(A Ī»w!) = A Ī»
w!, Ī·(A Ī»w!ā¢) = A Ī»
w!ā¢.
Now, assume that w ā QS and w ļæ½ v. Recall that BĪ»w ā MĪ»(Yw), and that
BĪ»w! ā MĪ»(X) can be viewed as an object of MĪ»(Xv). We define similarly the
objects BĪ»w ā MHM(Yw) and BĪ»
w!, BĪ»w!ā¢ ā MHM0(Xv) such that
Ī·(BĪ»w!) = BĪ»
w!, Ī·(BĪ»w!ā¢) = BĪ»
w!ā¢.
The object BĪ»w! has a canonical weight filtration W ā¢. We set JkBĪ»
w! = BĪ»w! for
k < 0. The following proposition is due to Gabber and Beilinson-Bernstein [BB,Theorem 5.1.2, Corollary 5.1.3].
Proposition 6.1. We have Ī·(W l(w)ākBĪ»w!) = JkBĪ»
w! in MĪ»0 (Xv) for all k ā Z.
So the problem that we posed at the beginning of the section reduces to calculatethe multiplicities of BĪ»
x!ā¢ in BĪ»w! in the category MHM0(Xv). Let q be a formal
parameter. The Hecke algebra Hq(S) of S is a Z[q, qā1]-algebra with a Z[q, qā1]-basis {Tw}wāS whose multiplication is given by
Tw1Tw2
= Tw1w2, if l(w1w2) = l(w1) + l(w2),
(Tsi + 1)(Tsi ā q) = 0, 0 ļæ½ i ļæ½ mā 1.
On the other hand, the Grothendieck group [MHM0(Xv)] is a Z[q, qā1]-modulesuch that
qk[M ] = [M (āk)], k ā Z, M ā MHM0(Xv).
For x ā S with x ļæ½ v consider the closed embedding
cx : pt ā Xv, pt ļæ½ā xB/B.
There is an injective Z[q, qā1]-module homomorphism; see e.g., [KT2, (5.4)],
ĪØ : [MHM0(Xv)] āā Hq(S),
[M ] ļæ½āāāxļæ½v
ākāZ
(ā1)k[Hkcāx(M )]Tx.
JANTZEN FILTRATION 247
The desired multiplicities are given by the following lemma.
Lemma 6.2. For w ā QS we have
ĪØ([BĪ»w!ā¢]) =
āxāQS,xļæ½w
(ā1)l(w)āl(x)Px,wĪØ([BĪ»x!]),
where Px,w ā Z[q, qā1] is the Kazhdan-Lusztig polynomial.
Proof. Since the choice for the element v above is arbitrary, we may assume thatw ļæ½ v. By the definition of ĪØ we have
(6.1) ĪØ([A Ī»w!]) = (ā1)l(w)Tw.
By [KL1], [KT1], we have
(6.2) ĪØ([A Ī»w!ā¢]) = (ā1)l(w)
āxāS
Px,wTx.
Next, for x ā QS with x ļæ½ v we have
ĪØ([BĪ»x!]) =
āyāS0
ākāZ
(ā1)k[Hkcāyx(BĪ»x!)]Tyx
= (ā1)l(x)āyāS0
Tyx.(6.3)
Since by Lemma 4.5 we have
A Ī»w!ā¢ = BĪ»
w!ā¢,
the following equalities hold
ĪØ([BĪ»w!ā¢]) = ĪØ([A Ī»
w!ā¢])
= (ā1)l(w)ā
xāQS,xļæ½w
āyāS0
Pyx,wTyx
= (ā1)l(w)ā
xāQS,xļæ½w
Px,w
āyāS0
Tyx
=ā
xāQS,xļæ½w
(ā1)l(w)āl(x)Px,wĪØ([BĪ»x!]).
Here the third equality is given by the well-known identity,
Pyx,w = Px,w, y ā S0, x ā QS, x ļæ½ w. ļæ½
Remark 6.3. Let x ā QS. Since ĪØ is injective, the equation (6.3) yields that
[BĪ»x!] =
āyāS0
(ā1)l(y)[A Ī»yx!].
By applying the functor Ī· we get the following equality in [MĪ»0 (X)]:
(6.4) [BĪ»x!] =
āyāS0
(ā1)l(y)[A Ī»yx!].
248 PENG SHAN
6.3. Proof of Theorem 0.1. Recall from (2.6) that we view Pn as a subset ofĪ+. By Corollary 2.14, Theorem 0.1 is a consequence of the following theorem.
Theorem 6.4. Let Ī», Ī¼ be partitions of n. Then for any negative integer Īŗ wehave
(6.5) dĪ»ā²Ī¼ā²(q) =āiļæ½0
[J iMĪŗ(Ī»)/Ji+1MĪŗ(Ī») : LĪŗ(Ī¼)]q
i.
Here dĪ»ā²Ī¼ā²(q) is the polynomial defined in the introduction with v = exp(2Ļi/Īŗ).
Proof. Let Ī½ ā Ī such that Ī½+Ļ is antidominant. We may assume that Ī¼, Ī» belongto the same orbit of Ī½ under the dot action of S; see (A.7). For any Ī¼ ā Ī+ā©(S Ā·Ī½)let w(Ī¼)Ī½ be the shortest element in the set
w(Ī¼)Ī½S(Ī½) = {w ā S |Ī¼ = w Ā· Ī½}.Note that w(Ī¼)Ī½S(Ī½) is contained in QS by Lemma 4.3. We fix v ā S such thatv ļæ½ w(Ī³)Ī½ for any Ī³ ā Pn. Let q1/2 be a formal variable. We identify q = (q1/2)2.
Let Px,w be the Kazhdan-Lusztig polynomial normalized as follows:
Px,w(q) = q(l(w)āl(x))/2Px,w(qā1/2).
Let Qx,w be the inverse Kazhdan-Lusztig polynomial given byāxāS
Qx,z(āq)Px,w(q) = Ī“z,w, z, w ā S.
Then by (6.1), (6.2) we have
[A Ī½x!] =
āwāS
q(l(x)āl(w))/2Qx,w(qā1/2)[A Ī½
w!ā¢], ā x ā S.
By Remark 6.3 we see that
(6.6) [BĪ½x!] =
āwāQS
( āsāS0
(ā1)l(s)q(l(sx)āl(w))/2Qsx,w(qā1/2)
)[BĪ½
w!ā¢], ā x ā QS.
Now, let
[MĪ½0(Xv)]q = [MĪ½
0(Xv)]āZ Z[q1/2, qā1/2], [OĪŗ]q = [OĪŗ]āZ Z[q1/2, qā1/2].
We have a Z[q, qā1]-module homomorphism
Īµ : [MHM0(Xv)] āā [MĪ½0(Xv)]q,
[M ] ļæ½āāāiāZ
[Ī·(W iM /W iā1M )]qi/2.
Note that Īµ([BĪ½x!ā¢]) = ql(x)/2[BĪ½
x!ā¢] and by Proposition 6.1 we have
Īµ([BĪ½x!]) =
āiāN
[J iBĪ½x!/J
i+1BĪ½x!]q
(l(x)āi)/2, x ā QS, x ļæ½ v.
Next, let
[MĪŗ(Ī»)]q =āiāN
[J iMĪŗ(Ī»)/Ji+1MĪŗ(Ī»)]q
āi/2.
Then by Proposition 5.5, we have
ĪĪµ([BĪ½w(Ī»)Ī½ !
]) = ql(w(Ī»)Ī½)/2[MĪŗ(Ī»)]q.
JANTZEN FILTRATION 249
On the other hand, by (6.6) we have
ĪĪµ([BĪ½w(Ī»)Ī½ !
])
=ā
wāQS
( āsāS0
(ā1)l(s)q(l(sw(Ī»)Ī½)āl(w))/2Qsw(Ī»)Ī½ ,w(āq1/2))ĪĪµ[BĪ½
w!ā¢]
=āĪ¼āPn
( āsāS0
(ā1)l(s)q(l(sw(Ī»)Ī½)āl(w(Ī¼)Ī½))/2Qsw(Ī»)Ī½ ,w(Ī¼)Ī½ (qā1/2)
)ql(w(Ī¼)Ī½)/2[LĪŗ(Ī¼)].
Here in the second equality we have used Proposition 4.6(c) and the fact that Pn isan ideal in Ī+. Note that l(sw(Ī»)Ī½) = l(w(Ī»)Ī½)ā l(s) for s ā S0 by Lemma 4.3(b).We deduce that
[MĪŗ(Ī»)]q =āĪ¼āPn
( āsāS0
(āqā1/2)l(s)Qsw(Ī»)Ī½ ,w(Ī¼)Ī½ (qā1/2)
)[LĪŗ(Ī¼)];
By [L, Proposition 5], we have
dĪ»ā²,Ī¼ā²(qā1/2) =āsāS0
(āqā1/2)l(s)Qsw(Ī»)Ī½ ,w(Ī¼)Ī½ (qā1/2),
see also the beginning of the proof of Proposition 6 in loc. cit., and [LT2, Lemma2.2] for instance. We deduce thatā
iāN
[J iMĪŗ(Ī»)/Ji+1MĪŗ(Ī»)]q
i =āĪ¼āPn
dĪ»ā²,Ī¼ā²(q)[LĪŗ(Ī¼)].
The theorem is proved. ļæ½Remark 6.5. The q-multiplicities of the Weyl modules Wv(Ī») have also been con-sidered in [Ar] and [RT]. Both papers are of combinatorial nature, and are verydifferent from the approach used here. In [Ar] Ariki defined a grading on the q-Schuralgebra and he proved that the q-multiplicities of the Weyl module with respect tothis grading is also given by the same polynomials dĪ»ā²,Ī¼ā² . However, it not clear tous how to relate this grading to the Jantzen filtration.
Remark 6.6. The radical filtration Cā¢(M) of an object M in an abelian category Cis given by putting C0(M) = M and Ci+1(M) to be the radical of Ci(M) for i ļæ½ 0.It follows from [BB, Lemma 5.2.2] and Proposition 5.5 that the Jantzen filtrationof B! coincides with the radical filtration. If Ī» ā Ī such that Ī»+ Ļ is antidominantand regular, then the exact functor Ī is faithful; see [BD, Theorem 7.15.6]. In thiscase, we have
Ī(Cā¢(B!)) = Cā¢(Ī(B!)) = Cā¢MĪŗ(Ī»).
So the Jantzen filtration on MĪŗ(Ī») coincides with the radical filtration. If we havefurther Ī» ā Pn and Īŗ ļæ½ ā3, then by the equivalence in Proposition 2.13 we deducethat the Jantzen filtration of Wv(Ī») also coincides with the radical filtration. Thisis compatible with recent result of Parshall-Scott [PS], where they computed theradical filtration ofWv(Ī») under the same assumption of regularity here but withoutassuming Īŗ ļæ½ ā3. We conjecture that for any Ī» the Jantzen filtration on MĪŗ(Ī»)coincides with the radical filtration.
Remark 6.7. The results of Sections 4, 5, 6 hold for any standard parabolic subgroupQ of G with the same proof. In particular, it allows us to calculate the graded de-composition matrices associated with the Jantzen filtration of the parabolic Vermamodules in more general cases.
250 PENG SHAN
Appendix A. Kashiwara-Tanisakiās construction, translation
functors and Proof of Proposition 4.2
The goal of this appendix is to prove Proposition 4.2. We first consider the casewhen Ī» + Ļ is regular. In this case, the result is essentially due to Kashiwara andTanisaki [KT1]. However, the setting of loc. cit. is slightly different from the oneused here. So we will first recall their construction and adapt it to our setting tocomplete the proof of the proposition in the regular case. Next, we give a geometricconstruction of the translation functor for the affine category O inspired from [BG],and apply it to deduce the result for singular blocks. We will use the same notationas in Section 4.
A.1. The Kashiwara affine flag variety. Recall that Ī is the root system of gand Ī + is the set of positive root. Write Ī ā = āĪ +. For Ī± ā Ī we write
gĪ± = {x ā g | [h, x] = Ī±(h)x, ā h ā t}.For any subset Ī„ of Ī +, Ī ā we set respectively
n(Ī„) =āĪ±āĪ„
gĪ±, nā(Ī„) =
āĪ±āĪ„
gĪ±.
For Ī± =āmā1
i=0 hiĪ±i ā Ī we write ht(Ī±) =āmā1
i=0 hi and for l ā N we set
Ī āl = {Ī± ā Ī ā | ht(Ī±) ļæ½ āl}, n
āl = nā(Ī ā
l ).
Consider the group scheme LāG0 = G0(C[[tā1]]). Let Bā be the preimage of Bā
0
by the map
LāG0 ā G0, tā1 ļæ½ā 0,
where Bā0 is the Borel subgroup of G0 opposite to B0. Let N
ā be the prounipotentradical of Bā. Let Nā
l ā Bā be the group subscheme given by
Nāl = limāā
k
exp(nāl /nāk ).
Let X be the Kashiwara affine flag variety; see [K]. It is a quotient schemeX = Gā/B, where Gā is a coherent scheme with a locally free left action ofBā and a locally free right action of B. The scheme X is coherent, prosmooth,nonquasi-compact, locally of countable type, with a left action of Bā. There is aright T -torsor
Ļ : Xā = Gā/N ā X.
For any subscheme Z of X let Zā be its preimage by Ļ. Let
X =āwāS
ā¦Xw.
be the Bā-orbit decomposition. Then X is covered by the following open sets
Xw =āvļæ½w
ā¦Xv.
For each w there is a canonical closed embedding Xw ā Xw. Moreover, for anyinteger l that is large enough, the group Nā
l acts locally freely on Xw, Xwā , thequotients
Xwl = Nā
l \Xw, Xwā l = Nā
l \Xwā
JANTZEN FILTRATION 251
are smooth schemes3, and the induced morphism
(A.1) Xw ā Xwl
is a closed immersion. See [KT1, Lemma 2.2.1]. Furthermore, we have
Xā w = Xw ĆXw
lX
wā l .
In particular, we get a closed embedding of Xā w ā Xw into the T -torsor Xwā
l ā Xwl .
This implies that the T -torsor Ļ : Xā ā X is admissible. Finally, let
pl1l2 : Xwā l1
ā Xwā l2
, pl : Xwā ā X
wā l , l1 ļæ½ l2
be the canonical projections. They are affine morphisms.
A.2. The category HĪ»(X). Fix w, y ā S with y ļæ½ w. For l1 ļæ½ l2 large enough,the functor
(pl1l2)ā¢ : MĪ»h(X
yl1, Xw) ā MĪ»
h(Xyl2, Xw)
yields a filtering projective system of categories, and we set
HĪ»(Xy, Xw) = 2limāāl
MĪ»h(X
yl , Xw).
For z ļæ½ y let jyz : Xyā ā Xzā be the canonical open embedding. It yields a map
jyz : Xyā l ā X
zā l for each l. The pull-back functors by these maps yield, by base
change, a morphism of projective systems of categories
(MĪ»h(X
zl , Xw))l ā (MĪ»
h(Xyl , Xw))l.
Hence we get a map
HĪ»(Xz, Xw) ā HĪ»(Xy, Xw).
As y, z varies these maps yield again a projective system of categories and we set
HĪ»(Xw) = 2 limāāyļæ½w
HĪ»(Xy, Xw).
Finally, for w ļæ½ v the categoryHĪ»(Xw) is canonically a full subcategory ofHĪ»(Xv).We define
HĪ»(X) = 2limāāw
HĪ»(Xw).
This definition is inspired from [KT1], where the authors considered the categories
MĪ»h(X
yl , Xw) instead of the categories MĪ»
h(Xyl , Xw). Finally, note that since the
category HĪ»(Xw) is equivalent to MĪ»h(X
yl , Xw) for y, l large enough, and since
the latter is equivalent to MĪ»h(Xw) (see Section 3.1), we have an equivalence of
categories
HĪ»(X) ā¼= MĪ»h(X).
3For l large enough the scheme Xwl is separated (hence quasi-separated). To see this, one uses
the fact that Xw is separated and applies [TT, Proposition C.7].
252 PENG SHAN
A.3. The functors Ī and Ī. For an object M of HĪ»(X), there exists w ā S
such that M is an object of the subcategory HĪ»(Xw). Thus M is represented by a
system (M yl )yļæ½w,l, with M y
l ā MĪ»h(X
yl , Xw) and l large enough. For l1 ļæ½ l2 there
is a canonical map
(pl1l2)ā(Myl1) ā (pl1l2)ā¢(M
yl1) = M y
l2.
It yields a map (see (3.6) for the notation)
Ī(Xyl1,M y
l1) ā Ī(Xy
l2,M y
l2).
Next, for y, z ļæ½ w and l large enough, we have a canonical isomorphism
Ī(Xyl ,M
yl ) = Ī(Xz
l ,Mzl ).
Following [KT1], we choose a y ļæ½ w and we set
Ī(M ) = limāāl
Ī(Xyl ,M
yl ).
This definition does not depend on the choice of w, y. Now, regard M as an
object of MĪ»h(X). Recall the object M ā ā O(X) from Section 3.4. Suppose that
M ā is represented by a system (M Oy )yļæ½w with M O
y ā O(Xy). By definition we
have M Oy = (i!M y
l )ā , where i denotes the closed embedding X
ā y ā X
yā l ; see (A.1).
Therefore we have
Ī(Xy,MOy ) = Ī(Xy, i
!(M yl )
ā )
ā Ī(Xyl ,M
yl ).(A.2)
Next, recall that we have
Ī(M ) = Ī(X,M ā ) = limāāy
Ī(Xy,MOy ).
So by first taking the projective limit on the right hand side of (A.2) with respectto l and then taking the inductive limit on the left hand side with respect to y weget an inclusion
Ī(M ) ā Ī(M ).
It identifies Ī(M ) with the subset of Ī(M ) consisting of the sections supported onsubschemes (of finite type) of X.
The vector space Ī(M ) has a g-action; see [KT1, Section 2.3]. The vector spaceĪ(M ) has also a g-action by Section 4.3. The inclusion is compatible with theseg-actions. Following loc. cit., let
Ī(M ) ā Ī(M )
be the set of t-finite elements. It is a g-submodule of Ī(M ).
A.4. The regular case. In this subsection we prove Proposition 4.2 in the regularcase. More precisely, we prove the following result.
Proposition A.1. Let Ī» ā Ī be such that Ī»+Ļ is antidominant and regular. Thenwe have isomorphisms of g-modules
(A.3) Ī(A Ī»v!) = NĪŗ(vĀ·Ī»), Ī(A Ī»
vā¢) = DNĪŗ(vĀ·Ī»), Ī(A Ī»v!ā¢) = LĪŗ(vĀ·Ī»), ā v ā S.
JANTZEN FILTRATION 253
Proof. By [KT1, Theorem 3.4.1] under the assumption of the proposition we haveisomorphisms of g-modules.
Ī(A Ī»v!) = NĪŗ(v Ā· Ī»), Ī(A Ī»
vā¢) = DNĪŗ(v Ā· Ī»), Ī(A Ī»v!ā¢) = LĪŗ(v Ā· Ī»), āv ā S.
We must check that for ļæ½ =!, ā¢, or !ā¢, the g-submodules Ī(A Ī»v ) and Ī(A Ī»
v ) of
Ī(A Ī»v ) are equal. Let us first prove this for ļæ½ = ā¢. We will do this in several steps.
Step 1. Following [KT1] we first define a particular section Ļ in Ī(A Ī»vā¢). Let Ļ be
a nowhere vanishing section of Ī©Xv. It is unique up to a nonzero scalar. Let tĪ»
be the nowhere vanishing section of L Ī»Xv
such that tĪ»(uvb) = eāĪ»(b) for u ā N ,
b ā B. Then Ļ ā tĪ» is a nowhere vanishing section of A Ī»,ā v over Xv. Now, for
y ļæ½ v and l large enough, let ivl : Xv ā Xyl be the composition of the locally
closed embedding Xv ā Xy and the closed embedding Xy ā Xyl in (A.1). We will
denote the corresponding embedding Xā v ā X
yā l again by ivl . Note that
(ivlā¢(A
Ī»v )
)l
represents the object A Ī»vā¢ in HĪ»(X). Therefore we have
Ī(A Ī»vā¢) = limāā
l
Ī(Xyl , i
vlā¢(A
Ī»v )).
Consider the canonical inclusion of OXyl-modules
ivlā(AĪ»v )ā āā ivlā¢(A
Ī»v )ā .
Let Ļl ā Ī(Xyl , i
vlā¢(A
Ī»v )ā ) be the image of Ļ ā tĪ» under this map. The family (Ļl)
defines an element
Ļ ā Ī(A Ī»vā¢).
Step 2. Let V y = yBāĀ·B/B. It is an affine open set in Xy. For l large enough, let V yl
be the image of V y in Xyl via the canonical projection Xy ā X
yl . Write jyl : V y
l ā Xyl
for the inclusion. Note that V ylā¼= Nā/Nā
l as affine spaces. Therefore, if l is large
enough such that Ī āl ā Ī ā ā© vĪ ā, then the right Dā
Xv-module structure on A Ī»,ā
v
yields an isomorphism of sheaves of C-vector spaces over V yl ,
jyāl (ivlāOXv)ā U(nā(Ī ā ā© v(Ī ā))/nāl )
ā¼ā jyāl(ivlā¢(A
Ī»v )ā
),
f ā p ļæ½ā (Ļ Ā· f) Ā· Ī“l(p).This yields an isomorphism of t-modules
(A.4) Ī(Xyl , i
vlā¢(A
Ī»v )) = U(nā/nāl )ā CvĀ·Ī»;
see [KT1, Lemma 3.2.1]. By consequence we have an isomorphism of t-modules
(A.5) Ī(A Ī»vā¢) = (limāā
l
U(nā/nāl ))ā CvĀ·Ī».
Step 3. Now, let us prove Ī(A Ī»vā¢) = Ī(A Ī»
vā¢). First, by (A.4) the spaceĪ(Xy
l , ivlā¢(A
Ī»v )) is t-locally finite. So (A.2) implies that Ī(A Ī»
vā¢) is the inductivelimit of a system of t-locally finite submodules. Therefore it is itself t-locally finite.Hence we have
Ī(A Ī»vā¢) ā Ī(A Ī»
vā¢).
To see that this is indeed an equality, note that if m ā Ī(A Ī»vā¢) is not t-locally finite,
then by (A.5) the section m is represented by an element in
limāāl
U(nā/nāl )ā CvĀ·Ī»
254 PENG SHAN
which does not come from U(nā)ā CvĀ·Ī» via the obvious map. Then one sees thatm cannot be supported on a finite dimensional scheme, i.e., it cannot belong toĪ(A Ī»
vā¢). This proves that
Ī(A Ī»vā¢) = Ī(A Ī»
vā¢).
Now, we can prove the other two equalities in the proposition. Since Ī» + Ļ isantidominant, by Proposition 4.1(a) the functor Ī is exact on MĪ»(X). So Ī(A Ī»
v!ā¢)is a g-submodule of Ī(A Ī»
vā¢). Therefore all the elements in Ī(A Ī»v!ā¢) are t-finite, i.e.,
we have
Ī(A Ī»v!ā¢) ā Ī(A Ī»
v!ā¢).
On the other hand, by [KT1, Theorem 3.4.1] we have
Ī(A Ī»v!ā¢) ā Ī(A Ī»
vā¢).
Therefore, Step 3 yields that each section in Ī(A Ī»v!ā¢) is supported on a finite di-
mensional scheme, and hence belongs to Ī(A Ī»v!ā¢). We deduce that
(A.6) Ī(A Ī»v!ā¢) = Ī(A Ī»
v!ā¢).
Finally, since A Ī»v! has a finite composition series whose constituents are given by
A Ī»w!ā¢ for w ļæ½ v. Since both Ī and Ī are exact functors on MĪ»
0 (X); see Proposition4.1 and [KT1, Corollary 3.3.3, Theorem 3.4.1]. We deduce from (A.6) that Ī(A Ī»
v!)
is t-locally finite, and the sections of Ī(A Ī»v!) are supported on finite dimensional
subschemes. Therefore we have
Ī(A Ī»v!) = Ī(A Ī»
v!).
The proposition is proved. ļæ½
A.5. Translation functors. In order to compute the images of A Ī»v! and A Ī»
vā¢ inthe case when Ī» + Ļ is not regular, we need the translation functors. For Ī» ā Īŗt
ā
such that Ī»+ Ļ is anti-dominant, we define OĪŗ,Ī» to be the Serre subcategory of OĪŗ
generated by LĪŗ(w Ā·Ī») for all w ā S. The same argument as in the proof of [DGK,
Theorem 4.2] yields that each M ā OĪŗ admits a decomposition
(A.7) M =ā
MĪ», MĪ» ā OĪŗ,Ī»,
where Ī» runs over all the weights in Īŗtā such that Ī» + Ļ is antidominant. The
projection
prĪ» : OĪŗ ā OĪŗ,Ī», M ļæ½ā MĪ»,
is an exact functor. Fix two integral weights Ī», Ī¼ in tā such that Ī» + Ļ, Ī¼ + Ļ areantidominant and the integral weight Ī½ = Ī»ā Ī¼ is dominant. Assume that Ī» ā Īŗt
ā,then Ī¼ belongs to Īŗā²tā for an integer Īŗā² < Īŗ. Let V (Ī½) be the simple g-module of
highest weight Ī½. Then for any M ā OĪŗā² the module M ā V (Ī½) belongs to OĪŗ.Therefore we can define the following translation functor:
ĪøĪ½ : OĪŗā²,Ī¼ ā OĪŗ,Ī», M ļæ½ā prĪ»(M ā V (Ī½));
see [Ku1]. Note that the subcategory OĪŗ,Ī» of O is stable under the duality D,because D fixes simple modules. We have a canonical isomorphism of functors
(A.8) ĪøĪ½ ā¦D = D ā¦ ĪøĪ½ .Indeed, it follows from (4.5) that D(M ā V (Ī½)) = D(M)āD(V (Ī½)) as g-modules.Since V (Ī½) is simple, we have DV (Ī½) = V (Ī½). The equality (A.8) follows.
JANTZEN FILTRATION 255
On the geometric side, recall the T -torsor Ļ : Xā ā X. For any integral weightĪ» ā tā the family of line bundles L Ī»
Xw(see Section 3.5) with w ā S form a projective
system of O-modules under restriction, yielding a flat object L Ī» of O(X). Notethat Ļā(L Ī») is a line bundle on Xā . For integral weights Ī», Ī¼ in tā the translationfunctor
ĪĪ»āĪ¼ : MĪ¼0 (X) ā MĪ»
0 (X), M ļæ½ā M āOXā Ļā(L Ī»āĪ¼),
is an equivalence of categories. A quasi-inverse is given by ĪĪ¼āĪ». By the projectionformula we have
(A.9) ĪĪ»āĪ¼(AĪ¼w ) = AĪ»
w , for ļæ½ =!, !ā¢, ā¢.Now, assume that Ī¼+ Ļ is antidominant. Consider the exact functor
Ī : MĪ¼(X) ā M(g), M ļæ½ā Ī(M )
as in Proposition 4.1. Note that if Ī¼+ Ļ is regular, then Ī maps A Ī¼v!ā¢ to LĪŗ(v Ā· Ī¼)
by Proposition A.1. Since the subcategory OĪŗ of M(g) is stable under extension,the exact functor Ī restricts to a functor
Ī : MĪ¼0 (X) ā OĪŗ,Ī¼.
The next proposition is an affine analogue of [BG, Proposition 2.8].
Proposition A.2. Let Ī», Ī¼ be integral weights in tā such that Ī» + Ļ, Ī¼ + Ļ areantidominant and Ī½ = Ī» ā Ī¼ is dominant. Assume further that Ī¼ + Ļ is regular.Then the functors
ĪøĪ½ ā¦ Ī : MĪ¼0 (X) ā OĪŗ,Ī» ā M(g) and Ī ā¦ĪĪ½ : MĪ¼
0 (X) ā M(g)
are isomorphic.
Proof. We will prove the proposition in several steps.
Step 1. First, we define a category Sh(X) of sheaves of C-vector spaces on X andwe consider g-modules in this category. To do this, for w ā S let Sh(Xw) bethe category of sheaves of C-vector spaces on Xw. For w ļæ½ x we have a closedembedding iw,x : Xw ā Xx, and an exact functor
i!w,x : Sh(Xx) ā Sh(Xw), F ļæ½ā i!w,x(F ),
where i!w,x(F ) is the subsheaf of F consisting of the local sections supported set-
theoretically on Xw. We get a projective system of categories
(Sh(Xw), i!w,x).
Following [BD, 7.15.10] we define the category of sheaves of C-vector spaces on Xto be the projective limit
Sh(X) = 2limāāSh(Xw).
This is an abelian category. By the same arguments as in the second paragraph ofSection 3.7, the category Sh(Xw) is canonically identified with a full subcategoryof Sh(X), and each object F ā Sh(X) is a direct limit
F = limāāFw, Fw ā Sh(Xw).
The space of global sections of an object of Sh(X) is given by
Ī(X,F ) = limāāĪ(Xw,Fw).
256 PENG SHAN
Next, consider the forgetful functor
O(Xw) ā Sh(Xw), N ļæ½ā N C.
Recall that for M ā O(X) we have M = limāāMw with Mw ā O(Xw). The tupleof sheaves of C-vector spaces
limāāxļæ½w
i!w,x(MCx ), w ā S,
gives an object in Sh(X). Let us denote it by M C. The assignment M ļæ½ā M C
yields a faithful exact functor
O(X) ā Sh(X)
such that
(A.10) Ī(X,M ) = Ī(X,M C);
see (3.12) for the definition of the left hand side. Now, let F = (Fw) be an objectin Sh(X). The vector spaces End(Fw) form a projective system via the maps
End(Fx) ā End(Fw), f ļæ½ā i!w,x(f).
We set
(A.11) End(F ) = limāāw
End(Fw).
We say that an object F of Sh(X) is a g-module if it is equipped with an algebrahomomorphism
U(g) ā End(F ).
For instance, for M ā MT (Xā ) the object (M ā )C of Sh(X) is a g-module via thealgebra homomorphism
(A.12) Ī“l : U(g) ā Ī(Xā ,DXā ).
See the beginning of Section 4.3.
Step 2. Next, we define G-modules in O(X). A standard parabolic subgroup of Gis a group scheme of the form P = Q ĆG0
P0 with P0 a parabolic subgroup of G0.Here the morphism Q ā G0 is the canonical one. We fix a subposet PS ā S suchthat for w ā PS the subscheme Xw ā X is stable under the P -action and
X = limāāwāPS
Xw.
We say that an object F = (Fw) of O(X) has an algebraic P -action if Fw hasthe structure of a P -equivariant quasicoherent OXw
-module for w ā PS and if theisomorphism iāw,xFx
ā¼= Fw is P -equivariant for w ļæ½ x. Finally, we say that F is aG-module if it is equipped with an action of the (abstract) group G such that forany standard parabolic subgroup P , the P -action on F is algebraic.
We are interested in a family of G-modules V i in O(X) defined as follows. Fix abasis (mi)iāN of V (Ī½) such that each mi is a weight vector of weight Ī½i and Ī½j > Ī½iimplies j < i. By assumption we have Ī½0 = Ī½. For each i let V i be the subspace ofV (Ī½) spanned by the vectors mj for j ļæ½ i. Then
V 0 ā V 1 ā V 2 ā Ā· Ā· Ā·
JANTZEN FILTRATION 257
is a sequence of B-submodule of V (Ī½). We write V ā = V (Ī½). For 0 ļæ½ i ļæ½ ā wedefine a OX-module V i
X on X such that for any open set U ā X we have
Ī(U,V iX) = {f : pā1(U) ā V i | f(gbā1) = bf(g), g ā Gā, b ā B},
where p : Gā ā X is the quotient map. Let V iw be the restriction of V i
X to Xw.
Then (V iw)wāS is a flat G-module in O(X). We will denote it by V i. Note that
since V (Ī½) admits a G-action, the G-module V ā ā O(X) is isomorphic to theG-module OX ā V (Ī½) with G acting diagonally. Therefore, for M ā MĪ¼
0 (X) theprojection formula yields a canonical isomorphism of vector spaces
Ī(M )ā V (Ī½) = Ī(X,M ā )ā V (Ī½)
= Ī(X,M ā āOXV ā).(A.13)
On the other hand, we have
Ī(ĪĪ½(M )) = Ī(X, (M āO
Xā Ļā(L Ī½))ā )
= Ī(X,M ā āOXL Ī½).(A.14)
Our goal is to compare the g-modules Ī(ĪĪ½(M )) and the direct factor ĪøĪ½(Ī(M ))of Ī(M )āV (Ī½). To this end, we first define in Step 3 a g-action on (M ā āOX
V i)C
for each i, then we prove in Steps 4-6 that the inclusion
(A.15) (M ā āOXL Ī½)C ā (M ā āOX
V ā)C
induced by the inclusion L Ī½ = V 0 ā V ā splits as a g-module homomorphism inSh(X).
Step 3. Let P be a standard parabolic subgroup of G, and let p be its Lie al-gebra. Let PS ā S be as in Step 2. The P -action on V i yields a Lie algebrahomomorphism
p ā End(V iw), ā w ā PS.
Consider the g-action on (M ā )C given by the map Ī“l in (A.12). Note that for w ļæ½ xin PS, any element Ī¾ ā p maps a local section of M ā
x supported on Xw to a localsection of M ā
x with the same property. In particular, for w ā PS we have a Liealgebra homomorphism
(A.16) p ā End((M ā
w āOXwV iw)
C), Ī¾ ļæ½ā
(mā v ļæ½ā Ī¾mā v +mā Ī¾v
),
where m denotes a local section of M ā w, v denotes a local section of V i
w. Thesemaps are compatible with the restriction
End((M ā
x āOXxV ix )
C)ā End
((M ā
w āOXwV iw)
C), f ļæ½ā i!w,x(f).
They yield a Lie algebra homomorphism
p ā End((M ā āOX
V i)C).
As P varies, these maps glue together yielding a Lie algebra homomorphism
(A.17) g ā End((M ā āOX
V i)C).
This defines a g-action on (M ā āOXV i)C such that the obvious inclusions
(M ā āOXV 0)C ā (M ā āOX
V 1)C ā Ā· Ā· Ā·are g-equivariant. So (A.15) is a g-module homomorphism. Note that the flatnessof V i yields an isomorphism in O(X):
(A.18) M ā āOXV i/M ā āOX
V iā1 ā¼= M ā āOXL Ī½i .
258 PENG SHAN
Step 4. In order to show that the g-module homomorphism (A.15) splits, we con-sider the generalized Casimir operator of g. Identify t and tā via the pairing ćā¢ : ā¢ć.Let ĻāØ ā t be the image of Ļ. Let hi be a basis of t0, and let hi be its dual basisin t0 with respect to the pairing ćā¢ : ā¢ć. For Ī¾ ā g0 and n ā Z we will abbreviateĪ¾(n) = Ī¾ā tn and Ī¾ = Ī¾(0). The generalized Casimir operator is given by the formalsum
(A.19) C = 2ĻāØ+āi
hihi+2ā1+āi<j
ejieij+ānļæ½1
āi ļæ½=j
e(ān)ij e
(n)ji +
ānļæ½1
āi
hi,(ān)h(n)i ;
see e.g., [Ka, Section 2.5]. Let Ī“l(C) be the formal sum given by applying Ī“l term byterm to the right hand side of (A.19). We claim that Ī“l(C) is a well defined elementin Ī(Xā ,DXā ), i.e., the sum is finite at each point of Xā . More precisely, let
Ī£ = {eij | i < j} āŖ {e(n)ji , h(n)i | i = j, n ļæ½ 1},
and let e be the base point of Xā . We need to prove that the sets
Ī£g = {Ī¾ ā Ī£ | Ī“l(Ī¾)(ge) = 0}, g ā G,
are finite. To show this, consider the adjoint action of G on g
Ad : G ā End(g), g ļæ½ā Adg,
and the G-action on DXā ā O(Xā ) induced by the G-action on Xā . The map Ī“l isG-equivariant with respect to these actions. So for Ī¾ ā g and g ā G we have
Ī“l(Ī¾)(ge) = 0 āā Ī“l(Adgā1(Ī¾))(e) = 0.
Furthermore, the right hand side holds if and only if Adgā1(Ī¾) /ā n. Therefore
Ī£g = {Ī¾ ā Ī£ | Adgā1(Ī¾) /ā n}is a finite set, the claim is proved. By consequence C acts on the g-module (M ā )C
for any M ā MT (Xā ). Next, we claim that the action of C on the g-module(M ā āOX
V i)C is also well defined. It is enough to prove this for (M āOXV ā)C.
By (A.16) the action of C on (M āOXV ā)C is given by the operator
Cā 1 + 1ā Cāā
nāZ,i ļæ½=j
e(ān)ij ā e
(n)ji ā
ānāZ,i
hi,(ān) ā h(n)i .
Since for both M ā and V ā, at each point, there are only finitely many elementsfrom Ī£ which act nontrivially on it, the action of C on the tensor product is welldefined.
Step 5. Now, let us calculate the action of C on (M ā āOXL Ī½i)C. We have
Adgā1(C) = C, ā g ā G.
Therefore the global section Ī“l(C) is G-invariant and its value at e is
Ī“l(C)(e) = Ī“l(2ĻāØ +
āi
hihi + 2ā1)(e).
On the other hand, the right T -action on Xā yields a map
Ī“r : t ā Ī(Xā ,DXā ).
Since the right T -action commutes with the left G-action, for any h ā t the globalsection Ī“r(h) is G-invariant. We have Ī“r(h)(e) = āĪ“l(h)(e) because the left andright T -actions on the point e are inverse to each other. Therefore the global
JANTZEN FILTRATION 259
sections Ī“l(C) and Ī“r(ā2ĻāØ +ā
i hihi + 2ā1) takes the same value at the point e.
Since both of them are G-invariant, we deduce that
Ī“l(C) = Ī“r(ā2ĻāØ +āi
hihi + 2ā1).
Recall from Section 3.4 that for Ī» ā tā and M ā MĪ»(X) the operator Ī“r(ā2ĻāØ +āi h
ihi + 2ā1) acts on M ā by the scalar
āĪ»(ā2ĻāØ +āi
hihi + 2ā1) = ||Ī»+ Ļ||2 ā ||Ļ||2.
Therefore C acts on M ā by the same scalar. In particular, for M ā MĪ¼(X)and i ā N, the element C acts on M ā āOX
L Ī½i by ||Ī¼ + Ī½i + Ļ||2 ā ||Ļ||2. Notethat the isomorphism (A.18) is compatible with the g-actions. So C also acts by||Ī¼+ Ī½i + Ļ||2 ā ||Ļ||2 on (M ā āOX
V i/M ā āOXV iā1)C.
Step 6. Now, we can complete the proof of the proposition. First, we claim that
(A.20) ||Ī»+ Ļ||2 ā ||Ļ||2 = ||Ī¼+ Ī½i + Ļ||2 ā ||Ļ||2 āā Ī½i = Ī½.
The āifā part is trivial. For the āonly ifā part, we have by assumption
||Ī¼+ Ī½ + Ļ||2 = ||Ī¼+ Ī½i + Ļ||2
= ||Ī¼+ Ī½ + Ļ||2 + ||Ī½ ā Ī½i||2 ā 2ćĪ¼+ Ī½ + Ļ : Ī½ ā Ī½ić.Since Ī½āĪ½i ā NĪ + and Ī¼+Ī½+Ļ = Ī»+Ļ is antidominant, the term ā2ćĪ»+ Ļ : Ī½ ā Ī½ićis positive. Hence the equality implies that ||Ī½ ā Ī½i||2 = 0. So Ī½ ā Ī½i belongs to NĪ“.But ćĪ»+ Ļ : Ī“ć = Īŗ < 0. So we have Ī½ = Ī½i. This proves the claim in (A.20). Adirect consequence of this claim and of Step 5 is that the g-module monomorphism(A.15) splits. It induces an isomorphism of g-modules
(A.21) Ī(M ā āOXL Ī½) = prĪ» Ī(M
ā āOXV ā), M ā MĪ¼
0 (X).
Finally, note that the vector spaces isomorphisms (A.13) and (A.14) are indeedisomorphisms of g-modules by the definition of the g-actions on (M ā āOX
V ā)C
and (M ā āOXL Ī½)C. Therefore (A.21) yields an isomorphism of g-modules
Ī(ĪĪ½(M )) = ĪøĪ½(Ī(M )). ļæ½Remark A.3. We have assumed Ī¼ + Ļ regular in Proposition A.2 in order to haveĪ(MĪ¼
0 (X)) ā OĪŗ,Ī¼. It follows from Proposition A.2 that this inclusion still holdsif Ī¼ + Ļ is not regular. So Proposition A.2 makes sense without this regularityassumption, and the proof is the same in this case.
A.6. Proof of Proposition 4.2. By Proposition A.1 it is enough to prove theproposition in the case when Ī» + Ļ is not regular. Let Ļi, 0 ļæ½ i ļæ½ m ā 1, be thefundamental weights in tā. Let
Ī½ =ā
Ļi,
where the sum runs over all i = 0, . . . ,m ā 1 such that ćĪ»+ Ļ : Ī±ić = 0. Theweight Ī½ is dominant. Let Ī¼ = Ī»ā Ī½. Then Ī¼+ Ļ is an antidominant weight. It is,moreover, regular, because we have
ćĪ¼+ Ļ : Ī±ić = ćĪ»+ Ļ : Ī±ić ā ćĪ½ : Ī±ić < 0, 0 ļæ½ i ļæ½ mā 1.
Let Īŗā² = ćĪ¼+ Ļ : Ī“ć. So Propositions A.1, A.2 and the equation (A.9) imply that
Ī(A Ī»w!) = ĪøĪ½(NĪŗā²(w Ā·Ī¼)), Ī(A Ī»
w!ā¢) = ĪøĪ½(LĪŗā²(w Ā·Ī¼)), Ī(A Ī»wā¢) = ĪøĪ½(DNĪŗā²(w Ā·Ī¼)).
260 PENG SHAN
So parts (a) and (c) follow from the properties of the translation functor ĪøĪ½ givenin [Ku1, Proposition 1.7]. Part (b) follows from (a) and the equality (A.8). ļæ½
Appendix B. Proof of Proposition 5.4
In this appendix, we prove Proposition 5.4. We will first study localization ofdeformed Verma modules; see Lemmas B.2 and B.3. Then we use them to deducethe parabolic version. In this appendix we will keep the notation of Section 5. Inparticular, recall that j = jw, f = fw, etc.
B.1. Deformed Verma modules. Fix Ī» ā Ī and w ā QS. Let n > 0 be aninteger. For v = xw ā S with x ā S0, let rx : Xā
v ā Y ā w be the canonical inclusion.
We have iv = j ā¦ rx. The tensor product
(B.1) A (n)v = A Ī»
v āOX
ā v
rāxfāI (n)
is equal to rāx(B(n)). By Lemma 5.3 it is an object of MĪ»
h(Xv). We consider the
following objects in MĪ»h(X):
A(n)v! = iv!(A
(n)v ), A
(n)v!ā¢ = iv!ā¢(A
(n)v ), A
(n)vā¢ = ivā¢(A
(n)v ).
For Ī¼ ā Īŗtā we have defined the Verma module NĪŗ(Ī¼) in Section 4.4. The
deformed Verma module is the Uk-module given by
Nk(Ī¼) = U(g)āU(b) RĪ¼+sĻ0.
Here the b-module RĪ¼+sĻ0is a rank one R-module over which t acts by Ī¼ + sĻ0,
and n acts trivially. The deformed dual Verma module is
DNk(Ī¼) =āĪ»ākt
ā
HomR(Nk(Ī¼)Ī», R),
see (2.5). We will abbreviate
N(n)k (Ī¼) = Nk(Ī¼)(ā
n), DN(n)k (Ī¼) = DNk(Ī¼)(ā
n).
For any R(n)-module (resp. R-module) M let Ī¼(si) : M ā M be the multiplicationby si and write siM for the image of Ī¼(si). We define a filtration
F ā¢M = (F 0M ā F 1M ā F 2M ā . . .)
on M by putting F iM = siM. We say that it is of length n if FnM = 0 andFnā1M = 0. We set
grM =āiļæ½0
gri M, gri M = F iM/F i+1M.
For any gR(n)-module M let ch(M) be the tR(n)-module image of M by the forgetfulfunctor.
Lemma B.1. If Ī» + Ļ is antidominant, then we have an isomorphism of tR(n)-modules
ch(Ī(A(n)vā¢ )) = ch(DN
(n)k (v Ā· Ī»)).
Proof. The proof is very similar to the proof of Proposition A.1. We will use thenotation introduced there.
JANTZEN FILTRATION 261
Step 1. Consider the nowhere vanishing section fs of (fāI (n))ā over Yw. Its re-striction to Xv yields an isomorphism
(A (n)v )ā ā¼= Ī©Xv
āOXvL Ī»
XvāR(n).
Let Ļ be a nowhere vanishing section of Ī©Xv, and let tĪ» be the nowhere vanishing
section of L Ī»Xv
over Xv such that tĪ»(uvb) = eāĪ»(b) for u ā N , b ā B. Then the
global section Ļ ā tĪ» ā fs of A(n)v defines an element
Ļs ā Ī(A(n)vā¢ )
in the same way as Ļ is defined in the first step of the proof of Proposition A.1.
Step 2. Let us show that
ch(Ī(A(n)vā¢ )) = ch(DN
(n)k (v Ā· Ī»)).
The proof is the same as in the second step of the proof of Proposition A.1. The
right Dā Xv
-module structure on (A(n)v )ā yields an isomorphism of sheaves of C-vector
spaces over V yl ,
jyāl (ivlāOXv)ā U(nā(Ī ā ā© v(Ī ā))/nāl )āR(n) ā¼ā jyāl
(ivlā¢(A
(n)v )ā
),
f ā pā r ļæ½ā((Ļs Ā· f) Ā· Ī“l(p)
)r.
This yields an isomorphism of tR(n) -modules
ch(Ī(Xy
l , ivlā¢(A
(n)v ))
)= ch
(U(nā/nāl )āR
(n)vĀ·Ī»+sĻ0
).
Therefore we have
(B.2) ch(Ī(A(n)vā¢ )) = ch
((limāā
l
U(nā/nāl ))āR(n)vĀ·Ī»+sĻ0
)and
ch(Ī(A(n)vā¢ )) = ch
(U(nā)āR
(n)vĀ·Ī»+sĻ0
)= ch(DN
(n)k (v Ā· Ī»)).(B.3)
Step 3. In this step, we prove that Ī(A(n)vā¢ ) = Ī(A
(n)vā¢ ) as gR(n)-modules. Since
both of them are gR(n)-submodules of Ī(A(n)vā¢ ). It is enough to prove that they are
equal as vector spaces. Consider the filtration F ā¢(A(n)v ) on A
(n)v . It is a filtration
in MĪ»(Xv) of length n and
gri(A (n)v ) = A Ī»
v , 0 ļæ½ i ļæ½ nā 1.
Since ivā¢ is exact and
RiĪ(A Ī»vā¢) = RiĪ(A Ī»
vā¢) = 0, ā i > 0,
the functor Ī ā¦ ivā¢ commutes with the filtration. Therefore both the filtrations
F ā¢Ī(A(n)vā¢ ) and F ā¢Ī(A
(n)vā¢ ) have length n and
gri Ī(A(n)vā¢ ) = Ī(A Ī»
vā¢), gri Ī(A(n)vā¢ ) = Ī(A Ī»
vā¢), 0 ļæ½ i ļæ½ nā 1.
262 PENG SHAN
By Step 3 of the proof of Proposition A.1, we have Ī(A Ī»v ) = Ī(A Ī»
v ). We deduce
that all the sections in Ī(A(n)vā¢ ) are t-finite and all the sections in Ī(A
(n)vā¢ ) are
supported on finite dimensional subschemes. This proves that
Ī(A(n)vā¢ ) = Ī(A
(n)vā¢ ).
We are done by Step 2. ļæ½Lemma B.2. If Ī»+ Ļ is antidominant there is an isomorphism of gR(n)-modules
Ī(A(n)vā¢ ) = DN
(n)k (v Ā· Ī»).
Proof. Note that
DN(n)k (v Ā· Ī») =
āĪ¼ākt
ā
HomR(Nk(v Ā· Ī»)Ī¼, R)(ān)
=āĪ¼ākt
ā
HomR(n)(N(n)k (v Ā· Ī»)Ī¼, R(n)).
For Ī¼ ā ktā let Ī(A
(n)vā¢ )Ī¼ be the weight space as defined in (2.2). By Lemma B.1
we haveĪ(A
(n)vā¢ ) =
āĪ¼ākt
ā
Ī(A(n)vā¢ )Ī¼,
because the same equality holds for DN(n)k (v Ā·Ī»). So we can consider the following
gR(n)-module
(B.4) DĪ(A(n)vā¢ ) =
āĪ¼ākt
ā
HomR(n)(Ī(A(n)vā¢ )Ī¼, R
(n)).
It is enough to prove that we have an isomorphism of gR(n)-modules
DĪ(A(n)vā¢ ) = N
(n)k (v Ā· Ī»).
By (B.4) we have
(B.5) ch(DĪ(A
(n)vā¢ )
)= ch
(Ī(A
(n)vā¢ )
).
Together with Lemma B.1, this yields an isomorphism of R(n)-modules
N(n)k (v Ā· Ī»)vĀ·Ī»+sĻ0
=(DĪ(A
(n)vā¢ )
)vĀ·Ī»+sĻ0
.
By the universal property of Verma modules, such an isomorphism induces a mor-phism of gR(n)-module
Ļ : N(n)k (v Ā· Ī») ā DĪ(A
(n)vā¢ ).
We claim that for each Ī¼ ā ktā the R(n)-module morphism
ĻĪ¼ : N(n)k (v Ā· Ī»)Ī¼ ā
(DĪ(A
(n)vā¢ )
)Ī¼
given by the restriction of Ļ is invertible. Indeed, by Lemma B.1 and (B.5), wehave
ch(DĪ(A(n)vā¢ )) = ch(DN
(n)k (v Ā· Ī»)) = ch(N
(n)k (v Ā· Ī»)).
SoN
(n)k (v Ā· Ī»)Ī¼ =
(DĪ(A
(n)vā¢ )
)Ī¼
as R(n)-modules. On the other hand, Proposition 4.2 yields that the map
Ļ(ā) = ĻāR(n) (R(n)/āR(n)) : NĪŗ(v Ā· Ī») ā DĪ(A Ī»vā¢)
JANTZEN FILTRATION 263
is an isomorphism of g-modules. So ĻĪ¼(ā) is also an isomorphism. Since the
R(n)-modules N(n)k (v Ā· Ī»)Ī¼ and
(DĪ(A
(n)vā¢ )
)Ī¼are finitely generated, Nakayamaās
lemma implies that ĻĪ¼ is an isomorphism. So Ļ is an isomorphism. The lemma isproved. ļæ½
Lemma B.3. If Ī»+ Ļ is antidominant and v is a shortest element in vS(Ī»), thenthere is an isomorphism of gR(n)-modules
Ī(A(n)v! ) = N
(n)k (v Ā· Ī»).
Proof. We abbreviate Ī½ = v Ā· Ī». The lemma will be proved in three steps.
Step 1. Recall the character map from (4.4). Note that since Ī and i! are exact,and
grA (n)v = (A Ī»
v )ān,
we have an isomorphism of t-modules
(B.6) gr Ī(A(n)v! ) = Ī(A Ī»
v!)ān.
Next, since the action of s on Ī(A(n)v! ) is nilpotent, for any Ī¼ ā tā we have
dimC(Ī(A(n)v! )Ī¼) = dimC
((gr Ī(A
(n)v! ))Ī¼
).
We deduce that as a t-module Ī(A(n)v! ) is a generalized weight module and
(B.7) ch(Ī(A(n)v! )) = ch(gr Ī(A
(n)v! )) = n chĪ(A Ī»
v!).
On the other hand, we have the following isomorphism of t-modules
(B.8) grN(n)k (Ī½) = NĪŗ(Ī½)
ān.
Therefore we have
ch(N(n)k (Ī½)) = n ch(NĪŗ(Ī½)).
Since Ī(A Ī»v!) = NĪŗ(Ī½) as g-modules by Proposition 4.2, this yields
(B.9) ch(Ī(A(n)v! )) = ch(N
(n)k (Ī½)).
Further, we claim that there is an isomorphism of R(n)-module
(B.10) Ī(A(n)v! )Ī¼ = N
(n)k (Ī½)Ī¼, ā Ī¼ ā t
ā.
Note that Ī(A(n)v! )Ī¼ is indeed an R(n)-module because the action of s on Ī(A
(n)v! )
is nilpotent. To prove the claim, it suffices to notice that for any finitely generatedR(n)-modules M , M ā² we have that M is isomorphic to M ā² as R(n)-modules if andonly if gri M = gri M ā² for all i. So the claim follows from the isomorphisms oft-modules (B.6), (B.8) and Proposition 4.2.
Step 2. In this step, we prove that as a tR(n) -module
Ī(A(n)v! )Ī½ = R
(n)Ī½+sĻ0
where R(n)Ī½+sĻ0
is the rank one R(n)-module over which t acts by the weight Ī½+ sĻ0.
Let us consider the canonical morphisms in MĪ»h(X):
A(n)v!
ļæ½ļæ½ ļæ½ļæ½ A (n)v!ā¢
ļæ½ ļæ½ ļæ½ļæ½ A (n)vā¢ .
264 PENG SHAN
Since Ī is exact on MĪ»(X), we deduce the following chain of gR(n)-module mor-phisms
Ī(A(n)v! )
Ī± ļæ½ļæ½ ļæ½ļæ½ Ī(A (n)v!ā¢ ) ļæ½
ļæ½ Ī² ļæ½ļæ½ Ī(A (n)vā¢ ).
Consider the following tR(n)-morphisms given by the restrictions of Ī±, Ī²
Ī(A(n)v! )Ī½
Ī±Ī½ ļæ½ļæ½ ļæ½ļæ½ Ī(A (n)v!ā¢ )Ī½
ļæ½ ļæ½ Ī²Ī½ ļæ½ļæ½ Ī(A (n)vā¢ )Ī½ .
We claim that Ī±Ī½ and Ī²Ī½ are isomorphisms. Note that by (B.7) we have
dimCĪ(A(n)v! )Ī½ = ndimCĪ(A
Ī»v!)Ī½ = n.
By Lemma B.2 we also have dimCĪ(A(n)vā¢ )Ī½ = n. Next, consider the exact sequence
in MĪ»h(Xv),
0 ā F i+1A (n)v ā F iA (n)
v ā gri A (n)v ā 0.
Applying the functor iv!ā¢ to it yields a surjective morphism
iv!ā¢(FiA (n)
v )/iv!ā¢(Fi+1A (n)
v ) ā iv!ā¢(gri A (n)
v ).
Since iv!ā¢(FiA
(n)v ) = F i(iv!ā¢(A
(n)v )) and gri A
(n)v = A Ī»
v , we deduce a surjectivemorphism
gri A(n)v!ā¢ ā A Ī»
v!ā¢, 0 ļæ½ i ļæ½ nā 1.
Applying the exact functor Ī to this morphism and summing over i gives a surjectivemorphism of g-modules
Ī³ : gr Ī(A(n)v!ā¢ ) ā Ī(A Ī»
v!ā¢)ān.
Since v is minimal in vS(Ī»), by Proposition 4.2(c) the right hand side is equal toLĪŗ(Ī½). We deduce from the surjectivity of Ī³ that
dimCĪ(A(n)v!ā¢ )Ī½ = dimC gr Ī(A
(n)v!ā¢ )Ī½
ļæ½ dimC(LĪŗ(Ī½)Ī½)ān
= n.
It follows that the epimorphism Ī±Ī½ and the monomorphism Ī²Ī½ are isomorphisms.The claim is proved. So we have an isomorphisms of tR(n) -modules
Ī²Ī½ ā¦ Ī±Ī½ : Ī(A(n)v! )Ī½ ā Ī(A
(n)vā¢ )Ī½ .
In particular, we deduce an isomorphisms of tR(n)-modules
Ī(A(n)v! )Ī½+sĻ0
ā Ī(A(n)vā¢ )Ī½+sĻ0
,
because
Ī(A(n)v )Ī½+sĻ0
ā Ī(A(n)v )Ī½ , for ļæ½ =!, ā¢.
By Lemma B.2, we have
Ī(A(n)vā¢ )Ī½+sĻ0
= R(n)Ī½+sĻ0
.
We deduce an isomorphism of tR(n)-modules
Ī(A(n)v! )Ī½+sĻ0
= R(n)Ī½+sĻ0
.
JANTZEN FILTRATION 265
Step 3. By the universal property of Verma modules and Step 2, there exists amorphism of gR(n)-modules
Ļ : N(n)k (Ī½) ā Ī(A
(n)v! ).
For any Ī¼ ā tā this map restricts to a morphism of R(n)-modules
ĻĪ¼ : N(n)k (Ī½)Ī¼ ā Ī(A
(n)v! )Ī¼.
By Step 1, the R(n)-modules on the two sides are finitely generated and they areisomorphic. Further, the induced morphism
Ļ(ā) : NĪŗ(Ī½) ā Ī(A Ī»v!)
is an isomorphism by Proposition 4.2. So by Nakayamaās lemma, the morphism ĻĪ¼
is an isomorphism for any Ī¼. Therefore Ļ is an isomorphism. The lemma is proved.
ļæ½
Remark B.4. The hypothesis that v is a shortest element in vS(Ī») is probably notnecessary but this is enough for our purpose.
B.2. Proof of Proposition 5.4. Consider the canonical embedding r : Xā w ā Y ā
w.We claim that the adjunction map yields a surjective morphism
(B.11) r!rā(B(n)) ā B(n).
Indeed, an easy induction shows that it is enough to prove that gri(r!rā(B(n))) ā
gri B(n) is surjective for each i. Since the functors r!, rā are exact and gri B(n) = B,
this follows from Lemma 4.4(a). Note that rā(B(n)) = A(n)w . So the image of (B.11)
by the exact functor Ī ā¦ j! is a surjective morphism
(B.12) Ī(A(n)w! ) ā Ī(B
(n)! ).
By Lemma B.3 we have Ī(A(n)w! ) = N
(n)k . Since the gR(n)-module Ī(B
(n)! ) is q-
locally finite and M(n)k is the largest quotient of N
(n)k in Ok, the morphism (B.12)
induces a surjective morphism
Ļ : M(n)k ā Ī(B
(n)! ).
Further, by Proposition 4.6(b) the map
Ļ(ā) : MĪŗ ā Ī(B!)
is an isomorphism. The same argument as in Step 1 of the proof of Lemma B.3
shows that for each Ī¼ ā tā the generalized weight spaces (M(n)k )Ī¼ and Ī(B
(n)! )Ī¼ are
isomorphic as R(n)-modules. We deduce that Ļ is an isomorphism by Nakayamaāslemma. This proves the first equality. The proof for the second equality is similar.We consider the adjunction map
(B.13) B(n) ā rā¢rā(B(n)).
It is injective by Lemma 4.4(b) and the same arguments as above. So by applyingthe exact functor Ī ā¦ jā¢, we get an injective morphism
Ļā² : Ī(B(n)ā¢ ) ā DM
(n)k .
Again, by using Proposition 4.6(b) and Nakayamaās lemma, we prove that Ļā² is anisomorphism. ļæ½
266 PENG SHAN
Index of notation
1.1: C ā©R -proj, Rā²C.1.2: CĪ, Ī, ā, DāØ.
1.3: R = C[[s]], ā, K, M(āi), MK , f(āi), fK , C(ā), CK .
1.4: J iD(ā).
1.5: F (ā).
2.1: G0, B0, T0, g0, b0, t0, g, t, 1, ā, c, Īŗ = c + m, aR, UĪŗ, MĪ», tā0, t
ā, Ī“, Ļ0,Īµi, ćā¢ : ā¢ć, ||h||2, Īŗt
ā, a, z, Ī , Ī 0, Ī +, Ī +
0 , Ī±i, S, S0, w Ā· Ī», Ļ0, Ļ, S(Ī»),l : S ā N.
2.2: q, l, Ī+, MĪŗ(Ī»), LĪŗ(Ī»), c, k, Uk, Mk(Ī»), Ī»s, ktā.
2.3: Ļ, DMk(Ī»), JiMĪŗ(Ī»).
2.4: Ok, OĪŗ,rĪŗtā, r
ktā, rOĪŗ,
rOk,rĪ+, rPĪŗ(Ī»), Lk(Ī»),
rPk(Ī»).
2.5: Pn, ļæ½, ļæ½, E, EĪŗ, gā², Ek, Pk(E), PĪŗ(E).
2.6: Ak, AĪŗ, Īk, ĪĪŗ.
2.7: D.
2.9: Hv, Sv, Av, Wv(Ī»), Īv.
2.10: v = exp(2Ļi/Īŗ), v = exp(2Ļi/k), J iWv(Ī»).
2.11: H1/Īŗ, BĪŗ, BĪŗ(Ī»), EĪŗ, H1/k, Bk, Ek, Bk(Ī»).
3.1: OZ , O(Z), fā, fā, f !, DZ , M(Z), Ī©Z , DYāZ , M O , M(Z,Z ā²), iā, iā¢, i
!.
3.2: Mh(Z), D, i!, i!ā¢.
3.3: MT (Z), MT (X,Z).
3.4: M ā , Dā Z , Ī“r, mĪ», M
Ī»(Z), MĪ»(Z), MĪ»(X,Z), MĪ»(X,Z), M(Dā Z).
3.5: MTh (Z
ā ), MĪ»h(Z), MĪ»
h(Z), Dā², L Ī»Z , ĪĪ», D = DĪ», DĪ»
Z .
3.6: 2limāāCĪ±, 2limāāCĪ±.3.7: X = limāāXĪ±, O(X), Ī(X,M ) (for M ā O(X)), O(X), OX , ā āOX
F ,
Ī(X,F ) (for F ā O(X)).
3.8: M(X) (with X an ind-scheme), M O , Ī(X,M ) (for M ā M(X)), MT (X),
MĪ»(X), MĪ»(X), Ī(M ) (for M ā MĪ»(X)), MĪ»h(X), MĪ»
h(X), i!, iā¢.
3.9: DX (with X a formally smooth ind-scheme).
4.1: G, B, Q, T , N , g, b, n.
4.2: X = G/B, Xā = G/N , Ļ : Xā ā X, Xw, w, Xw.
4.3: Ī“l : U(g) ā Ī(Xā ,DXā ), M(g), Ī
4.4: MĪ», ch(M), O, OĪŗ, NĪŗ(Ī»), LĪŗ(Ī»), Ī, A Ī»w , iw, A Ī»
w!, A Ī»w!ā¢, A Ī»
wā¢.
4.5: QS, w0, Yw, Y w, jw, BĪ»w, BĪ»
w!, BĪ»w!ā¢, BĪ»
wā¢, r : Xā w ā Y ā
w.
5.1: CR, Ī¼M , for, FR.
5.2: Qā², fw.
5.3: j, f , B, B!, I (n), xs, fs, B(n), B(n)! , B
(n)!ā¢ , B
(n)ā¢ .
5.4: MĪŗ, Mk, M(n)k , DM
(n)k .
5.5: Ļ(a, n), Ļa(B), J i(B!).
JANTZEN FILTRATION 267
6.1: MHM(Z), W ā¢M , (k), Perv(Z), ļæ½.
6.2: MĪ»0 (Xv), DR, MHM0(Xv), Ī·, A Ī»
w , A Ī»w!, A Ī»
w!ā¢, BĪ»w, BĪ»
w!, BĪ»w!ā¢, q, Hq(S),
Tw, ĪØ, Px,w,
A.1: Ī ā, n(Ī„), nā(Ī„), Ī āl , n
āl , N
ā, Nāl , X, Xā , Xw, Xw
l , Xwā l , pl1l2 , pl.
A.2: HĪ»(Xy, Xw), HĪ»(Xw), H
Ī»(X).
A.3: Ī(M ), Ī(M ).
A.5: OĪŗ,Ī», prĪ», V (Ī½), ĪøĪ½ , L Ī», ĪĪ»āĪ¼.
B.1: rx, A(n)v , A
(n)v! , A
(n)v!ā¢ , A
(n)vā¢ , Nk(Ī¼), RĪ¼+sĻ0
, DNk(Ī¼), N(n)k (Ī¼), DN
(n)k (Ī¼),
siM , F ā¢M , grM , ch(M).
Acknowledgement
The author deeply thanks her advisor Eric Vasserot for many helpful discussions,lots of useful remarks and for his substantial support.
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