centers and translation functors for the category

29
DOI: 10.1007/s00209-002-0462-2 Math. Z. 243, 689–717 (2003) Mathematische Zeitschrift Centers and translation functors for the category O over Kac–Moody algebras Peter Fiebig Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, USA Received: 29 November, 2001; in final form: 20 October 2002 / Published online: 24 February 2003 – c Springer-Verlag 2003 1 Introduction Let g be a symmetrizable Kac–Moody algebra as defined in [Kac] (note that in the affine situation we add the action of vector fields to the centrally extended loop algebra). We will study the structure of its category O. There are three types of special objects in O, the simple objects L(λ), the Verma modules M (λ) and the projective covers P (λ) (the latter only exist in certain truncated subcategories). All of them are parametrized by linear forms λ h on the Cartan subalgebra h of g and they are linked by the Bernstein- Gelfand-Gelfand-Humphreys reciprocity (P (λ): M (µ)) = M (µ): L(λ) , where the left hand side is the multiplicity of M (µ) in a Verma flag of P (λ) and the right hand side is the Jordan-H ¨ older multiplicity. Let {P Λ } Λ be a maximal decomposition of the set {P (λ)} λ such that Hom(P, P )=0 whenever P ∈P Λ , P ∈P Λ and Λ = Λ . Then we have the following block decomposition O = Λ O Λ , where O Λ is the subcategory generated by P Λ . We can view Λ as a subset of h . A classical theorem of Kac and Kazhdan ([KK]) shows that –outside the critical hyperplanes– Λ is an orbit of the integral Weyl group W(Λ) Partially supported by the EC TMR network “Algebraic Lie Representations”, EC- contract no ERB FMRX-CT97-0100.

Upload: peter-fiebig

Post on 10-Jul-2016

224 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Centers and translation functors for the category

DOI: 10.1007/s00209-002-0462-2

Math. Z. 243, 689–717 (2003) Mathematische Zeitschrift

Centers and translation functorsfor the category O over Kac–Moody algebras

Peter Fiebig�

Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley,CA 94720-5070, USA

Received: 29 November, 2001; in final form: 20 October 2002 /Published online: 24 February 2003 – c© Springer-Verlag 2003

1 Introduction

Let g be a symmetrizable Kac–Moody algebra as defined in [Kac] (notethat in the affine situation we add the action of vector fields to the centrallyextended loop algebra). We will study the structure of its categoryO. Thereare three types of special objects in O, the simple objects L(λ), the Vermamodules M(λ) and the projective covers P (λ) (the latter only exist in certaintruncated subcategories). All of them are parametrized by linear forms λ ∈h� on the Cartan subalgebra h of g and they are linked by the Bernstein-Gelfand-Gelfand-Humphreys reciprocity

(P (λ) : M(µ)) =[M(µ) : L(λ)

],

where the left hand side is the multiplicity of M(µ) in a Verma flag of P (λ)and the right hand side is the Jordan-Holder multiplicity.

Let {PΛ}Λ be a maximal decomposition of the set {P (λ)}λ such thatHom(P, P ′) = 0 whenever P ∈ PΛ, P ′ ∈ PΛ′ and Λ �= Λ′. Then we havethe following block decomposition

O =∏Λ

OΛ,

where OΛ is the subcategory generated by PΛ. We can view Λ as a subsetof h�. A classical theorem of Kac and Kazhdan ([KK]) shows that –outsidethe critical hyperplanes– Λ is an orbit of the integral Weyl group W(Λ)

� Partially supported by the EC TMR network “Algebraic Lie Representations”, EC-contract no ERB FMRX-CT97-0100.

Page 2: Centers and translation functors for the category

690 P. Fiebig

(we define the critical hyperplanes by the integrality conditions given byKac-Kazhdan’s formula with respect to imaginary roots). The integral WeylgroupW(Λ) is generated by the reflections sα corresponding to the integralreal roots ∆(Λ) with respect to Λ. We will see that the categoriesOΛ outsidethe critical hyperplanes are amalgamations of categories equivalent to theprincipal block for sl2, i.e. to the block containing the trivial representa-tion of sl2. The representation theory inside a critical hyperplane changesdrastically and has no finite dimensional counterpart (cp. [FF]).

For finite dimensional semisimple Lie algebras Soergel proved that thecategorical structure of OΛ can be expressed purely in terms ofW(Λ) andthe singularity of Λ, i.e. the stabilizer of one point (cp. [Soe]).OΛ is equiva-lent to the category of right representations of a finite dimensional algebraA,which, moreover, is a self-dual Koszul algebra. The present article providesthe first steps towards a generalization of Soergel’s results to the symmetriz-able Kac–Moody case. It will be completed in a forthcoming article.

The two most important ingredients in Soergel’s proof are the center ofthe enveloping algebra of g and the construction of projective objects viatranslation functors. In the Kac–Moody case, however, the center of the uni-versal enveloping algebra is far too small to generate the categorical centerof the blocks. Moreover, due to the infinite dimensionality of the integrablemodules of g, it is not obvious that translation functors are biadjoint, henceit is not obvious that they preserve projectivity.

In this article we will address both problems. As a substitute for the centerof the universal enveloping algebra we calculate the centers of the blocksOΛ outside the critical hyperplanes, i.e. the rings of endotransformationsof the identity functors. In the second part we prove the biadjointness oftranslation functors.

We will make extensive use of Jantzen’s deformation theory. Let T bean algebra over the universal enveloping algebra S of h. By deforming cha-racers of h in the tautological direction we arrive at a deformed versionOT

of O. It is an analogously defined subcategory of the category of g ⊗C T -modules. We will show that these deformations behave coherently underbase change T → T ′, i.e. that for a projective module P in OT the naturalmap

Hom(P, ·)⊗T T ′ → Hom(P ⊗T T ′, · ⊗T T ′)

is an isomorphism of functors. We will also show that if T is a local do-main the base change functor induces a bijection between the simple andprojective isomorphism classes ofOT andOK, where K is the residue fieldof T . This together with the coherence of deformation immediately impliesa block decomposition of OT =

∏ΛOT,Λ analogous to the block decom-

position of OK. We can view OK as a certain direct summand of the usualcategory O over the Kac–Moody algebra g ⊗C K, hence the theorem of

Page 3: Centers and translation functors for the category

Centers and translation functors 691

Kac and Kazhdan gives a description of the sets Λ via certain integralityconditions with respect to the roots of g.

Let R = S(0) be the localization of S at the maximal ideal generated byh ⊂ S (we call this the universal local deformation) and let Q be its quotientfield.OQ is a semisimple category, i.e. simple, projective and Verma modulescoincide, hence every block is equivalent to the category of Q-vector spaces.Let Rp be the localization at a hyperplane p. If it is not integral on any rootthen the blocks of ORp are equivalent to the category of Rp-modules. If itis integral on a real root α, the blocks of ORp are either isomorphic to thecategory of Rp-modules or to the deformed principal block of sl2 . We callthe later the subgeneric case.

The reason for the use of deformation theory is the following. We willexplicitly calculate the structure in the generic and subgeneric cases and acombination then gives information on the structure of OR and hence onO = OC.

We get the following description of the center in the deformed case.

Theorem. Let OR,Λ be a block outside the critical hyperplanes. Then theaction of the center ZR,Λ of OR,Λ on deformed Verma modules gives anisomorphism

ZR,Λ∼=

{{tν} ∈

∏ν∈Λ

R | tν ≡ tsα.ν mod α∨ ∀α ∈ ∆(Λ)

}.

One can show that the center in the non-deformed case is the image ofZR,Λ under the natural projection

∏Λ R → ∏

Λ C. It turns out that it iscanonically isomorphic to a completion of the cohomology ring of the flagmanifold GL/BL for Langlands dual Kac–Moody algebra gL (cp. [KoKu]).Note that in the finite dimensional case, the center of the principal block isalso canonically isomorphic to Soergel’s algebra of coinvariants.

In the second part of this article we choose certain pairs of blocks of thedeformed category and define translation functors in both directions on thesubcategories of modules with finite Verma flag and show their biadjoint-ness. There is a canonical adjunction in one direction, which was alreadydescribed in [Nei]. However, there is no canonical adjunction in the seconddirection. Instead we will give in Theorem 5.9 a description of all possi-ble transformations by describing their action on Verma modules. Over thegeneric point Q the translation functors are equivalences of categories. Wewill see that this equivalence has a pole of order one along all hyperplaneswhich stabilize exactly one of the chosen blocks.

The article is arranged as follows. In Sect. 2 we define the deformed ver-sionOT of categoryO and introduce the base change functor correspondingto any map of deformation rings T → T ′. We prove that the deformed cate-gories behave coherently under base change and that, if T is a local domain

Page 4: Centers and translation functors for the category

692 P. Fiebig

with residue field K, the base change functor induces a bijection betweensimple and projective isomorphism classes of OT and OK. We also provea generalized BGGH-reciprocity theorem and describe the correspondingblock decomposition (we closely follow [RCW]). Finally we show that basechange functors respect block decompositions.

In Sect. 3 we define the centers of the deformed categories and show thatany base change naturally induces a map between the corresponding centers.We then define what we call the “universal local deformation” and give amore explicit description of the sets Λ corresponding to its localizations athyperplanes. We will see that outside the critical hyperplanes this leads togeneric and subgeneric cases and we explicitly compute their structure bydescribing all projective objects. Intersecting the hyperplanes then gives adescription of the center in the universally deformed case.

In Sect. 4 we define a pair of translation functors between the subcate-gories of modules with Verma flag of certain blocks OΛ and OΛ′ .

In Sect. 5 we construct an adjunction in the first direction which we thenuse to show that over generic localizations the translation functors definean equivalence of blocks. This is used to describe all possible adjunctionsin the other direction, first for the subgeneric cases and then for the uni-versally locally deformed case. It follows that translation functors preserveprojectivity.

My most sincere thanks go to Wolfgang Soergel who always shared hisdeep insight into the representation theory of Lie algebras. I would also liketo thank Fred van Oystaeyen, who pointed out an error in an earlier version.

2 Deformed category O and its simple and projective objects

Let g be a Kac–Moody algebra as defined in [Kac] with Cartan subalgebra hand Borel subalgebrab. LetU = U(g), B = U(b) andS = U(h) = S(h)bethe corresponding universal enveloping algebras. Let b→ h be the splittingof the inclusion h → b. Then every linear form λ ∈ h� defines a one-dimensional B-module Cλ. Let M(λ) = U ⊗B Cλ be the correspondingVerma-module and L(λ) its unique simple quotient. Both are modules ofcategory O, which is the full subcategory of all g-modules on which h actsdiagonally and b locally finitely. In order to study the structure of O, wenow define relative versions of these objects.

2.1 Deformed category OLet T be a commutative, associative, unital, noetherian S-algebra with struc-ture map τ : S → T . We will call T a deformation ring in the sequel. For

any λ ∈ h�, the composition B → Sλ+τ−→ T defines a B ⊗C T -module

Page 5: Centers and translation functors for the category

Centers and translation functors 693

structure on T which we call Tλ. The deformed Verma module with highestweight λ is defined as

MT (λ) := U ⊗B Tλ.

It is an object of the deformed category OT , which we define as follows.Let M be a U ⊗C T -module. For any λ ∈ h� let

Mλ := {m ∈M | H.m = (λ + τ)(H)m ∀H ∈ h} .

Mλ is a T -submodule of M . We call any U ⊗C T -module M with M =⊕λ∈h� Mλ a weight module. The deformed category OT is the category of

all U ⊗C T -modules M such that

1. M is a weight module,2. B ⊗C T.m is finitely generated as a T -module for all m ∈M .

OT is an abelian category.

Example 2.1. Any µ ∈ h� defines a structure of an S-algebra on C withstructure map µ : S → C. Then OC = O and MC(λ) = M(λ + µ). Moregenerally, if K is any field with the structure of an S-algebra, then OK

is the subcategory of the usual category O over the Kac–Moody algebrag⊗C K which consists of all objects whose weights lie in the complex affinesubspace τ + h� = τ + HomC(h, C) ⊂ HomK(h ⊗C K, K). Hence thesimple objects in this category are parametrized by h�.

Let T → T ′ be any morphism of deformation rings. Then there is a basechange functor

· ⊗T T ′ : OT → OT ′ .

Obviously MT (λ)⊗T T ′ = MT ′(λ).

2.2 Deformed simple objects

We will now determine the simple objects in OT for a local deformationring T .

Proposition 2.1. Let T be a local deformation ring with residue field K. Thefunctor · ⊗T K gives a bijection{

simple isomorphism classesof OT

}←→

{simple isomorphism classes

of OK

}.

Page 6: Centers and translation functors for the category

694 P. Fiebig

Proof. Let m ⊂ T be the maximal ideal. If L ∈ OT is simple, then byNakayama’s Lemma mL = 0, hence L⊗T K is simple. On the other hand,let Res: OK → OT be the right adjoint functor. It is clear that Res L issimple in OT whenever L is simple in OK. These two functors are clearlyinverse on the sets of simple isomorphism classes. �Corollary 2.2. If T is local, then the simple isomorphism classes ofOT areparametrized by their highest weights, i.e. by elements of h�.

For any λ ∈ h� let LT (λ) be the simple module in OT with highestweight λ. It is a quotient of MT (λ).

2.3 Deformed projective objects

Our next goal is to give a similar description for projective isomorphismclasses (for a local deformation domain T ). Unfortunately, both the de-formed and the non-deformed categories do not have enough projectiveobjects. But both are filtered in a standard way by so-called truncated subca-tegories with enough projectives. In the following we generalize argumentsof [RCW] to the deformed case.

Let ∆+ ⊂ ∆ ⊂ h� be the positive roots in the root system of g withrespect to h and b. Define a partial order on h� by µ ≤ λ iff λ− µ ∈ N∆+.For any λ ∈ h� let O�λ

T be the full subcategory of OT consisting of allM ∈ OT such that Mµ �= 0 implies µ ≤ λ. Then every finitely generatedobject of OT lies in some O�λ

T .

Lemma 2.3. Let T be a deformation ring and let λ, µ ∈ h�. There is anobject Q�λ

T (µ) in O�λT which represents the functor M → Mµ. Moreover,

Q�λT (µ) has a Verma flag, i.e. a finite filtration whose subquotients are

isomorphic to Verma modules.

Proof. For ν ∈ h� the subspace B ��ν :=⊕

γ �≤ν Bγ is an ideal in B. Define

B�ν := B/B ��ν .

For λ, µ ∈ h� with µ ≤ λ let

Q�λT (µ) := U ⊗B (B�λ−µ ⊗S Tµ).

As in [RCW] one shows that Q�λT (µ) is a module inO�λ

T , has a Verma flagand that the map

HomOT(Q�λ

T (µ), M)→Mµ,

given by evaluation at 1⊗ 1⊗ 1, is an isomorphism if M ∈ O�λT . �

Page 7: Centers and translation functors for the category

Centers and translation functors 695

By definition M → Mµ is exact on OT , hence Q�λT (µ) is projective in

O�λT . In particular, there are enough projective objects in O�λ

T . The nextproposition shows that the base change functor respects projectivity andthat the deformed categories behave coherently under base change. Thiswill allow us to compare categories over different deformation rings.

Proposition 2.4. Let T → T ′ be a morphism of deformation rings and Pbe projective in O�λ

T . Then P ⊗T T ′ is projective in O�λT ′ . If P is finitely

generated, then the natural map

HomOT(P, ·)⊗T T ′ → HomOT ′ (P ⊗T T ′, · ⊗T T ′)

is an isomorphism of functors from OT to T ′ -mod.

Proof. From the proof of Lemma 2.3 we see that if a module M ∈ O�λT

is generated by a weight vector m ∈ Mµ, then there exists a surjectionQ�λ

T (µ)→→M . Hence every module inO�λT is a quotient of a module of the

form⊕

Q�λT (µ), hence every projective module P is a direct summand of

such a module. Then P⊗T T ′ is a direct summand of(⊕

Q�λT (µ)

)⊗T T ′ =⊕

Q�λT ′ (µ), hence is projective.

It suffices to show the second part of the proposition for P = Q�λT (µ)

for some µ. Then P ⊗T T ′ = Q�λT ′ (µ). We have natural isomorphisms

HomOT(P, M)⊗T T ′ ∼= Mµ ⊗T T ′,

HomOT ′ (P ⊗T T ′, M ⊗T T ′) ∼= (M ⊗T T ′)µ.

The natural map Mµ ⊗T T ′ → (M ⊗T T ′)µ is certainly an isomorphism.�In the proof we have seen that every projective object inO�λ

T is a directsummand of a sum of Q�λ

T (µ)’s. From the next lemma it follows that everyfinitely generated projective object has a (finite) Verma flag. The proof isvery similar to the proof of the corresponding statement in the non-deformedsituation, which can be found in [RCW].

Lemma 2.5. Let T be a local deformation ring and let M ∈ OT be a modulewith Verma flag. Then every direct summand of M has a Verma flag.

Since every module with Verma flag is free over T , we get in particularthat every finitely generated projective object inO�λ

T is free as a T -module.In the following we add the assumption that T is a local domain.

Proposition 2.6. Let T be a local deformation domain with residue field K.Then the base change functor · ⊗T K induces a bijection{

projective isomorphism classesof O�λ

T

}←→

{projective isomorphism classes

of O�λK

}.

Page 8: Centers and translation functors for the category

696 P. Fiebig

Proof. Every projective object is a direct summand of a direct sum of someQ�λ(µ)’s. Since Q�λ

T (µ) ⊗T K = Q�λK

(µ) and since every Q�λT (µ) is

completely decomposable, we only have to show that P ⊗T K does notdecompose for an indecomposable projective object P ∈ O�λ

T .Let m ⊂ T be the maximal ideal and suppose that P ⊗T K decomposes,

i.e. that there is a non-trivial idempotent in End(P ⊗T K) = End(P )⊗T K.By the idempotent lifting lemma, we can successively lift this idempotentto End(P ) ⊗T T/mn for any n > 0. Taking the projective limit gives anidempotent in End(P )⊗T T = End(P ⊗T T ), where T is the completionof T , hence P ⊗T T splits.

Now let Q and Q be the quotient fields of T and T , resp. The moduleP⊗T Q = (P⊗T T )⊗

TQ splits. Since there is a Z-form of the Kac–Moody

algebra, this splitting must be induced by a splitting of P ⊗T Q. Hence bothP ⊗T Q and P ⊗T T decompose. We claim that P is the intersection ofthese two modules inside P ⊗T Q, hence we get a decomposition of P ,contrary to the assumption. Since P is free as a T -module we have to proveT = T ∩Q ⊂ Q.

Let ab ∈ Q ∩ T with a, b ∈ T . The inclusion Tb ↪→ Ta + Tb becomes

an isomorphism after completion since a ∈ T b. By [Ma], Theorem 8.14,T is faithfully flat over T , hence already Tb = Ta + Tb. So a ∈ Tb,i.e. a

b ∈ T . �We are now able to prove the existence of projective covers in trun-

cated categories and a generalization of the Bernstein-Gelfand-Gelfand-Humphreys-reciprocity formula.

Theorem 2.7. Let T be a local deformation domain and K its residue field.Choose λ ∈ h� and a simple object LT ∈ O�λ

T . Then there exists a projective

cover P�λT of LT inO�λ

T . It has a Verma flag and for the multiplicities holdsthe BGGH-reciprocity formula(

P�λT : MT

)=

[MK : LK

]for all Verma modules MT in O�λ

T .

Proof. In [RCW] it was proved that LK has a projective cover P�λK

inO�λK

.Let P�λ

T be its projective preimage under base change. Then it is a projectivecover of LT in O�λ

T by Nakayama’s lemma.P�λ

T has a Verma flag by the remark following Lemma 2.5 and themultiplicities satisfy (

P�λT : MT

)=

(P�λ

K: MK

)

Page 9: Centers and translation functors for the category

Centers and translation functors 697

for any Verma module MT in O�λT . Again by [RCW](

P�λK

: MK

)=

[MK : LK

]. �

For µ ≤ λ let P�λT (µ) be the projective cover of LT (µ) in O�λ

T .

2.4 Block decomposition

Next we introduce the block decomposition of category OT . We keep theassumption that T is a local domain with residue field K. Let ∼T be theequivalence relation on h� generated by λ ∼T µ if there is a ν ∈ h� suchthat P�ν

T (λ) contains MT (µ) as a subquotient, hence, by BGGH-reciprocity,if

[MK(λ) : LK(µ)

] �= 0. Hence ∼T =∼K. For any union of equivalenceclasses θ ⊂ h�/∼T defineOT,θ as the subcategory generated by all P�ν

T (λ)for λ ∈ θ and ν ∈ h�. It is the full subcategory of OT consisting of allmodules M such that every highest weight of a subquotient of M lies in θ.

Example 2.2 (generic case). Take an equivalence class Λ ∈ h�/∼T whichis trivial, i.e. Λ = {λ}. Then the block OT,Λ contains a unique Vermamodule MT (λ) which is projective, even a projective generator. Since T ∼=End(MT (λ)) we have an equivalence of categories

OT,Λ∼= T -mod.

In general we get the following block decomposition.

Proposition 2.8. Let T be a local deformation domain. The functor∏Λ∈h�/∼T

OT,Λ → OT

{MΛ} →⊕

is an equivalence of categories.

Proof. For any module N of OT and any equivalence class Λ ∈ h�/∼Twe define NΛ ⊂ N to be the submodule generated by the images of allmorphisms P�λ

T (µ) → N for µ ∈ Λ and λ > µ. Then the functor N →{NΛ} is inverse to the functor above. �

The next lemma shows that base change respects the block decomposi-tion.

Lemma 2.9. Let T → T ′ be a morphism of local deformation domains andµ, ν ∈ h�. If µ ∼T ′ ν, then µ ∼T ν.

Page 10: Centers and translation functors for the category

698 P. Fiebig

Proof. Let K and K′ be the residue fields of T and T ′. Suppose

[MK′(ν) :

LK′(µ)] �= 0. Take any λ > µ. Then the module P�λ

T (µ)⊗T T ′ is projective,

hence P�λT ′ (µ) is isomorphic to a direct summand. Hence(

P�λT (µ) : MT (ν)

)≥

(P�λ

T ′ (µ) : MT ′(ν))≥ 1,

hence ν ∼T µ. �Corollary 2.10. Let T → T ′ be a morphism of local deformation domainsand θ ⊂ h�/∼T a union of equivalence classes. Then the functor

· ⊗T T ′ : OT → OT ′

induces a functor· ⊗T T ′ : OT,θ → OT ′,θ.

3 Deformed centers

Let T be a deformation ring (not necessarily a local domain) and let ZT (resp.Z�λ

T ) be the center of OT (resp. O�λT ), i.e. the ring of endotransformations

of the identity functor.

3.1 Base change

We will describe the effect of base change on the center.

Proposition 3.1. Every morphism T → T ′ of deformation rings induces acanonical map ZT → ZT ′ .

Proof. Let {Pi}i∈I be a set of objects ofOT such that for any pair λ, µ ∈ h�

with µ ≤ λ there is a Pi which is projective inO�λT and surjects onto LT (µ).

This set generatesOT in the sense that every object is a quotient of a directsum of Pi’s and every morphism is a quotient of a morphism between directsums of Pi’s.

An element of the center determines an endomorphism of Pi for anyi ∈ I . This collection of endomorphisms has the property that it commuteswith every morphism Pi → Pj . On the other hand, since {Pi} generatesOT , a collection of endomorphisms of Pi’s which commutes with everymorphism Pi → Pj uniquely determines an element of the center.

Via base change we get a collection {Pi ⊗T T ′}i∈I of objects ofOT ′ ofthe same nature and since Hom(Pi⊗T T ′, Pj⊗T T ′) = Hom(Pi, Pj)⊗T T ′,the above description of the center shows that the natural map End(Pi)→End(Pi⊗T T ′) induces a map ZT → ZT ′ . It does not depend on the choiceof the set {Pi}i∈I . �

If T and T ′ are local domains, we analogously get a map ZT,θ → ZT ′,θfor any collection of equivalence classes θ ⊂ h�/∼T .

Page 11: Centers and translation functors for the category

Centers and translation functors 699

3.2 Orbits and equivalence classes

In this section we will examine more closely the equivalence classes under∼T . For this we have to assume that g is symmetrizable, i.e. we assumethat there is a non-degenerate invariant bilinear form (·, ·) : g × g → C.Then the restriction to h × h is also non-degenerate and induces a bilinearform (·, ·) : h� × h� → C. For any deformation ring T let h�

T = h� ⊗C

T = HomC(h, T ) and (·, ·)T : h�T × h�

T → T the T -bilinear continuation.We consider h� ∼= h� ⊗ 1 ⊂ h�

T as a subspace. The structure morphismτ |h : h → S → T is an element of h�

T . Let ρ ∈ h� be an element such that(ρ, α) = 1 for every simple root α.

Let T be a local deformation domain with residue field K. Let ↑T be thepartial order on h� generated by µ ↑T λ if there exists n ∈ N and a positiveroot β ∈ ∆+ with 2(λ+ρ+τ, β)K = n(β, β)K and λ−µ = nβ. Recall thefollowing theorem of Bernstein, Gelfand and Gelfand and Jantzen, whichwas proven in the infinite dimensional case by Kac and Kazhdan.

Theorem 3.2 ([KK], Theorem 2). Let λ, µ ∈ h�. Then[MK(λ) : LK(µ)

] �= 0

if and only if µ ↑T λ.

Corollary 3.3. The equivalence relation ∼T is generated by ↑T .

We will give a more explicit description of equivalence classes. Supposeλ, µ and β are as in the definition of ↑T . If β is a real root, then there is acorresponding reflection sβ in the Weyl groupW of g and µ = sβ.λ underthe ρ-shifted action ofW on h�, i.e. sβ.λ = sβ(λ + ρ)− ρ.

Let Λ be an equivalence class under ∼T . Define

∆T (Λ) := {β ∈ ∆ | 2(λ + ρ + τ, β)K ∈ Z(β, β)K for some λ ∈ Λ}.Let ∆re ⊂ ∆ be the subset of real roots. Assume ∆T (Λ) ⊂ ∆re. LetWT (Λ)be the subgroup of W generated by all sβ with β ∈ ∆T (Λ). By what weobserved above

Λ =WT (Λ).λ

for any λ ∈ Λ. Conjecturally, it isWT (Λ) which governs the structure of theblockOT,Λ, i.e. there should be a description of the categorical structure ofOT,Λ which only depends on the combinatorics ofWT (Λ), the singularity ofthe orbit Λ, i.e. on the stabilizer of one point, and on the “level”of Λ, i.e. onthe existence of a highest or lowest weight in the orbit. This description willbe given in a forthcoming paper. The corresponding statement in the finitedimensional case was proved in [Soe].

Page 12: Centers and translation functors for the category

700 P. Fiebig

3.3 “Universal” local deformation and the subgeneric case

From now on let R = S(0) be the localization of S at the maximal idealgenerated by h ⊂ S. Let Q be its quotient field. For any prime ideal p ⊂ Rlet Rp be the localization at p and Kp = Rp/pRp the residue field. For anyroot α ∈ ∆ let hα = (τ, α)S ∈ S and let Rα be the localization at Rhα

and Kα its residue field. We will use the base change functor to comparethe deformed categories over different localizations of R. We view C asthe residue field of R, hence it inherits a structure of a deformation ring.Note that the equivalence relations ∼R and ∼C are the same. We will firstdetermine the equivalence classes for the deformation rings Rp.

Let p ⊂ R be a prime ideal, λ ∈ h� and β ∈ ∆. Then 2(λ+ρ+τ, β)Kp ∈Z(β, β)Kp is equivalent to 2(λ + ρ, β) ∈ Z(β, β), i.e. β ∈ ∆C(Λ), andhβ = (τ, β) = 0 in Kp, i.e. hβ ∈ p. In particular, for any real root β ∈∆ and any equivalence class Λ ∈ h�/∼Rβ

, ∆Rβ(Λ) is either empty or

∆Rβ(Λ) = {±β}, hence Λ = {λ} or Λ = {λ, sβ.λ}.

We already know the structure of generic blocks, i.e. of blocks corres-ponding to trivial equivalence classes (example 2.2). Now we calculate thestructure of subgeneric blocks, i.e. of blocks corresponding to equivalenceclasses with two elements. The next proposition shows that they are equi-valent to the deformed principal block of categoryO for sl2. Note that if anequivalence class Λ is finite, the corresponding block OT,Λ is equivalent toO�λ

T,Λ if λ is big enough. Hence there are projective covers PT (µ) for anysimple LT (µ) in OT,Λ.

Proposition 3.4. Let α be a real root and Λ ∈ h�/∼Rαbe an equivalence

class consisting of two elements λ and µ = sα.λ and suppose λ > µ. ThenPRα(λ) ∼= MRα(λ) and there is a short exact sequence

0→MRα(λ)→ PRα(µ)→MRα(µ)→ 0.

The block ORα,Λ is isomorphic to the category of right representations of

the quiver over Rα with relation j ◦ i = hα, i.e. to the

category of right representations of the Rα-algebra generated by the pathsof the quiver with relation j ◦ i = hαeλ, where eλ is the trivial path at thevertex λ.

Proof. Let us abbreviate PRα , MRα by P, M and PKα , MKα by P ′, M ′.From the BGGH-reciprocity we directly learn that P (λ) is isomorphic

to M(λ). We will take now a closer look at the structure of M(λ) by usingthe Jantzen filtration. Let (·, ·) : M(λ)×M(λ)→ Rα be a non-degenerate

Page 13: Centers and translation functors for the category

Centers and translation functors 701

and invariant form (with respect to a Chevalley involution σ : g → g). LetM(λ) = M0 ⊃M1 ⊃ . . . be the Jantzen filtration, i.e.

Mi := {m ∈M(λ) | (m, M(λ)) ∈ hiαRα}.

Let Mi ⊂M ′(λ) be the image of Mi under the projection M(λ)→M ′(λ).Then the Jantzen sum formula holds:∑

i>0

ch Mi =∑

α∈∆+,n∈N

(λ+ρ+τ,α)Kα=n(α,α)

ch M ′(λ− nα) = ch M ′(µ).

The right hand side is the character of a simple module, hence M1 ∼= M ′(µ)and M2 = 0. Since M1 is the unique maximal submodule of M ′(λ), wehave (M ′(λ) : L′(µ)) = 1 and hence[

P (µ) : M(λ)]

= 1,

hence P (µ) has a Verma flag with subquotients M(µ) and M(λ). Since thedominant Verma module has to appear as a submodule there is a short exactsequence

0→M(λ)→ P (µ)→M(µ)→ 0,

as claimed.Now we will determine the categorical structure of ORα,Λ. The module

P = P (λ) ⊕ P (µ) is a projective generator of ORα,Λ and hence ORα,Λ

is isomorphic to the category of right modules over the ring End(P ). Sowe have to calculate End(P ). Choose an isomorphism P (λ) ∼= M(λ)and let i : P (λ) → P (µ) be the inclusion of the short exact sequenceabove. Over the closed point, there is an inclusion M ′(µ) → M ′(λ).Take a lift j : P (µ) → M(λ) ∼= P (λ). From the above descriptionof the projective covers it follows that over the closed point the mapsi′ : P ′(λ) → P ′(µ) and j′ : P ′(µ) → P ′(λ) together with the idempo-tents generate End(P ′(λ) ⊕ P ′(µ)) with relation j′ ◦ i′ = 0. Hence weonly have to describe j ◦ i ∈ End(P (λ)). It is multiplication with a scalarx ∈ Rα, since P (λ) is isomorphic to a Verma module. We may assumex = hn

α, since we can still multiply i or j by an invertible scalar. Fromj′ ◦ i′ = 0 follows n ≥ 1. We now prove n = 1, which finishes the proof.

Suppose n ≥ 2. Then, by definition of the Jantzen filtration, the image ofj is contained in M2. Since M2 = 0, j′ is trivial, contrary to its construction,hence n = 1. �

Evaluation of the center ZRα,{λ,µ} on Verma modules gives a map

ZRα,{λ,µ} → End(MRα(λ))⊕ End(MRα(µ)) = Rα ⊕Rα,

z → (z|MRα (λ), z|MRα (µ)).

Page 14: Centers and translation functors for the category

702 P. Fiebig

The preceding proposition together with the description of the center inSect. 3.1 gives the following

Corollary 3.5. The map above is injective and induces an isomorphism

ZRα,{λ,µ}∼→ {(tλ, tµ) ∈ Rα ⊕Rα | tλ ≡ tµ mod hα} .

3.4 The center outside the critical hyperplane

Let T be any deformation ring and Λ ∈ h�/∼T an equivalence class. Eva-luating the center on Verma modules gives a map

ZT,Λ →∏ν∈Λ

End(MT (ν)) =∏ν∈Λ

T.

This map commutes with base change morphisms T → T ′ and the inducedmap ZT,Λ → ZT ′,Λ.

Theorem 3.6. Let R be the localization of S at 0, i.e. at the ideal generatedby h ⊂ S, and let Λ ∈ h�/∼R be an equivalence class such that ∆R(Λ) ⊂∆re. Then the evaluation on Verma modules induces an isomorphism

ZR,Λ∼=

{{tν} ∈

∏ν∈Λ

R | tν ≡ tsα.ν mod hα ∀α ∈ ∆R(Λ)

}.

Proof. In the proof of Proposition 3.1 we showed that for any deformationringT the center can be viewed as a subring ofEnd(

∏i∈I Pi), where{Pi}i∈I

is an appropriate set of objects in OT . For any morphism of deformationrings T → T ′ the base change map ZT → ZT ′ is then induced by the naturalmap End(

∏Pi)→ End(

∏Pi ⊗T T ′).

For any prime ideal p ⊂ R consider the morphism Rp→ Q = QuotR.Then the induced map End(

∏Pi)→ End(

∏Pi ⊗T Q) is injective, so we

can view ZRp as a subring in ZQ. Moreover, since R is the intersection ofall its localizations at prime ideals of height one ([Ma], Theorem 11.4), weget

ZR =⋂p⊂R

ht p=1

ZRp ⊂ ZQ.

Page 15: Centers and translation functors for the category

Centers and translation functors 703

So let p ⊂ R be a prime ideal of height 1. We calculate ZRp,Λ. If hα �∈ pfor all α ∈ ∆R(λ), then Λ decomposes under ∼Rp into trivial equivalenceclasses by what we observed in Sect. 3.3. Hence

ZRp,Λ∼=

∏ν∈Λ

Rp.

If hα ∈ p for some α ∈ ∆R(Λ), then p = Rhα since ht p = 1. By Corollary3.5,

ZRα,Λ∼=

{{tν} ∈

∏ν∈Λ

Rα | tν ≡ tsα.ν mod hα

}.

The intersection of these spaces now proves the theorem. �

4 Translation functors

For any deformation domain T let MT be the subcategory of OT con-sisting of modules with Verma flag, i.e. modules which admit a finite fil-tration whose subquotients are isomorphic to Verma modules. We anal-ogously define MT,Λ. In this section we will define translation functorsMT,Λ → MT,Λ′ andMT,Λ′ → MT,Λ for certain choices of equivalenceclasses Λ and Λ′. We will prove that translation functors respect short ex-act sequences and are biadjoint in the case T = R. This implies that theypreserve projectivity, hence they allow us to construct projective objects.

4.1 Definition of translation functors

Let T be a local deformation domain and K its residue field. For any U⊗CT -module M and any U -module N there is a natural structure of a U ⊗C

T -module on M ⊗C N . If moreover M is a module of OT and N is amodule ofO, then M ⊗C N is a module ofOT . For any morphism T → T ′of deformation rings there is a natural isomorphism of U ⊗C T ′-modules(M ⊗T T ′)⊗C N ∼= (M ⊗C N)⊗T T ′.

For any U ⊗C T -module N which is a weight module and any γ ∈ h�

we define N�γ to be the quotient modulo the submodule generated by allweight spaces Nγ′ with γ′ �≤ γ. We get a right exact functor ·�γ from thecategory of weight modules to O�γ

T . For γ′ > γ we have a natural mapN�γ′ → N�γ .

Now let λ, λ′ ∈ h� and let Λ, Λ′ be their equivalence classes under ∼T .Assume the following:

1. λ − λ′ is integral and there is a dominant weight in the linearW-orbitof λ− λ′, i.e. ν = w(λ− λ′) > 0 for some w ∈ W .

Page 16: Centers and translation functors for the category

704 P. Fiebig

2. ∆T (Λ) ⊂ ∆re. From 1 it follows that ∆T (Λ′) = ∆T (Λ) and henceWT (Λ) =WT (Λ′) and

Λ =WT (Λ).λ, Λ′ =WT (Λ′).λ′.

3. λ and λ′ lie in the closure of the same Weyl chamber, i.e. (λ+ρ+τ, α)K ≥0 iff (λ′ + ρ + τ, α)K ≥ 0 for all α ∈ ∆T (Λ).

4. StabT (λ) ⊂ StabT (λ′) with finite index, i.e. λ′ “lies on the walls” andλ “lies off the walls” (StabT denotes the stabilizer under the ρ-shiftedaction ofWT (Λ) =WT (Λ′)).

Let L = L(ν) be the simple integrable g-module with highest weightν and L� its restricted dual, i.e. the simple integrable module with lowestweight −ν. For any module M ∈ OT let MΛ and MΛ′ be the projectionsonto the Λ-block and the Λ′-block. For M ∈MT,Λ′ we define

ϑout(M) = ϑλλ′(M) := (M ⊗C L)Λ

and for N ∈MT,Λ we set

ϑon(M) = ϑλ′λ (N) := lim←−

γ

((N ⊗C L�)�γ)Λ′ .

Proposition 4.1. 1. ϑout and ϑon define functors

ϑout : MT,Λ′ →MT,Λ

andϑon : MT,Λ →MT,Λ′

which map short exact sequences to short exact sequences.2. Letw ∈ WT (Λ). ThenϑoutMT (w.λ′)has a Verma flag with subquotients

MT (wx.λ), where x ∈ StabT (λ′)/StabT (λ), each occuring once. Onthe other hand, ϑonMT (w.λ) is isomorphic to MT (w.λ′).

Proof. We prove part 2 of the proposition first. So let w ∈ WT (Λ). As in[Jan], 2.2, one proves that MT (w.λ′) ⊗ L has an (increasing and in gene-ral infinite) filtration whose subquotients are Verma modules with highestweights w.λ′ + µ, where µ runs through the multiset of weights of L. Nowevery direct summand of a module with Verma flag also has a Verma flag.Hence ϑoutMT (w.λ′) has a Verma flag whose subquotients are parametrizedby the weights w.λ′ + µ which lie inWT (Λ).λ. As in [Jan], 2.9, one shows(using property 3 of the choice of λ and λ′) that these are exactly the weightswx.λ for x ∈ StabT (λ′)/StabT (λ). In particular, the Verma filtration isfinite.

Similarly one shows that MT (w.λ)⊗L� has a so-called reversed Vermaflag with subquotients MT (w.λ + µ), where µ is an element of the multiset

Page 17: Centers and translation functors for the category

Centers and translation functors 705

of weights of L�. A reversed Verma flag for a module N is a (possibly finite)filtration

N = N0 ⊃ N1 ⊃ N2 ⊃ . . .

such that

1.⋂∞

i=0 Ni = {0},2. Ni/Ni+1 ∼= MT (νi) for some νi ∈ h�, such that the set {i|νi ≤ ν} is

finite for all ν ∈ h�,3. for all i ≥ 0 there is ni ∈ Ni such that, for all j ≥ 0, the set {ni}i≥j

generates Nj .

Let N be a module with a reversed Verma flag and let b− ⊂ g be theopposite Borel subalgebra, i.e. the subalgebra generated by h and all negativeweight spaces. Let B− = U(b−) be its universal enveloping algebra. Thenthere is an isomorphism of B− ⊗C T -modules

N ∼=⊕

i

MT (νi).

This follows from the fact that a reversed Verma flag splits as a filtration ofB−⊗C T -modules, i.e. for any i ≥ 0 we have an isomorphism of B−⊗C T -modules

N ∼= ⊕

j=0,...,i

MT (νi)

⊕Ni+1.

Because of property 3 of the definition of a reversed Verma flag this inducesan isomorphism of the stated form.

The functor ·�γ then sorts out all Verma subquotients whose highestweight is lower or equal to γ. Hence ((MT (w.λ)⊗ L�)�γ)Λ′ has a Vermaflag whose subquotients are parametrized by all µ in the multiset of weightsof L� such that w.λ + µ ≤ γ and w.λ + µ ∈ WT (Λ′).λ′. Again one showsas in [Jan], 2.9, that w.λ+µ ∈ WT (Λ′).λ′ only if w.λ+µ = w.λ′, i.e. µ =w(λ − λ′) and hence the projective limit lim←− γ((MT (w.λ) ⊗C L�)�γ)Λ′

stabilizes to MT (w.λ′). Hence we proved assertion 2.We will prove now that translation functors respect short exact sequences.

For ϑout this immediately follows since it is the composition of exact func-tors. In the case of ϑon it is enough to show that for any γ ∈ h� and anyshort exact sequence

0→ N ′ → N → N ′′ → 0

of modules with reversed Verma flags the induced sequence

0→ (N ′)�ν → N�ν → (N ′′)�ν → 0

Page 18: Centers and translation functors for the category

706 P. Fiebig

is also exact. This follows from the fact that the above sequence splits as asequence of B− ⊗ T -modules and that the functor ·�γ sorts out the factorswith highest weights lower or equal to γ. Now the rest of assertion 1 followsfrom 2 by induction on the length of a Verma flag. �

5 Biadjointness of translation functors

We will show that the translation functors ϑon and ϑout are biadjoint. Weneed some generalities. LetA and B be two categories and F : A → B andG : B → A be functors. Then every natural transformation idA → G ◦ Finduces a natural transformation

HomB(F (·), ·)→ HomA(·, G(·))of functors fromAop×B to the category of sets. Analogously, every naturaltransformation F ◦G→ idB defines a transformation

HomA(·, G(·))→ HomB(F (·), ·).These transformations are inverse if and only if the compositions F →F ◦G ◦ F → F and G→ G ◦ F ◦G→ G are the identities.

5.1 Left adjointness

We will prove that ϑon is left adjoint to ϑout.

Proposition 5.1. There are natural transformations of functors

ϑonϑout → idMT,Λ′

andidMT,Λ

→ ϑoutϑon.

These transformations define an adjunction

HomMT,Λ′ (ϑon·, ·) ∼= HomMT,Λ(·, ϑout·).

Proof. Let M ∈MT,Λ′ . Then there is the natural evaluation map

M ⊗C L⊗C L� →M

which we may restrict to the direct summand (M ⊗C L)Λ ⊗C L� to get amap

ϑout(M)⊗C L� →M.

Page 19: Centers and translation functors for the category

Centers and translation functors 707

Now M has a finite Verma flag, hence its set of weights is restricted fromabove, so this map factors over a map

(ϑout(M)⊗C L�)�γ →M

if γ is big enough. We project this on the Λ′-block to get a map

((ϑout(M)⊗C L�)�γ)Λ′ →M

between modules inMT,Λ′ . The direct limit of the left hand side now sta-bilizes, hence we get a map

ϑonϑoutM →M,

which is natural in M . So we constructed a transformation ϑonϑout → id.Let N ∈ MT,Λ. Choose a basis {li} of L consisting of weight vectors

and the dual basis {l�i } of L�. The map

N → (N ⊗C L�)�γ ⊗C L,

n →∑

i

n⊗ l�i ⊗ li

is well-defined for all γ ∈ h�. We compose this map with the projectiononto ((N ⊗C L�)�γ)Λ′ ⊗C L and take the projective limit to get a map

N → ϑon(N)⊗C L.

The image of this map then lies in the Λ-block, so we get an induced mapN → ϑoutϑonN . Hence we constructed a transformation id→ ϑoutϑon.

Now it is not difficult to observe that the compositions

ϑon → ϑonϑoutϑon → ϑon

andϑout → ϑoutϑonϑout → ϑout

are identities. �We proved that ϑon is left adjoint to ϑout. We state an immediate conse-

quence.

Proposition 5.2 (generic case). Suppose thatStabT (λ) = StabT (λ′). Thenthe natural transformations

ϑonϑout → idMT,Λ′

andidMT,Λ

→ ϑoutϑon,

constructed in Proposition 5.1, are isomorphisms of functors. Hence ϑon

and ϑout are inverse equivalences of categories.

Page 20: Centers and translation functors for the category

708 P. Fiebig

Proof. By Proposition 4.1 we have isomorphisms ϑoutMT (w.λ′) ∼=MT (w.λ) and ϑonMT (w.λ) ∼= MT (w.λ′) for all w ∈ WT (Λ). We willfirst show that for all w ∈ WT (Λ) the natural maps

MT (w.λ)→ ϑoutϑonMT (w.λ),ϑonϑoutMT (w.λ′)→MT (w.λ′)

are isomorphisms. We only have to show that they are not zero over the closedpoint K. This follows from the fact that these maps induce adjunctions.

By the five lemma the canonical maps have to be isomorphisms on allmodules with a finite Verma flag, hence the transformations themselves areisomorphisms. �

In the situation of the proposition, take any natural transformation id→ϑonϑout. Then composition with the isomorphism ϑonϑout → id induces anatural transformation of the identity functor, hence an element of the centerofMT,Λ′ and we get the following corollary.

Corollary 5.3. Suppose StabT (λ) = StabT (λ′). The above constructiongives an isomorphism of T -modules{

natural transformationsidMT,Λ′ → ϑonϑout

}∼−→ ZT,Λ′ .

In particular, the above proposition and its corollary apply to the genericcases Q and Rp, where p ⊂ R is a prime ideal which does not contain anyhα for α ∈ ∆. Before we deal with the subgeneric case, we have to examinethe behaviour of translation functors under base change.

5.2 Base change

Let T → T ′ be a homomorphism of local deformation domains. Then Λand Λ′ split under ∼T ′ into equivalence classes which are the orbits inΛ and Λ′ under the subgroup WT ′(Λ) = WT ′(Λ′). Hence, for any w ∈WT ′(Λ)\WT (Λ)/StabT (λ) we have translation functors

ϑon(w) = ϑw.λ′w.λ : MT ′,[w.λ] →MT ′,[w.λ′]

andϑout(w) = ϑw.λ

w.λ′ : MT ′,[w.λ′] →MT ′,[w.λ],

where [w.λ] denotes the respective equivalence classes under∼T ′ . Since thefunctor ·�γ and the projection onto blocks commute with base change, weget the following

Page 21: Centers and translation functors for the category

Centers and translation functors 709

Lemma 5.4. We have natural equivalences of functors

ϑout(·)⊗T T ′ ∼=∏w∈D

ϑout(w)(· ⊗T T ′)

andϑon(·)⊗T T ′ ∼=

∏w∈D

ϑon(w)(· ⊗T T ′),

where D =WT ′(Λ)\WT (Λ)/StabT (λ).

The next lemma shows that we have direct images of natural transfor-mations under base change.

Lemma 5.5. Every natural transformation

idMT,Λ′ → ϑonϑout

induces a natural transformation

idMT ′,Λ′ →∏w∈D

ϑon(w)ϑout(w).

We get a map{natural transformations

idMT,Λ′ → ϑonϑout

}→

∏w∈D

{natural transformations

idMT ′,Λ′ → ϑon(w)ϑout(w)

}.

Proof. Let {Pi} be a system of objects ofMT,Λ′ which generatesMT,Λ′ ,i.e. which has the property that for any truncation O�ν

T,Λ′ and any simple

object L in O�νT,Λ′ there is a Pi which maps onto L and is projective in

O�νT,Λ′ . A natural transformation idMT,Λ′ → ϑonϑout is uniquely determined

by its action on these objects and, vice versa, every collection of mapsPi → ϑonϑoutPi which respects all morphisms Pi → Pj uniquely definesa natural transformation idMT,Λ′ → ϑonϑout. So the proposition followsfrom the fact that {Pi⊗T T ′} is a system inMT ′,Λ′ of the same nature andfrom the preceding lemma. �

Now let T = R = S(0) and Q its quotient field. The preceding lemmatogether with Corollary 5.3 provides a map{

natural transformationsidMR,Λ′ → ϑonϑout

}→

∏w∈WR(Λ)/StabR(λ)

Q.

From the proof of the lemma we see that it is injective.The meaning of this map is the following. Let MR be a Verma module in

MR,Λ′ . Then ϑonϑoutMR is isomorphic to |StabR(λ′)/StabR(λ)| copies

Page 22: Centers and translation functors for the category

710 P. Fiebig

of MR, though there is no canonical isomorphism. Over the generic pointQ, however, the translation functor ϑout canonically splits into a direct sumof |StabR(λ′)/StabR(λ)| functors, hence there is a canonical isomorphism

ϑonϑoutMQ∼=

⊕MQ.

We will see that this isomorphism has a pole of order one along each hyper-plane hα = 0 on which λ′ lies, but not λ.

5.3 Subgeneric case

Let α ∈ ∆re be a real root. In this section we will deal with the case T = Rα.Suppose the stabilizers of λ and λ′ are not equal, i.e. StabRα(λ) = {e} andStabRα(λ′) = {e, sα}. Hence Λ = {λ, sα.λ} and Λ′ = {λ′}, so ORα,Λ

is equivalent to the principal deformed block of sl2 by Proposition 3.4 andORα,Λ′ is equivalent to Rα -mod.

Consider the base change Rα → Q. It induces an injective map{natural transformationsidMRα,Λ′ → ϑonϑout

}→ Q⊕Q.

Proposition 5.6. The above map induces an isomorphism{natural transformationsidMRα,Λ′ → ϑonϑout

}∼−→

{(te, tsα) ∈ Q⊕Q

∣∣∣∣te, tsα ∈ h−1α Rα

te + tsα ∈ Rα

}.

Remark 5.1. The pair (te, tsα) lies in the set on the right hand side if andonly if (hαte,−hαtsα) is an element of ZRα,Λ.

Proof. Let us abbreviate PRα , MRα by P , M and PKα , MKα by P ′, M ′ andlet f ′ denote the specialization at Kα of a map f between Rα-modules. LetX denote the space on the right hand side in the assertion of the proposition.We show that the image of any natural transformation lies inside X . So lett : idMRα,Λ′ → ϑonϑout be a transformation and (te, tsα) its image in Q⊕Q.Composition with the natural transformation can: ϑonϑout → idMRα,Λ′ isgiven by the scalar te + tsα , hence te + tsα ∈ Rα.

Consider ϑoutM(λ′). By Proposition 4.1 it has a Verma flag with sub-quotients M(λ) and M(sα.λ). Suppose λ > sα.λ. Then there is a shortexact sequence

0→M(λ) i→ ϑoutM(λ′) π→M(sα.λ)→ 0.

By Corollary 3.4 there is an element of the center of ORα,Λ which inducesthe scalars 0 and hα on the Verma modules M(λ) and M(sα.λ), resp. Itsaction on ϑoutM(λ′) then induces a map

π : M(sα.λ)→ ϑoutM(λ′).

Page 23: Centers and translation functors for the category

Centers and translation functors 711

Similarly there is an endomorphism of ϑoutM(λ′), inducing the scalars hα

and 0 on M(λ) and M(sα.λ), hence inducing a map

i : ϑoutM(λ′)→M(λ).

By applying the functor ϑon we get the following diagram

ϑonM(λ)

ϑoni��

M(λ′) t �� ϑonϑoutM(λ′)

ϑon i

��

ϑonπ��

can �� M(λ′)

ϑonM(sα.λ)

ϑonπ

��

where t and can are evaluations of the chosen transformation id→ ϑonϑout

and the natural transformation ϑonϑout → id of Proposition 5.1 at the Vermamodule. By definition π ◦ π = hα · id and i ◦ i = hα · id. Hence, on thegeneric point Q, π and i are hα times the splitting maps of the short exactsequence. It follows that the composition

can ◦ ϑoni ◦ ϑoni ◦ t : M(λ′)→M(λ′)

is given by multiplication with the scalar hαte, so hαte ∈ Rα. Analogously,the composition

can ◦ ϑonπ ◦ ϑonπ ◦ t : M(λ′)→M(λ′)

is given by multiplication with hαtsα , hence hαtsα ∈ Rα. So we proved

im({

natural transformationsidMRα,Λ′ → ϑonϑout

})⊂ X.

On the other hand, take any pair (te, tsα) in X . Hence te = h−1α te and

tsα = h−1α tsα with te, tsα ∈ Rα and te + tsα ∈ hαRα. Now we prove

that there is a natural transformation inducing the scalars (te, tsα). Sinceevery module inMRα,Λ′ is isomorphic to a direct sum of copies of M(λ′),giving a natural transformation id → ϑonϑout is the same as giving a mapM(λ′)→ ϑonϑoutM(λ′).

We will need the following technical lemma.

Lemma 5.7. 1. There is an isomorphism ϑoutM(λ′) ∼= P (sα.λ).2. The compositions

can ◦ ϑoni : ϑonM(λ)→M(λ′)

andcan ◦ ϑonπ : ϑonM(sα.λ)→M(λ′)

are isomorphisms.

Page 24: Centers and translation functors for the category

712 P. Fiebig

3. The maps

(ϑoni)′ : ϑonM ′(λ)→ ϑonϑoutM′(λ′)

and

(ϑonπ)′ : ϑonM ′(sα.λ)→ ϑonϑoutM′(λ′)

have the same image.

Proof. We already know that ϑoutM(λ′) and P (sα.λ) have isomorphicVerma subquotients. So we only need to show that they are isomorphic overthe special point Kα. The description of the category OKα,Λ in Proposition3.4 shows that ϑoutM

′(λ′) is either isomorphic to the direct sum of its Vermasubquotients or to P ′(sα.λ).

In the first case we would get dim Hom(M ′(sα.λ), ϑoutM′(λ′)) = 2,

but, by the canonical adjunction of Proposition 5.1, this space is isomorphicto Hom(ϑonM ′(sα.λ), M ′(λ′)) ∼= Hom(M ′(λ′), M ′(λ′)) = Kα whichyields a contradiction. So ϑoutM

′(λ′) ∼= P ′(sα.λ′), hence ϑoutM(λ′) ∼=P (sα.λ′) and we proved 1. It follows that the maps i and π are non-zeroover the special point Kα, because otherwise h−1

α i and h−1α π would be

well-defined splittings of ϑoutM(λ′).Now ϑonM(λ), ϑonM(sα.λ) and M(λ′) are isomorphic. In order to

prove 2, it is enough to show that the maps (can ◦ ϑoni)′ and (can ◦ ϑonπ)′are non-zero. But they are the images of i′ and π′ under the adjunctions

Hom(M ′(λ), ϑoutM′(λ′)) ∼= Hom(ϑonM ′(λ), M ′(λ′))

Hom(M ′(sα.λ), ϑoutM′(λ′)) ∼= Hom(ϑonM ′(sα.λ), M ′(λ′))

of Proposition 5.1, hence non-zero.Finally we prove 3. By construction, the composition π ◦ π ∈

End(M(sα.λ)) is multiplication with hα. Hence it is zero over the spe-cial point Kα, i.e. (π ◦ π)′ = π′ ◦ π′ = 0. So π′ factors over the inclusioni′, hence we get

π′ : M ′(sα.λ)→M ′(λ) i′−→ ϑoutM′(λ′).

Now we apply the functor ϑon to this sequence to get

ϑonπ′ : ϑonM ′(sα.λ)→ ϑonM ′(λ) ϑoni′−→ ϑonϑoutM′(λ′).

The first two modules are isomorphic Verma modules, and, since π′ is non-zero, the first map is an isomorphism. Hence the image of ϑoni′ is the sameas the image of the composition, i.e. the image of ϑonπ′. �

Page 25: Centers and translation functors for the category

Centers and translation functors 713

We finish the proof of the proposition. In view of the preceding lemma we canconstruct maps fe : M(λ′) → ϑonM(λ) and fsα : M(λ′) → ϑonM(sα.λ)such that the compositions

can ◦ ϑoni ◦ fe : M(λ′)→M(λ′)

andcan ◦ ϑonπ ◦ fsα : M(λ′)→M(λ′)

are given by multiplication with te and tsα , resp. We define f := ϑoni ◦fe + ϑonπ ◦ fsα .

ϑonM(λ)

ϑoni��

M(λ′)

fe

�������������f ��

fsα �������������ϑonϑoutM(λ′) can �� M(λ′)

ϑonM(sα.λ)

ϑonπ

��

By construction, f induces the scalars (te, tsα). Hence we have to provethat h−1

α f is a well-defined map. The composition can ◦ f is given bymultiplication with tsα + te ∈ hαRα, hence is zero over the special pointKα. Since by the preceding lemma the images of (ϑoni)′ and (ϑonπ)′ arethe same and since can′, restricted to this image, is an isomorphism, eventhe map f ′ is zero. Hence f := h−1

α f : M(λ′) → ϑonϑoutM(λ′) is well-defined. �So we have determined all possible transformations id→ ϑonϑout over Rα.We will now identify those which induce an adjunction.

Proposition 5.8. Let idMRα,Λ′ → ϑonϑout be a transformation given byscalars (te, tsα). Then the induced map

Hom(ϑout·, ·)→ Hom(·, ϑon·)is an isomorphism if and only if hαte and hαtsα are invertible in Rα.

Proof. Take any transformation

(�) Hom(ϑout·, ·)→ Hom(·, ϑon·).It is an equivalence if and only if Hom(ϑout·, M) → Hom(·, ϑonM) is anisomorphism for all M ∈MRα,Λ. This is the case if and only if the identityis in the image of

Hom(ϑoutϑonM, M)→ Hom(ϑonM, ϑonM),

Page 26: Centers and translation functors for the category

714 P. Fiebig

since a preimage ϑoutϑonM →M of the identity defines a transformation

Hom(·, ϑonM)→ Hom(ϑout·, M),

which is right inverse to the transformation (�). Since we already now thatover the generic point Q the translation functors are biadjoint, every rightinverse is also left inverse.

Hence the natural transformation (�) is an equivalence if and only if theidentity is in the image of Hom(ϑoutϑonM, M) → Hom(ϑonM, ϑonM)for any M . We claim that it is enough to check this for Verma modules M .So assume it for all Verma modules. It suffices to show that it then followsfor the two indecomposable projective objects P (λ) and P (sα.λ). Supposeλ > sα.λ. Then P (λ) is already isomorphic to a Verma module and P (sα.λ)fits into a short exact sequence

0→M(λ)→ P (sα.λ)→M(sα.λ)→ 0.

Now ϑonP (sα.λ) is isomorphic to M(λ′)⊕M(λ′), hence ϑoutϑonP (sα.λ)is, by Lemma 5.7, isomorphic to P (sα.λ)⊕ P (sα.λ), hence projective. Sowe get a diagram whose columns are exact and the first and the third rowsare isomorphisms:

0 0↓ ↓

Hom(ϑoutϑonP, M(λ)) ∼→ Hom(ϑonP, ϑonM(λ))↓ ↓

Hom(ϑoutϑonP, P ) → Hom(ϑonP, ϑonP )↓ ↓

Hom(ϑoutϑonP, M(sα.λ)) ∼→ Hom(ϑonP, ϑonM(sα.λ))↓ ↓0 0

Hence also the middle map is an isomorphism and, in particular, the identityis in its image.

So we proved that the transformation (�) is an equivalence if and onlyif the identity is in the image of the maps

Hom(ϑoutϑonM(λ), M(λ))→ Hom(ϑonM(λ), ϑonM(λ))

and

Hom(ϑoutϑonM(sα.λ), M(sα.λ))→ Hom(ϑonM(sα.λ), ϑonM(sα.λ)).

We now prove that this is the case if and only if hαte and hαtsα are in-vertible in Rα. Choose isomorphisms ϑonM(λ) ∼= ϑonM(sα.λ) ∼= M(λ′)and consider the maps i and π of the proof of Proposition 5.6. They generatethe spaces on the left hand side and their images are hαte · id and hαtsα · id,hence the identities are in the images if and only if hαte and hαtsα areinvertible. �

Page 27: Centers and translation functors for the category

Centers and translation functors 715

5.4 The most singular case

The base change R→ Q induces an injective map{natural transformations

idMR,Λ′ → ϑonϑout

}→

∏w∈D

Q,

where D =WR(Λ)/StabR(λ). Define

c :=∏

hα ∈ Q,

where the product is taken over all positive roots α ∈ ∆R(Λ) such thatStabRαλ �= StabRαλ′, in other words, such that sα stabilizes λ′, but not λ.Under the linear action ofW on Q, c is stabilized by StabR(λ), hence forany w ∈ D, cw := w(c) is well-defined.

Theorem 5.9. The above map is injective and induces an isomorphism{natural transformations

idMR,Λ′ → ϑonϑout

}∼−→

{(tw) ∈

∏w∈D

Q

∣∣∣∣∣ (cwtw) ∈ ZR,Λ

}.

The induced transformation Hom(ϑout·, ·) ∼= Hom(·, ϑon·) is an equiva-lence if and only if (cwtw) is an invertible element of the center.

Proof. The left hand side is the intersection of all its localizations at primeideals of height one in the sense that any collection of transformationsidMRp,Λ′ → ϑonϑout for prime ideals p ⊂ R of height one, which in-duce the same transformation on the generic point Q, uniquely determinesa transformation idMR,Λ′ → ϑonϑout. Hence, by combining Corollary 5.3and the remark following Proposition 5.6, we get the first assertion. Thesecond part follows similarly from Proposition 5.8. �Corollary 5.10. The deformed translation functors

ϑon : MR,Λ →MR,Λ′

andϑout : MR,Λ′ →MR,Λ

are biadjoint. Analogously, the non-deformed translation functors

ϑon : MΛ →MΛ′

andϑout : MΛ′ →MΛ

are biadjoint.

Page 28: Centers and translation functors for the category

716 P. Fiebig

Proof. One direction is already given by Proposition 5.1 and (tw) := (c−1w )

defines an adjunction in the other direction for the deformed case. Lemma5.5, applied to the base change R → C, provides a natural transforma-tion idMΛ′ → ϑonϑout in the non-deformed case. Using the descriptionat the beginning of Sect. 5 we deduce that this transformation induces anadjunction. �

The next corollary shows that translation functors can be used to constructprojective objects.

Corollary 5.11. Let P ∈ OR,Λ′ or P ∈ OΛ be a projective object. ThenϑoutP is projective inOR,Λ orOΛ, resp. The analogous statement holds forϑon.

Proof. Choose any object Q ∈ MR,Λ (Q ∈ MΛ, resp.) which admits asurjection Q→→ϑoutP . We have to show that this surjection splits, i.e. thatthe identity is in the image of the induced map

Hom(ϑoutP, Q)→ Hom(ϑoutP, ϑoutP ).

By adjunction we get a commutative diagram

Hom(ϑoutP, Q)→ Hom(ϑoutP, ϑoutP )‖ ‖

Hom(P, ϑonQ) → Hom(P, ϑonϑoutP ).

Now Q→→ϑoutP a priori need not fit into a short exact sequence ofmodules with Verma flags. But in any case, due to the right exactness of·�γ , ϑon respects surjectivity, hence ϑonQ→ ϑonϑoutP is surjective. SinceP is projective, the bottom map is surjective, hence also the upper map issurjective. �Remark 5.2. We always assumed StabT (λ) ⊂ StabT (λ′). The oppositecase StabT (λ′) ⊂ StabT (λ) is not symmetric due to the essentially differentdefinition of the two translation functors. But analogous results can be provedsimilarly by determining the transformations ϑoutϑon → idMT,Λ

.

References

[DGK] Deodhar, V.V., Gabber, O., Kac, V.G.: Structure of some categories of representa-tions of infinite-dimensional Lie algebras. Adv. in Math. 45, 92–116 (1982)

[FF] Feigin, B., Frenkel, E.: Affine Kac–Moody algebras at the critical level andGelfand-Dikii algebras. Int. J. Mod. Phys. A 7(Suppl. 1A), 197–215 (1992)

[Jan] Jantzen, J.C.: Moduln mit einem hochsten Gewicht. Lecture Notes in Mathematics750, Springer 1979

[Kac] Kac, V.G.: Infinite dimensional Lie algebras, 3rd ed. Cambridge University Press1990

Page 29: Centers and translation functors for the category

Centers and translation functors 717

[KK] Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight ofinfinite-dimensional Lie algebras. Adv. in Math. 34, 97–108 (1979)

[KoKu] Kostant, B., Kumar, S.: The nil Hecke ring and cohomology of G/P for a Kac–Moody group G. Adv. in Math. 62, 187–237 (1986)

[Ma] Matsumura, H.: Commutative ring theory. Cambridge Studies in Advanced Math-ematics 8, Cambridge University Press 1986

[Nei] Neidhardt, W.: Translation to and fro over Kac–Moody algebras. Pacific Journalof Mathematics 139(1), 107–153 (1989)

[RCW] Rocha-Caridi, A., Wallach, N.R.: Projective modules over graded Lie algebras.Mathematische Zeitschrift 180, 151–177 (1982)

[Soe] Soergel, W.: Kategorie O, perverse Garben, und Moduln uber den Koinvariantenzur Weylgruppe. J. Am. Math. Soc. 3(2), 421–445 (1990)