Twisted Verma modules and

sheaves on moment graphs

Der Naturwissenschaftlichen Fakultat

der Friedrich-Alexander-Universitat Erlangen-Nurnberg

zur

Erlangung des Doktorgrades

Dr. rer. nat.

vorgelegt vonNarendiran Sivanesan

aus Backnang

Als Dissertation genehmigt

von der Naturwissenschaftlichen Fakultat

der Friedrich-Alexander-Universitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 25. Juli 2014

Vorsitzender des Promotionsorgans: Prof. Dr. Johannes Barth

Gutachter/in: Prof. Dr. Peter Fiebig

Prof. Dr. Friedrich Knop

Zusammenfassung

In der vorliegenden Arbeit beschaftigen wir uns mit der Darstellungstheo-rie symmetrisierbarer Kac-Moody-Algebren. Hierbei beschranken wir unsauf die Bernstein-Gel′fand-Gel′fand Kategorie O, deren einfache ObjekteHochstgewichtsmoduln sind. Wichtig zur Bestimmung von Charakteren ein-facher Objekte sind die sogenannten Vermamoduln in Kategorie O. UnserenFokus legen wir speziell auf w-getwistete Vermamoduln, wobei w ein Ele-ment der Weylgruppe bezeichnet: dies sind Objekte, die denselben Charak-ter wie Vermamoduln besitzen, sich aber durch eine andere Modulstrukturauszeichnen.Wir konstruieren projektive Decken in der vollen Unterkategorie der Modulnmit w-getwisteter Vermafahne. Insbesondere zeigen wir, dass die Kategorieder Moduln mit w-getwisteter Vermafahne genugend Projektive besitzt.Mithilfe dieser projektiven Objekte konnen wir das Zentrum eines deform-ierten Blocks der Kategorie der Moduln mit w-getwisteter Vermafahne bes-timmen. Wir sehen, dass dieses mit dem Zentrum eines deformierten Blocksder Kategorie O ubereinstimmt.Schlussendlich zeigen wir, dass im Falle eines deformierten Blocks im nega-tiven Level die Kategorie der Moduln mit w-getwisteter Vermafahne aquiva-lent ist zu der Kategorie von Garben mit Vermafahne auf einem Impuls-graphen mit w-getwisteter Ordnung. Diese Aquivalenz zeigt, dass fur be-liebige Weylgruppenelemente w die zugehorigen Kategorien von Moduln mitw-getwisteter Vermafahne zueinander und damit auch aquivalent zur Kat-egorie der Moduln mit Vermafahne sind. Dadurch konnen wir die Multi-plizitaten w-getwisteter Vermamoduln in w-getwisteten projektiven Deckenin Verbindung setzen zu Multiplizitaten einfacher Hochstgewichtsmoduln inVermamoduln. Dieses Phanomen kann man als eine w-getwistete Versiondes Bernstein-Gel′fand-Gel′fand Reziprozitatstheorems auffassen.

Abstract

In the present thesis, we study the representation theory of a symmetriz-able Kac-Moody algebra. We restrict ourselves to the Bernstein-Gel′fand-Gel′fand category O, in which the simple objects are highest weight modules.In determining the characters of simple objects, the so-called Verma modulesplay an important role in category O. We focus our attention on w-twistedVerma modules, where w denotes an element of the Weyl group: these areobjects which have the same character as ordinary Verma modules, but ex-hibit a different module structure.We construct projective covers in the full subcategory of modules with aw-twisted Verma flag. In particular, we show that the category of moduleswith a w-twisted Verma flag contains enough projectives. With the help ofthese projective objects, we can determine the center of a deformed blockof the category of modules with a w-twisted Verma flag. We notice that itcoincides with the center of a deformed block of the larger category O.Finally, we show that a deformed block of negative level of the categoryof modules with a w-twisted Verma flag is equivalent to the category ofsheaves with a Verma flag on a moment graph with w-twisted order. Thisequivalence shows that for arbitrary Weyl group elements w, the respectivecategories of modules with a w-twisted Verma flag are equivalent to eachother. In particular, they are also equivalent to the category of modules witha Verma flag. This way, we can relate the multiplicities of w-twisted Vermamodules in w-twisted projective covers to multiplicities of simple highestweight modules in Verma modules. This phenomenon can be considered asa w-twisted version of the Bernstein-Gel′fand-Gel′fand reciprocity theorem.

Contents

Introduction i

1 Kac-Moody algebras 1

2 Deformed Category O 7

2.1 The category OA . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Duality on OA . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Deformed Verma and Nabla modules . . . . . . . . . . 8

2.1.3 Characters . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.4 Simple objects in OA . . . . . . . . . . . . . . . . . . . 10

2.2 Projective objects in OA . . . . . . . . . . . . . . . . . . . . . 10

2.3 Block decomposition . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Subgeneric blocks . . . . . . . . . . . . . . . . . . . . . 12

2.3.3 Further description of blocks . . . . . . . . . . . . . . 13

2.4 Translation functors . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Twisting and Tilting functors . . . . . . . . . . . . . . . . . . 17

2.5.1 The semi-regular bimodule . . . . . . . . . . . . . . . 17

2.5.2 The Twisting functor . . . . . . . . . . . . . . . . . . 19

2.5.3 The Tilting functor . . . . . . . . . . . . . . . . . . . . 21

3 Modules with a twisted Verma flag 27

3.1 Twisted Verma modules . . . . . . . . . . . . . . . . . . . . . 27

3.2 Modules with a twisted Verma flag . . . . . . . . . . . . . . . 34

3.3 Projective objects . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Hom and Ext1 . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2 Projective covers . . . . . . . . . . . . . . . . . . . . . 49

3.4 The center of OV FS,Λ

. . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.1 The center of a category . . . . . . . . . . . . . . . . . 53

3.4.2 The subgeneric case . . . . . . . . . . . . . . . . . . . 57

3.4.3 The general case . . . . . . . . . . . . . . . . . . . . . 61

CONTENTS

4 Combinatorics 654.1 Sheaves on Moment Graphs . . . . . . . . . . . . . . . . . . . 654.2 Z-modules and Localization . . . . . . . . . . . . . . . . . . . 684.3 Projective objects . . . . . . . . . . . . . . . . . . . . . . . . . 724.4 The connection to Representation Theory . . . . . . . . . . . 764.5 Homological algebra of BMP sheaves . . . . . . . . . . . . . . 77

5 An equivalence of categories 815.1 A combinatorial Twisting functor . . . . . . . . . . . . . . . . 815.2 Tw preserves exactness . . . . . . . . . . . . . . . . . . . . . . 865.3 An equivalence of categories . . . . . . . . . . . . . . . . . . . 90

Bibliography 96

Introduction

Let h ⊆ b ⊆ g be a semisimple Lie algebra with a Borel subalgebra b anda Cartan subalgebra h. We denote by U := U(g) the universal envelopingalgebra of g. In [BGG76], Bernstein, Gel’fand and Gel’fand introduced acertain subcategory of U−mod, the so-called BGG category O. It is definedas the full subcategory of modules which are

1. finitely generated over U,

2. locally finite under the action of b,

3. semisimple under the action of h.

This category has very nice properties, in particular all simple objects in Oare highest weight modules. For every weight λ ∈ h∗ there exists a simplemodule L(λ), and in fact this correspondence is one-to-one. Very importantobjects in O are the “universal” highest weight modules, the so-called Vermamodules ∆(λ) for λ ∈ h∗. Their importance stems from the fact that due tothe universal nature of their construction, it is very easy to calculate theircharacter. On the other hand, the guiding problem in representation theoryis to determine the character of the simple objects L(λ). Now all objectsin O are of finite length, that means that they possess a Jordan-Holdercomposition series of finite length. If we denote by

[∆(λ) : L(µ)]

the multiplicity of L(µ) in such a series of ∆(λ), we can relate the characterof L(µ) to the character of a Verma module in the following way:

ch ∆(λ) =∑σ∈h∗

[∆(λ) : L(σ)] ch L(σ).

Hence, we have to determine the multiplicities [∆(λ) : L(µ)] to calculatech L(µ). This seemingly harmless problem lead to the development of avariety of techniques and machinery. We need to introduce some notation.The highest weights of simple modules that can occur in a composition seriesof a Verma module lie in the orbit of the dot-action of the Weyl group Wof a weight λ. Let x, y ∈ W. In [KL79], Kazhdan and Lusztig formulatedthe famous

i

Conjecture (Kazhdan-Lusztig Conjecture, cf. [KL79]).

[∆(x · 0) : L(y · 0)] = Px,y(1),

where the right-hand side is a Kazhdan-Lusztig polynomial evaluated at 1.

This conjecture (respectively an equivalent formulation) was proved inde-pendently by Beilinson and Bernstein in [BB81] and Brylinski and Kashi-wara in [BK81]. The idea behind the proof is to relate the coefficients ofKazhdan-Lusztig polynomials to certain intersection cohomology groups ofSchubert varieties in the corresponding dual flag variety G/B (cf. [KL80b]).Then the Riemann-Hilbert correspondence states that under the de Rhamfunctor, these intersection cohomology complexes correspond to irreducibleD-modules on the flag variety.Finally, the connection between Verma mod-ules and Kazhdan-Lusztig polynomials is established by localizing Vermamodules to certain D-modules on the flag variety (Beilinson-Bernstein lo-calization).A different approach was presented by Soergel in [Soe90]. Here one exploitsthe fact that in O there exists a projective cover P(x ·λ) L(x ·λ) for everysimple object L(x ·λ). Furthermore, every P(x ·λ) admits a Verma flag, thatmeans a finite filtration by submodules where the subquotients at each stepare isomorphic to a Verma module. Now the mediator role of the Vermamodule ∆(x · λ) between P(y · λ) and L(y · λ) is explained in the following

Theorem (BGG Reciprocity Theorem, cf. [BGG76]). We have

(P(y · 0) : ∆(x · 0)) = [∆(x · 0) : L(y · 0)],

where (· : ·) on the left-hand side denotes the multiplicity of a Verma modulein a Verma flag.

So one can shift the classical problem to the question of determining themultiplicities of Verma modules in a Verma flag of a projective cover. So-ergel showed that in the language of the algebra of coinvariants of the Weylgroup W of g, projective covers and intersection cohomology complexes ofSchubert varieties in G/B yield the same combinatorial objects. In [Soe92],equivariant analogues of these combinatorial objects are introduced whichlater would be called “Soergel bimodules”.In [BM01], Braden and MacPherson introduced a new technique to com-pute the equivariant intersection cohomology of complex, projective vari-eties equipped with an equivariant formal torus action. Namely, they aredescribed by the so-called canonical sheaf on the associated moment graph.Now let us consider a symmetrizable Kac-Moody algebra g. In this set-ting, Deodhar, Gabber and Kac formulated a generalization of the Kazhdan-Lusztig conjectures (cf. [DGK82]) which was proven in [Kas90]. In [Fie08a],Fiebig picks up the machinery developed in [BM01] to localize projective

ii

covers as canonical sheaves on a moment graph. For this, he has to workwith a deformed version of category O, but since the deformation does notchange multiplicities in modules with a Verma flag, the important informa-tion is preserved. Fiebig formulates a conjecture on the rank of the stalksof canonical sheaves, of which we give a non-graded version:

Conjecture (Multiplicity conjecture, cf. [Fie10], Conjecture 4.4.).

rk B(y)x = Px,y(1),

where B(y)x denotes the stalk of the canonical sheaf B(y) at x.

The main result in [Fie08a] is the following

Theorem (cf. [Fie08a], cf. Theorem 7.1. and Proposition 7.2.). We havean equivalence of categories

V : OVFS,Λ→ Z(Λ)−modVF,

under which projective covers are in one-to-one relation to canonical mod-ules.

Here, the left-hand side denotes the equivariant version of modules with aVerma flag (which includes the full subcategory of projectives) and the right-hand side is a category of combinatorial objects with a certain cofiltration.This combinatorial category includes the full subcategory of canonical mod-ules, which are global sections of canonical sheaves. Without going intofurther details, the theorem above yields a remarkable equality:

(PS(y · 0) : ∆S(x · 0)) = rk B(y)x.

Since the deformation does not change multiplicities in projective covers,the left-hand side equals

[∆(x · 0) : L(y · 0)].

Hence, the multiplicity conjecture is equivalent to the Kazhdan-Lusztig con-jecture. The multiplicity conjecture holds if the characteristic of the groundfield is zero. In positive characteristic, it holds if the the characteristic isbigger than an explicit bound which was found by Fiebig (cf. [Fie12b]).However, the proof still makes use of the decomposition theorem by Beilin-son, Bernstein, Deligne and Gabber.Only recently, Elias and Williamson proved that Hodge theoretic proper-ties hold for Soergel bimodules and deduced Soergel’s conjecture on thecharacter of indecomposable Soergel bimodules (which is equivalent to themultiplicity conjecture in characteristic zero) from which they were able tofinally give a purely algebraic proof of the Kazhdan-Lusztig conjecture (cf.[EW12]).Hence, the theory of sheaves on moment graphs provides a purely algebraicapproach to obtain character formulas for simple highest weight modules.

iii

The present thesis

In this thesis, we generalize the main result of [Fie08a] to modules witha w-twisted Verma flag. Hereby we obtain a method to determine multi-plicity formulas for w-twisted projective covers by linking this problem tocalculating the rank of the stalks of w-twisted canonical sheaves. Since wefurthermore prove that the category of modules with a w-twisted Verma flagis equivalent to the category of modules with a Verma flag, one could namethis phenomenon “w-twisted BGG reciprocity principle”.A w-twisted Verma module ∆w(λ) is an indecomposable object in O whichhas the same character as ∆(λ), but exhibits a different module structure.In the case of a semisimple Lie algebra g, we have

∆w(λ) ∼=

∆(λ) if w = e

∇(λ) if w = w0,

where e denotes the neutral element and w0 denotes the longest elementin the Weyl group W. With ∇(λ) we denote the dual of ∆(λ). Thus,∆w(λ) is an indecomposable module “between a Verma and its dual”. Onthe combinatorial level, namely on the level of moment graphs which aregraphs together with a partial order on their set of vertices, one cannotdistinguish between a Verma sheaf and a w-twisted Verma sheaf. However,representation theoretically they display a different extension behavior. Wereflect this via associating the same moment graph to both categories, butin the latter case we twist the order on the moment graph by the elementw. Now the sheaves on this “w-twisted moment graph” model the extensionstructure of w-twisted Verma modules, and we obtain an equivalence ofcategories which reproduces the result in [Fie08a] in the case w = e.

The present thesis is organized as follows:

Chapter 1

The first chapter deals with symmetrizable Kac-Moody algebras and theirWeyl groups. We will further recall some statements on the combinatoricsof Coxeter groups. Details can be found in [Kac90], [Car05] and [BB05].

Chapter 2

We define the deformed categoryOS for a symmetrizable Kac-Moody algebra

g. Here, S := S(h)(0) denotes the symmetric algebra of h localized at themaximal ideal corresponding to 0. We recall results on the existence ofprojective covers and the block decomposition of OS. In particular, wedescribe the structure of generic and subgeneric blocks (cf. [Fie03]).In the second part of Chapter 2, we introduce three important functors

iv

on OS. Firstly, we recall results by Fiebig on translation functors on thecategory of modules with a Verma flag OVF

S(cf. [Fie03]). These functors

can be used to construct projective objects in an inductive way. Then wedefine the twisting functor Tw associated with w ∈ W, describe its propertiesand construct its right adjoint Gw. Lastly, we define the tilting functor t (cf.[Soe98]). It induces an equivalence between OVF

Sand its opposed category

(OVFS

)opp, which allows us to pass from blocks of negative level to blocks ofpositive level. In general, it is the latter case where projective objects exist.

Chapter 3

Oriented towards [AL03] we define formal properties of a family of twistedVerma modules ∆w

S(λ). We show existence and uniqueness of a family of

twisted Verma modules (Theorem 3.1.1 and Theorem 3.1.3). For this, weheavily exploit the subgeneric situation, since in those cases a twisted Vermamodule is either a Verma module or its dual (Theorem 3.1.2).Then we investigate the extension behavior of w-twisted Verma modules(Theorem 3.2.1) and give a criterion for modules to admit a w-twisted Vermaflag (Theorem 3.2.2).We show that the Hom-spaces and Ext1-groups of w-twisted Verma mod-ules exhibit good base change behavior (Theorem 3.3.1 and Theorem 3.3.2).This shows that localizing preserves projectivity, a very crucial property toreconstruct a general situation from generic and subgeneric ones (Corollary3.3.1).By constructing maximal extensions of w-twisted Verma modules we obtainw-twisted projective covers (Theorem 3.3.3 and Corollary 3.3.3).Finally, we use our results to determine the categorical center of Ow-VF

S,Λ,

which is denoted by ZwS,Λ

. It is defined as the ring of endotransformations of

the identity functor on Ow−V FS

. We approach this problem by determiningthe center in generic and subgeneric blocks (Proposition 3.4.3 and Theorem3.4.1) at first, and then the general case is obtained by taking the intersec-tion over generic and subgeneric situations (Proposition 3.4.4). As the firstmain result of this thesis, we obtain the following

Theorem. ZwS,Λ∼=xλ ∈

∏λ∈Λ S |xλ ≡ xsα·λ mod α ∀α ∈ R+(Λ)

(cf. Theorem 3.4.2). We directly see that the center of Ow-VF

S,Λdoes not

depend on the Weyl group element w. Furthermore, it is equal to the centerof the larger category OS,Λ.

Chapter 4

Here, we define sheaves on moment graphs. As a first example we definethe structure sheaf on an arbitrary moment graph G. We define the global

v

section functor Γ and see that it is a functor from the category of sheaves onmoment graphs to the category of modules over the global sections of thestructure sheaf, in formulae

Γ : SHS(G) → ZS −modf

(note that we have imposed some finiteness conditions on both sides). Thenwe construct its adjoint functor, the Localization functor. We see that if werestrict ourselves to the essential images of both functors, we have an equiv-alence of categories, so we can think of the objects of this category (denotedby C(G)) as objects having “both a local and a global nature”. Within thissubcategory, we identify the objects which are reflexive over the base ringC(G)ref and the objects which admit a so-called Verma flag C(G)VF.We recall Theorem 4.1 in [Fie08a] which shows that we can define an ex-act structure on ZS − modf in the sense of Quillen (cf. [Qui73]). Finally,we state an algorithm to construct indecomposable projective sheaves, theso-called (indecomposable) Braden-MacPherson sheaves B(x) (cf. Theorem4.3.2).We explain the connection of these combinatorial objects to representationtheory by assigning a moment graph to a block of negative level. Then wedefine a duality on C(G)VF and the work of Fiebig (cf. [Fie08a], Section4.6) shows that the duality induces an equivalence between C(G)VF and itsopposed category on the same moment graph, but with the order reversed.Then we recall the main statement of [Fie08a] (cf. [Fie08a], Theorem 7.1and Proposition 7.2) which establishes an equivalence between the moduleswith a Verma flag in a block of negative level and C(G)VF. With the equiv-alence induced by the duality above and with the help of the tilting dualityexplained in Chapter 2.5.3 one deduces the according statement for a blockof positive level.Lastly, we prove B(x)δz ∼= Ext1

Z(B(x)<z,V(z)) (cf. Theorem 4.5.2). Thissupports the idea that the Braden-MacPherson algorithm works analogouslyto the representation theoretic method of constructing projective covers asmaximal extensions, and could serve as a starting point for further researchon the homological properties of Braden-MacPherson sheaves.

Chapter 5

We define a combinatorial twist functor Tw on Z − modf . Then we showthat Tw takes canonical modules in Z − modVF to canonical modules inZ −modw-VF (Theorem 5.1.1).In the representation theoretic setting, we prove that Tw takes projectivecovers in OVF

Sto projective covers in Ow-VF

S(Theorem 5.2.1).

Now we define the w-twisted structure functor

Vw : Ow-VFS,Λ

→ Z −modw-VF

vi

(cf. Corollary 5.3.1.). We show that we have a natural isotransformation offunctors

Tw ⇒ Vw Tw V−1

(cf. Proposition 5.3.1), from which we deduce the main result of this thesis,namely (cf. Theorem 5.3.1)

Theorem. Let Λ ⊂ h∗/ ∼ be a block of negative level and w ∈ W. Thenthe functors

Tw : OVFS,Λ→ Ow-VF

S,Λ

and

Vw : Ow-VFS,Λ

→ Z −modw-VF

are equivalences of exact categories under which projective covers correspondto projective covers.

As a corollary, we obtain a w-twisted version of the BGG reciprocity (cf.Corollary 5.3.3):

Corollary (“w-twisted BGG reciprocity”). Let Λ be a block of negativelevel, σ ∈ Λ antidominant, ν ∈ Λ and x, y ∈ W(Λ) such that x ·σ, y ·σ ≤w ν.Let Pw

S(x · σ) denote the projective cover of ∆w

S(x · σ) in Ow-VF

S.Λ≤wν. We have

(PwS

(x · σ) : ∆wS

(y · σ)) = [∆(w−1y · σ) : L(w−1x · σ)],

and the multiplicities can be determined by calculating the rank of Bw(x)y.

The classical BGG Reciprocity Theorem requires the projective covers to beprojective in O (respectively in OS), not just in a full subcategory. Via thetheory of sheaves on moment graphs, one can circumvent this problem andobtain the result above. This concludes our project to determine multiplicityformulas for w-twisted projective covers.

vii

viii

Chapter 1

Kac-Moody algebras

In this chapter we provide the definition of a symmetrizable Kac-Moodyalgebra. A detailed exposition of the subject can be found in [Car05] and in[Kac90].

Definition 1.0.1. An (n×n)-matrix C = (cij)i,j=1,...n is called a generalizedCartan matrix (abbreviated to GCM), if

• cii = 2 for all i ∈ 1, ..., n,

• cij ∈ Z≤0 if i 6= j,

• cij = 0 implies cji = 0.

Definition 1.0.2. Let C be a GCM. C is called symmetrizable, if thereexists an invertible diagonal matrix D = diag(d1, ..., dn) and a symmetrizablematrix B = (bij)i,j=1,...,n such that

C = DB.

Definition 1.0.3. Let A be a (n × n)-matrix over C. A realization ofA = (aij)ij is a triple (h,Π, Π) where

• h is a finite dimensional vector space over C,

• Π = α1, ..., αn is a set of linear independent elements of h∗ (the dualof h),

• Π = α1, ..., αn is a set of linear independent elements of h such that

αi(αj) = aij

for all i, j ∈ 1, ..., n.

1

Definition 1.0.4. A minimal realization of A is a realization where

dimC h = 2n− rank A

holds.

Proposition 1.0.1 (cf. [Car05], Proposition 14.2). Any (n×n)-matrix overC has a minimal realization.

Now let C be a symmetrizable GCM of rank l with a minimal realization(h,Π, Π). Denote by g the Kac-Moody algebra associated with C which weget by constructing the Lie algebra L(C) according to the Serre relations ofa set of Chevalley generators corresponding to C, and then by taking thequotient by the unique maximal ideal in L(C) which intersects h trivially.

Definition 1.0.5. The set Π is called the set of simple roots, and Π theset of simple coroots.

The vector space h inherits the structure of a Cartan subalgebra in g, andunder its adjoint action we get a decomposition into root spaces

g =⊕α∈h∗

gα,

where the subspaces on the right-hand side are defined as

gα := g ∈ g | [h, g] = α(h)g ∀h ∈ h.

We get a root system

R := α ∈ h∗ | gα 6= 0, α 6= 0.Within this root system, the simple roots Π determine the subset of positiveroots of R which is denoted by R+.

Definition 1.0.6. We define a partial order on h∗ as follows: let λ, µ ∈ h∗,then

µ ≤ λ ⇐⇒ λ− µ ∈ 〈R+〉Z≥0.

We define n− := ⊕α<0gα and n+ := ⊕α>0gα. By this we get a triangulardecomposition of g

g = n− ⊕ h⊕ n+.

We set b := n+ ⊕ h as the Borel subalgebra of g with respect to Π.

Now fix a set of Chevalley generators e1, ...en ∪ f1, ..., fn of g. TheChevalley involution σ : g→ g is given by

σ(ei) = −fiσ(fi) = −eiσ(h) = −h

2

for h ∈ h.

We want to define a Z-grading on g:The multiplicity of any α ∈ R is defined as mult(α) := dimC gα. For α ∈ Rsuch that α =

∑i liαi for αi ∈ Π, li ∈ Z, we define its height by htα =

∑i li.

Definition 1.0.7. The principal grading on g is a Z-grading

g =⊕i∈Z

gi

wheregi :=

⊕α∈R,htα=i

gα.

Denote by h′ :=∑

iCαi the subspace of h generated by the simple corootsand denote by h′′ the vector space complement of h′ in h. For C symmetriz-able we can define a C-valued form (·|·) on h by

(αi|h) := 〈αi, h〉di for h ∈ h, i = 1, ..., n

(h′|h′′) := 0 for h′, h′′ ∈ h′′,

where 〈·, ·〉 : h∗ × h → C denotes the natural pairing. We can extend thisform to a bilinear form (·, ·) on g such that

• (·|·) is symmetric,

• (·|·) non-degenerate on g and non-degenerate if restricted to h,

• invariant, that means ([x, y]|z) = (x|[y, z]) for all x, y, z ∈ g.

Since (·|·) is non-degenerate, it defines an isomorphism h→h∗ and henceallows us to define a symmetric, non-degenerate bilinear form on h∗ whichwe also denote by (·|·).

For αi ∈ Π we define the fundamental reflection sαi by

sαi : h∗ → h∗

λ 7→ λ− 〈λ, αi〉αi.

We further define the Weyl group W ⊂ GL(h∗) by

W := 〈sαi |i = 1, ..., n〉.

We denote by S := sα |α ∈ Π the set of simple reflections. Then (W,S)is a Coxeter system. The set W(α1, ..., αn) is called the set of real rootsRre, and accordingly Rim := R \Rre the set of imaginary roots. We define

Rre,+ := Rre ∩R+

Rim,+ := Rim ∩R+

3

to be the set of real, positive roots (respectively imaginary, positive roots).A root is imaginary if and only if (α|α) ≤ 0. For any positive root α thevector space

[gα, g−α]

is one-dimensional. Its coroot α is the element in h which is uniquely definedby the two properties

α ∈ [gα, g−α],

〈α, α〉 = 2.

We denote by T ⊂ W the set of reflections defined as

T := sα |α ∈ Rre,+ = wsw−1 |w ∈ W, s ∈ S.

We will denote by αt the positive real root corresponding to the reflectiont ∈ T .

We choose an element ρ ∈ h∗ such that 〈ρ, αi〉 = 1 for any simple corootαi ∈ Π. In the finite-dimensional case, ρ := 1/2

∑α∈R+ α provides such an

element (and it is hereby uniquely determined). Now (·|·) is invariant underthe action of W on h∗. From now on, we will work with the action of Wthat is shifted by −ρ. We define the dot-action of W on h∗ by

w · λ := w(λ+ ρ)− ρ.

Definition 1.0.8. The Kac-Moody algebra g is called indecomposable if itcorresponds to an indecomposable, symmetrizable GCM C.

Definition 1.0.9 (cf. [Car05], Definition 15.1). A GCM C has finite type,if

• detC 6= 0,

• there exists u > 0 with Cu > 0,

• Cu > 0 implies u = 0 or u > 0.

C has affine type, if

• rkC = n− 1,

• there exists u > 0 such that Cu = 0,

• Cu ≥ 0 implies Cu = 0.

C has indefinite type, if

• there exists u > 0 such that Cu < 0,

4

• Cu ≥ 0 and u ≥ 0 imply u = 0.

All vectors u are meant to be in Rn.

We conclude with a classification of indecomposable symmetrizable Kac-Moody algebras:

Proposition 1.0.2 (cf. [Car05], Proposition 15.14). Assume g is an inde-composable, symmetrizable Kac-Moody algebra. We can classify g to be inone of the following three classes:

1. g is of finite type if and only if C is positive definite,

2. g is of affine type if and only if C is positive semidefinite of corank 1,

3. g is of indefinite type if and only if C fulfills none of the above.

5

6

Chapter 2

Deformed Category O

2.1 The category OAIn this chapter we introduce the equivariant or deformed version of the BGGcategory O. All the details and proofs can be found in [Fie03], [Fie06],[Fie08a] and [Fie12a]. Let g be a symmetrizable Kac-Moody algebra. Fix aBorel subalgebra b and a Cartan subalgebra h. By S := S(h) we mean thesymmetric algebra of h, and in the following any unital, associative, commu-tative, finitely generated and noetherian S-algebra is called a deformationalgebra. The structure map of a deformation algebra A is denoted by

τ : S→ A.

By U := U(g) we denote the universal enveloping algebra of g. The tensorproduct gA := g ⊗C A has an induced structure of a Lie algebra, and wedenote its enveloping algebra by UA. We identify (hA)∗ with h∗⊗CA. Henceevery λ ∈ h∗ can be viewed as an element λ ⊗C 1A in h∗ ⊗C A ∼= h∗A. Byg−A−mod we denote the category of g-A-bimodules. Since we require A tobe commutative, this category is equivalent to the category of gA-modulesgA −mod.Let M ∈ gA −mod and let λ ∈ h∗. Then

Mλ := m ∈M |h.m = (λ+ τ)(h)m for all h ∈ h

is the deformed weight space of M of weight λ. Here, we view λ + τ as anelement of h∗A. M is called a weight module if it decomposes as

M = ⊕λ∈h∗Mλ.

We say M is locally bA-finite if for every m ∈ M the submodule 〈m〉bA isfinitely generated over A.

Definition 2.1.1. The deformed category O is the full subcategory ofgA −mod which consists of modules M such that

7

1. M is a weight module,

2. M is locally bA-finite.

The category OA is abelian. If A = K is a field, then OK is the subcategoryof the usual category O over g⊗C K consisting of all objects whose weightslie in the complex affine subspace τ + h∗ ⊂ HomK(h ⊗C K,K). We get theusual category O over C by taking the canonical map

S S /S h

as the structure map τ .

2.1.1 Duality on OAIn the preceding chapter we defined the Chevalley automorphism σ : g→ g.For M ∈ OA we define

d(M) :=⊕λ∈h∗

M∗λ ,

where M∗λ denotes the A-dual of Mλ. Let f ∈ d(M), g ∈ g and m ∈ M .The A-module d(M) gets the structure of a g-module by setting

g.f(m) := f(σ(g).m).

This yields a contravariant functor

d : OA → OA.

We have a natural map M → d(d(M)) which is an isomorphism if M is freeover A with weight spaces of finite rank.

2.1.2 Deformed Verma and Nabla modules

For any λ ∈ h∗, the composition

U(b)→ U(h) = S(h)λ+τ−−→ A

defines a bA-module on A which we denote by Aλ. The deformed Vermamodule with highest weight λ is defined as

∆A(λ) := U⊗U(b)Aλ.

We denote its dual d(∆A(λ)) by ∇A(λ) and call it the deformed Nabla mod-ule. Due to their construction, they possess a weight space decompositionsuch that

∆A(λ)µ 6= 0⇒ µ ≤ λ(the same holds for ∇A(λ)). Furthermore, since both objects are free overA with weight spaces of finite rank, we have d(∇A(λ)) ∼= ∆A(λ).We have the following

8

Lemma 2.1.1 (cf. [Soe08], Proposition 2.12.). Let A be a deformationalgebra and let λ, µ ∈ h∗.

1. The restriction on the highest weight space together with the canonicalidentifications ∆A(λ)λ ∼= A and ∇A(λ)λ ∼= A induce an isomorphism

HomOA(∆A(λ),∇A(µ)) ∼=

A if µ = λ

0 if µ 6= λ

2. Ext1OA(∆A(λ),∇A(µ)) = 0.

Remark 2.1.1. Even though the proof in [Soe08] is given for g semisimple,it generalizes to symmetrizable Kac-Moody algebras.

Definition 2.1.2. Let M ∈ OA. M has a deformed Verma flag if thereexists a finite sequence of submodules

0 = M0 ⊂ ... ⊂Mn = M,

such that the subquotient Mi+1/Mi∼= ∆A(λi) for all i ∈ 0, ..., n − 1 and

λi ∈ h∗.

Any module with a deformed Verma flag is free over A. Any direct summandof a module with a deformed Verma flag admits a deformed Verma flag.

Remark 2.1.2. In the case where A = Q is a field, the simple head LQ(λ)of ∆Q(λ) is the socle of ∇Q(λ) for all λ ∈ h∗. If τ : S→ Q is such that

(λ+ τ)(α) 6∈ Z ⊂ Q

for every weight λ ∈ h∗ and every coroot α ∈ R∗, the category OQ becomessemisimple. Hence the composition of the maps

∆Q(λ) LQ(λ)→LQ(λ) → ∇Q(λ)

induces an isomorphism ∆Q(λ)→∇Q(λ).

2.1.3 Characters

Denote by Z[h∗] the group ring of h∗ (considered as an abelian group). Let

Z[h∗] ⊂ Z[[h∗]] be the subring which consists of formal sums∑λ∈h∗

aλeλ ∈ Z[[h∗]]

such that there exists a finite subset λ1, ..., λn ⊂ h∗ with the followingproperty: if aλ 6= 0, then there exists i ∈ 1, ..., n with λ ≤ λi. Denote byOfreeA the full subcategory of OA which consists of all modules which are free

over A with weight spaces of finite rank.

9

Definition 2.1.3. Let M ∈ OfreeA . We define the character of M by

ch AM :=∑λ∈h∗

rk AMλ · eλ

where rk A denotes the rank of a free module over A.

2.1.4 Simple objects in OALet A → A′ be a morphism of deformation algebras. Then we can define abase change functor

(· ⊗A A′) : OA → OA′M 7→ M ⊗A A′,

which sends ∆A(λ) to ∆A′(λ). Let A be a local deformation algebra withmaximal ideal m ⊂ A and residue field K := A/Am.

Proposition 2.1.1 (cf. [Fie03], Proposition 2.1.). The functor (· ⊗A K) :OA → OK induces a bijection between isomorphism classes of simple objectsin OA and isomorphism classes of simple objects in OK.

We denote by LA(λ) the simple object in OA with highest weight λ ∈ h∗.The simple objects in OA, where A is local, are parametrized by their highestweights, i.e. by elements in h∗ (cf. [Fie03], Corollary 2.2).

Definition 2.1.4 (cf. [Soe98], Definition 4.1.). For any simple object in anabelian category A and any object M ∈ A we define

[M : L] ∈ N ∪ ∞

to be the supremum over all finite filtrations of M of the multiplicity of Lin these filtrations.

2.2 Projective objects in OALet A be a local deformation algebra again, and denote its residue field byK. We collect some statements about projective objects in OA. Over aninfinite-dimensional Lie algebra, the existence of enough projectives in O isnot true in general. So we have to pass to certain truncated versions. Theproofs can be found in [Fie03] and [Fie12a].

Definition 2.2.1. A subset J ⊆ h∗ is called open, if µ ≤ λ and λ ∈ Jalready implies µ ∈ J . J is called bounded if for any µ ∈ J the setλ ∈ J |λ ≥ µ is finite. Denote by OJA ⊂ OA the full subcategory ofmodules M such that Mλ 6= 0 implies λ ∈ J .

10

Theorem 2.2.1 (cf. [Fie03], Proposition 2.6 and [Fie12a], Theorem 4.2.).Let J ⊆ h∗ be bounded and open and let λ ∈ J . Then the simple objectLA(λ) has a projective cover PJA (λ) in OJA . Furthermore,

PJA (λ)⊗A K ∼= PJK (λ)

and for any other open and bounded J ′ ⊆ J with λ ∈ J ′ we have

(PJA (λ))J′ ∼= PJ

′

A (λ).

Hence there is a one-to-one correspondence between projective isomorphismclasses in OJA and OJK via (· ⊗A K). The following proposition is usedextensively by Fiebig in his work. It is called the base change property :

Proposition 2.2.1 (cf. [Fie03], Proposition 2.4.). Let A → A′ be a mor-phism of deformation algebras, P,M ∈ OJA , P projective. If P is finitelygenerated, then the natural map

HomOA(P,M)⊗A A′ → HomOA′ (P ⊗A A′,M ⊗A A′)

is an isomorphism.

The following theorem, the deformed BGG Reciprocity Theorem, “justifies”the use of deformation theory. If A is a deformation algebra and local, thendeforming does not change multiplicities in projective covers:

Theorem 2.2.2 (cf. [Fie03], Theorem 2.7). Let A be a local deformationalgebra with residue field K, J an open and bounded subset and λ ∈ J .Then PJA (λ) admits a deformed Verma flag and we have

(PJA (λ) : ∆A(µ)) =

[∆K(µ) : LK(λ)] if µ ∈ J0 otherwise,

where (· : ·) denotes the multiplicity of a deformed Verma module in a de-formed Verma flag.

2.3 Block decomposition

2.3.1 Blocks

Let A be a local deformation algebra with structure map τ : S → A andresidue field K. Consider the equivalence relation ∼A=∼K on h∗ which isgenerated by λ ∼A µ if [∆K(λ) : LK(µ)] 6= 0. Such an equivalence classΛ ⊂ h∗/ ∼A is called A-block (K-block), or, for a fixed local deformationalgebra, we just call it a block. Denote by (·|·) : g× g→ C the bilinear formon h∗ which was introduced in Chapter 1. Denote by (·|·)A the inducedA-bilinear form on h∗A. The equivalence classes are described in the following

11

Theorem 2.3.1 (cf. [KK79], Theorem 2). The equivalence relation ∼A=∼Kis generated by λ ∼A µ if

(λ+ ρ+ τ |α)K = n(α|α)K

for n ∈ N and α ∈ R such that λ− µ = nα.

Definition 2.3.1. Let λ ∈ h∗. The integral root system corresponding toλ is defined as

RA(λ) := α ∈ R | (λ+ ρ+ τ |α)K ∈ Z(α|α)K.

The integral Weyl group corresponding to λ is defined as

WA(λ) := 〈sα|α ∈ RA(λ)〉 ⊂ W .

Let µ ∈ h∗ such that λ ∼A µ, one verifies RA(λ) = RA(µ). So if Λ is a blockwith λ ∈ Λ, we define RA(Λ) := RA(λ) and this is indeed well-defined. Notethat for A = S and K = S / S h ∼= C we omit the subscript and write R(Λ)and W(Λ).Let Λ ⊂ h∗/ ∼A be a block. We define OA,Λ to be the full subcategory ofmodules generated by PJA (λ) for λ ∈ Λ and J ⊆ h∗ an open and boundedsubset. Denote by MΛ the submodule of a module M ∈ OA generated byall images PJA (λ)→M with λ ∈ Λ and J an open and bounded subset.

Proposition 2.3.1 (cf. [Fie03], Proposition 2.8). The functor∏Λ∈h∗/∼A OA,Λ → OA

MΛ 7→⊕

ΛMΛ

is an equivalence of categories.

Remark 2.3.1. Consider the case where Λ is generic, i.e. Λ = λ fora single λ ∈ h∗. Then the Verma module ∆A(λ) is simple and projective,hence the category OA,Λ is semisimple and isomorphic to A − mod due tothe fact that EndOA(∆A(λ)) ∼= A.

Lemma 2.3.1 (cf. [Fie03], Lemma 2.9). Let A→ A′ be a morphism of localdeformation algebras, λ, µ ∈ h∗. If λ ∼A′ µ, then λ ∼A µ.

2.3.2 Subgeneric blocks

Define S := S(0) to be the localization of S at the maximal ideal correspond-

ing to 0. Let p ⊂ S be a prime ideal. For certain prime ideals, we want tounderstand how a S-block splits under the functor (·⊗S Sp) : OS → OSp

into

Sp-blocks:

12

Lemma 2.3.2 (cf. [Fie06], Lemma 3.5). Let Λ ⊂ h∗/ ∼S be a block and

p ⊂ S a prime ideal.

1. If α 6∈ p for all α ∈ RS(Λ), then Λ splits under ∼Spinto generic

equivalence classes.

2. If p = Sα, then Λ splits under ∼Spinto generic and subgeneric equiv-

alence classes, the latter having the form λ, sα · λ.

Proposition 2.3.2 (cf. [Fie03], Proposition 3.4). Let p = Sα for α ∈ Rre

and Sp the localization of S at p. Furthermore, let λ, µ = sα · λ be asubgeneric block under ∼Sp

and assume λ > µ. Then PSp(λ) ∼= ∆Sp

(λ) andthere is a short exact sequence

0→ ∆Sp(λ)→ PSp

(µ)→ ∆Sp(µ)→ 0.

The block OSpis equivalent to the category of right representations of the

quiver

λ•i

++ •µj

kk

over Sp = Sα with relation j i = α.

We get the following

Corollary 2.3.1 (cf. [Fie03], Corollary 3.5). We have an isomorphism

EndOSα,λ,sα·λ(PSα

(sα · λ)) ∼= (xλ, xsα·λ) ∈ Sα ⊕ Sα |xλ ≡ xsα·λ mod α

These results will be very important in the following and we will use themextensively.

2.3.3 Further description of blocks

Let A be a local deformation algebra with structure map τ : S → A andresidue field K.

Definition 2.3.2. A weight λ ∈ h∗ is called (ρ-)dominant, if

〈λ+ ρ, α〉 6∈ Z<0 for all α ∈ R+ .

Similarly, λ is called antidominant, if

〈λ+ ρ, α〉 6∈ Z>0 for all α ∈ R+ .

13

In this thesis, we consider only blocks which lie outside the critical hy-perplanes, that means blocks which correspond to equivalence classes notintersecting the hyperplanes defined by

2(λ+ ρ+ τ, β)K = n(β, β)K

for any n ∈ N and an imaginary root β ∈ Rim. Hence, a block Λ liesoutside the critical hyperplanes if and only if RA(Λ) ⊂ Rre. For such a Λthe Kac-Kazhdan Theorem 2.3.1 implies Λ =W(Λ) · λ.

Definition 2.3.3 (cf. [Fie06], Section 3.4.). For an equivalence class Λoutside the critical hyperplanes, we say Λ is of negative level if it containsan antidominant weight, and of positive level if it contains a dominantweight.

2.4 Translation functors

This chapter mainly summarizes the results of [Fie03], chapter 4. We providethe reference of the precise result in the title of each statement. Let A be alocal S-algebra and KA its residue field.

Remark 2.4.1. For any M ∈ OA and N ∈ O the module M ⊗C N lies inOA. If A→ A′ is a morphism of S-algebras, there is a natural isomorphismof U⊗CA

′-modules

(M ⊗A A′)⊗C N ∼= (M ⊗C N)⊗A A′

Definition 2.4.1. For any U⊗CA-module N which is a weight module andµ ∈ h∗ we define

N≤µ := N/(∑µ′ 6≤µ

(U⊗CA).Nµ′).

This defines a right-exact functor (·)≤µ from the category of weight modulesto O≤µA . For µ′ > µ we have a natural map N≤µ

′ → N≤µ.

Now let λ, λ′ ∈ h∗ and denote their respective equivalence classes under ∼Aby Λ and Λ′. We assume the following:

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

2. RA(Λ) ⊂ Rre. Now, from the first assumption we deduce that

RA(Λ) = RA(Λ′),

WA(Λ) =WA(Λ′)

and that

Λ =WA(Λ) · λ,Λ′ =WA(Λ′) · λ′.

14

3. We have

(λ+ ρ+ τ |α)KA ≥ 0 ⇐⇒ (λ′ + ρ+ τ |α)KA ≥ 0

for all α ∈ RA(Λ), i.e. λ and λ′ lie in the closure of the same Weylchamber.

4. StabA(λ) ⊂ StabA(λ′) with finite index, where StabA denotes the sta-bilizer under the ρ-shifted action of WA(Λ) =WA(Λ′). This means λ“lies off the walls” and λ′ “lies on the walls”.

Definition 2.4.2. Let L(ν) be the simple integrable g-module with highestweight ν and let L(ν)∗ be its restricted dual, i.e. the simple integrable g-module with lowest weight −ν. For M ∈ OA denote by M[Λ] (respectivelyM[Λ′]) the projection on the Λ-block (respectively the Λ′-block).

For M ∈ OV FA,Λ we define

θon(M) = θλ′λ (M) := lim←−

µ

((M ⊗C L(ν)∗)≤µ)[Λ′]

and for N ∈ OV FA,Λ′ we set

θout(N) = θλλ′(N) := (M ⊗C L(ν))[Λ].

Proposition 2.4.1 (cf. [Fie03], Proposition 4.1.). 1. θon and θout definefunctors

θon : OV FA,Λ → OV FA,Λ′

and

θout : OV FA,Λ′ → OV FA,Λ

which map short exact sequences to short exact sequences.

2. Let w ∈ WA(Λ). Then θout∆A(w · λ′) has a Verma flag with sub-quotients ∆A(wx · λ′), where x ∈ StabA(λ′)/StabA(λ), each occurringonce. On the other hand, θon(∆A(w · λ)) is isomorphic to ∆A(w · λ′).

Let A→ A′ be a morphism of local S-algebras. Then Λ and Λ′ split under∼A′ into equivalence classes which are the orbits in Λ and Λ′ under thesubgroup WA′(Λ) =WA′(Λ

′) of WA(Λ). So for any

w ∈ WA′(Λ)\WA(Λ)/ StabA(λ)

we have translation functors

θout(w) = θw·λw·λ′ : OV FA′,[w·λ′] → OV FA′,[w·λ]

15

and

θon(w) = θw·λ′

w·λ : OV FA′,[w·λ] → OV FA′,[w·λ′]

where [w ·λ] denotes the respective equivalence class under ∼A′ . Since (·)≤µand the projection onto blocks commute with base change, we have thefollowing

Lemma 2.4.1 (cf. [Fie03], Lemma 5.4.). We have natural equivalences offunctors

θout(·)⊗A A′ ∼=∏w∈D

θout(w)(· ⊗A A′)

and

θon(·)⊗A A′ ∼=∏w∈D

θon(w)(· ⊗A A′)

where D :=WA′(Λ)\WA(Λ)/StabA(λ).

Finally one proves

Theorem 2.4.1 (cf. [Fie03], Corollary 5.10.). Let A = S, Sp for p ⊂ S aprime ideal, or one of their residue fields. Then the deformed translationfunctors

θon : OV FA,Λ → OV FA,Λ′

and

θout : OV FA,Λ′ → OV FA,Λ

are biadjoint. In particular, the non-deformed translation functors

θon : OV FΛ → OV FΛ′

and

θout : OV FΛ′ → OV FΛ

are biadjoint for A = S/mS ∼= C.

Definition 2.4.3. Let λ, λ′ be as in definition 2.8.1. and assume further-more that λ is regular and StabA(λ′) = e, s for s ∈ W a simple reflection.Then we define the functor

Θs := θλλ′θλ′λ ,

the so-called translation through the s-wall.

16

2.5 Twisting and Tilting functors

2.5.1 The semi-regular bimodule

In this chapter, we define the semi-regular bimodule. For the details werefer to [AL03], [Ara13], [Ark96] and [AS03]. The approach is strongly ori-ented towards [AL03], but we have to work with a deformed version of thesemi-regular bimodule.Consider the triangular decomposition of a symmetrizable Kac-Moody al-gebra g = n+ ⊕ h ⊕ n− and recall the principal Z-grading from Chapter 1which makes g into a graded Lie algebra, i.e.

g =⊕i∈Z

gi

wheregi :=

⊕α∈R,htα=i

gα.

This Z-grading induces a ZR-grading on U = U(g) via

U =⊕λ∈ZR

Uλ.

Using the Z-linear height function ht : ZR→ Z, we get a Z-grading

U = ⊕n∈Z U(n),

whereU(n) :=

⊕ht(λ)=n

Uλ, n ∈ Z, λ ∈ ZR .

Denote by N := U(n−). Then N ⊂ U becomes a negatively graded subalge-bra. Let w ∈ W. We define a set

Rw := α ∈ R− |w(α) ∈ R+.

Note that this set contains as many elements as α ∈ R+ |w(α) ∈ R−,which is exactly the length of w and therefore Rw is finite. Hence we get afinite-dimensional nilpotent subalgebra in g by setting

nw := n− ∩ w−1(n+) =⊕α∈Rw

gα.

The corresponding enveloping algebra Nw := U(nw) is then a negativelygraded subalgebra of U with (Nw)(0) = C. Its graded dual

N~w :=⊕n∈Z

HomC((Nw)(n),C)

17

is a Z-graded bimodule over Nw, where

(N~w)(n) := HomC((Nw)(−n),C), n ∈ Z.

The left action of Nw on N~w is defined by

x.f : n 7→ f(nx)

for f ∈ N~w, x, n ∈ Nw. One defines the right action similarly.

Definition 2.5.1. We define the semi-regular bimodule Sw by

Sw := U⊗Nw N~w .

By definition Sw is a left U-module and a right Nw-module. Yet we needsome more notation to justify the term “bimodule”:Let e ∈ n− \ 0. Let U(e) := U⊗C[e]C[e, e−1]. Consider in particular thecase where e is equal to the Chevalley generator e−α ∈ g−α where α is asimple root. Then U(e) is the Ore localization of U at the set 1, e±1

−α.

Theorem 2.5.1 (cf. [Ark96]). 1. For each 0 6= e ∈ n−1 we have thatU(e) is an associative algebra which contains U as a subalgebra. SetSe := U(e) /U.

2. For each simple root α with corresponding simple reflection s ∈ W wehave an isomorphism of left U-modules Ss ∼= Seα.

3. (cf. Theorem 1.3 in [Soe98]) For each w ∈ W we have a bijection

Sw ∼= N~w⊗Nw U,

which makes Sw into a right U-module (hence a U-bimodule).

4. Let w ∈ W and choose a filtration nw = F 0 ⊃ F 1 ⊃ ... ⊃ F r ⊃ 0consisting of ideals F p ⊂ nw of codimension p, p = 0, 1, ..., r = l(w).If ep ∈ F p−1 \ F p, then we have an isomorphism of U-bimodules

Sw ∼= Se1 ⊗U ...⊗U Ser .

Definition 2.5.2. Let A ⊂ Q be a localization of S. By abuse of notation,we tensor Nw with A and denote the resulting algebra

Nw⊗CA

with Nw again. It becomes a negatively graded subalgebra of UA := U⊗CA.Analogously, its graded dual is defined as

N~w :=⊕n∈Z

HomA((Nw)(n), A).

Then the deformed semi-regular bimodule is defined as

Sw := UA⊗Nw N~w .

18

2.5.2 The Twisting functor

For a more detailed exposition of Twisting functors, we refer to [AL03],[Ara13] and [AS03].

Definition 2.5.3. Denote by Q := Quot(S) the quotient field of S and letA ⊂ Q be a localization of S. Let w ∈ W. Denote by φw ∈ Aut(g) theautomorphism corresponding to

w : gα 7→ gw(α).

For M ∈ g−mod−A, g acts on φw(M) via the action twisted by φw. TheTwisting functor Tw relative to w on g−mod−A is given on objects by

M 7→ φw(Sw⊗UAM)

and on morphisms by

f 7→ (1⊗UA f).

Clearly, Tw is a right-exact functor by definition. Moreover, Theorem 2.5.1(3) guarantees

Tws = Tw Ts

whenever s is a simple reflection for which ws > w holds. Due to Theorem2.5.1 (4) we directly see that Tw is even exact for modules which are freeover Nw. We have the following

Theorem 2.5.2 (cf. [AL03], [AS03]). Tw preserves each individual blockOΛ and hence O. We have

[Tw ∆(λ)] = [∆(w · λ)]

in the Grothendieck group of O.

Observe that for a morphism A → A′ of S-algebras and M ∈ OA we havean isomorphism

Tw(M ⊗A A′) ∼= Tw(M)⊗A A′.

We will extensively make use of the fact that twisting is compatible withlocalization.

Definition 2.5.4. Let M ∈ g−mod−A. The right UA-module structure ofSw defines a left UA-module structure on

HomUA(Sw, φw−1(M)).

Denote by

19

Gw(M) ⊂ HomUA(Sw, φw−1(M))

the maximal h-diagonalizable submodule in HomUA(Sw, φw−1(M)). Then

M 7→ Gw(M)

defines a functor from OA to g −mod−A. Via the tensor-hom adjunctionwe directly see that Gw is right-adjoint to Tw, hence Gw defines a left-exactendofunctor of OA.

Lemma 2.5.1. Let M ∈ OA.

1. Suppose M is free over Nw. Then M ∼= Gw Tw(M).

2. Suppose M is cofree over Nw. Then M ∼= Tw Gw(M).

Proof. Ad (1): Choose an isomorphism

f : M→(Nw)⊕r

(note that r =∞ is allowed) as Nw-A-modules. By definition, we have

Gw Tw(M) ⊆ HomUA(Sw, φw−1(φw(Sw⊗UAM))).

Using f and Sw ∼= N~w⊗Nw UA (cf. [Soe98], Theorem 1.3) we compute

HomUA(Sw, φw−1(φw(Sw⊗UAM))) ∼= HomUA(Sw, Sw⊗UA(Nw)⊕r)∼= HomUA(Sw,N

~w⊗Nw UA⊗UA(Nw)⊕r)

and the last space is isomorphic to HomUA(N~w⊗Nw UA, N~w ⊗Nw (Nw)⊕r).

We continue with

HomUA(N~w⊗Nw UA,N~w⊗Nw(Nw)⊕r) ∼= HomUA(UA⊗Nw N~w, (N

~w)⊕r).

By Frobenius reciprocity,

HomUA(UA⊗Nw N~w, (N~w)⊕r) ∼= HomNw(N~w,N

~w)⊕r

NowHomNw(N~w,N

~w)⊕r ∼= (Nw)⊕r→M

where the last isomorphism is given by f−1.Ad (2): Choose an isomorphism g : M→(N~w)⊕r as N~w-A-modules. Thenwe have

Sw⊗UA HomUA(Sw,M) ∼= Sw⊗UA HomUA(Sw, (N~w)⊕r)

20

(note that we omitted the automorphism φw respectively φw−1 , since theydo not play a role here). Again, Frobenius reciprocity yields

Sw⊗UA HomUA(Sw, (N~w)⊕r) ∼= Sw⊗UA HomNw(N~w, (N

~w)⊕r).

Hence we compute

Sw⊗UA HomNw(N~w, (N~w)⊕r) ∼= N~w⊗Nw UA⊗UA(Nw)⊕r

∼= N~w⊗Nw(Nw)⊕r

∼= (N~w)⊕r.

Via g−1 the latter space is isomorphic to M .

2.5.3 The Tilting functor

In this section we want to discuss a categorical equivalence called the tiltingequivalence (cf. [Ark97], [IK11], [Soe98]). Let K be a field of characteristiczero. For this, we work with a special version of the semi-regular bimodule.Recall the principal Z-grading on g and the decomposition g = n− ⊕ b.Denote by π− : g n− the projection and by i− : n− → g the inclusionwith respect to the aforementioned decomposition of g. We define

π := i− π− : g→ g.

Definition 2.5.5. The critical cocycle (or critical 2-cocycle)

ω ∈ HomK(g ∧ g,K)

of a Z-graded Lie algebra g is defined by

ω(x, y) := trg([π adx, π ady]− π [adx, ady]),

where trg denotes the trace on g.

It is not clear whether ω(x, y) is well-defined. This is shown in thefollowing

Lemma 2.5.2 (cf. [IK11], Lemma 7.3.). For x ∈ gn1 , y ∈ gn2 the followinghold:

1. If n1 + n2 6= 0, then ω(x, y) = 0.

2. If n1 = n2 = 0, then ω(x, y) = 0.

3. If n1 = n and n2 = −n for n ∈ Z>0, then

ω(x, y) = tr⊕nm=1g−mady adx.

21

In a general setting, a semi-infinite character should be a a character of g0

whose differential coincides with ω. For our purposes, it is enough to workwith a weaker definition:

Definition 2.5.6 (cf. [Soe98], Definition 1.1). We call an element γ ∈HomK(g0,K) a semi-infinite character of g, if

1. g is generated as a Lie algebra by the homogeneous pieces g0, g1 andg−1,

2. γ([x, y]) = tr(ad(x) ad(y)) : g0 → g0 for all x ∈ g1and y ∈ g−1.

Proposition 2.5.1 (cf. [Soe08], Lemma 7.1.). 2ρ is a semi-infinite charac-ter for g.

Now we are ready to define the semi-regular bimodule associated to thesemi-infinite character 2ρ. We set

S2ρ := N~⊗K U(b).

Just from the definition, it is not clear where the 2ρ plays a role. But we

explain this in the following: Denote by K2ρ = K(0)2ρ the one-dimensional

graded U(b)-module (concentrated in degree zero) such that

b+ h ∈ [b, b]⊕ h = b

acts via multiplication with 2ρ(h).

Definition 2.5.7. We denote by

HomK(N,U(b))(n)

(respectively HomU(b)(U,K(0)2ρ ⊗KU(b))(n)) the homogeneous homomorphisms

of degree n between N and U(b) (respectively between U and K(0)2ρ ⊗K U(b)).

Lemma 2.5.3 (cf. [IK11], Lemma 7.6.). We have isomorphisms⊕n∈Z HomK(N,U(b))(n) → S2ρ and⊕n∈Z HomK(N,U(b))(n) →

⊕n∈Z HomU(b)(U,K

(0)2ρ ⊗K U(b))(n)

of Z-graded left N-modules.

Since⊕

n∈Z HomU(b)(U,K(0)2ρ ⊗K U(b))(n) carries a left g-module structure,

we get a left g-module structure on S2ρ by composing the isomorphismsabove. For the right g-module structure we use the following

Lemma 2.5.4 (cf. [IK11], Lemma 7.6.). We have an isomorphism of Z-graded right U(b)-modules

N~⊗K U(b)→N~⊗N U .

22

Remark 2.5.1. Let us assume for one moment that g is a semisimple Liealgebra over K. Denote by w0 the longest element in the Weyl group W.Then Nw0

∼= N, hence

Sw0 = N∗w0⊗Nw0

U ∼= N∗ ⊗N U →N∗ ⊗K U(b).

Hereby, we see the relation between S2ρ and the semi-regular bimodule wedefined in Chapter 2.5.1.

Recall the category of modules with a finite Verma flag OVF and let M ∈OVF. Then M is a free module of finite rank over N. Thus, we find afinite-dimensional K-vector space F such that we have an isomorphism

M ∼= N⊗KF

as N-modules. Hence we deduce

S2ρ⊗UM ∼= N~⊗NM ∼= N~ ⊗K F

as N-modules. Take a short exact sequence

0→ L→M → P → 0

in OVF. Since all considered modules are free over N, this sequence splitson the level of N-modules. Now

S2ρ⊗UL ∼= N~⊗NL

(the same holds for M and P ). Hence the sequence

0→ N~⊗NL→ N~⊗NM → N~⊗NP → 0

splits as well on the level of N-modules. From this, we deduce that thesequence of g-modules

0→ S2ρ⊗UL→ S2ρ⊗UM → S2ρ⊗UP → 0

is exact.Denote by K the category of g-modules that are cofree of finite rank over N.Then we have

Theorem 2.5.3 (cf. [Soe98], Theorem 2.1.). The functor

S2ρ⊗U· : OVF → K

defines an equivalence of categories such that short exact sequences corre-spond to short exact sequences.

23

Since M =⊕

λ∈h∗Mλ holds for M ∈ O, its graded dual is given by

M~ =⊕λ∈h∗

M∗λ ,

where the g-action is defined as

(x.f)(m) := −f(x.m) for f ∈M~, x ∈ g,m ∈M.

Hence we can define a functor

t := (S2ρ⊗U·)~ : OVF → (OVF)opp

where (OVF)opp denotes the opposed category of OVF.

Definition 2.5.8. The functor

t : OVF → (OVF)opp

is called the tilting functor.

The properties of t are summarized in the following

Theorem 2.5.4 (cf. [Soe98], Corollary 2.3.). The functor

t : OVF → (OVF)opp

is an equivalence of categories under which short exact sequences correspondto short exact sequences. Moreover, the Verma module ∆K(λ) is mapped to∆K(−2ρ− λ) for all λ ∈ h∗.

For the following the details can be found in [Fie06], Section 2.6. Let Abe an S-algebra, and denote by OVF

A the category of modules with a finitedeformed Verma flag. If either

1. A is a localization of S at a prime ideal p which is stable under themap on S which is induced by

h 7→ −h

for h ∈ h, or

2. A is a quotient of S which corresponds to a subvariety of h∗ beingstable under the dual of aforementioned map,

we can extend the tilting functor t to an A-linear equivalence

tA := (S2ρ⊗U·)~ : OVFA → (OVF

A )opp,

where (·)~ is the graded dual in the category of g-A-bimodules. We alsodeduce that ∆A(λ) is mapped to ∆A(−2ρ− λ).

24

Remark 2.5.2. Let A → A′ be a homomorphism of S-algebras that ful-fills the stability condition under h 7→ −h from above. We have a naturalisomorphism of functors

tA′ (· ⊗A A′)⇒(· ⊗A A′) tA .

Hence the deformed tilting functor commutes with base change.

Furthermore, tA even respects the block structure, that means for a blockΛ ∈ h∗/ ∼ we get an equivalence

tA : OVFA,Λ → (OVF

A,t(Λ))opp,

where t(Λ) := −2ρ− λ |λ ∈ Λ. By calculating

w · (−2ρ− λ) = w(−2ρ− λ+ ρ)− ρ = w(−ρ− λ)− ρ= −ρ− w(λ+ ρ)

= −2ρ− w(λ+ ρ) + ρ

= −2ρ− w · λ

we directly deduce that t(Λ) is again an equivalence class. The most im-portant application of the tilting equivalence is the fact that Λ is of positivelevel if and only if t(Λ) is of negative level. Hence we get the following

Corollary 2.5.1. The tilting equivalence

tA : OVFA → (OVF

A )opp

induces equivalences between blocks of negative level and blocks of positivelevel.

Remark 2.5.3. This will become very important in Chapter 4, for in gen-eral, projective objects only exist in blocks of positive level (in blocks of neg-ative level, one has to work with truncated subcategories). Then again, todefine the structure functor V in Chapter 4, the block is required to have anantidominant weight. The tilting functor is a tool which allows us to switchbetween blocks of negative and blocks of positive level.

25

26

Chapter 3

Modules with a twistedVerma flag

3.1 Twisted Verma modules

Before we begin with stating the formal properties of a family of twistedVerma modules, we introduce a new functor. Recall that Gw takes an objectfrom OS which is cofree as an Nw-module to an object in OS which is freeconsidered as an Nw-module. The latter category contains the category ofmodules with a Verma flag. Motivated by this, we state the following

Definition 3.1.1. Let M ∈ OS such that Gw(M) ∈ OV FS

. We define thew-twisted translation functors via

θwon(·) := Tw θon Gw(·),θwout(·) := Tw θout Gw(·),θws (·) := Tw θs Gw(·)

(cf. Chapter 2.4 for the non-twisted translation functors). Obviously theimage of these functors is contained in

K ∈ OS |K is cofree over Nw.

Let A be a localization of S in Q. Let Λ ⊆ h∗/ ∼ be a block of negative leveland let W(Λ) denote its integral Weyl group .

Definition 3.1.2 (cf. [AL03]). A family of twisted Verma modules inOA,Λ is a collection of Nw-cofree modules (∆w

A(λ))w∈W(Λ),λ∈Λ such that thefollowing properties are fulfilled:

1. ∆eA(ν) = ∆A(ν) for some antidominant regular weight ν ∈ Λ.

27

2. Let ν ∈ Λ be antidominant and let w, y, s ∈ W(Λ), where s is a simplereflection. If ws > w and w−1y < sw−1y, then we have an isomor-phism

∆wA(y · ν)→∆ws

A (y · ν).

3. Let ν ∈ Λ be antidominant and regular and let ν ′ ∈ h∗ be such that(ν, ν ′) is a pair of compatible weights as in Chapter 2.4. Assume fur-thermore that StabA(ν ′) = e, s. Let w, y, s ∈ W, where s is a simplereflection. If w−1y > w−1ys, then we have a short exact sequence

0→ ∆wA(y · ν)→ Θw

s ∆wA(y · ν)→ ∆w

A(ys · ν)→ 0.

4. Let ν ∈ Λ be antidominant and regular and let ν ′ ∈ h∗ be such that(ν, ν ′) is a pair of compatible weights as in Chapter 2.4. Then we have

θwon∆wA(y · ν) = ∆w

A(y · ν ′)

for all w, y ∈ W(Λ).

Remark 3.1.1. One can also define families of twisted Verma modules ina block of positive level. In that case,“antidominant” has to be interchangedwith “dominant” and the “<” signs have to be reversed accordingly.

Definition 3.1.3. Let Λ be a block of negative level, w ∈ W(Λ) and λ ∈ Λ.We define

∆wA(λ) := Tw(∆A(w−1 · λ)).

The following theorem we will prove for the deformation algebra A = S.

Theorem 3.1.1. (Tw(∆S(w−1·λ)))λ∈Λ is a family of twisted Verma modulesin OS,Λ.

Proof. 1. This is clear by the definition of the Twisting functor (cf. Chap-ter 2.5.2). In fact,

∆eS(y · ν) ∼= ∆S(y · ν)

for all y ∈ W(Λ).

2. We postpone the proof of property (2) and prove it as a separateproposition (Proposition 3.1.2).

3. We have a short exact sequence (cf. [Fie03], Proposition 4.1.)

0→ ∆S(w−1y · ν)→ Θs∆S(w−1y · ν)→ ∆S(w−1ys · ν)→ 0

for ν, ν ′ and w, y, s as above. Since Tw is exact on N -free modules,the sequence

0→ Tw(∆S(w−1y · ν))→ Tw(Θs∆S(w−1y · ν))→ Tw(∆S(w−1ys · ν))

→ 0

28

is exact. Since ∆S(w−1y · ν) ∼= Gw(Tw(∆S(w−1y · ν))) (cf. Lemma2.5.1), we see that

Tw Θs∆S(w−1y · ν) ∼= (Tw Θs Gw)(Tw(∆S(w−1y · ν)))∼= Θw

s (∆wS

(y · ν)),

which proves the statement.

4. Since

θon∆S(w−1y · ν) = ∆S(w−1y · ν ′),

we have an isomorphism

Tw θon∆S(w−1y · ν) = Tw ∆S(w−1y · ν ′) = ∆wS

(y · ν ′).

Again by using ∆S(w−1y · ν) ∼= Gw(Tw(∆S(w−1y · ν))) we deduce

Tw θon∆S(w−1y · ν) ∼= (Tw θon Gw)(Tw(∆S(w−1y · ν)))∼= θwon∆w

S(y · ν).

Before we are able to prove property (2), we need some further results in thesubgeneric case. Assume that Λ = µ, sα · µ =: λ, λ > µ. Then we haveW(Λ) = e, sα. Note that the indecomposable modules in OSα,Λ

which are

free over Sα are exactly ∆Sα(µ) ∼= ∇Sα

(µ), ∆Sα(λ), ∇Sα

(λ) and PSα(µ) (cf.

[Siv10], Theorem 5.2.8.). For ws > w and w−1y < sw−1y in property (2)to hold, we solely have the possibility of w = e and y = e. Ergo, in thesubgeneric case, showing property (2) amounts to prove the following

Lemma 3.1.1. ∆Sα(µ) ∼= ∆sα

Sα(µ).

Proof. Since Tsα is exact and fully faithful on Nsα-free modules, ∆sαSα

(µ) is

indecomposable. Then again, since

ch ∆sαSα

(µ) = ch ∆Sα(µ) = ch ∇Sα

(µ)

and due to the fact that we have a list of the indecomposable modules inOSα,Λ

which are free over Sα, we conclude ∆sαSα

(µ) ∼= ∆Sα(µ) ∼= ∇Sα

(µ).

Recall that we have already proven property (4) in the general case.

Lemma 3.1.2. ∆sαSα

(λ) ∼= ∇Sα(λ).

29

Proof. For w = sα, y has to be the neutral element e such that the inequalityw−1y > w−1ysα is fulfilled. Due to property (4), we have a short exactsequence

0→ ∆sαSα

(µ)→ Θsαsα∆sα

Sα(µ)→ ∆sα

Sα(λ)→ 0.

Now

Θsαsα∆sα

Sα(µ) := (Tsα Θsα Gsα)(Tsα(∆Sα

(sα · µ))) ∼= Tsα Θsα∆Sα(sα · µ).

Θsα∆Sα(sα ·µ) is isomorphic to the antidominant projective PSα

(µ). Again,since Tsα is exact and fully faithful on Nw-free modules, Tsα(PSα

(µ)) isindecomposable. By using

ch ∆sαSα

(λ) = ch ∆Sα(λ) = ch ∇Sα

(λ)

and

Ext1OSα

(∆Sα(λ),∆Sα

(µ)) ∼= Ext1OSα

(∆Sα(λ),∇Sα

(µ)) = 0,

we deduce ∆sαSα

(λ) ∼= ∇Sα(λ).

Now let us return to the general case where Λ is a block of negative level andA = S. Assume that λ > sα · λ =: µ. Since Tw commutes with localization,we derive

∆wS

(λ)⊗S Sα ∼= Tw(∆S(w−1 · λ))⊗S Sα

∼= Tw(U⊗U(b)S(w−1·λ))⊗S Sα

∼= Tw(U ⊗U(b) (Sα)(w−1·λ))

∼= ∆wSα

(λ).

By construction, ∆wSα

(λ) has the same character as ∆Sα(λ) and ∇Sα

(λ). We

first state the following

Proposition 3.1.1 (cf. [BB06], Proposition 4.4.6.). Let w ∈ W, α ∈ R+.Then we have

w(α) ∈ R+ ⇐⇒ l(wsα) > l(w).

Theorem 3.1.2. Let w ∈ W(Λ) and α ∈ R+. Then

∆wS

(λ)⊗S Sα ∼=

∆Sα

(λ) if w−1(α) ∈ R+,

∇Sα(λ) if w−1(α) ∈ R− .

Remark 3.1.2. This statement can also be found in [Siv10], but without acomplete proof.

30

Proof. Take y = sα. If w−1sα > w−1sαsα = w−1, property (4) ensures ashort exact sequence

0→ ∆wSα

(λ)→ Θws ∆w

Sα(λ)→ ∆w

Sα(µ)→ 0.

With the same arguments as before we see that ∆wSα

(µ) ∼= ∆Sα(µ). Since

d2 ∼= idOSα

for modules which are free over Sα with weight spaces of finite rank, wededuce

Ext1OSα

(∆Sα(µ),∇Sα

(λ)) ∼= Ext1OSα

(∆Sα(λ),∇Sα

(µ)) = 0.

Hence ∆wSα

(λ) ∼= ∆Sα(λ).

On the contrary, if y = e and w−1e > w−1esα, then property (3) ensures ashort exact sequence

0→ ∆wSα

(µ)→ Θsα∆wSα

(µ)→ ∆wSα

(λ)→ 0.

Since Ext1OSα

(∆Sα(λ),∆Sα

(µ)) = 0 this is only possible if ∆wSα

(λ) ∼= ∇wSα

(λ).

Now we are finally ready to prove property (2) in the general setting. Topresent it more clearly, we formulate it as a

Proposition 3.1.2. Let ν be antidominant and w, y, t ∈ W, where t is asimple reflection. If wt > w and w−1y < tw−1y, we have an isomorphism

∆wS

(y · ν)→∆wtS

(y · ν).

Proof. Let β be the simple root corresponding to t. Now wt > w implies

l(w) = l(w−1) = l(wt)− 1 = l(tw−1)− 1.

This means, that for all α ∈ R+ \β

w−1(α) ∈ R+ ⇐⇒ (tw−1)(α) ∈ R+

(respectively w−1(α) ∈ R− ⇐⇒ (tw−1)(α) ∈ R−). Hence, by using theprevious theorem, localizing at p = Sα yields

∆wS

(y · ν)⊗S Sα→∆wtS

(y · ν)⊗S Sα

for all α ∈ R+ \β. Now we calculate

wt > w ⇐⇒ (wtw−1)w > w ⇐⇒ sw(β)w > w.

31

Keep in mind that wt > w implies w(β) ∈ R+. So it remains to verify that

∆wS

(y · ν)⊗S Sw(β)→∆wtS

(y · ν)⊗S Sw(β) (3.1)

holds. By our observations, the isomorphism in 3.1 is only possible if y · ν is(ρ)-antidominant in the equivalence class y · ν, sw(β)y · ν (since this is theonly case in a subgeneric block where ∆Sβ

(y · ν) ∼= ∇Sβ(y · ν) holds). The

assumption tw−1y > w−1y is equivalent to y−1wt > y−1w, which in turn isequivalent to

y−1w(β) ∈ R+ .

Now y · ν is antidominant in y · ν, sw(β)y · ν

⇐⇒ (y · ν) < (sw(β)y · ν)

⇐⇒ sw(β)(y · ν)− (y · ν) ∈ Z>0 R+

⇐⇒ y · ν − 〈y · ν, w(β)〉w(β)− y · ν ∈ Z>0 R+

⇐⇒ −〈y · ν, w(β)〉w(β) ∈ Z>0 R+

⇐⇒ (−〈y(ν + ρ), w(β)〉+ 〈ρ, w(β)〉)w(β) ∈ Z>0 R+

where we used theW-invariance of 〈·, ·〉 in the second last line. Now 〈ρ, w(β)〉is positive (the height of w(β), to be precise). Due to ν being ρ-antidominantand y−1w(β) ∈ R+, the expression −〈ν+ρ, y−1w(β)〉 is in Z>0, which yieldsthat y · ν is (ρ)-antidominant in the equivalence class y · ν, sw(β)y · ν.

Now we have proven the existence of a family of twisted Verma modules.Before we show uniqueness, we have to collect some statements:

Lemma 3.1.3. Let ν ∈ Λ be antidominant, w ∈ W(Λ). Then

∆S(ν) → ∆wS

(ν).

Proof. If we take y = e, the two assertions in property (2) are always fulfilledfor ws > w (since this is equivalent to w−1e = w−1 < sw−1 = sw−1e). Sofor w = e and s a simple reflection, we have an isomorphism

∆S(ν) ∼= ∆sS(ν).

Now an obvious induction over the length of w proves the statement.

Lemma 3.1.4. Return to the setting of property (4) of the formal propertiesof a family of twisted Verma modules. We have

HomOS(∆w

S(y · ν),Θw

s ∆wS

(y · ν)) ∼= S.

32

Proof. We make heavy use of the adjunctions we have constructed so far:

HomOS(∆w

S(y · ν),Θw

s ∆wS

(y · ν)) ∼= HomOS(∆S(w−1y · ν),Θs Gw ∆w

S(y · ν))

∼= HomOS(∆S(w−1y · ν),Θs∆S(w−1y · ν))

∼= EndOS(θon∆S(w−1y · ν))

∼= EndOS((∆S(w−1y · ν ′))

∼= S.

Remark 3.1.3. The image of 1S ∈ EndOS(θon∆S(w−1y · ν)) in

HomOS(∆S(w−1y · ν),Θs∆S(w−1y · ν))

under the adjunction isomorphism is called the adjunction morphism. Hencethe lemma tells us that the map

0→ ∆wS

(y · ν)→ Θws ∆w

S(y · ν)

in property (3) has to be a S-multiple of the adjunction morphism. Since

Θs∆S(w−1y · ν)) ∼= Θs∆S(w−1ys · ν)),

analogous arguments show that the morphism

Θs∆S(w−1y · ν)) ∼= Θs∆S(w−1ys · ν))→ ∆S(w−1ys · ν))→ 0

is a multiple of the adjunction morphism.

Theorem 3.1.3. Fix w ∈ W(Λ). A family of w-twisted Verma modules inOS,Λ is unique up to isomorphism.

Proof. Let (ΩwS

(y · ν))y∈W(Λ),w∈W(Λ) be another family of w-twisted Vermamodules in OS,Λ. We have already seen that

∆wS

(ν) ∼= ∆S(ν) ∼= ΩwS

(ν)

by applying property (2). Now let y = s be a simple reflection such thatw−1s > w−1. Now

Θws ∆w

S(s · ν) ∼= Tw Θs∆S(w−1s · ν) ∼= Tw Θs∆S(w−1 · ν) ∼= Θw

s ∆wS

(ν)

yields Θws Ωw

S(s · ν) ∼= Θw

s ΩwS

(ν) with the same arguments. Now property (3)ensures two short exact sequences

33

0 // ΩwS

(s · ν) // Θws Ωw

S(ν) // Ωw

S(ν)

// 0

0 // ∆wS

(s · ν) // Θws ∆w

S(ν) // ∆w

S(ν) // 0

Since the vertical arrow is an isomorphism and since the map

Θws ∆w

S(ν))→ ∆w

S(ν)

is a multiple of the adjunction morphism, we get a commutative diagram

0 // ΩwS

(s · ν) // Θws Ωw

S(ν)

// ΩwS

(ν)

// 0

0 // ∆wS

(s · ν) // Θws ∆w

S(ν) // ∆w

S(ν) // 0

where the vertical arrows are isomorphisms. This forces ΩwS

(s·ν) ∼= ∆wS

(s·ν).Now an induction over the length of y completes the proof.

3.2 Modules with a twisted Verma flag

Let Λ ⊂ h∗/ ∼ be a block outside the critical hyperplanes. Fix w ∈ W(Λ).

Definition 3.2.1. We define a partial order on Λ by setting

λ ≤w µ := w−1 · λ ≤ w−1 · µ

for λ, µ ∈ Λ.

Let OfreeS

be the subcategory of OS consisting of all modules which are free

over the deformation ring S. Denote by Q := Quot(S) the quotient field ofS. Let M ∈ Ofree

S,Λ. Since M is (torsion) free over S, we have an inclusion

M →M ⊗S Q =: MQ.

Over Q, the category OQ,Λ is semisimple with generic blocks, i.e. blockscontaining a single element. The deformed Verma modules ∆Q(λ) are simpleand projective within this category (cf. [Fie03], Example 2.2). Thus MQ

decomposes into the so-called generic decomposition

MQ =⊕λ∈Λ

MλQ,

where each MλQ is isomorphic to a direct sum of deformed Verma modules

∆Q(λ) over Q.

34

Definition 3.2.2. Let I ⊆ Λ. We define

MI := M ∩⊕λ∈I

MλQ.

Definition 3.2.3. Let J ⊆ Λ. We say J is w-open, if J is w-downwardlyclosed, that means

µ ≤w λ, λ ∈ J ⇒ µ ∈ J .

Analogously, we say I ⊆ Λ is w-closed, if I is w-upwardly closed.

Definition 3.2.4. Let J ⊆ Λ be w-open. We define

MJ := M/MJ c .

where Jc := Λ \ J . Note that we have natural inclusions MI → MI′ forI ⊆ I ′ ⊆ Λ w-closed and natural surjections MJ MJ

′for J ′ ⊆ J ⊆ Λ

w-open.

Lemma 3.2.1. Let J ′ ⊆ J ⊆ Λ be w-open subsets. We have a canonicalisomorphism

MJ′ ∼−→ (MJ )J

′.

Proof. Define I ′ := J ′c and I := J c. Denote by f the composition of themaps M →MJ and MJ → (MJ )J

′. Since the composition

MI′ →Mf−→ (MJ )J

′

is zero, we get an injective mapMI′ → (ker f). But since (ker f) is generatedby weights in I ′, this has to be an isomorphism. Hence the kernels of themap M →MJ ′ and (ker f) coincide, which yields the claim.

Definition 3.2.5. Let M ∈ OS. We say M has a w-twisted Verma flag ifthere exists a finite sequence of submodules

0 = M0 ⊂ ... ⊂Mn = M

such that

Mi/Mi−1∼= ∆w

S(λi)

for some λi ∈ h∗. Denote the full subcategory of OS consisting of moduleswith a w-twisted Verma flag by Ow-VF

S.

Remark 3.2.1. Note that since Ow-VFS

is closed under extensions, it inheritsan exact structure by OS. Hence, it becomes an exact category. It is notabelian in general.

Definition 3.2.6. Denote by P(S) := p ∈ Spec(S) | ht p = 1.

35

Proposition 3.2.1. Let λ, µ ∈ Λ with λ 6<w µ. Consider a short exactsequence

∆wS

(µ) →M ∆wS

(λ).

Then the short exact sequence

∆wS

(µ)⊗S Sp →M ⊗S Sp ∆wS

(λ)⊗S Sp

splits for all prime ideals p ∈ P(S) of height one.

If p is a prime ideal for which @α ∈ R+ such that p = Sα, the blocks becomegeneric and there is nothing left to show.So suppose we have α ∈ R+ such that λ = sα · µ. We first need to prove acombinatorial

Lemma 3.2.2. Keep the assumption that w−1sα · µ 6< w−1 · µ.

1. Suppose sα · µ > µ. Then

w−1sα(α) ∈ R−,

i.e. w−1(α) ∈ R+.

2. Suppose sα · µ < µ. Then

w−1sα(α) ∈ R+,

i.e. w−1(α) ∈ R−.

Proof.

w−1sα · µ 6< w−1 · µ ⇐⇒ w−1sα(µ+ ρ)− ρ 6< w−1(µ+ ρ)− ρ⇐⇒ w−1sα(µ+ ρ)− w−1(µ+ ρ) 6< 0

⇐⇒ w−1(sα(µ+ ρ)− (µ+ ρ)) 6< 0

⇐⇒ w−1(−〈µ+ ρ, α)〉α 6< 0

⇐⇒ −〈µ+ ρ, α〉w−1(α) 6< 0.

Now it is enough to prove 1. because 2. follows from 1. if we interchangethe roles of sα · µ and µ. Now

sα · µ > µ ⇐⇒ sα(µ+ ρ)− ρ > µ

⇐⇒ −〈µ+ ρ, α〉α > 0.

We have already shown −〈µ+ ρ, α〉w−1(α) 6< 0. But

36

−〈µ+ ρ, α〉w−1(α) 6< 0 ⇐⇒ w−1(α) 6< 0

⇐⇒ w−1(α) ∈ R+

⇐⇒ w−1(−α) ∈ R−

⇐⇒ w−1sα(α) ∈ R−,

where we used that W preserves the root system R in going from the firstto the second line.

Now we are finally able to prove Proposition 3.2.1:

Proof. In the case that sα · µ > µ holds, our short exact sequence

∆wS

(µ)⊗S Sα →M ⊗S Sα ∆wS

(λ)⊗S Sα

transforms itself, with the help of the preceding Lemma and Theorem 3.1.2,into

∆Sα(µ) →M ⊗S Sα ∆Sα

(λ).

Now the preimage of a primitive vector of weight λ in ∆Sα(λ) is a primitive

vector in M ⊗S Sα, generating a submodule mapping isomorphically onto∆Sα

(λ), which splits the sequence. In the case sα · µ < µ, our short exactsequence has the form

∇Sα(µ) →M ⊗S Sα ∆Sα

(λ),

due to Theorem 3.1.2 again. By dualizing this sequence, we get

∆Sα(λ) → d(M ⊗S Sα) ∆Sα

(µ)

since d∆Sα(λ) ∼= ∆Sα

(λ). With the same arguments as above, we get asplitting of d(M⊗SSα). Dualizing this splitting, we get a splitting of M⊗SSαwhich shows the statement. Here we used that d is an autoequivalence onOfree

Swith d2 ∼= id .

Theorem 3.2.1. Let λ, µ ∈ Λ. Suppose λ 6<w µ. Then

Ext1OS

(∆wS

(λ),∆wS

(µ)) = 0.

Proof. Consider a short exact sequence

∆wS

(µ) →M ∆wS

(λ).

As an extension of w-twisted Verma modules, M is free over S, hence wehave an inclusion

M →M ⊗S Q .

37

The w-twisted Verma flag of M yields a canonical splitting over Q

M ⊗S Q = ∆Q(µ)⊕∆Q(λ).

We define MµQ := ∆Q(µ) and denote by πµ : MQ := M ⊗S Q Mµ

Q the

canonical projection (and analogously MλQ and πλ). By πµM we mean the

image of the composition

M →MQ MµQ.

Clearly, we have an inclusion

M → πµM ⊕ πλM.

Since πµM ⊕ πλM is torsion free over S, we get an inclusion

πµM ⊕ πλM →⋂

p∈P(S)

(πµM ⊕ πλM)p.

Now the generic decompositions of M and Mp coincide and thus we have

(πµM ⊕ πλM)p = πµMp ⊕ πλMp

for all p ∈ P(S). The preceding proposition shows

Mp∼= ∆w

S(µ)⊗S Sp ⊕∆w

S(λ)⊗S Sp,

hence πµMp = ∆wS

(µ)⊗S Sp (analogously, we determine πλMp). This shows(πµM ⊕ πλM)p ∼= Mp for all prime ideals of height one. We conclude

πµM ⊕ πλM →⋂p

(πµM ⊕ πλM)p ∼=⋂p

Mp = M

which yields M ∼= πµM ⊕ πλM .

Lemma 3.2.3. Let M have a w-twisted Verma flag

0 = M0 ⊂ ... ⊂Mi ⊂ ... ⊂Mn = M

with subquotients Mi/Mi−1∼= ∆w

S(λi). Assume further that there exists

an i such that λi+1 6<w λi. Then, we can interchange the correspondingsubquotients in this Verma flag.

Proof. We factor out Mi−1 and get a two step Verma flag

∆wS

(λi) →Mi+1/Mi−1 ∆wS

(λi+1).

By Theorem 3.2.1 this sequence splits. Let

π : Mi+1 Mi+1/Mi−1∼= ∆w

S(λi)⊕∆w

S(λi+1)

38

be the canonical projection. Define

M ′i := π−1(∆wS

(λi+1)) ⊂Mi+1.

Then we have

M ′i/Mi−1∼= ∆w

S(λi+1)

and

Mi+1/M′i∼= Mi+1/Mi−1

/M ′i/Mi−1

∼= Mi+1/Mi−1/∆wS

(λi+1)

∼= ∆wS

(λi).

Definition 3.2.7. Let λ ∈ Λ. We define the module

M[λ]w := cok(M>wλ →M≥wλ).

Lemma 3.2.4. Let M ∈ Ofree

S,Λand assume MJ free over S for all w-open

J ⊆ Λ. For two w-closed subsets I ⊆ I ⊆ Λ, we claim that MI/MI is free

over S.

Proof. Define J := Ic, J := Ic. Then J ⊆ J . Consider the diagram

MI // M // //

M J

MI?

OO

// M

id

OO

// // MJ

π

OOOO

We have a map MI →MJ . By factoring out MI we get an injection

MI/MIi→MJ .

Since π i = 0, we get a map MI/MIf→ kerπ, which has to be a monomor-

phism since i is one. kerπ is generated by weights in J \ J = I \ I, so f isactually an isomorphism. This yields a short exact sequence

MI/MI →MJ M J .

M J is free over S, so this sequence splits as S-modules. As a direct sum-mand of a free module (MJ is free by assumption), MI/MI is projective,hence every weight space of MI/MI is projective. Now a finitely generatedprojective module over a local ring is free.

39

Lemma 3.2.5. Let M ∈ Ofree

Sand let A be a localization of S with respect

to a multiplicative subset. For every w-closed I ⊆ Λ we have

(MI)A ∼= (MA)I ,

where MA := M ⊗S A.

Proof. Since (MI)A →MA, we get ((MI)A)I → (MA)I , but

((MI)A)I = (MI)A.

Let m ∈ (MA)I . Then we find a ∈ A invertible such that (am) ∈ MI . Itfollows that m ∈ a−1MI ⊂ (MI)A, which yields the claim.

Theorem 3.2.2. Suppose M ∈ Ofree

S,Λand assume additionally that MJ is

free over S for all w-open J ⊆ Λ. Then M has a w-twisted Verma flag if andonly if M[λ]w

∼= ∆wS

(λ)⊕mλ for all λ ∈ Λ. Here, mλ denotes the multiplicityof ∆w

S(λ) in a w-twisted Verma flag of M . We require furthermore that

mλ 6= 0 for only finitely many λ ∈ Λ.

Proof. Assume that M has a w-twisted Verma flag with ∆wS

(µ) occurringwith the multiplicity mµ. Define

n :=∑

λ>wµmλ,

k := mµ

l :=∑

λ 6≥wµmλ.

Repeatedly using lemma 3.2.3, we can arrange a Verma flag

0 = M0 ⊂ ... ⊂Mn ⊂ ... ⊂Mn+k ⊂ ... ⊂Mn+k+l = M

in a such a way that the subquotients Mi+1/Mi fulfill Mi+1/Mi∼= ∆w

S(λi)

where

λi >w µ for 0 ≤ i ≤ n,

λi = µ for n < i ≤ n+ k,λ 6 ≥w µ for n+ k < i ≤ n+ k + l.

Now, Mn has subquotients ∆wS

(λi) where

λi >w µ (3.2)

holds, hence Mn is generated by its weight spaces corresponding to weightswhich satisfy (3.2). So we get

Mn →M>wµ.

40

Consider the diagram

Mn

// M

// // M/Mn

M>wµ // M // // M/M>wµ

M/Mn has a Verma flag with subquotients corresponding to weights whichare not strictly w-larger than µ. Hence, the last vertical arrow has to bean isomorphism. Applying the Five Lemma yields Mn

∼= M>wµ and analo-gously, we obtain Mn+k

∼= M≥wµ. We deduce

M≥wµ/M>wµ∼= Mn+k/Mn,

where the latter has a Verma flag where the subquotients are all isomorphicto ∆w

S(µ). Due to

Ext1OS

(∆wS

(λ),∆wS

(λ)) = 0

for all λ ∈ h∗, we obtain

M≥wµ/M>wµ∼= ∆w

S(µ)⊕mµ .

For the other direction, recall the assumption that the set

supp(M) := λ ∈ Λ |mλ 6= 0

is finite. We prove the statement by induction over the number of elementsin supp(M). Take µ ∈ supp(M) maximal with respect to ≤w. Then

M≥wµ = M[µ]w∼= ∆w

S(µ)⊕mµ

and we get a short exact sequence

∆wS

(µ)⊕mµ →M K, (3.3)

where K := M/∆wS

(µ)⊕mµ . Since K = M 6≥wµ and 6≥w µ ⊆ Λ is w-open,

K is free over S by assumption. By noting that KJ = MJ∩6≥wµ due to

Lemma 3.2.1 for all w-open J ⊆ Λ, we deduce that K fulfills the assumptionof the Theorem. By the induction hypothesis, we just have to show that

K[λ]w∼= ∆w

S(λ)⊕r,

because then K admits a w-twisted Verma flag, and due to (3.3), M as welladmits a w-twisted Verma flag. Since (·)I is right exact for I ⊆ Λ w-closed,we get a commutative diagram

41

M>wλ

// M≥wλ

// // M[λ]

K>wλ // K≥wλ // // K[λ]

Now suppose λ = µ. Then

K[µ] ⊂ K[µ] ⊗S Q ∼= (KQ)[µ]∼= (

⊕σ∈Λ,σ 6=µ

∆Q(σ)mσ)[µ] = 0

If λ 6= µ, we get the following diagram

M[λ] ⊗S Q // K[λ] ⊗S Q

M[λ]

?

OO

// // K[λ]

?

OO

Since

K[λ] ⊗S Q ∼= (KQ)[λ]∼= (

⊕ν∈Λ,ν 6=µ

∆Q(ν)mν )[λ]∼= ∆Q(λ)mλ ∼= (MQ)[λ],

the upper horizontal is an isomorphism, forcing the lower horizontal to beinjective. Hence,

K[λ]w∼=

∆w

S(λ)⊕mλ if λ 6= µ,

0 if λ = µ.

By the induction hypothesis, K admits a w-twisted Verma flag.

Corollary 3.2.1. Let M ∈ Ofree

S,Λ, I be a w-closed subset of Λ and let J :=

Ic.

1. M admits a w-twisted Verma flag if and only if both MI and MJ

admit w-twisted Verma flags.

2. If either of the two equivalent properties in (1) hold, we have for allµ ∈ h∗

(MI : ∆wS

(µ)) =

(M : ∆w

S(µ)), if µ ∈ I

0, else

and

(MJ : ∆wS

(µ)) =

(M : ∆w

S(µ)), if µ ∈ J

0. else

42

Proof. By definition we have a short exact sequence

0→MI →M →MJ → 0.

Clearly M admits a w-twisted Verma flag if both MI and MJ admit a w-twisted Verma flag. So suppose M admits a w-twisted Verma flag. By thepreceding theorem, we can find a filtration

0 = M0 ⊂ ... ⊂Mn = M

such that λ1, ..., λk ⊂ I and λk+1, ..., λn ⊂ J for some k ≥ 0. Hence wehave MI = Mk, since Mk is generated by its vectors of weights λ1, ..., λk andthe weights of M/Mk belong to J . This shows MJ = M/Mk. Hence bothMI and MJ admit a w-twisted Verma flag. The multiplicity statements in(2) follow easily.

We can also prove

Corollary 3.2.2. Suppose M ∈ Ofree

S,Λand assume furthermore that MJ is

free over S for all w-open J ⊆ Λ. Then M has a w-twisted Verma flag ifand only if M [λ]w := ker(M≤

wλ M<wλ) ∼= ∆wS

(λ)⊕mλ for all λ ∈ Λ.

Proof. Denote by I := <w λc and by I ′ := ≤w λc. Obviously I ′ ⊂ I.We have a commutative diagram

0 _

0 _

M [λ]w _

MI′ //

_

M // // M≤wλ

MI //

M // //

M<wλ

M[λ]w 0 0

Applying the Snake lemma yields M [λ]w ∼= M[λ]w . Now Theorem 3.2.2 yieldsthe claim.

Lemma 3.2.6. A,B ∈ Ofree

S,Λsuch that AJ , BJ are free over S for all w-

open J ⊆ Λ. Then A ⊕ B admits a w-twisted Verma flag if and only if Aand B admit a w-twisted Verma flag.

Proof. If A⊕B admits a w-twisted Verma flag, then (A⊕B)[λ] ∼=⊕

∆wS

(λ)for all λ ∈ Λ. Since the localization functor is additive we get

ker(A⊕B)≤wλ (A⊕B)<

wλ ∼= ker(A≤wλ A<

wλ)⊕ ker(B≤wλ B<wλ),

43

hence (A ⊕ B)[λ]w ∼= A[λ]w ⊕ B[λ]w . The Krull-Remak-Schmidt theoremshows that indeed A[λ]w and B[λ]w are isomorphic to direct sums of ∆w

S(λ)

and hence, by Corollary 3.2.2, A and B admit a w-twisted Verma flag. Theother direction is obvious.

3.3 Projective objects

3.3.1 Hom and Ext1

Let Λ be a block outside the critical hyperplanes. We will work with atruncation Λ≤

wν for ν ∈ Λ to ensure that for every weight λ ∈ Λ≤wν , there

are only finitely many weights w-above λ. For the following, we will needinformation about Hom-spaces and Ext1-spaces of modules with a w-twistedVerma flag. We will orient ourselves towards [Jan08]. Let A ⊂ Q be alocalization of S with respect to a multiplicative subset. So A is noetherianand integrally closed. Denote by P(A) := p ∈ Spec(A) | ht p = 1. EveryM ∈ Ow−V F

A,Λ≤wνdecomposes into weight spaces which are free of finite rank

over A. For λ ∈ Λ we identify Mλ with the A-submodule

Mλ ⊗A 1Q ⊂Mλ ⊗A Q

and, since Mλ is free and hence reflexive,

Mλ =⋂

p∈P(A)

Mλ ⊗A Ap.

Note that also Mλ ⊗A Ap is identified with a submodule of Mλ ⊗A Q. Sum-ming up, we identify M and each M ⊗A Ap with a submodule of M ⊗A Qand get

M =⋂

p∈P(A)

M ⊗A Ap.

Let N ∈ Ow−V FA,Λ≤wν

. Observe that we have

HomOA(M,N) = ψ ∈ HomOQ(M ⊗A Q, N ⊗A Q) |ψ(M) ⊆ N,

and for all prime ideals p ⊂ A and M ′, N ′ ∈ Ow−V FAp,Λ≤

wν

HomOAp(M ′, N ′) = ψ ∈ HomOQ

(M ′ ⊗Ap Q, N ′ ⊗Ap Q) |ψ(M ′) ⊆ N ′.

Now

44

HomOA(M,N) = HomOA(⋂

p∈P(A)

M ⊗A Ap,⋂

p∈P(A)

N ⊗A Ap)

=⋂

p∈P(A)

HomOAp(M ⊗A Ap, N ⊗A Ap),

and this equality has to be understood as an equality of subspaces in

HomOQ(M ⊗A Q, N ⊗A Q).

Lemma 3.3.1. Let λ, µ ∈ Λ. We have

HomOS(∆w

S(λ),∆w

S(µ)) = 0 if λ 6= µ

and

EndOS(∆w

S(λ)) ∼= S.

Proof. Let f : ∆wS

(λ)→ ∆wS

(µ), λ 6= µ. Both modules are free over S, so weget a diagram

∆wS

(λ) _

f// ∆w

S(µ) _

∆wS

(λ)⊗S QfQ

// ∆wS

(µ)⊗S Q

Now ∆wS

(λ) ⊗S Q ∼= ∆Q(λ) and ∆wS

(µ) ⊗S Q ∼= ∆Q(µ) are simple, whichmeans fQ is zero and hence f is zero. In case λ = µ we use the observationsabove and get

EndOS(∆w

S(λ)) =

⋂p∈P(S)

EndOSp(∆w

S(λ)⊗S Sp).

In each case EndOSp(∆w

S(λ) ⊗S Sp) ∼= Sp ⊂ Q, since ∆w

S(λ) ⊗S Sp is either

isomorphic to ∆Sp(λ) or to ∇Sp(λ) (cf. Theorem 3.1.2).

Now ⋂p∈P(S)

Sp = S.

Lemma 3.3.2. Let λ ∈ Λ. Every ∆wS

(λ) is generated by finitely manyweight spaces.

45

Proof. Since ∆wS

(λ) is free over S, it is the intersection over its localizations

at prime ideals of height one. For λ > sα · λ, α ∈ R+ we know that

∆wS

(λ)⊗S Sα ∼=

∆Sα

(λ) if w−1(α) ∈ R+

∇Sα(λ) if w−1(α) ∈ R−

(cf. Theorem 3.1.2). ∆Sα(λ) is generated by its λ-weight space. From the

subgeneric situation we know that ∇Sα(λ) appears in a short exact sequence

0 → ∆Sα(sα · λ) → d(PSα

(sα · λ)) ∇Sα(λ) 0.

Using that d(PSα(sα ·λ)) ∼= PSα

(sα ·λ) and that PSα(sα ·λ) is generated by its

(sα ·λ)-weight space, we derive that ∇Sα(λ) is generated by its (sα ·λ)-weight

space. Since the set

L := α ∈ R+ |w−1(α) ∈ R−

is finite (to be precise, |L| = l(w−1)), we conclude with

∆wS

(λ) = U(g).(∆wS

(λ)λ +∑β∈L

∆wS

(λ)sβ ·λ).

Theorem 3.3.1. Let M,N ∈ Ow−V FS,Λ≤wν

and A a localization of S. Then

HomOS(M,N)⊗S A

∼= HomOA(M ⊗S A,N ⊗S A)

Proof. Due to the preceding lemma we find finitely many weights λ1, ..., λnsuch that

M =n∑i=1

U(g) ·Mλi .

As seen above, we have

HomOS(M,N) = ψ ∈ HomOA(M ⊗S A,N ⊗S A) |ψ(M) ⊆ N.

For ψ ∈ HomOA(M ⊗S A,N ⊗S A) we find t ∈ S, t 6= 0 such that

tψ(Mλi) ⊆ Nλi ∀i.

Hence tψ(M) ⊆ N . So

ψ = tψ ⊗S t−1 ∈ HomOS

(M,N)⊗S A.

46

Lemma 3.3.3. Let M,N ∈ Ow−V FS,Λ≤wν

, f ∈ HomOS(M,N) and denote by

m ⊂ S the maximal ideal. Recall that S/mS = C. If

f ⊗S 1C : M ⊗S C→ N ⊗S C

is an isomorphism, then so is f .

Proof. f is surjective: We have an isomorphism M/mM→M ⊗S C which

gives an isomorphism f : M/mM→N/mN . The commutative diagram

M //

N

M/mMf

// N/mN

yields N = mN+f(M). Applying Nakayama’s Lemma on the weight spaces,we obtain Nλ = f(Mλ) ∀λ ∈ Λ≤

wν . By taking the sum over all λ ∈ Λ≤wν ,

we get the desired result.f is injective: Since every Nλ is free of finite rank, the sequence

(ker f)λ →Mλ Nλ

splits. So (ker f)λ is a direct summand of Mλ, hence a finitely generatedprojective module over S. Since the latter is local, we see that (ker f)λis even free over S and again, by taking the sum over all λ ∈ Λ≤

wν , weconclude that ker f is free. By assumption we have ker f ⊗S C = 0, whichforces ker f = 0.

Lemma 3.3.4. Let λ, µ ∈ Λ. Then Ext1OS

(∆wS

(λ),∆wS

(µ)) is finitely gener-

ated over S.

Proof. Since ∆wS

(λ) and ∆wS

(µ) are generated by finitely many weight spaces,

we find γ ∈ h∗ such that ∆wS

(λ),∆wS

(µ) ∈ O≤γS

. Let

...→ P 2 → P 1 ∆wS

(λ)→ 0

be a projective resolution of ∆wS

(λ) in O≤γS

. Every P i is a direct sum of

projective covers PJS

(σj), where J is an open and bounded subset of Λ (cf.[Fie03], Theorem 2.7). These objects are direct summands of objects QS(σj)which represent the exact functors

(·)σj : OJS→ S−mod

M 7→ Mσj

(cf. [Fie03], Lemma 2.3). Hence

47

HomOS(PS(σj),∆

wS

(µ)) ⊂ ∆wS

(µ)σj ,

which shows that HomOS(PS(σj),∆

wS

(µ)) is finitely generated as ∆wS

(µ)σj isa finitely generated module over a noetherian ring. Hence taking cohomologyat the first step in

0→ HomOS(P 1,∆w

S(µ))→ HomOS

(P 2,∆wS

(µ))→ ...

yields a finitely generated S-module.

Theorem 3.3.2. Let M ∈ OS,Λ≤wν be finitely generated over S and A a

localization of S. Then

Ext1OS

(M,∆wS

(λ))⊗S A∼= Ext1

OA(M ⊗S A,∆wA(λ))

for all λ ∈ Λ≤wν .

Proof. Since M is finitely generated over S, we can assume that it is alreadycontained in a subcategory O≤γ

Sfor suitable γ ∈ h∗. We take a projective

resolution

...π2→ P 2 π1→ P 1 π0→M → 0

in O≤γS

. Applying the functor HomOS(·,∆w

S(λ)) yields a sequence

0→ HomOS(P 1,∆w

S(λ))

d1→ HomOS(P 2,∆w

S(λ))

d2→ ...

where di := (· πi). By definition

Ext1OS

(M,∆wS

(λ)) = ker d2/ im d1.

Now localizing is exact and preserves projectives (cf. [Fie03], Proposition2.4), so

...π2⊗SidA→ P 2 ⊗S A

π1⊗SidA→ P 1 ⊗S Aπ0⊗SidA→ M ⊗S A→ 0

is a projective resolution of M ⊗S A in O≤γA . We get the sequence

0→ HomOA(P 1⊗S A,∆wS

(λ)⊗S A)t1→ HomOA(P 2⊗S A,∆

wS

(λ)⊗S A)t2→ ...

where ti = (· πi ⊗S idA). Since

HomOS(P i,∆w

S(λ))⊗S A

∼= HomOA(P i ⊗S A,∆wA(λ))

(cf. [Fie03], Proposition 2.4), we conclude ti = di ⊗S idA. Using exactnessof (· ⊗S A) we derive

48

Ext1OA(M ⊗S A,∆

wA(λ)) = ker t2/ im t1 ∼= (ker d2/ im d1)⊗S A,

the latter one being isomorphic to Ext1OS

(M,∆wS

(λ))⊗S A.

Definition 3.3.1. We call P ∈ Ow−V FS

projective if

Ext1OS

(P,M) = 0

for all M ∈ Ow−V FS

. Note that we calculate Ext1OS

(P,M) in the usual

deformed category OS, as we have enough projectives there (when we passto truncated subcategories).

Corollary 3.3.1. Let A be a localization of S. Suppose P ∈ Ow−V FS,Λ≤wν

is

projective. Then P ⊗S A is projective in Ow−V FA,Λ≤wν

.

Proof. If P ∈ Ow−V FS,Λ≤wν

is projective, this in particular means

Ext1OS

(P,∆wS

(λ)) = 0

for all λ ∈ Λ. Theorem 3.3.2 shows

Ext1OA(P ⊗S A,∆

wA(λ)) = 0

for all λ ∈ Λ. For an arbitrary M ∈ Ow−V FA,Λ≤wν

, induction on the length of a

w-twisted Verma flag yields the desired projectivity.

3.3.2 Projective covers

In the following, we construct indecomposable projective objects in Ow−V FS,Λ≤wν

as maximal extensions of w-twisted Verma modules. The technique is anal-ogous to the method of Soergel to construct indecomposable tilting modulesin [Soe99].

Theorem 3.3.3. For every λ ∈ Λ≤wν there exists a unique (up to non-

unique isomorphism) indecomposable projective object PwS

(λ) satisfying

1. (PwS

(λ) : ∆wS

(µ)) 6= 0⇒ µ ≥w λ,

2. (PwS

(λ) : ∆wS

(λ)) = 1.

Proof. Firstly, we prove existence by induction on the number of elementsw-above λ. If λ = ν, we set Pw

S(λ) := ∆w

S(λ). If not, ν >w λ. By induction,

we find an indecomposable P0 ∈ Ow−V FS,Λ≤wνsuch that

49

1. (P0 : ∆wS

(µ)) 6= 0⇒ µ ≥w λ,

2. Ext1OS

(P0,∆wS

(µ)) = 0 for all µ ∈ Λ≤wν , µ 6= ν,

3. (P0 : ∆wS

(λ)) = 1.

Now if Ext1OS

(P0,∆wS

(ν)) = 0, set PwS

(λ) := P0. If not, then at least

Ext1OS

(P0,∆wS

(ν))

is finitely generated over S since P0 has a finite ∆w-flag. As a finitely gener-ated module over a local ring, Ext1

OS(P0,∆

wS

(ν)) has a minimal set of gener-

ators (any other minimal set of generators contains the same amount of ele-ments). We fix a minimal set of generators v1, ..., vs of Ext1

OS(P0,∆

wS

(ν)).Now we represent v1 via a short exact sequence

0 → ∆wS

(ν) → P1 P0 0.

in OS. A segment of the corresponding long exact sequence of Ext1 groupsis

HomOS(∆w

S(ν),∆w

S(ν))

δ→ Ext1OS

(P0,∆wS

(ν))f→ Ext1

OS(P1,∆

wS

(ν))→ 0,

since Ext1OS

(∆wS

(ν),∆wS

(ν)) = 0. Now δ maps id∆wS

(ν) on v1, hence the

module ExtOS(P1,∆

wS

(ν)) is generated by the images of v2, ..., vs under f .Similarly, by looking at

Ext1OS

(P0,∆wS

(µ))→ Ext1OS

(P1,∆wS

(µ))→ Ext1OS

(∆wS

(ν),∆wS

(µ)),

we still have ExtOS(P1,∆

wS

(µ)) = 0 for all µ ∈ Λ≤wν \ν since ∆w

S(ν) is pro-

jective in Ow−V FS,Λ≤wν

. Continuing this way we inductively construct P2,P3, ...

until we arrive at a Ps that fits into a short exact sequence

0 → ∆wS

(ν)s → Ps P0 0

and satisfies Ext1OS

(Ps,∆wS

(µ)) = 0 for all µ ∈ Λ≤wν .

We want to show that if s is the smallest possible integer such that thereexists a short exact sequence

0 → ∆wS

(ν)s → P P0 0

with P satisfying Ext1OS

(P,∆wS

(µ)) = 0 for all µ ∈ Λ≤wν , then this P is

indecomposable and hence our sought for PwS

(λ). We have a short exactsequence

0 → P≥wν = ∆wS

(ν)s → P P<wν = P0 0.

50

Now a decomposition P = A⊕B yields a decomposition

A≥wν ⊕B≥wν → A⊕B A<wν ⊕B<wν .

Without loss of generality we may assume B<wν = 0 since P0 is indecom-posable by assumption. Due to

Ext1OS

(A⊕B, ·) ∼= Ext1OS

(A, ·)⊕ Ext1OS

(B, ·),

and A having ∆wS

(λ) as a quotient, A≥wν ∼= ∆wS

(ν)t with t < s would giverise to a short exact sequence

∆wS

(ν)t → A A<wν ∼= P0

contradicting the minimality of s. So we necessarily have t = s which forcesB≥wν = B to be zero.

Corollary 3.3.2. Let M ∈ Ow−V FS,Λ≤wν

. Then Ext1OS

(PwS

(λ),M) = 0 for all

λ ∈ Λ≤wν .

Proof. Induction over the length of a ∆w-flag of M .

Remark 3.3.1. In other words, the corollary states that for every λ ∈ Λ≤wν

the object PwS

(λ) is projective in Ow−V FS,Λ≤wν

.

To show uniqueness, let P′ be an indecomposable object in Ow−V FS,Λ≤wν

satisfy-

ing

1. Ext1OS

(P′,∆wS

(µ)) = 0 for all µ ∈ Λ and

2. P′ ∆wS

(λ).

Now the Ext1-vanishing condition of both objects ensures morphisms

PwS

(λ)→ P′

andP′ → Pw

S(λ),

hence an endomorphism f ∈ EndOS(Pw

S(λ)) which induces an isomorphism

on the Verma quotient. Since P′ is indecomposable, it suffices to prove thefollowing

Lemma 3.3.5. Let f ∈ EndOS(Pw

S(λ)) be an endomorphism which induces

an isomorphismf [λ]w : ∆w

S(λ)→ ∆w

S(λ).

Then f itself is an isomorphism.

51

Proof. Again by induction over the number of elements w-above λ: So sup-pose λ = ν. Then by construction Pw

S(ν) = ∆w

S(ν) and the claim follows.

Now let λ <w ν as before. Due to construction we have a commutativediagram

∆wS

(ν)s //

f≥wν

PwS

(λ) // //

f

PwS

(λ)<ν

f<ν

∆wS

(ν)s // PwS

(λ) // // PwS

(λ)<ν

By the induction hypothesis f<wν is an isomorphism. We take a further

look at a segment of the long exact cohomology sequence

HomOS(∆w

S(ν)⊕s,∆w

S(ν))

HomOS

(·,∆wS

(ν))(f≥wν)

π // // Ext1OS

(PwS

(λ)<ν ,∆wS

(ν))

Ext1O

S(·,∆w

S(ν))(f<

wν)

HomOS(∆w

S(ν)⊕s,∆w

S(ν)) π // // Ext1

OS(Pw

S(λ)<ν ,∆w

S(ν))

Observe that Ext1OS

(·,∆wS

(ν))(f<wν) is an isomorphism. Since we are work-

ing over a local ring

(HomOS(∆w

S(ν)⊕s,∆w

S(ν)) ∼= S

⊕s, π)

is a projective cover of the S-module

Ext1OS

(PwS

(λ)<ν ,∆wS

(ν)),

which implies that HomOS(·,∆w

S(ν))(f≥wν) is surjective. Since the S-module

HomOS(∆w

S(ν)⊕s,∆w

S(ν)) is free we get a splitting, hence

HomOS(·,∆w

S(ν))(f≥wν)

and consequently f≥wν are isomorphisms. Now the Five Lemma applied to

∆wS

(ν)s //

f≥wν

PwS

(λ) // //

f

PwS

(λ)<ν

f<ν

∆wS

(ν)s // PwS

(λ) // // PwS

(λ)<ν

yields the statement.

Corollary 3.3.3. PwS

(λ) is a projective cover of ∆wS

(λ) in Ow−V FS,Λ≤wν

.

52

Proof. Let M ∈ Ow−V FS,Λ≤wν

and Mg→ Pw

S(λ) be such that the composition

Mg→ Pw

S(λ) ∆w

S(λ)

is surjective. Then the projectivity of PwS

(λ) ensures a map PwS

(λ)h→ M

such that the following diagram commutes

PwS

(λ)

((QQQQQQQQQQQQQQQh // M

g// Pw

S(λ)

∆wS

(λ)

.

(g h) is an endomorphism of PwS

(λ) which induces an isomorphism on∆w

S(λ), thus (g h) is an isomorphism by the preceding lemma. This forces

g to be surjective.

3.4 The center of OV FS,Λ

3.4.1 The center of a category

In this chapter we want to determine the categorical center of Ow-VFS,Λ

for

a block Λ outside the critical hyperplanes. We orient ourselves towards[Fie01] and [Fie03]. Recall the construction of the projective covers Pw

S(λ)

in truncated subcategories from the preceding chapter. We will work withthe truncated block Λ≤

wν for ν ∈ Λ.

Lemma 3.4.1. Consider the exact sequence

kerπ → PwS

(λ)π ∆w

S(λ)

in OS,Λ≤wν . Then kerπ ∈ Ow−V FS,Λ≤wν

.

Proof. Due to the construction of PwS

(λ), the above sequence can be rewrit-ten as

PwS

(λ)>wλ → PwS

(λ)≥wλ PwS

(λ)[λ]w .

We immediately deduce

(PwS

(λ)>wλ)[µ]w∼=

Pw

S(λ)[µ]w if µ 6= λ,

0 else ,

which is equivalent to admitting a w-twisted Verma flag due to Theorem3.2.2.

53

Definition 3.4.1. We define

Pw := ⊕λi∈I

PwS

(λi)⊕mλi | I ⊆ Λ≤

wν finite,mλi ∈ N.

Lemma 3.4.2. For every M ∈ Ow−V FS,Λ≤wν

there exists P ∈ Pw with

P M.

Proof. We argue by induction over the length of a twisted Verma flag: sup-pose

N →M ∆wS

(µ)

with P ′ N for a P ′ ∈ Pw. Set P := P ′ ⊕ PwS

(µ). Projectivity of P yieldsa map

P →M

and the Five lemma yields that this map is surjective.

Lemma 3.4.3. For every M ∈ Ow−V FS,Λ≤wν

and P ∈ Pw with Pπ M con-

structed as above, the kernel

kerπ → PπM

is already contained in Ow−V FS,Λ≤wν

.

Proof. Again by induction over the length of a twisted Verma flag: Supposewe have a short exact sequence

N →M ∆wS

(µ).

Taking kernels in OS,Λ we get a diagram

kerπ′ // _

kerπ // _

kerπ′′ _

P ′ //

π′

P ′ ⊕ PwS

(µ)

π

// // PwS

(µ)

π′′

N // M // // ∆w

S(µ)

where kerπ′, kerπ′′ ∈ Ow−V FS,Λ≤wν

. Now the Nine Lemma forces the first hori-

zontal to be exact, hence kerπ ∈ Ow−V FS,Λ≤wν

.

54

By combining the two statements above, we deduce the following

Corollary 3.4.1. For every M ∈ Ow−V FS,Λ≤wν

we find a resolution

...→ P 2 → P 1 M → 0

where P i ∈ Pw.

Remark 3.4.1. The statements above show that the exact category Ow-VFS,Λ≤wν

contains enough projectives.

Definition 3.4.2. We denote by ZwS,Λ

the categorical center of Ow−V FS,Λ

, i.e.

the ring of endotransformations of the identity functor. By Z≤wν

S,Λwe denote

the center of Ow−V FS,Λ≤wν

.

For ν ′ ≤w ν we have a natural restriction Z≤wν

S,Λ→ Z≤

wν′

S,Λ, hence the Z≤

wν

S,Λ

form a directed poset.

Proposition 3.4.1. The restriction maps ZwS,Λ→ Z≤

wν

S,Λinduce an isomor-

phismZw

S,Λ∼= lim←−Z

≤wνS,Λ

.

Proof. An element of the center is determined by its action on the finitelygenerated modules. Since ∆w

S(λ) is finitely generated for any λ ∈ Λ, and

Ow−V FS,Λ

consists of modules with a finite ∆w-flag, the map above is injective

and surjective.

Now every z ∈ Z≤wν

S,Λdefines an endomorphism zλ : Pw

S(λ) → Pw

S(λ). We

get a mapφS : Z≤

wν

S,Λ→

∏λ≤wν EndOS

(PwS

(λ))

z → zλλ≤wν .

We define

B≤wν

S,Λ:=

zλ ∈ ∏λ≤wν

EndOS(Pw

S(λ))

∣∣∣∣ f zλ = zµ f ∀λ, µ ≤w ν∀f ∈ HomOS

(PwS

(λ),PwS

(µ))

.

Remark 3.4.2. The arguments for the next two statements are basicallythe same as in [Fie01], Proposition 4.4. and Lemma 4.6.. For the sake ofcompleteness, we will state them nevertheless.

Proposition 3.4.2. φS induces an isomorphism

φS : Z≤wν

S,Λ→ B≤

wν

S,Λ⊂∏λ≤wν EndOS

(PwS

(λ)).

55

Proof. Obviously, we have imφS ⊂ B≤wν

S,Λ. In case φS(z) = 0, z induces the

trivial endomorphism on all PwS

(λ). Since every M ∈ Ow−V FS,Λ≤wν

is a quotient

of an object in Pw, the naturalness of z forces it to act trivially on each suchM . This shows injectivity.Now let z = zλ ∈ B≤

wν

S,Λ, M ∈ Ow−V F

S,Λ≤wν. We take a resolution of M

P ′ → P M → 0

with P, P ′ ∈ Pw. Since the diagram

P ′ //

z

P // //

z

M

P ′ // P // // M

commutes, we get an induced map M →M . By choice of z, the morphismsin the diagram commute with morphisms between objects in Pw, so themorphism M →M does not depend on the specific choice of a resolution

P ′ → P M → 0.

Now let M,M ′ ∈ Ow−V FS,Λ≤wν

with two resolutions

P ′ → P M → 0,

Q′ → QM ′ → 0

as above and let M →M ′ be a morphism. Suppose that

M //

z

M ′

z

M // M ′

does not commute. Then

Pf

//

z

Q

z

Pf

// Q

would not commute where we obtain the map f : P → Q due to the projec-tivity of P . This contradicts z ∈ B≤

wν

S,Λ, hence we see that z indeed defines

an endotransformation of the identity functor.

Let A be a localization of S. We have seen that PwS

(λ)⊗S A is a projective

object in Ow−V FA,Λ≤wν

with quotient ∆wA(λ) (cf. Corollary 3.3.1). Since every

∆wS

(λ) is finitely generated over S, each PwS

(λ) is finitely generated over S.

56

Lemma 3.4.4. The canonical map

ψ :∏λ≤wν EndOS

(PwS

(λ)) →∏λ≤wν EndOA(Pw

S(λ)⊗S A)

induces maps

Z≤wν

S,Λ→ Z≤

wνA,Λ

and

ZwS,Λ

→ ZwA,Λ.

Proof. We have the base change property

HomOS(Pw

S(λ),Pw

S(µ))⊗S A

∼= HomOA(PwS

(λ)⊗S A,PwS

(µ)⊗S A)

(cf. Theorem 3.3.1). So for z ∈ Z≤wν

S,Λ,

ψ(z) ∈∏λ≤wν

EndOA(PwS

(λ)⊗S A)

commutes with each morphism PwS

(λ)⊗S A → PwS

(µ)⊗S A, hence it lies in

Z≤wν

A,Λ . Since every M ∈ Ow−V FS,Λ

lies in a suitable Ow−V FS,Λ≤wν

we get

ZwS,Λ→ ZwA,Λ.

3.4.2 The subgeneric case

Remark 3.4.3. In this chapter, we determine the center of a subgenericblock. Here, we strongly follow [Fie03]. For modules with a w-twisted Vermaflag, one apriori has to include the case of modules a with Nabla flag. By ap-plying the duality functor, we can essentially reduce it to the case of moduleswith a Verma flag, which was treated by Fiebig.

The action of the center on twisted Verma modules yields a map

φS : ZwS,Λ

→∏λ∈Λ S

z 7→ z|∆wS

(λ).

Proposition 3.4.3. For every prime ideal p ⊂ S such that α /∈ p for allα ∈ R+ the map

φSp: Zw

Sp,Λ→

∏λ∈Λ Sp

is an isomorphism. In general, the map above is injective.

57

Proof. For a prime ideal p ⊂ S such that α /∈ p for all α ∈ R+, Λ decomposesinto generic equivalence classes. In each of these blocks ∆w

S(λ) ∼= ∆S(λ) is

projective and there are no morphisms between Verma modules of differenthighest weights. Hence the condition in

B≤wν

Sp,Λ:=

zλ ∈ ∏λ≤wν

EndOSp(Pw

Sp(λ))

∣∣∣∣ f zλ = zµ f ∀λ, µ ≤w νf ∈ HomOSp

(PwSp

(λ),PwSp

(µ))

becomes trivial and we get the desired result. For the general case, thecanonical embedding Sp → Q induces an injective map

EndOSp(Pw

Sp(λ)) → EndOSp

(PwSp

(λ))⊗SpQ,

which in turn gives an embedding ZwSp,Λ

→ ZwQ,Λ. The diagram

ZwSp,Λ

//

ZwQ,Λ

∏λ∈Λ Sp

//∏λ∈Λ Q

is commutative with injective horizontals and isomorphism ZwQ,Λ→∏λ∈Λ Q,

which forces ZwSp,Λ→∏λ∈Λ Sp to be injective.

Lemma 3.4.5. Assume Λ = λ, sα · λ where µ := sα · λ. Further supposeλ <w µ. Then

Ext1OSα

(∆wSα

(λ),∆wSα

(µ)) ∼= Sα/αSα ∼= Kα.

Proof. First we assume λ < µ. Then ∆wSα

(µ) ∼= ∆Sα(µ) and

∆wSα

(λ) ∼= ∇Sα(λ) ∼= ∆Sα

(λ)

(cf. Theorem 3.1.2). We have short exact sequence

0→ ∆Sα(µ)

i→ PSα(λ)→ ∆Sα

(λ)→ 0

(cf. [Fie03], Proposition 3.4.). Then again, since ∆Sα(µ) is projective, this

sequence is a projective resolution of ∆Sα(λ). Now the map

HomOSα(PSα

(λ),∆Sα(µ))

(·i)→ HomOSα(∆Sα

(µ),∆Sα(µ))

induces an isomorphism

im (· i) ∼= α ·HomOSα(∆Sα

(µ),∆Sα(µ))

58

by the description of the subgeneric block in [Fie03], Proposition 3.4. Hence,

Ext1OSα

(∆Sα(λ),∆Sα

(µ)) ∼= Sα/αSα ∼= Kα.

Now if λ > µ, we have ∆wSα

(λ) ∼= ∇Sα(λ) and

∆wSα

(µ) ∼= ∇Sα(µ) ∼= ∆Sα

(µ)

(cf. Theorem 3.1.2). Since the exact functor

d : OSα→ OSα

is fully faithful on Sα-free objects, we deduce

Ext1OSα

(∇Sα(λ),∇Sα

(µ)) ∼= Ext1OSα

(∆Sα(µ),∆Sα

(λ)).

Again we have a short exact sequence

0→ ∆Sα(λ)

i→ PSα(µ)→ ∆Sα

(µ)→ 0,

due to Proposition 3.4. in [Fie03], thus we can argue as above.

Corollary 3.4.2. Let Λ = λ, sα ·λ and assume λ > sα ·λ, but λ <w sα ·λ.Define µ := sα · λ. Then there is a unique non-split extension of ∆w

Sα(λ) by

∆wSα

(µ), and it is isomorphic to PSα(µ).

Proof. If λ > µ, we have ∆wSα

(λ) ∼= ∇Sα(λ) and

∆wSα

(µ) ∼= ∇Sα(µ) ∼= ∆Sα

(µ).

Dualizing the short exact sequence

0→ ∆Sα(λ)→ PSα

(µ)→ ∆Sα(µ)→ 0

yields a non-split extension

0→ ∇Sα(µ)→ d(PSα

(µ))→ ∇Sα(λ)→ 0.

Observing that PSα(µ) is self-dual, PSα

(µ) has to be isomorphic to theunique non-split extension of ∆w

Sα(λ) by ∆w

Sα(µ).

Let Λ = λ, sα · λ, λ > sα · λ but λ <w sα · λ. By construction, we have ashort exact sequence

∆wSα

(sα · λ) → PwSα

(λ) → ∆wSα

(λ). (3.4)

With the preceding results, we deduce

59

PwSα

(λ) ∼= PSα(sα · λ).

Define µ := sα · λ. The short exact sequence (3.4) can then be rewritten as

PwSα

(λ)[µ]w → PwSα

(λ)≤wµ Pw

Sα(λ)<

wµ.

Now every f ∈ EndOSα(Pw

Sα(λ)) induces a morphism

f [µ]w ∈ EndOSα(Pw

Sα(λ)[µ]w)

and a morphism

f<wµ ∈ EndOSα

(PwSα

(λ)<wµ).

Since these endomorphism rings are endomorphism rings of twisted Vermamodules, we get a map

h : EndOSα(Pw

Sα(λ)) → Sα ⊕ Sα

f 7→ (f [µ]w , f<wµ).

Due to the naturalness of the action of ZwSα,Λ

, we get a commutative diagram

ZwSα,Λ

// Sα ⊕ Sα

EndOSα(Pw

Sα(λ))

h

77oooooooooooo

Since the horizontal as well as h are injective, the vertical arrow has to beinjective as well. We are left to show that every f ∈ EndOSα

(PwSα

(λ)) induces

an element of ZwSα,Λ

. For f ∈ EndOSα(Pw

Sα(λ)) we get a pair

(f, f [µ]w) ∈ EndOSα(Pw

Sα(λ))⊕ EndOSα

(∆wSα

(µ)).

EndOSα(∆w

Sα(µ)) is commutative. Using that Pw

Sα(λ) ∼= PSα

(sα · λ), we see

that

EndOSα(Pw

Sα(λ)) ∼= EndOSα

PSα(sα · λ) ⊂ Q⊕Q

is commutative. So (f, f [µ]w) commutes with all morphisms

∆wSα

(sα · λ)→ ∆wSα

(sα · λ)⊕ (PwSα

(λ)

and all endomorphisms PwSα

(λ)→ PwSα

(λ). If we have PwSα

(λ)→ ∆wSα

(sα ·λ),

we concatenate with ∆wSα

(sα·λ) → PwSα

(λ) which results in an endomorphism

PwSα

(λ) → PwSα

(λ). This shows that for each f ∈ EndOSα(Pw

Sα(λ)) the pair

(f, f [µ]w) lies in B≤wν

Sα,Λ, which then yields

60

ZwSα,Λ∼= EndOSα

(PwSα

(λ)).

Using PwSα

(λ) ∼= PSα(sα · λ) and the explicit description of

EndOSα(PSα

(sα · λ))

(cf. [Fie06], Lemma 3.1.), we get the following

Theorem 3.4.1. ZwSα,Λ∼= (xµ, xλ) ∈ Sα ⊕ Sα |xµ ≡ xλ mod α.

3.4.3 The general case

Let p ⊂ S be a prime ideal. With the preceding results, we can view ZwS,Λ

and ZwSp,Λ

as subrings of∏λ∈Λ Q (cf. Proposition 3.4.3). Denote by

P(S) := p ∈ Spec(S) | ht p = 1.

Proposition 3.4.4. We have

ZwS,Λ

=⋂

p∈P(S)

ZwSp,Λ

in∏λ∈Λ Q.

Proof. Since we just deal with finitely generated objects, it suffices to showthe corresponding statement for Z≤

wν

S,Λ. Take projective objects

PwS

(λ)λ≤wν

and identify Z≤wν

S,Λ, Z≤

wν

Sp,Λand Z≤

wνQ,Λ with subrings in

∏λ≤wν

EndOS(Pw

S(λ)),

∏λ≤wν

EndOSp(Pw

S(λ)⊗S Sp)

and ∏λ≤wν

EndOQ(Pw

S(λ)⊗S Q).

We have a commutative diagram

61

Z≤wν

S,Λ//

∏λ≤wν EndOS

(PwS

(λ))

Z≤wν

Sp,Λ//

∏λ≤wν EndOSp

(PwS

(λ)⊗S Sp)

Z≤wν

Q,Λ//∏λ≤wν EndOQ

(PwS

(λ)⊗S Q)

The canonical embedding S → Sp ensures Z≤wν

S,Λ⊂⋂

p∈P(S)Z≤wνSp,Λ

. Now take

z ∈⋂

p∈P(S)

Z≤wν

Sp,Λ⊂

⋂p∈P(S)

∏λ≤wν

EndOSp(Pw

S(λ)⊗S Sp)

and observe that

⋂p∈P(S)

∏λ≤wν

EndOSp(Pw

S(λ)⊗S Sp) =

⋂p∈P(S)

∏λ≤wν

EndOSp(Pw

S(λ))⊗S Sp

=∏λ≤wν

⋂p∈P(S)

EndOS(Pw

S(λ))⊗S Sp

=∏λ≤wν

EndOS(Pw

S(λ)),

where we used that

EndOS(Pw

S(λ)) =

⋂p∈P(S)

EndOS(Pw

S(λ))⊗S Sp.

Hence z ∈∏λ≤wν EndOS

(PwS

(λ)). Due to

HomOS(Pw

S(λ),Pw

S(µ)) ⊂ HomOSp

(PwS

(λ)⊗S Sp,PwS

(µ))⊗S Sp)

for all prime ideals p ⊂ S, z commutes with all morphisms PwS

(λ)→ PwS

(µ).

So it lies in B≤wν

S,Λ∼= Z≤

wν

S,Λ.

Theorem 3.4.2.

ZwS,Λ∼=

xλ ∈

∏λ∈Λ

S |xλ ≡ xsα·λ mod α ∀α ∈ R+(Λ)

.

62

Proof. Let p ∈ P. If α /∈ p for all α ∈ R+(Λ), then ZwSp,Λ∼=∏λ∈Λ Sp. If

α ∈ p, then it generates a prime ideal Sα ⊂ p forcing p = Sα since p is ofheight one. For α ∈ R+(Λ) we have

ZwSα,Λ∼= (xµ, xλ) ∈ Sα ⊕ Sα |xµ ≡ xλ mod α

(cf. Theorem 3.4.1). Taking the intersection of these subrings in∏λ∈Λ Q

proves the theorem.

Remark 3.4.4. Firstly, we notice that the algebraic description of the centeris independent of the Weyl group element w. Secondly, Zw

S,Λhas the same

algebraic description as Z(OS,Λ), the categorical center of OS,Λ (cf. [Fie03],Theorem 3.6.), even though OS,Λ is a larger category. In [Fie03], Fiebig usesthe category of modules with a Verma flag to determine the center, since theprojective objects of OS,Λ admit a Verma flag. Apparently, when it comesto determining the categorical center, it does not make a difference whetheryou consider modules with a Verma flag or modules with a w-twisted Vermaflag. This will become clear in Chapter 5, where we prove that the categoryof modules with a Verma flag is equivalent to the category of modules witha w-twisted Verma flag.

63

64

Chapter 4

Combinatorics

4.1 Sheaves on Moment Graphs

In this chapter, we introduce the combinatorial language we use to describea block of OS. We recall the main theorem in [Fie08a] and provide all themachinery we use in the following chapter to generalize Fiebig’s result to thecase of modules with a w-twisted Verma flag. The details to this chaptercan be found in [BM01], [Fie08a] and [Jan08].

Definition 4.1.1. Let k be a field and V a finite-dimensional vector spaceover k. A moment graph G = (V, E ,≤, l) is given by

• a graph (V, E) with a set of vertices V and a set of edges E,

• a partial order “ ≤′′ on V where two vertices x, y ∈ V are comparableif they are linked by an edge,

• a labeling function

l : E → P(V )

which labels an edge by a one-dimensional subspace of V .

Remark 4.1.1. We write x −→ y if there is an edge E between x and yand if x ≤ y holds. If the order is not important, we will denote the arrowabove by E : x−−y. For α = α(E) ∈ V \ 0 a generator of the line l(E) wewrite x

α−→ y.

Let G be a moment graph and denote by S := S(V ) the symmetric algebraof the vector space V . We give S a Z-grading by setting deg V = 2. We willnow consider graded S-modules and graded S-morphisms, i.e. morphisms ofdegree 0. For M a graded S-module, k ∈ Z, we denote by M [k] the gradedS-module shifted by degree k, i.e. M [k](n) := M(n+k).

Definition 4.1.2. A sheaf M = (Mx, ME, ρx,E) is given by

65

• an S-module Mx for each vertex x ∈ V ( the stalk at the vertex x),

• an S-module ME for each edge E ∈ E with the property that

l(E) · ME = 0,

• and S-morphisms ρx,E :Mx →ME and ρy,E :My →ME if x and yare the endpoints of E.

A morphism of sheaves f :M→N is a collection of maps fx :Mx → N xand fE :ME → NE such that the diagram

Mx

ρx,E

fx// N x

ρ′x,E

MEfE

// NE

commutes for all x and E such that x is an endpoint of E. So the sheaveson G obviously form a category which we denote by GS −mod.

As a first example we give the definition of the structure sheaf.

Definition 4.1.3. The structure sheaf A is given by

• Ax = S for all x ∈ V,

• AE = S /α S for an edge E labeled by α,

• ρx,E : S→ S /α S the natural quotient map.

Definition 4.1.4. By a subgraph I ⊆ G we mean a full subgraph, i.e. anyedge of G connecting two vertices of I will also be contained in I.

Definition 4.1.5. Let I ⊆ G be a subgraph andM a sheaf on G. The spaceof sections of M over I denoted by Γ(M, I) :=M(I) is defined as

M(I) := (mx) ∈∏x∈IMx | ρx,E(mx) = ρy,E(my) for (E : x− y) ∈ I.

The space of sections of the structure sheaf A over I denoted by Γ(A, I) :=Z(I) is defined as

Z(I) := (zx) ∈∏x∈I

S | zx ≡ zy mod α for any edge (xα− y) ∈ I.

66

We denote the global sections of M by Γ(M) := Γ(M, I) = M(G). Thespace of global sections of the structure sheafA is called the structure algebraof G, and it is given by

Z(G) := Γ(A) = (zx) ∈∏x∈G

S | zx ≡ zy mod α for any edge xα− y.

The name is justified since pointwise addition and multiplication define analgebra structure on Z(G). Now for M ∈ GS − mod, its space of globalsections Γ(M) becomes a module over Z(G) under pointwise action. Hencetaking global sections defines a functor

Γ : GS −mod → Z(G)−mod .

Remark 4.1.2. Instead of S, we can consider the localization of S at aprime ideal p ⊂ S, which we denote by Sp. Let I ⊆ G be a subgraph. Wedefine the structure algebra over I over the base Sp as

Zp(I) := (zx) ∈∏x∈I

Sp | zx ≡ zy mod α for any edge (xα− y) ∈ I.

Now if E is an edge such that l(E) = α ∈ Sp is invertible, the relation abovebecomes trivial. Hence it suffices to look at the graph Ip where the verticesof I are kept, but the edges whose labels are invertible in Sp are deleted.The canonical map S→ Sp induces a functor

(· ⊗S Sp) : GS −mod → GSp −modM 7→ M⊗S Sp,

where

(M⊗S Sp)x :=Mx ⊗S Sp,

(M⊗S Sp)E :=ME ⊗S Sp,

(ρ⊗S idSp)x,E := ρx,E ⊗S idSp .

With the observations above,M⊗SSp can be viewed as a sheaf of Sp-moduleson Gp. If Gp decomposes into a disjoint union of subgraphs,M⊗S Sp becomesa direct sum of sheaves on the subgraphs of Gp.

For J ⊆ I ⊆ G subgraphs, we get restriction maps

Z(G)→ Z(I)→ Z(J ).

These maps do not have to be surjective in general. Denote by ZI the imageof the restriction map Z(G)→ Z(I).

67

Definition 4.1.6. For M∈ GS −mod we define its support as

supp (M) := x ∈ V |Mx 6= 0.

Definition 4.1.7. Let SHS(G) ⊂ GS−mod be the full subcategory of sheavesM which have finite support and for which Mx is finitely generated andtorsion free over S for every x ∈ V.

We abbreviate Z := Z(G).

Definition 4.1.8. Let Z − modf ⊂ Z − mod be the full subcategory ofmodules which are finitely generated and torsion free as modules over S andfor which there exists a finite subgraph I ⊆ G such that the action of Zfactors over ZI .

The global section functor restricts to a functor

Γ : SHS(G) → Z −modf

(cf. [Fie08a]). Everything works equally well if we replace S by one of itslocalizations at a prime ideal Sp.

We want to define an adjoint functor to Γ. For this, we need to imposefurther assumptions on the moment graph G. For α ∈ V \ 0 and I ⊆ G asubgraph, we define

Zα(I) := ZSα(I),

where Sα denotes the localization of S at the prime ideal generated by α in Sand ZSα(I) is the structure algebra over I with respect to the base ring Sα.Now the map S → Sα induces a map ZI → Z(I)→ Zα(I) which providesan injective map ZI ⊗S Sα → Zα(I).

Definition 4.1.9. A moment graph G is called quasi-finite if the map

ZI ⊗S Sα → Zα(I)

is surjective for every finite subgraph I ⊆ G and every α ∈ V \ 0.

4.2 Z-modules and Localization

Assume that G is a quasi-finite moment graph. Let Q := Quot(S) be thequotient field of S. For any finite subgraph I ⊆ G we have an isomorphism

ZI ⊗S Q ∼=⊕x∈I

Q .

68

Let M ∈ Z − modf . Define MQ := M ⊗S Q. Let I be a finite subgraphsuch that the action of Z factors over ZI . The isomorphism above inducesa canonical decomposition of MQ

MQ =⊕x∈I

MxQ.

As an S-module, M is torsion free, hence we have an inclusion

M →MQ =⊕x∈I

MxQ

via which we can view any element m ∈ M as a tuple m = (mx), wheremx ∈Mx

Q. For J ⊆ G an arbitrary subgraph, define

MJ := M ∩⊕x∈J

MxQ

and

MJ := im(M →MQ ⊕x∈J

MxQ)

where the last map is the projection from the canonical decomposition ontothe vertices in J (onto the vertices in I ∩ J to be precise). For a vertexx ∈ V, we just write Mx (respectively Mx).

At this point we are ready to define the localization functor L. In thefollowing, we continue working over the base ring S, but everything worksequally well for a localization of S at a prime ideal. For M ∈ Z −modf , wedefine a sheaf L(M) by setting L(M)x := Mx for a vertex x ∈ V. For an

edge E : xα− y consider its local structure algebra

Z(E) := (zx, zy) ∈ S⊕S | zx ≡ zy mod α.

We have an inclusion Mx,y := Mx,y ⊂Mx⊕My. Let M(E) := Z(E)·Mx,y

be the Z(E)-submodule generated by Mx,y and take the projections ontothe vertices M(E) → Mx and M(E) → My. Then we define the S-moduleL(M)E as the pushout of the following diagram

M(E) //

Mx

My // L(M)E

considered in the category S−mod. Let

ρx,E : Mx → L(M)E ,

ρy,E : My → L(M)E

69

be the pushout maps. We still have to check that

l(E) · L(M)E = 0

holds. Now the maps M(E) → Mx and M(E) → My are surjective, hencewe have isomorphisms

Mx/M(E)x→L(M)E ,

My/M(E)y→L(M)E ,

where M(E)x = ker(M(E) → My) ∼= Mx ∩ M(E) (and analogously forM(E)y).Now (α, α) = (α, 0) + (0, α) ∈ Z(E), hence αMx = (α, 0) ·M(E) ⊂M(E)x(and analogously αMy ⊂M(E)y). This shows α · L(M)E = 0, hence L(M)is a sheaf. It is finitely supported and by construction its stalks are finitelygenerated and torsion free over S. Hereby we have constructed the localiza-tion functor

L : Z −modf → SH(G).

We have the following

Theorem 4.2.1 (cf. [Fie08a], Theorem 3.5.). Γ and L induce a pair ofadjoint functors

SH(G)

Γ**

Z −modf ,Loo

and the canonical maps Γ(M) → ΓLΓ(M) and L(M) → LΓL(M) are iso-morphisms.

Denote by SH(G)glob the essential image of the functor L and by Z−modloc

the essential image of Γ. The theorem implies that (L,Γ) are mutuallyinverse equivalences between SH(G)glob and Z−modloc. Following [Fie08a],we will denote these categories by C(G) and think of them as a categorythat is simultaneously embedded in Z −modf and SH(G). Its objects arecharacterized by having both a local and global structure.

Definition 4.2.1. A subgraph I ⊆ G is called open if it is downwardlyclosed, that means, if y ∈ I and x ≤ y, then x ∈ I.

Definition 4.2.2. A module M ∈ Z − modf admits a Verma flag, if MI

is graded free for all open subgraphs I ⊆ G. Denote by

Z −modVF ⊂ Z −modf

the full subcategory of modules admitting a Verma flag.

70

Remark 4.2.1. In case we work with a localization of S as base ring, werequire the module MI to be free.

Denote by Z −modref ⊂ Z −modf the full subcategory of modules M suchthat MI is reflexive for all open subgraphs I ⊆ G.

Definition 4.2.3. A moment graph G is called a GKM-graph-graph (GKMstands for Goresky, Kottwitz and MacPherson), if it is quasi-finite and iffor any two E,E′ ∈ E with E 6= E′ and E ∩ E′ 6= ∅ l(E) 6= l(E′) holds, i.e.the labels attached to two different edges which share a vertex are pairwiselinearly independent.

Proposition 4.2.1 (cf. [Fie08a], Proposition 4.6.). Suppose G is a GKM-graph, then we have

Z −modref ⊂ Z −modloc = C(G).

Since any free S-module is reflexive, we have

Z −modVF ⊂ Z −modref ⊂ Z −modloc

and each of these categories forms a category of sheaves on G. We willdenote them by

C(G)VF ⊂ C(G)ref ⊂ C(G).

Now fix a moment graph G = (V, E ,≤, l). Let x ∈ V. We define

Ux := E ∈ E |E : x→ y

and

Dx := E ∈ E |E : y → x.

Definition 4.2.4. Let x ∈ V and M ∈ SH(G). We define the partialcostalk at x by

M[x] :=⋂

E∈Dx

ker(ρx,E)

and the costalk at x by

Mx :=⋂

E∈Ux∪Dx

ker(ρx,E).

Definition 4.2.5. The skyscraper sheaf V(x) at the vertex x ∈ V is definedvia

V(x)y =

S if x = y

0 otherwise

and V(x)E = 0 for all E ∈ E.

71

Definition 4.2.6. We call the global sections of the skyscraper sheaf

V(x) := Γ(V(x),G)

the Verma module at the vertex x. It can be identified with the free S-moduleof rank one, generated in degree zero, on which Z acts via the evaluation

Z → S(zy) 7→ zx.

Let M ∈ Z −modref and x ∈ G. We define

M [x] := ker(M≤x →M<x).

Then we get a nice characterization of modules with a Verma flag:

Lemma 4.2.1 ([Fie08a], Lemma 4.7). Let M ∈ Z−modref. M ∈ Z−modVF

if and only if

M [x] ∼= ⊕ni=1 V(x)[ki]

for some k1, ..., kl ∈ Z and any x ∈ W(Λ). Note that for non-graded basering, we require the module above to be free, i.e.

M [x] ∼= ⊕ni=1 V(x)

for any x ∈ W(Λ).

4.3 Projective objects

We want to give the category C(G) an exact structure, so that we can talkabout short exact sequences and projective objects. Let

A→ B → C

be a sequence in Z−modf . It is called short exact if for any open subgraphI ⊆ G the sequence

0→ AI → BI → CI → 0

is short exact as a sequence of abelian groups.

Theorem 4.3.1 (cf. [Fie08a], Theorem 4.1.). The definition above gives anexact structure in the sense of Quillen (cf. [Qui73]) on Z −modf .

Any subcategory of Z −modf which is closed under extensions inherits anexact structure from Z −modf . In particular, we deduce that Z −modref

and Z −modVF are exact categories.

72

Remark 4.3.1. Before we continue by constructing projective objects, weintroduce a notion which is inspired by classical sheaf theory. In classicalsheaf theory, one wants to know if one can extend the sections over an opensubset to global sections, or, to be more precise: Let X be a topological space,U ⊆ V ⊆ X two open subsets, F a sheaf on X with values in the categoryof abelian groups AB. Then F is called flabby, if the restriction map

ρVU : Γ(F , V ) → Γ(F , U)

is surjective.

Definition 4.3.1. Let M ∈ Z − modf . M is called flabby, if MI is de-termined by local relations, i.e. MI ∈ Z − modloc for any open subgraphI ⊆ G.

Definition 4.3.2. Let M ∈ SH(G), x ∈ G. The module Mδx is defined asthe image of the composition

M(< x) ⊂⊕y<x

My π→⊕y→xMy ⊕ρy,E→

⊕E∈Dx

ME ,

where π denotes the projection.

We give a characterization of flabbiness on the sheaf level:

Proposition 4.3.1 (cf. [Fie08a], Proposition 4.2.). Let M ∈ SH(G)glob.Then the following are equivalent:

1. Γ(M) is flabby.

2. For any open I ⊆ G, the restriction map

Γ(M,G)→ Γ(M, I)

is surjective.

3. For any vertex x ∈ G, the restriction map

Γ(M, ≤ x)→ Γ(M, < x)

is surjective.

4. For any vertex x ∈ G, the map⊕E∈Dx

ρx,E :Mx →Mδx

is surjective.

73

In the following, we assume that G is a GKM-graph.For an exact category A we say that P ∈ A is projective, if the functor

HomA(P, ·) : A → AB

is exact. Again, AB denotes the category of abelian groups. On the levelof Z-modules, one could call this a “global” definition of projective objects.The next proposition gives a more “local” characterization of projectiveobjects:

Proposition 4.3.2 (cf. [Fie08a], Proposition 5.1.). Let P be in C(G)ref.Suppose that

1. For any vertex x of G, Px is projective (in the category of S-modules)and

2. for any edge E : xα→ y the map ρx,E : Px → PE induces an isomor-

phism

Px/αPx→PE .

Then P is projective in C(G)ref.

We give a definition of a certain class of objects that are universal withrespect to extending local sections to global sections (cf. [BM01], [FW10]).

Definition 4.3.3. A sheaf B on the moment graph G is called a Braden-MacPherson sheaf (BMP sheaf) if it satisfies the following properties:

1. Bx is a graded free S-module of finite rank for any x ∈ G,

2. for a directed edge E : x −→ y, the map ρx,E : Bx → BE is surjectivewith kernel l(E)Bx,

3. for any open subgraph J ⊆ G the map

Γ(B,G)→ Γ(B,J )

is surjective, and

4. the composition

Γ(B,G) ⊂⊕z∈GBz → Bx

is surjective for any x ∈ G (where the map on the right is the projectionalong the decomposition).

74

In the following, we will classify the indecomposable BMP sheaves in C(G)ref

from the sheaf theoretic point of view. They form a class of projective andflabby objects. However, we do need to impose a further assumption on theGKM-graph G, namely that it is bounded from above, which means that forany vertex x ∈ V the set y ∈ V | y ≥ x is finite. The next theorem can befound in [Fie08a], Theorem 5.2. and [Jan08], Section 3.5:

Theorem 4.3.2. 1. For any x ∈ V, there exists a graded S-sheaf

B(x) ∈ C(G)ref

with

• B(x) is indecomposable and projective,

• supp(B(x)) ⊂ y | y ≥ x and B(x)x ∼= S.

2. If P is a graded projective sheaf in C(G)ref, then there exist verticesx1, ..., xn ∈ V, integers l1, ..., ln and an isomorphism of graded S-sheafs

P ∼= B(x1)[l1]⊕ ...⊕ B(xn)[ln].

Remark 4.3.2. If we work with S := S(0), the localization of S at themaximal ideal corresponding to zero, there exists a sheaf BS(x) with theproperties in Theorem 4.3.2 (1), and if P is projective in CS(G)ref, then

there exist x1, ..., xn ∈ V and an isomorphism of S-sheaves

P ∼= BS(x1)⊕ ...⊕ BS(xn).

Remark 4.3.3. We explain the algorithm to construct B(x).Set B(x)x = S and B(x)y = 0 for y < x. Now Proposition 4.3.2 states thatfor an edge E : y → z, B(x)E and ρy,E are already determined by the stalkB(x)y and the label of E. Let us assume that we have inductively constructedall the stalks and restriction maps on the subgraph < z for a z ∈ V. Sowe are left with constructing B(x)z and the restriction maps

ρz,E : B(x)z → B(x)E

for all E ∈ Dz. Again, let B(x)δz ⊂ ⊕E∈DzB(x)E be the image of thecomposition of maps

B(x)(< z) ⊂⊕y<z

B(x)yπ→⊕y→zB(x)y

⊕ρy,E→⊕E∈Dz

B(x)E .

Then take a projective cover of B(x)δz in the category of graded S-modules,which we define as B(x)z. Finally, the restriction maps

ρz,E : B(x)z → B(x)E

for E ∈ Dz are defined as the composition of maps

B(x)z → B(x)δz → B(x)E .

75

4.4 The connection to Representation Theory

We explain how one assigns a moment graph to a block of OS outside thecritical hyperplanes.Let Λ ⊂ h∗/ ∼ be an equivalence class outside the critical hyperplanes.We assume furthermore that it is of negative level, i.e. that it contains anantidominant weight. The associated moment graph G = GΛ over h is definedas follows: Let σ be antidominant in Λ, and denote by W(Λ)σ its stabilizerin W(Λ). We set V := W(Λ)/W(Λ)σ and identify the vertices of GΛ withthe set of minimal representatives according to the length function given bythe Bruhat order. Let T (Λ) ⊂ W(Λ) be the set of reflections in W(Λ). Fort ∈ T (Λ) denote by αt ∈ h the corresponding coroot. Then λ, µ ∈ V arelinked by an edge if there is a reflection t ∈ T (Λ) such that λ = t · µ andλ 6= µ. We label this edge by αt ∈ h. The order on GΛ is the induced Bruhatorder on W(Λ)/Wσ.Now assume Λ ⊂ h∗/ ∼ is of positive level and let ν be the dominant weight.Then we define GΛ as above, but reverse the order. In general, we denoteby t(G) the tilted moment graph of a moment graph G, which has the samevertices, edges and labels as G, but carries the reversed order.

We work with S as base ring now and consider the category ZS(G)−modVF.We want to define a duality on ZS(G)−modVF. For M ∈ ZS(G)−modVF,we define

DS(M) := HomS(M, S)

to be the S-dual of M . It becomes a ZS(G)-module via

z.f(m) := f(z.m)

for z ∈ ZS(G), f ∈ HomS(M, S) and m ∈M .

Proposition 4.4.1 (cf. [Fie08a], Section 4.6). The duality induces an equiv-alence of categories

DS : ZS(G)−modVF → (ZS(t(G))−modVF)opp.

Note that since the structure algebra is defined independently of the order,we have ZS(G) ∼= ZS(t(G)).

Let OVFS,Λ

be a block corresponding to an equivalence class of negative level,

and let σ ∈ Λ be antidominant. Let J be an open and bounded subset, letλ ∈ J and denote by PJ

S(λ) a projective cover of ∆S(λ) in the truncated

category OJS,Λ

. Now for J ′ an open and bounded subset with J ⊆ J ′

we have a natural surjection PJ′

S(λ) PJ

S(λ), and hence we can take a

completionPS(σ) := lim←−PJ

S(σ)

76

of the truncated antidominant projectives (cf. [Fie08a], Section 7.1). Fiebigdefines the structure functor in [Fie06]

V : OVFS,Λ

→ Z(OS,Λ)−modf ,

where Z(OS,Λ) is the categorical center of OS,Λ. It is shown that

V = HomOS(PS(σ), ·)

if Λ contains an antidominant weight σ (cf. [Fie06], Section 3.4) and thedescription of Z(OS,Λ) (cf. [Fie03], Theorem 3.6) yields

Z(OS,Λ) ∼= ZS(GΛ).

Hence, we get a functor

V : OVFS,Λ

→ ZS(GΛ)−modf

M → HomOS(PS(σ),M).

The main statement of [Fie08a] is the following

Theorem 4.4.1 (cf. [Fie08a], Theorem 7.1. and Proposition 7.2.). We havean equivalence of exact categories

V : OVFS,Λ

→ ZS(GΛ)−modVF

under which a projective cover PJS

(λ) of ∆S(λ) in OJS,λ

corresponds to the

global sections of the Braden-MacPherson sheaf BJ (x), where λ = x · σ forx ∈ W(Λ)/Wσ.

Now if Λ contains a dominant weight ν, we get an equivalence of categoriesby composing

OVFS,Λ

tS→ (OVFS,t(Λ)

)opp Vopp

→ (ZS(Gt(Λ))−modVF)opp DS→ ZS(GΛ)−modVF,

where t(Λ) = −2ρ− Λ is of negative level (cf. Chapter 2.5.3).

4.5 Homological algebra of BMP sheaves

In this section, we want to give a different characterization of the S-moduleB(x)δz:Let Λ ⊂ h∗/ ∼ be a regular block of positive level. Let x, z ∈ W(Λ). Weorient ourselves towards [Fie08b]. To Λ we assign the moment graph GΛ asexplained in the previous section. As base ring, we take S but we can alsowork with a localization of S at a prime ideal. For every vertex x ∈ GΛ, wehave described how to get an indecomposable BMP-sheaf B(x) (cf. Theorem4.3.2). For the global sections of B(x), there is a similar description:

77

Theorem 4.5.1 (cf. [Fie08b], Theorem 6.1.). For all x ∈ W(Λ) there existsan object B(x) ∈ Z −modVF, unique up to isomorphism, with the followingproperties:

1. B(x) is indecomposable and projective in Z(Λ)−modVF,

2. supp B(x) ⊂ y | y ≥ x and B(x)x ∼= S[2l(x)].

Moreover, B(x) is self-dual, i.e. D(B(x)) ∼= B(x) as a Z-module. Eachprojective object in Z−modVF is isomorphic to a finite direct sum of modulesof the form B(x)[k] for x ∈ W(Λ) and k ∈ Z.

Remark 4.5.1. Note that in [Fie08b], the “ ≥ “ sign is reversed, sinceFiebig defines BMP-sheaves using the reversed order. In the present contextthis does not make a difference.Furthermore, note that is not known if the objects B(x) are self-dual if Λ isof negative level.

Recall the definition of costalks, partial costalks and stalks in Chapter 4.1and 4.2.

Proposition 4.5.1 (cf. [Fie08b], Proposition 7.1.). Suppose that y ≥ x.

1. The S-modules B(x)y, B(x)[y] and B(x)y are graded free over S.

2. Let α1, . . . , αl(y) be the labels of the edges starting at y (note thatl(y) = |t ∈ T | ty < y| where T denotes the set of reflections). ThenB(x)y = α1 · · ·αl(y) · B(x)[y].

3. There is an isomorphism B(x)y ∼= D(B(x)y) of graded S-modules.

4. Let B(x)y ∼=⊕

i S[ki]. Then B(x)[y] ∼=⊕

i S[2l(y)− ki].

5. B(x)y → B(x)y/B(x)[y] is a projective cover.

Let z ∈ W(Λ). Since B(x) is flabby, B(x)I = Γ(B(x), I) for any opensubgraph I ⊆ GΛ, hence we have a surjection

B(x)≤z B(x)<z.

By definition, we have a short exact sequence of Z-modules

B(x)[z] → B(x)≤z B(x)<z.

Then again, we have a short exact sequence of S-modules

0→ B(x)[z] = B(x)[z] i→ B(x)z B(x)δz → 0

(cf. [Fie08a], Section 4.3.). We apply the functor HomZ(·,V(z)) on the firstsequence and obtain a short exact sequence

78

HomZ(B(x)≤z,V(z)) → HomZ(B(x)[z],V(z)) Ext1Z(B(x)<z,V(z))

due to the fact that HomZ(B(x)<z,V(z)) = 0 and Ext1Z(B(x)≤z,V(z)) = 0

since B(x)≤z is projective in Z(Λ≤z)−modVF as one notices in the inductiveconstruction of B(x). This short exact sequence is isomorphic to the shortexact sequence of S-modules

HomS((B(x)≤z)z,S)j→ HomS((B(x)[z])z,S) Ext1

S(B(x)<z,S).

We have a sequence of isomorphisms

HomS((B(x)≤z)z, S) ∼= HomS(B(x)z, S) ∼= D(B(x)z) ∼= B(x)z,

where we used Proposition 4.5.1 (3). Then again, we have

HomS((B(x)[z])z,S) ∼= HomS(B(x)[z], S) ∼= D(B(x)[z])

∼= D(α−11 · ... · α

−1l(z) B(x)z)

∼= α1 · ... · αl(z) D(B(x)z)

∼= α1 · ... · αl(z) B(x)z,

where we have used Proposition 4.5.1 (2) and (3). By multiplying withα−1

1 · ... · α−1l(z) on both sides of the map

j : HomS((B(x)≤z)z, S) → HomS((B(x)[z])z, S)

the observations above yield that this inclusion of Hom-spaces equals theinclusion

i : B(x)[z] → B(x)z

up to shift by [2l(z)]. Hence the (shifted) cokernels of both maps coincide,which results in the following

Theorem 4.5.2. We have an isomorphism of graded S-modules

B(x)δz ∼= Ext1Z(B(x)<z,V(z))[2l(z)].

Remark 4.5.2. This isomorphism is quite surprising since it provides ev-idence that the BMP algorithm mirrors the representation theoretic methodof constructing indecomposable projective objects as maximal extensions ofw-twisted Verma modules.Let us recall the construction of the projective covers Pw

S(x · λ) in Chapter

79

3.3.2: by induction, we started with the object PwS

(x · λ)<wz·λ and looked at

the Ext1-groupExt1

OS(Pw

S(x · λ)<

wz·λ,∆wS

(z · λ)).

Then we took a projective cover

HomOS(∆w

S(z · λ)r,∆w

S(z · λ)) Ext1

OS(Pw

S(x · λ)<

wz·λ,∆wS

(z · λ))

in the category of S-modules. In the BMP algorithm, you look at the S-module B(x)δz and take a projective cover

B(x)z B(x)δz

in the category of S-modules. But B(x)δz ∼= Ext1Z(B(x)<z,V(z))!

We will see this isomorphism in a new light as soon as we have provenTheorem 5.3.1. Let us assume that the block Λ is finite, since in the negativelevel it is not known whether the objects BS(x) are self-dual: due to

Vw(PwS

(x · λ)) ∼= BwS

(x)

(cf. Theorem 5.3.1) and ∆wS

(z · λ)r ∼= PwS

(x · λ)[z·λ]w , where

r = (PwS

(x · λ) : ∆wS

(z · λ)),

we have

HomOS(∆w

S(z · λ)r,∆w

S(z · λ)) ∼= HomOS

(PwS

(x · λ)[z·λ]w ,∆wS

(z · λ))

∼= HomOS(Vw(Pw

S(x · λ)[z·λ]w ,Vw(∆w

S(z · λ)))

∼= HomZS(Bw

S(x)[z]w ,Vw

S(z))

∼= BwS

(x)z,

where we used self-duality of BwS

(x) in the last isomorphism. Thus, not onlyhave we taken a projective cover of the same module, but also the projectivecovers were canonically isomorphic to each other.This observation also holds in positive level as soon as one can transfer Theo-rem 5.3.1 to blocks of positive level. In the negative level, we expect a similarresult for indecomposable tilting modules (cf. [Kub12]). Nevertheless, at thisstage Theorem 4.5.2 is not sufficient to prove

Vw(PwS

(x · λ)) ∼= BwS

(x).

for a finite block Λ. We still need more results to show that.

80

Chapter 5

An equivalence of categories

5.1 A combinatorial Twisting functor

Let Λ ⊂ h∗/ ∼ be a block of negative level, σ ∈ Λ antidominant. Inthis chapter, we want to establish an equivalence between Ow-VF

S,Λand the

combinatorial category ZS(GΛ) − modw-VF. Recall that the set of verticesof the quasi-finite moment graph GΛ was defined by V := W(Λ)/W (Λ)σ.Denote by W(Λ)σ the set of minimal representatives of W(Λ)/W (Λ)σ. Forx ∈ W(Λ), denote by xσ its minimal representative in W(Λ)σ. In thefollowing, we will just write Z for ZS(GΛ). We start with an equivalenceon the Z-module level. For sheaves on moment graphs, this was done byLanini (cf. [Lan12]).

Definition 5.1.1. Let w ∈ W(Λ). We define an automorphism tw of Z via

tw : Z → Z(zx)x∈W(Λ)σ 7→ (tw(z)x)x∈W(Λ)σ ,

where tw(z)x := w−1(z(w−1x)σ). Note that with w−1 we mean the S-moduleautomorphism

w−1 : S → S1S 7→ 1Sα 7→ w−1(α),

for α ∈ R∗.

Remark 5.1.1. This map is indeed well-defined: Let zx, zy be such thaty = sαx and

zx ≡ zy mod α.

Noww−1sαx = w−1sαww

−1x,

andw−1sαw = sw−1(α),

81

where either w−1(α) or −w−1(α) is a positive root. Without loss of gener-ality we assume that w−1(α) is positive and define β := w−1(α). Since

zx ≡ zy mod α,

we deduce

w−1(z(w−1x)σ)) ≡ w−1(z(sβw−1x)σ) mod β.

Hence tw is a well-defined endomorphism of Z with inverse tw−1

.

Definition 5.1.2. Let w ∈ W(Λ) and M ∈ Z − modf . We define thecombinatorial Twisting functor

Tw : Z −modf → Z −modf

via Tw(M) := M , and

zx · Tw(M) := tw(zx) ·M

for zx ∈ Z. It is obviously an equivalence of categories with inverse functorTw−1.

We want to define a w-twisted order on GΛ. W(Λ)σ is a parabolic subgroupof W(Λ).

Definition 5.1.3. For xσ, yσ ∈ W(Λ)σ and w ∈ W(Λ) we define

xσ ≤w yσ :⇐⇒ (w−1xσ)σ ≤ (w−1yσ)σ.

Let J ⊆ W(Λ)σ and w ∈ W(Λ). We say J is w-open if

xσ ∈ J , yσ ≤w xσ ⇒ yσ ∈ J .

Proposition 5.1.1. Let w ∈ W(Λ) and let J ⊆ W(Λ)σ be w-open. Then(w−1J )σ is e-open.

Proof. Let yσ ∈ W(Λ)σ and xσ ∈ (w−1J )σ. Assume furthermore yσ ≤ xσ.Now yσ = w−1(wyσ), where wyσ ∈ W(Λ). If we write

wyσ = yσ1 v,

where yσ1 ∈ W(Λ)σ and v ∈ W(Λ)σ, we deduce

yσ1 = wyσv−1

and

(w−1yσ1 )σ = (w−1wyσv−1)σ = (yσv−1)σ = yσ.

82

Since xσ ∈ (w−1J )σ, we find xσ1 ∈ J such that xσ = (w−1xσ1 )σ. Thus,

(w−1yσ1 )σ = yσ ≤ xσ = (w−1xσ1 )σ.

Since J is w-open and yσ1 ≤w xσ1 , we see that yσ1 ∈ J . But this yields thatyσ = (w−1yσ1 )σ ∈ (w−1J )σ. Hence, (w−1J )σ is e-open.

Definition 5.1.4. Let w ∈ W(Λ). Analogously to Chapter 4.3, we candefine an exaxt structure on Z(GΛ) − modf with respect to the w-twistedorder: a sequence

A→ B → C

in Z(GΛ)−modf is called w-exact (or w-short exact), if the sequence

0→ AJ → BJ → CJ → 0

is short exact as a sequence of abelian groups for all w-open J ⊆ GΛ. Sincethe proof of Theorem 4.3.1 does not depend on the specific type of the orderon GΛ, this defines indeed an exact structure on Z(GΛ)−modf .

Remark 5.1.2. Let M ∈ Z−modf , x ∈ W(Λ) and denote by xσ its minimalrepresentative in W(Λ)σ. The idempotent exσ ∈ Z ⊗S Q acts via

exσ · Tw(M) = e(w−1x)σ ·M,

so we have Tw(M)xσ ∼= M (w−1x)σ . Hence, for J ⊆ W(Λ)σ we get

Tw(M)J = M (w−1J )σ .

Corollary 5.1.1. Let M,N,P ∈ Z −modf and let

M → N → P

be a short exact sequence in Z −modf . Then

Tw(M)→ Tw(N)→ Tw(P )

is short exact in Z −modf with respect to the w-twisted order.

Definition 5.1.5. Let x ∈ W(Λ)σ. Recall the Verma module

V(x) := Γ(V(x))

we introduced in Chapter 4.4. We define the w-twisted Verma module Vw(x)at x by

Vw(x) := Tw(V((w−1x)σ)).

83

Since

zx ·Vw(x) = w−1(z(w−1x)σ) ·V((w−1x)σ) = w−1S ∼= S,

Vw(x) is isomorphic to V(x) as a Z-module. Thus, we cannot distinguishbetween them as Z-modules. However, they exhibit a different extensionbehavior depending on the underlying order, which we will further examinein the following.

Definition 5.1.6. We say M ∈ Z −modf admits a w-twisted Verma flagif MI is graded free for all w-open I ⊆ W(Λ)σ. We denote the category ofZ-modules with a w-twisted Verma flag by Z −modw-VF.

Lemma 5.1.1. Let x ∈ W(Λ)σ. Then (w−1≤w x)σ = ≤ (w−1x)σ.

Proof. “⊆” is clear. For the other direction, argue as in the proof of Propo-sition 5.1.1.

Definition 5.1.7. Let M ∈ Z −modref and x ∈ W(Λ)σ. We define M [x]w

by

M [x]w := ker(M≤wx M<wx).

Analogously to the non-twisted case (cf. Lemma 4.2.1) one gets the following

Lemma 5.1.2. Let M ∈ Z −modref. M ∈ Z −modw-VF if and only if

M [x]w ∼= ⊕ni=1 Vw(x)

for any x ∈ W(Λ)σ. Note that for the base ring S, we require the moduleabove to be graded free, i.e. that

M [x]w ∼= ⊕ni=1 Vw(x)[ki]

for some k1, ..., kl ∈ Z and any x ∈ W(Λ)σ.

Combining the results above we deduce the following

Corollary 5.1.2. Tw restricts to a functor

Tw : Z −modVF → Z −modw-VF

which transforms e-exact sequences to w-exact sequences.

Definition 5.1.8. Let x ∈ W(Λ)σ. Again, we denote the global sections ofthe BMP-sheaf B(x) by B(x). We define

Bw(x) := Tw(B((w−1x)σ)).

84

Theorem 5.1.1. Let x ∈ W(Λ)σ. Bw(x) is isomorphic to the global sec-tions of Bw(x), the Braden-MacPherson sheaf in SHS(GwΛ ), where GwΛ is themoment graph GΛ with the order twisted by w.

Proof. We use the fact that we have an equivalence of categories

L : ZS(GwΛ )−modw-ref → SHS(GwΛ )w-ref.

Consequently it suffices to check that L(Bw(x)) has the properties of anindecomposable BMP sheaf with respect to the w-twisted order.

• suppL(Bw(x)) := y ∈ W(Λ)σ | L(Bw(x))y = Bw(x)y 6= 0. We haveseen that Bw(x)y ∼= B((w−1x)σ)(w−1y)σ . Now

supp B((w−1x)σ) ⊂ y ∈ W(Λ)σ | y ≥ (w−1x)σ .

By rewriting each y as y = (w−1y′)σ for suitable y′ ∈ W(Λ)σ as inProposition 5.1.1, we conclude

suppL(Bw(x)) ⊂ y′ ∈ W(Λ)σ | y′ ≥w x.

• L(Bw(x))x = Bw(x)x ∼= B((w−1x)σ)(w−1x)σ ∼= S.

• Keeping in mind that Tw is an equivalence of categories, we get

EndSH(L(Bw(x))) = EndZ(Tw(B((w−1x)σ))) ∼= EndZ(B((w−1x)σ)).

Since B((w−1x)σ) is indecomposable, we deduce L(Bw(x)) is indecom-posable.

• Let y ∈ W(Λ)σ. We have

L(Bw(x))y = Bw(x)y ∼= B((w−1x)σ)(w−1y)σ .

Now the stalks of BMP sheaves are free at every vertex, hence thestalk L(Bw(x))y is free for every y ∈ W(Λ)σ.

We are left to show property (2) in Proposition 4.3.2.So suppose we have y ∈ W(Λ)σ and z = (sαy)σ such that y >w z. Thus,there is an arrow F : z −→ y ∈ Ew labeled by α. In the e-twisted order, wehave

(w−1y)σ > (w−1z)σ.

Since w−1sαy = w−1sαww−1y and w−1sαw = sw−1(α), (w−1y)σ and (w−1z)σ

are connected by an edge E : (w−1z)σ −→ (w−1y)σ ∈ E labeled by

β := w−1(α)

85

(the root system and the dual root system are W-invariant) on the momentgraph equipped with the e-twisted order. The local structure algebra forF : z −→ y is given by

Z(F ) = (zy, zz) ∈ S⊕ S | zy ≡ zz mod α .

Define a := (w−1y)σ and b := (w−1z)σ. Obviously

Z(E) = (za, zb) ∈ S⊕ S | za ≡ zb mod β

is isomorphic to Z(F ) via the S-module isomorphism

w : Z(E) → Z(F ).

Hence w induces an S-module isomorphism

B((w−1x)σ)(E) → Bw(x)(F ).

Define c := (w−1x)σ. We get a commutative cuboid

B(c)(E) // //

''PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

B(c)a

((QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

B(c)bρb,E

// //

''PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP L(B(c))E

((

Bw(x)(F ) // //

Bw(x)y

Bw(x)zρ′z,F

// // L(Bw(x))F

where the edges of the cuboid are S-module isomorphisms induced by w.Now the S-module isomorphism w identifies ker ρb,E with ker ρ′z,F , whichresults in

ker ρ′z,F∼= w(ker ρb,E) ∼= w(β B(c)b) ∼= αBw(x)z.

Hence L(Bw(x)) is isomorphic to the BMP-sheaf at x ∈ W(Λ)σ with respectto the w-twisted order.

5.2 Tw preserves exactness

Let Λ ⊂ h∗/ ∼ be a block of negative level, σ ∈ Λ an antidominant weight.Let w ∈ W(Λ).

86

Lemma 5.2.1. The twisting functor Tw : OS → OS is exact on the categoryof modules with a Verma flag.

Proof. By Theorem 1.3. in [Soe98] we have an isomorphism of functors

Sw ⊗U (·) ∼= N∗w⊗Nw(·).

A module with a Verma flag is free over N. Since N is free over Nw, weobtain the desired result.

Since we defined ∆wS

(λ) by setting ∆wS

(λ) := Tw(∆S(w−1 ·λ), we obtain thefollowing

Corollary 5.2.1. Tw restricts to an exact functor

Tw : OVFS→ Ow-VF

S.

Remark 5.2.1. Note that this also implies that the right adjoint Gw isexact.

Lemma 5.2.2. Tw takes indecomposables to indecomposables.

Proof. Let M ∈ OVFS

be indecomposable. Then we have

HomOS(Tw(M),Tw(M)) ∼= HomOS

(M, (Gw Tw)(M)) ∼= HomOS(M,M),

hence Tw(M) is indecomposable.

Lemma 5.2.3. Let A be a localization of S and M ∈ OVFS

. Then

Tw(M)⊗S A∼= Tw(M ⊗S A).

Proof. This amounts to verifying the associativity of tensor products:

(Sw ⊗U M)⊗S A∼= Sw ⊗U (M ⊗S A).

Let M ∈ Ow-VFS,Λ

. Thanks to Corollary 3.2.1, the object MJ lies again in

Ow-VFS,Λ

for every w-open subset J ⊆ Λ. Analogously to Proposition 4.15 in

[Jan08], one shows that the functor

(·)J : Ow-VFS,Λ

→ Ow-VFS,Λ

is exact for all w-open subsets J ⊆ Λ. Observing that Λ ⊆ Λ is w-open, weget the following

87

Proposition 5.2.1. Let M,N,P ∈ Ow-VFS,Λ

. The sequence

M → N → P

is short exact in Ow-VFS,Λ

if and only if the sequence

MJ → NJ → PJ

is short exact in Ow-VFS,Λ

for all w-open subsets J ⊆ Λ.

Definition 5.2.1. Motivated by this statement we call such a sequence aw-exact sequence in Ow-VF

S,Λ. Note that this is analogous to Definition 5.1.4,

which defines a w-exact structure on the level of Z-modules.

Lemma 5.2.4. Let w ∈ W(Λ) and J ⊆ Λ be a w-open subset. Then w−1 ·Jis e-open.

Proof. Since we have a bijection

W(Λ)/W (Λ)σ → Λ

of partially ordered sets, this follows from Proposition 5.1.1.

Lemma 5.2.5. Let M ∈ OVFS,Λ

and take J ⊆ Λ w-open. Then we have

Tw(Mw−1·J ) ∼= Tw(M)J .

Proof. w−1 · J is e-open. This means, that Mw−1·J ∈ OVFS,Λ

. We calculate

Tw(Mw−1·J ⊗S Q) ∼= Tw(⊕µ∈J

∆Q(w−1 · µ)mµ) ∼=⊕µ∈J

Tw(∆Q(w−1 · µ)mµ)

∼=⊕µ∈J

∆Q(µ)mµ

∼= (Tw(M)⊗S Q)J

∼= Tw(M)J ⊗S Q .

Since Tw is exact on OVFS,Λ

, M Mw−1·J yields a surjection

Tw(M) Tw(Mw−1·J ).

Then again, since Tw(Mw−1·J ) is free over S, we have

Tw(Mw−1·J ) ⊂ Tw(Mw−1·J )⊗S Q ∼= Tw(Mw−1·J ⊗S Q) ∼= Tw(M)J ⊗S Q .

Thus, the composition

88

Tw(M)J c → Tw(M) Tw(Mw−1·J )

is zero, so we have a commutative diagram

Tw(M) // //

&& &&NNNNNNNNNNN Tw(M)J

Tw(Mw−1·J ),

where Tw(Mw−1·J ) ⊗S Q ∼= Tw(M)J ⊗S Q forces the vertical arrow to bean isomorphism.

Proposition 5.2.2. Let

M → N → P

be e-exact in OVFS

. Then

Tw(M)→ Tw(N)→ Tw(P )

is w-exact in Ow-VFS

.

Proof. Let J ⊆ Λ be w-open. Then w−1 · J is e-open. By assumption

0→Mw−1·J → Nw−1·J → Pw−1·J → 0

is a short exact sequence in OVFS

. Since Tw is exact on OVFS

,

0→ Tw(Mw−1·J )→ Tw(Nw−1·J )→ Tw(Pw−1·J )→ 0

is short exact in Ow-VFS

. Due to the preceding lemma we have

Tw(Mw−1·J ) ∼= Tw(M)J ,

hence

0→ Tw(M)J → Tw(N)J → Tw(P )J

is short exact.

Remark 5.2.2. By noting that for each e-open J the set w ·J is w-open inΛ, and furthermore applying the fact that Gw commutes with base change,one proves analogously that the right adjoint Gw takes w-exact sequences toe-exact sequences.

89

Theorem 5.2.1. Let λ ∈ Λ and let

PJS

(w−1 · λ) ∆S(w−1 · λ)

be the projective cover of ∆S(w−1 · λ) in OVFS,Λ

for J an e-open and bounded

subset of Λ. Then

Tw(PS(w−1 · λ)) Tw(∆S(w−1 · λ))

is the projective cover of Tw(∆S(w−1·λ)) = ∆wS

(λ) in Ow-VFS,Λw·J

, where Λw·J is

the block truncated with respect to the w-open and bounded subset w ·J ⊆ Λ.

Proof. By what we have seen before, Tw(PS(w−1 ·λ)) is indecomposable andhas ∆w

S(λ) as a quotient. Due to remark 5.2.2, Gw takes w-exact sequences

to e-exact sequences. We have an isomorphism

HomOS(Tw(PS(w−1 · λ), ·) ∼= HomOS

(PS(w−1 · λ), ·) Gw(·),

hence the functor HomOS(Tw(PS(w−1·λ), ·) takes w-exact sequences to exact

sequences, which shows that Tw(PS(w−1 · λ)) is projective in Ow-VFS,Λw·J

. By

the uniqueness statement in Theorem 3.3.3, we conclude

Tw(PS(w−1 · λ)) ∼= PwS

(λ).

5.3 An equivalence of categories

Let Λ be a block of negative level and let σ ∈ Λ be antidominant. Denoteby P≤ν

S(σ) the antidominant projective object in the truncated subcategory

O≤νS,Λ⊂ OS,Λ for ν ∈ Λ. Furthermore, we denote by Z(OS,Λ) the categorical

center of OS,Λ. The action of Z(OS,Λ) on P≤νS

(σ) induces a map

Z(OS,Λ) → EndOS(PS(σ)),

wherePS(σ) = lim←−

ν

P≤νS

(σ)

is a completion of truncated antidominant projective objects. Hence we geta functor

OS,Λ → Z(OS,Λ)−modf

M 7→ HomOS(PS(σ),M).

(cf. Chapter 4.4).

90

Definition 5.3.1. Let w ∈ W(Λ). We define the w-twisted structure func-tor Vw via

Vw : Ow-VFS,Λ

→ Z(GwΛ )−modf

M 7→ HomOS(PS(σ),M).

Here we have used that Z(OS,Λ) ∼= Z(GwΛ ) (Theorem 3.4.2 and Remark3.4.4).

Since Vw := V|Ow-VFS,Λ

, the functor Vw is exact for the exact structure on

Ow-VFS,Λ

induced by the exact structure on OS. We will investigate the be-

havior of Vw on w-twisted Verma modules.

Lemma 5.3.1. Let x ∈ W(Λ). Then

Vw(∆wS

(x · σ)) = Vw(xσ)

Proof. We work in the truncated subcategory O≤µS,Λ

, where µ := x · σ. Since

P≤µS

(σ) is projective in O≤µS,Λ

, we have an isomorphism

HomOS(P≤µ

S(σ),∆w

S(x · σ))⊗S C ∼= HomOS

(P≤µC (σ),∆w(x · σ)).

(cf. [Fie03], Proposition 2.4). Now

dimC HomO(P≤µC (σ),∆w(x · σ)) = [∆w(x · σ) : L(σ)]

(cf. [Hum08], Theorem 3.9.). Since the characters of ∆w(x · σ) and ∆(x · σ)coincide, we deduce

dimC HomO(P≤µC (σ),∆w(x · σ)) = 1

(cf. [Fie06], Lemma 3.8.). Hence we can lift a basis element of

HomO(P≤µC (σ),∆w(x · σ))

to a generator of

HomOS(P≤µ

S(σ),∆w

S(x · σ)).

Again, by using Lemma 3.8. in [Fie06] and Proposition 2.4 in [Fie03], wededuce that

HomOS(P≤µ

S(σ),∆w

S(x · σ))⊗S Q

is isomorphic to

HomOQ(P≤µ

S(σ)⊗S Q,∆Q(x · σ)) ∼= HomOQ

(∆Q(x · σ),∆Q(x · σ)),

hence

HomOS(P≤µ

S(σ),∆w

S(x · σ)) ∼= S.

91

Since

HomOS(P≤µ

S(σ),∆w

S(x · σ))⊗S Q ∼= HomOQ

(∆Q(x · σ),∆Q(x · σ)),

(zy)y ∈ Z acts via zxσ . This proves the statement.

For M ∈ Ow-VFS,Λ

, M ⊗S Q splits into a direct sum of Verma modules over Q.

Applying the twisted structure functor on M ⊗S Q results in the canonicaldecomposition of the Z-module Vw(M)⊗S Q. Hence Vw transforms w-exact

sequences into w-exact sequences in Z −modf . Combining our findings wedirectly deduce the following

Corollary 5.3.1. Vw restricts to a w-exact functor

Vw : Ow-VFS,Λ

→ Z −modw-VF .

Recall Theorem 7.1. in [Fie08a], where Fiebig proves for w = e that thefunctor V := Ve is an equivalence of exact categories. We will use this inthe following and denote the inverse functor of V by V−1.

Lemma 5.3.2. The composition of functors

Vw Tw V−1 : Z −modVF → Z −modw-VF

is a functor which transforms e-exact sequences into w-exact sequences.

Proof. This follows from combining Theorem 7.1 in [Fie08a], Proposition5.2.2 and Corollary 5.3.1.

Proposition 5.3.1. There is an isotransformation of functors

η : Tw ⇒ Vw Tw V−1,

and η is given on objects by the identity morphism.

Proof. Let x ∈ W(Λ). For V((w−1xσ)σ) we see

V−1(V((w−1xσ)σ)) = ∆S(w−1x · σ)

and by definition Tw(∆S(w−1x · σ)) = ∆wS

(x · σ). Using Lemma 5.3.1 wededuce

Vw(∆wS

(x · σ)) ∼= Vw(xσ) = Tw(V((w−1xσ)σ)).

Now let M ∈ Z −modVF. Note that supp(M) is finite. We argue by induc-tion over the number of elements in supp(M). Take x ∈ supp(M) maximalwith respect to “ ≤e=≤ “. We find y ∈ W(Λ)σ such that (w−1y)σ = x. Bydefinition, M admits a cofiltration at x

V(x)⊕mx →M≤x M<x.

92

Since Tw and Fw := Vw Tw V−1 transform e-exact sequences to w-exactsequences, we get a diagram

Vw(y)⊕mx // Tw(M)≤wy // Tw(M)<

wy

Vw(y)⊕mx // Fw(M)≤wy // Fw(M)<

wy

Now Tw(M)<wy,Fw(M)<

wy ∈ Z −modw-VF. Since

supp(M) = supp(Tw(M)) = supp(Fw(M)),

and due to the fact that y is w-maximal in supp(M), the induction hypoth-esis yields a commutative diagram

V(y)mx // Tw(M)≤wy // Tw(M)<

wy

V(y)mx // Fw(M)≤wy // Fw(M)<

wy

From this we deduce

Tw(M)≤wy = Fw(M)≤

wy.

Hence, the isotransformation

η : Tw ⇒ Vw Tw V−1

is given on an object M ∈ Z −modVF by

ηM = idM .

The proposition obviously implies

Tw V = Vw Tw .

Firstly, we notice that the left-hand side is a composition of equivalences.Secondly, Theorem 5.2.1 shows

Tw(PS(w−1x · σ)) ∼= PwS

(x · σ),

and furthermore, we have proven

Tw V(PS(w−1x · σ)) ∼= Bw(xσ)

(cf. Theorem 7.1 in [Fie08a] and Theorem 5.1.1). The equality of functorsthen yields

Vw(PwS

(x · σ)) ∼= Bw(xσ).

Thus, we have proven

93

Theorem 5.3.1. Let Λ ⊆ h∗ be of negative level and w ∈ W(Λ). Then thefunctors

Tw : OVFS,Λ→ Ow-VF

S,Λ

and

Vw : Ow-VFS,Λ

→ Z −modw-VF

are equivalences of exact categories under which projective covers correspondto projective covers.

Remark 5.3.1. Let Λ be a block of positive level. We get a functor

Ow-VFS,Λ

→ Ow-VFS,t(Λ)

,

where t(Λ) = −2ρ− λ |λ ∈ Λ, by composing

Ow-VFS,Λ

Gw→ OVFS,Λ

t→ (OVFS,t(Λ)

)opp→(Ow-VFS,t(Λ)

)opp.

At this point, it is not clear whether Gw (respectively Tw) are equivalences onblocks of positive level. Hence, either one has to show directly that these func-tors are equivalences, or one proves a generalization of [Soe98], Corollary2.3, hence a tilting equivalence on categories of modules with a w-twistedVerma flag, to obtain the equivalence in Theorem 5.3.1 for blocks of positivelevel. We believe that this should be feasible, unfortunately it exceeded thescope of this thesis.

Let Λ be a block of negative level, σ ∈ Λ an antidominant weight, x, y ∈W(Λ) and ν ∈ Λ such that x · σ, y · σ ∈ Λ≤

wν . Denote by PwS

(x · σ) the

projective cover of ∆wS

(x · σ) in Ow-VFS,Λ≤wν

. Due to Theorem 5.3.1, we have

HomOS(Pw

S(x · σ),∆w

S(y · σ)) ∼= HomZ(Λ)(Vw(Pw

S(x · σ)),Vw(∆w

S(y · σ)))

∼= HomZ(Λ)(Bw(xσ),Vw(yσ))

∼= D(Bw(xσ)yσ)

Since Bw(xσ)yσ ∼= B((w−1x)σ)(w−1y)σ is free of finite rank, we get as a first

Corollary 5.3.2. For all x, y ∈ W(Λ) such that x ·σ, y ·σ ∈ Λ≤wν the space

of homomorphisms

HomOS(Pw

S(x · σ),∆w

S(y · σ))

is free, and furthermore we have

rkS HomOS(Pw

S(x · σ),∆w

S(y · σ)) = rkS Bw(xσ)y

σ.

94

Since Tw is an exact equivalence of categories, we deduce for the multiplic-ities

(PwS

(x · σ) : ∆wS

(y · σ)) = (PS(w−1x · σ) : ∆S(w−1y · σ)).

By Theorem 7.1. in [Fie08a] we have

rkS B((w−1x)σ)(w−1y)σ = (PS(w−1x · σ) : ∆S(w−1y · σ))

= [∆(w−1y · σ) : L(w−1x · σ)].

Again, due to B((w−1x)σ)(w−1y)σ ∼= Bw(xσ)yσ, we deduce as a

Corollary 5.3.3 (“w-twisted BGG reciprocity”).

(PwS

(x · σ) : ∆wS

(y · σ)) = [∆(w−1y · σ) : L(w−1x · σ)],

and the multiplicities can be determined by calculating the rank of Bw(xσ)yσ.

Remark 5.3.2. Note that in order to relate the multiplicites above to eachother, we had to translate this question into the language of sheaves on mo-ment graphs. The classical BGG Reciprocity Theorem requires the projectivecovers to be projective in O (respectively in OS), not just in a full subcat-egory. However, this does not hold for the objects Pw

S(x · σ). Ergo, the

theory of sheaves on moment graphs provides a beautiful tool to circumventthis problem. Corollary 5.3.3 concludes our project to determine multiplicityformulas for w-twisted projective covers.

95

96

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