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Moment graphs in representation theory andtopology

Peter Fiebig, Universitat Erlangen-Nurnbergstill a preliminary version

Contents

Introduction 1

Lecture 1: Moment graphs 2

Lecture 2: Representation theory 8

Lecture 3: Topology 14

References 21

Introduction

One of the central problems in representation theory is the determinationof the simple characters of a given group or, more generally, of a given algebra.Often this problem turns out to be difficult to solve and there is an abundanceof situations in which we do not yet have an answer. The problem seems to beparticularly difficult if the base field of our theory is of prime characteristic.

A very successful approach towards the character problem is to find arelation between the representation theoretical structures and the geometry ofsome algebraic variety. If such a relation is established one might hope thatthe machinery of algebraic geometry is powerful enough to provide a solution.

There are at least two different ways to relate representation theory togeometry. The first, which is quite well-known and well established in several

areas, appeared for the first time in the article [BB] and is called the BeilinsonBernstein localization. It amounts to realizing a Lie algebra inside the spaceof differential operators on a complex algebraic variety and thus allows us torelate the category of D-modules (i.e. the category of sheaves of modules overthe sheaf of differential operators) to representations of the Lie algebra viathe global sections functor. This, however, assumes that the base field is ofcharacteristic 0. Analogous approaches for fields of positive characteristic arevery hard to study and are still not completely understood.

The second, less well-known alternative is to relate the geometry to therepresentation theory via an intermediate category, which often is of a linear-algebraic, combinatorial nature. These categories can sometimes be realized

1


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2 MOMENT GRAPHS

over arbitrary fields and even over the integers, which opens a way to comparethe structure over different fields using a Brauer type base change approach.

In these lectures we want to discuss the main ideas of the second approachin the particular example of representations of a semisimple complex Lie alge-bra, which is technically the least demanding instance. The associated com-binatorial category is then the category of sheaves on the associated Bruhatgraph, which is a finite, directed and labelled graph that can be read of from theunderlying root system and its Weyl group. We want to state the main stepsand explain the main ideas for a proof of the classical Kazhdan-Lusztig conjec-ture, which states a character formula for the simple highest weight modulesof the Lie algebra. A characteristic p version of this approach can be found inthe articles [F5, F6].

While both approaches explained above typically yield character formulasin various situations, they also relate categorical structures. Their real poweris revealed once one combines the two: in some cases this yields the celebrated

Koszul duality between representation theory and geometry.The first lecture is the most elementary one and deals with the basics ofmoment graph theory. In particular, we introduce the moment graph associ-ated to a finite root system. Then we motivate the construction of the princi-pal objects associated to a moment graph: the BradenMacPherson sheaves.We then present a conjecture on the graded rank of the stalks of BradenMacPherson sheaves on the moment graphs associated to root systems.

The second lecture describes the link between the representation theory ofa finite dimensional complex Lie algebra and the moment graph theory. Wepresent the most classical case of the KazhdanLusztig conjectures and showhow it follows from the moment graph conjecture presented earlier. Here we

mainly follow [F4].The third and final lecture gives the link between the topology of flag va-

rieties and moment graph theory. It is in this topological-geometric situationthat moment graphs made their first appearance. We present the localizationfunctor that associates a sheaf on the moment graph to a torus-equivariantsheaf on the variety following [FW]. The most important class of equivariantsheaves are the parity sheaves introduced recently by Juteau, Mautner andWilliamson (cf. [JMW]). We show that by localizing parity sheaves one ob-tains the BradenMacPherson sheaves on the graph. Deep results in algebraicgeometry show that the parity sheaves that we consider are intersection coho-mology sheaves if the characteristic of the field of coefficients is either zero or

large enough. A formula for the dimension of the stalks of the latter sheavesthen yields our moment graph conjecture for such fields and, in particular,proves the KazhdanLusztig conjecture we presented in the second lecture.

Lecture 1: Moment graphs

In this lecture we introduce the notion of a moment graph and then presentthe most important class (for us), the moment graphs associated to a rootsystem. We define sheaves on such graphs and the corresponding functorof local sections. A natural extension property then leads directly to thedefinition of the BradenMacPherson sheaves. In the case of a moment graph


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MOMENT GRAPHS 3

of a root system we state a multiplicity conjecture for the BradenMacPhersonsheaves. The last part of the lecture discusses the relation between sheaves andtheir spaces of global sections. Here we obtain another characterization of theBradenMacPherson sheaves: their global sections form a set of representativesof the projective objects in a certain exact category. This is the property that

latter allows us to give these sheaves a representation theoretic meaning.

Moment graphs. Let Y = Zr be a lattice of finite rank. An (unordered)moment graph over Y is the datum G = (V, E, ), where (V, E) is a finite graphwithout loops (i.e. edges connecting a vertex to itself) and no double edges(i.e. two vertices are connected by at most one edge), and : E Y is amap. We write E: x y to denote an edge connecting x and y and we write

E:

y if we also want to specify its label = (E).

Moment graphs probably made their first appearance in geometry as theimage under the moment map of the 0-1-skeleton of a variety acted upon by atorus. We discuss a more topological version of this in Lecture 3.

One obtains an important class of moment graphs from finite (more gen-erally, affine) root systems. Let us recall the notion of a root system.

The basic datum. Let V be a finite dimensional, rational vector space.Recall that a subset R of V is called a root systemif it satisfies the followingaxioms:

R is finite, does not contain 0 and generates V, for R we have 1/2 R. for all R there exists an element V such that , = 2

and such that the reflection

s, : V V,

,

stabilizes R.

For latter use we also fix a system R+ of positive roots. Recall that is asubset of R such that R is the disjoint union of R+ and R+ and such thatif for 1, . . . , n R the sum 1 + + n is a root, then it is positive, i.e.contained in R+.

The element is uniquely defined by the above property, so we can writes for s,. The s for R generate a finite group W inside GL(V), which

is called the Weyl group of R. The weight lattice isX := { V | , Z for all R}.

This is a lattice in V of rank dimQ V.

Moment graphs associated to root systems. To the root system Rwe associate the following graph GR over the weight lattice X:

its set of vertices is W and the elements x, y W are connected by an edge if there is a root

R+ with x = sy, we set (E) = , where is a root such that E connects x and sx.


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4 MOMENT GRAPHS

Note that the root appearing above is defined only up to a sign, but thefollowing theory does not depend on the choice we make.

Sheaves on moment graphs. Suppose that G is an arbitrary moment

graph defined over the lattice Y. Now let us fix a field k. We denote byYk = Y Z k the k-vector space spanned by Y. By S = S(Yk) we denote itssymmetric algebra. Recall that this is a polynomial algebra (over k) of rankdimk Yk.

A k-sheaf on G is the datum F = (Fx,FE, x,E), where

Fx is an S-module for any vertex x V, FE is an S-module with (E)FE = 0 for each edge E E, x,E: Fx FE is a homomorphism of S-modules for any vertex x

lying on the edge E.

A morphism f: F G of sheaves on G is given by components fx : Fx Gx

and fE: FE GE such that for any vertex x lying on the edge E the diagram

Fx

x,E

fx// Gxx,E

FEfE

//GE

commutes. We denote by G-modk the corresponding category of k-sheaves onG.

The most natural sheaf on G is probably the structure sheafZ that isdefined by Zx = S for all x V and ZE = S/(E)S for any edge E, andx,E: S S/(E)S the canonical quotient map.

Sections of sheaves. Let F be a sheaf on the moment graph G and letI be a a subset of V. The space of sections ofF over I is

(I,F) =

(fx)

xI

Fx

x,E(fx) = y,E(fy)for all edges E: x y with x, y I

In particular, we define (F) := (V,F) as the space of global sections.For each pair of subsets I, I of V with I I the projection

xIF

x

xIF

x along the decompositions induces a restriction map

(I,F) (I,F).

The space

Z = (Z) =

(fx)

xV

S

fx fy mod (E)for all edges E: x y

of global sections of the structure sheaf carries a canonical structure of an S-algebra. The addition and the multiplication are given componentwise. Moregenerally, the space of global sections of any sheaf F naturally forms a Z-module.


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A topology on the moment graph. A natural problem in topologicalsheaf theory is the question whether a sheaf is flabby, i.e. whether the re-striction map from the global sections to the sections over an open subset issurjective. We want to study the analogous question in our sheaf theory, sowe need a notion of an open subset.

For this we now assume that we are given an additional structure on themoment graph G, namely we assume that each edge ofG is directed. We writeE: x y or E: x

y for a directed edge. For x, y V we then set x y if

either x = y or if there is a directed path leading from x to y, i.e. if there arevertices x0, . . . , xn with x = x0, y = xn and directed edges xi1 xi for alli = 1, . . . , n. We assume that defines a partial order on V, i.e. that thereare no directed cycles.

In the case of the moment graph associated to the root system R we usethe following directions: For an edge E: x y there is a unique positive root with x = sy. We direct E towards y if l(x) < l(y) (l : W N is the length

function on W).Let us return to the case of a general moment graph. The partial ordercan be used to define a topology on the set V: a subset I of V is open if it isclosed under the partial order, i.e. if for any x I and y V with x y wehave x I, i.e. ifI =

yI{ y}.

Extending local sections. So now we want to ask whether a given sheafF is flabby, i.e. whether for any open subset I of V the restriction of theglobal sections

(F) (I,F)

is surjective.

For a vertex x of G we define

Vx := {y V | there is an edge E: x y}.

Accordingly, we let

Ex := {E E | E: x y}

be the set of the corresponding edges. For a sheafF and a vertex x we definethe map

ux : ({> x},F) EEx

FE

as the composition

({> x},F) y>x

Fy p

yVx

Fy

EEx

FE,

where p is the projection along the decomposition and =

yVxy,E. We

let

Fx := ux(({> x},F))

EEx

FE

be the image of this map. Finally, we define the map

dx := (x,E)TEEx

: Fx

EExF

E.


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6 MOMENT GRAPHS

The connection of the above definitions with the problem of extending localsections is the following. Suppose that m (> x,F) is a section and thatmx Fx. Then the concatenated element (mx, m)

yxF

y is a sectionover { x} if and only if ux(m) = dx(mx).

Lemma 1.1. For a sheafF on the moment graphG the following are equiv-alent:

(1) For any open subsets I J of V the restriction map (J,F) (I,F) is surjective.

(2) For any vertex x V, the restriction map ( x,F) (> x,F) issurjective.

(3) For any x V, the map ux : Fx

EEx

FE contains Fx in itsimage.

Proof. Clearly property (2) is a consequence of property (1). Let us provethe converse, so let us assume that (2) holds. It is enough to prove property

(1) in the special case that J = I {x} for a single element x, since we getthe general case from this by induction. So let m = (my) by a section in(J \ {x},F). Since {> x} J \ {x} we can restrict m and get a section m

in (> x, J). By assumption there is an element mx Fx such that (mx, m)

is a section over { x}. From the definition of a section it follows directly that(mx, m) is a section over J (which extends m). Hence (2) implies (1).

Let us show that (2) is equivalent to (3). Now (2) means that for anysection m over {> x} we can find mx Fx such that (mx, m) is a section over{ x}. But (mx, m) is a section over { x} if and only if ux(mx) = dx(m).Hence, a section m over {> x} can be extended to the vertex x if and only ofdx(m) is contained in the image of ux.

The Braden-MacPherson sheaf. The above discussion leads naturallyto the following definition of sheaves (cf. [BMP]) that are universal with re-spect to the problem of extension local sections:

Definition 1.2. A sheafB on the moment graph G is called a BradenMacPherson sheaf if it satisfies the following properties:

(1) Bx is a graded free Sk-module for any x V,(2) for a directed edge E: x y the map y,E: By BE is surjective

with kernel (E)By,

(3) for any open subset J of V the map (B

) (J,B

) is surjective.(4) The map (B) Bx is surjective for any x V.

The property (4) is a technical property that assures that the stalks are assmall as they need to be. One can quite easily deduce the following results onthe category of BradenMacPherson sheaves.

Theorem 1.3. (1) For any w V there is an up to isomorphismunique BradenMacPherson sheafB(w) onG with the following prop-erties:

We have B(w)w = S andB(w)x = 0 unless x w. B(w) is indecomposable in G-modk.


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MOMENT GRAPHS 7

(2) LetB be a BradenMacPherson sheaf. Then there are w1, . . . , wn Vand l1, . . . , ln Z such that

B = B(w1)[l1] B(wn)[ln].

The multiset (w1, l1), . . . ,(wn, ln) is uniquely determined byB.

A conjecture. Let w V be a vertex. We say that (w, k) is a GKM-pair,if char k = 2 and if for any two edges E: x y and Ex y with x,y,y wand y = y we have that (E) 1 k(E) 1 in Yk.

In the case of moment graphs associated to a root system the GKM-property is a restriction on the characteristic of the field. For char k = 0,each pair (k, w) is a GKM-pair. For finite root systems we only have to ex-clude the primes 2 and 3 (the latter only if R is of type G2). For the affinegraphs that appear in the application to modular representation theory and thecorresponding character conjecture of Lusztig, we have to exclude all primesbelow the Coxeter number of the root system.

Conjecture 1.4. Suppose that G is the moment graph associated to afinite or affine root system. Let w W and suppose that (w, k) is a GKM-pair. Then

rkB(w)x = hx,w(1)

for all x W.

For latter use we now study the connection between the category of sheavesand the category of Z-modules obtained as global sections of sheaves.

A generic decomposition of Z-modules. Again we return to the caseof a general moment graph. Let Q be the quotient field of S. The embedding

Z xVS induces an embeddingZ S Q

xV

Q.

The following is quite easy to prove in the case that the moment graph isfinite.

Lemma 1.5 ([F4, Lemma 3.2]). The latter embedding is a bijection, i.e.we have

Z S Q =xV

Q.

Let M be a Z-module. We denote by MQ the Q-vector space M S Qobtained from M by base change. Is is naturally acted upon by Z S Q =

xWQ, hence there is a natural decomposition

MQ =xW

MxQ,

such that (zx)

xWQ acts on MxQ as multiplication with zx.

Definition 1.6 ([F3, Definition 2.7]). LetI V be a subset and M a Z-module. We let MI be the image of the composition M MQ

xIM

xQ,

where the map on the left is the canonical map that sends m M to m 1 inMQ, and the map on the right is the projection with kernelxV\IMxQ.


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8 MOMENT GRAPHS

For a vertex x we write Mx instead ofM{x} and call this space the stalk ofM at x.

The following statement readily follows from the definitions. It allows usto determine the stalks of the BradenMacPherson sheaves from the spaces oftheir global sections

Lemma 1.7. LetB be a BradenMacPherson sheaf. For each x V wehave a canonical identification

(B)x = Bx.

We now put the sheaf theory in a more categorical context.

Z-modules admitting a Verma flag.

Definition 1.8 ([F4, Section 4]). LetM be a Z-module. We say that Madmits a Verma flag if for any open set J the module MJ is free over S.

We denote by CV erma Z-mod the full category that consists of all modulesadmitting a Verma flag. The category CV erma is not an abelian category. Inthe following we define an exact structure on CV erma. The notion of an exactstructure on an additive category A was introduced by Quillen in ..... (it isa collection of sequences A B C in A that satisfies some axioms). Weneed the following exact structure:

Definition 1.9 ([F4, Section 4.1]). Let A B C be a sequence inCV erma. We say that it is exact if for any open subset J of V the inducedsequence

0 AJ BJ CJ 0

is an exact sequence of abelian groups.

We have to check that this indeed satisfies Quillens axioms ....

As the final result of this lecture we obtain a categorical characterizationof the BradenMacPherson sheaves:

Theorem 1.10 ([F4, F3]). Suppose thatG is the moment graph associatedto the root system R and k is such that (k, w) is a GKM-pair for any w W.Then the following holds:

(1) IfB is a BradenMacPherson sheaf onG, then(B) admits a Vermaflag.

(2) induces an equivalence between the category of BradenMacPhersonsheaves and the subcategory of projective objects in CV erma.

In particular, the set {(B(w))}wW is a full set of representatives for theprojective isomorphism classes in CV erma.

Lecture 2: Representation theory

The aim of this lecture is to discuss the original KazhdanLusztig conjec-ture which served as a prototype for a series of similar conjectures in repre-sentation theory. Let g be a complex simple Lie algebra and h b g aCartan and a Borel subalgebra of g. We assume that we have identified the


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MOMENT GRAPHS 9

root system associated to h and g with R in such a way that R+ is the set ofroots ofb. Recall that a g-module M is a weight module if it is semi-simplefor the action ofh, i.e. if M =

h M, where

M = {m M | H.m = (H)m for all H h}.

There is a partial order on the set h given by if and only if isa sum of positive roots. Let h. A module M is called a module of highestweight if there is m M, m = 0, that generated M and such that M = 0implies .

It is not difficult to see that for any h there is an up to isomorphismunique simple module L() of highest weight and that L() = L() implies = .

Characters of highest weight modules. Let Z[h] be the lattice spannedby the set h. We denote the basis element associated to by e. An elementin Z[h] is hence a finite Z-linear combination h ae

. As the modules

L() are infinite dimensional in general, we need the following completion ofthe lattice Z[h]: we let Z[h] be the (abelian group) set of formal Z-linear com-binations

h ae

such that there exist 1, . . . , n in h such that a = 0

implies i for some i.

Now, a highest weight module M has finite dimensional weight spaces M,hence we can define its character as

char M =h

dimC M e Z[h].

Our principal aim is to calculate the simple highest weight characters ofg, i.e.to give a formula for char L() for any .

The KazhdanLusztig conjecture. Let h. The set of integralroots associated to is

R() := { R | , Z},

and the integral Weyl group of is

W() := s | R()

Then R() is a root system with Weyl group W().

We say that is regular, if the W()-dot-orbit of is regular, i.e. if thestabilizer of in W() is trivial. We say that is dominant is it is maximal

in its orbit. We set = 1/2R+

and define the dot-operation of W on h by shifting the linear action by ,i.e. such that for w W and h we have

w. = w( + ) .

Conjecture 2.11 ([KL1]). Suppose that is dominant and regular. Thenwe have for any w W()

char L(w ) =

xW()(1)l(w)l(x)hw0x,w0w(1)e

x0

R+(1 + e + e2 + . . . ).


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10 MOMENT GRAPHS

Here hx,y denotes the KazhdanLusztig polynomial for the Coxeter system(W(), S()) at the parameters x, y, and w0 W() is the longest element.

Using the translation principle one can show that it is enough to know thecharacters for regular highest weights.

In the case that is, in addition, integral (i.e. R() = R) and w = 1 onecan show that the above formula is equivalent to Weyls character formula:

char L() =

wW(1)

l(w)ew(+)wW(1)

l(w)ew().

Our next task is to rewrite the KazhdanLusztig conjecture as a multiplicityconjecture for Verma modules.

Verma modules. Recall that we have b = h [b, b] as vector spaces. Forany linear form h we let C be the 1-dimensional b-module on whichh b acts via the character and [b, b] acts trivially. Induction then yields

the Verma module() = U(g) U(b) C.

This is a module of highest weight and it even is universal in the sensethat any module of highest weight (in particular, L()) is isomorphic to aquotient of ().

From the PBW-theorem one deduces almost immediately the followingcharacter formula for the Verma modules.

Lemma 2.12. For any h we have

char () = e R+(1 + e + e2 + . . . ).

JordanHolder multiplicities. Each highest weight module M admitsa Jordan-Holder series, i.e. a filtration

0 = M0 M1 Mn = M

such that Mi/Mi1 = L(i) for some 1, . . . , n h. For each the numberof occurences of L() in such a filtration is independent of the choice of thefiltration and is denoted by [M : L()].

Using Lemma 2.12 and an inversion formula for the KazhdanLusztig poly-nomials one shows that the following conjecture is equivalent to the Kazhdan

Lusztig conjecture 2.11.Conjecture 2.13. Suppose that is dominant and regular. Then we have

[(w.) : L()] =

hw,x(1), if = w. for some w W(),

0, else.

Category O. In order to study the above multiplicity conjecture we needto define a category around our objects. This category is the highest weightcategory O defined originally by Bernstein, Gelfand and Gelfand. Here is itsdefinition:

Definition 2.14. LetM be ag-module.


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MOMENT GRAPHS 11

(1) M is called a weight module if M =

h M, where M = {m M | H.m = (H)m for all H h}.

(2) M is called locally b-finite, if every m M is contained in a finitedimensionalb-submodule of M.

We denote by O the full subcategory of the category of g-modules that consistsof locallyb-finite weight modules.

Note that wach highest weight module belongs to O, in particular, ()

and L() belong to O for any h.Verma flags. We say that an object M ofO admits a Verma flag if there

exists a finite filtration

0 = M0 M1 Mn = M

such that for each i = 1, . . . , n the subquotient Mi/Mi1 is isomorphic to (i)for some h. The number of occurences of () in a filtration as above is

in fact independent of the filtration and is denoted by (M : ()).

BGG-reciprocity. Let A be an abelian category and L a simple object inA. Recall that a projective cover of L in A is a non-zero morphism f: P Lthat has the property that P is projective and that any morphism g : Q Psuch that f g is non-zero, is surjective. Often one calls P a projective coverof L and assumes that the morphism f is given.

Theorem 2.15. Let h.

(1) There exists an up to isomorphism unique projective cover P() L() in O.

(2) The object P() admits a Verma flag and for the multiplicities holdsthe BGG-reciprocity formula

(P() : ()) = [() : L()].

Using the BGG-reciprocity we can now reformulate the KazhdanLusztigconjecture:

Conjecture 2.16. We have

(P(y 0) : ()) =

hx,y(1), if = x 0 for some x W,

0, otherwise.

The reason for this additional reformulation of the KazhdanLusztig con-jecture is that both P() and () admits deformed versions and can hencebe studied in a relative setting. This is not possible for the simple object L().

Deformed (or equivariant) representations. The main idea of defor-mation theory is to not only consider a certain Verma module () by itself,but to study families of Verma modules. This allows us to study a Vermamodule by first studying its nearby fibres, which hopefully are simpler tounderstand. Having understood all Verma modules in a neighbourhood of() we hope to obtain information on () itself. The method for this is asfollows.


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12 MOMENT GRAPHS

So let S = S(h) be the symmetric algebra of the vector space h and let Abe a commutative, unital, Noetherian, finitely generated S-algebra. We callsuch an algebra a deformation algebra in the following. We can now considergA := g C A as a A-Lie algebra and study gA-modules. Note that such anobject is a A-module M endowed with an A-linear action ofg.

As A is supposed to be unital it comes with the structure homomorphism: S A, f f 1A. For h we define the bA module A as thefree A-module of rank 1 on which H h acts as multiplication with the scalar(H)+(H) A and [b, b] acts trivially. By induction we obtain the deformedVerma module

A() := U(g) U(b) A.

The deformed category OA. We now introduce a relative setting of theabove structures using a deformation theory which was originally developedby Jantzen for the construction of the Jantzen filtration. As a reference forthe following one might consult the papers [?, F2, F4].

The straightforward generalization of the definition of O is the following:

Definition 2.17. LetM be agA-module.

(1) M is called a weight module if M =

h M, where M = {m M | H.m = ((H) + (H))m for all H h}.

(2) M is called locally bA-finite, if every m M is contained in a finitelygeneratedbA-submodule of M.

We denote by OA the full subcategory of the category of gA-modules that con-sists of locally bA-finite weight modules.

Simple objects in OA. From now on we assume that A is a local defor-mation algebra with maximal ideal m A and quotient field K = A/m. Thenwe can consider K as a deformation algebra as well. Note that gK is againa Lie algebra over a field of characteristic zero and all our non-deformedresults apply.

The corresponding category OK is just a direct summand of the usualcategory O over the Lie algebra gK that contains all modules whose set ofweights is contained in the affine space +h, which we consider as a subspaceof the K-linear dual of hK. In particular, each h parametrizes a simpleobject LK() in OK with highest weight + .

The locality of A allows us to apply the Nakayama lemma for the proof of

the following statement.

Theorem 2.18. The base change functor A K: OA OK induces abijection

simple isomorphismclasses of OA

simple isomorphism

classes of OK

.

We now need to find deformed versions of the projective covers.

Deformed projective objects. Almost as in the non-deformed case oneproves the first part of the following Theorem. The second part uses theidempotent lifting lemma.


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MOMENT GRAPHS 13

Theorem 2.19. Suppose that A is a local deformation algebra with residuefield K. Let h.

(1) There exists an up to isomorphism unique projective cover PA() LA() in OA.

(2) The objectPA

() admits a deformed Verma flag and for the multiplic-ities holds the BGG-reciprocity formula

(PA() : A()) = [K() : LK()].

Note that on the right of the BGG-reciprocity above the Verma module andthe simple object are defined over the residue field K. This allows us to giveyet another formulation of the KazhdanLusztig conjecture. Let S = S(h) be

the symmetric algebra of the vector space h and denote by S its localization atthe maximal ideal S h. As the residue field of S is C we obtain the followingequivalent formulation of the KazhdanLusztig conjecture:

Conjecture 2.20. Suppose that h is dominant and regular. Then

we have for any w W

(PeS(w.) : eS()) =

hx,y(1), if = x 0 for some x W(),

0, otherwise.

Finally, we can relate the deformed category O to the category of sheaveson a moment graph.

A functor into moment graph combinatorics. For notational simplic-ity we now assume that our dominant and regular element is in fact = 0.We have R(0) = R and W(0) = W. One can prove the following results forgeneral dominant (and not even regular) with some extra care.

Let w0 W be the longest element in the Weyl group. Then Soergelsstructure functor is given by

V := Hom(PA(w0 0), ) : OA mod-End(PA(w0 0)).

Let us consider End(PeS(w0 0). Each f End(PeS(w0 0) induces a homo-morphism fQ := f1 : PeS(w00)eSQ End(PeS(w00)eSQ. As PeS(w00)eSQ

admits a deformed Verma flag, it is torsion free as a S-module, hence the mapf fQ is injective. So we can consider End(PeS(w0 0)) as a submodule inEnd(PeS(w0 0) eS Q).

Proposition 2.21. (1) There is an isomorphism

PeS(w0 0) eS Q = xW

Q(x 0).

It induces a canonical identification

End(PeS(w0 0) eS Q) =xW

Q.

(2) The subspace End(PeS(w0 0)) End(PeS(w0 0) eS Q) =

xWQcoincides with

(zx)

xW S

zx zsx mod

for all x W and R+

.


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14 MOMENT GRAPHS

Now let us consider C-sheaves on the moment graph assocaited to R. Thedefinition of the structure algebra Z on this graph .... resembles greatly theresult we obtained for End)PeS(w0 0) except for two details (one is a minordetail, the other one quite important). First, the symmetric algebra is replacedby a completed symmetric algebra. But more importantly, the role of the roots

is now played by the coroots. Hence, if we define G, Z, B(w) in the sameway as before, but starting from the Langlands dual root system, then weobtain the following result:

Corollary 2.22. There is a canonical isomorphism End(PeS(w0 0))=

Z S S.

Hence we can now consider V as a functor from OeS to Z S S-mod. Let

us denote by OeS,[0] the principal block of OeS, i.e. the indecomposable block

containing (0), and by OV ermaeS,[0]

its subcategory of objects admitting a Verma

flag. It inherits an exact structure from the natural exact structure of the

abelian category OeS. Here is now our link to moment graph theory:

Theorem 2.23. (1) The functorV induces an exact equivalence be-tween the OV erma[0] and C

,V erma S S.

(2) For each w W there is an isomorphism

V(P(w 0)) = (B(w)) S S

of Z S S-modules.(3) We have for each x W

(P(w 0) : (x 0)) = rkB(w)x.

In particular, Conjecture 1.4 implies the KazhdanLusztig conjecture 2.11.In the following, final lecture we explain how one can use the equivariant topol-ogy of flag varieties in order to prove the characteristic 0 case of Conjecture1.4.

Lecture 3: Topology

In the third and final lecture we want to relate the topology of Schubertvarieties to the Braden-MacPherson sheaves on the underlying Bruhat graph.

A reference for the following is the upcoming paper [FW], which contains apositive characteristic analog of the main result in [GKM]. For the definitionof the equivariant derived category, see [BL].

In this lecture we have to assume some acquaintance with the formalismof derived categories of sheaves of vector spaces on a topological space X. Inparticular, we assume that the reader has an idea of the following structures:

For each topological space Y and each field k there is the derived cate-gory of sheaves ofk-vector spaces D+(Y, k) (with cohomology boundedfrom below).

For a suitable continuous map f: X Y there are the base changefunctors f, f! : D

+(X, k) D+(Y, k) and f, f! : D+(Y, k) D+(X, k).


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MOMENT GRAPHS 15

If j : U X is an open inclusion and i : A = X \ U X its closedcomplement, then we have for any sheaf F D+(X, k) the followingdistinguished triangles in D+(X, k):

i!i!F F jj

F[1],

j!j!F F ii

F[1] .

They are called the Gysin sequences associated to i and its comple-ment j.

H-spaces. Let H be a topological group. An H-space is a topologicalspace X together with a continuous H-action H X X. An H-space iscalled (topologically) free if the quotient map X X/H is an H-bundle (i.e.a locally trivial fibration with fiber H).

For the following we fix an H-space EH that is free and contractible onwhich H acts topologically freely. By a theorem of Dold (cf.[D]), an H-spaceEH together with a topological H-action is contractible if and only if theprojection EH EH/H is a universal H-bundle. Milnor has given a con-struction of a universal bundle for any topological group H. It is, of course,not uniquely defined, but nothing that follows depends on our choice for EH.

Now we can define for an arbitrary H-space the topological space

XH := XH EH,

which is the orbit space of X EH under the diagonal H-action. We thenhave a diagram

X EHq

yytttt

tttt

tt p

$$III

IIII

III

XH X,

where q is the canonical orbit map and p is the projection onto the first factor.

The equivariant derived category. For the sheaf theory we now alsoneed a field k of coefficients. To the data (H,X,k) one associates the followingcategory:

Definition 3.24. The equivariant derived category D+H(X, k) of sheavesonX with coefficients in k is the full subcategory of D(XH, k) that contains allsheaves F for which there is a sheaf FX D(X, k) such that q(F) = p(FX).

Suppose that X and X are H-spaces and that f: X X is an H-equivariant map. Then f id : X EH Y EH induces a continuous mapfG : XG YG and (for suitable f, cf.[BL]) we get base change functors fG,f!G, fG, and fG,!. One checks, again under suitable assumptions on the mapf, that these functors induce functors between the subcategories D+G(X, k) D+(XG, k) and D

+G(Y, k) D

+(YG, k). We denote these restrictions again byf, f!, f and f! in order to save indizes.


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16 MOMENT GRAPHS

The main example - flag varieties as T-spaces. Let G be a connectedreductive complex algebraic group with root system R, and let T B Gbe a maximal torus inside a Borel subgroup of G. The quotient X = G/Bcarries a canonical structure of an algebraic variety. It is naturally acted upon(algebraically) by the torus T.

Note that each algebraic variety can be viewed as a topological space withthe underlying metric topology. In particular, we can view T as a topologicalgroup and G/B as a T-space. Now the space (C \ {0})r = lim(Cn \ {0})r isa contractible space with a topologically free T-action, hence we can take thisas a model for ET. So for any field k we now have a category D+T(X, k).

Hypercohomology. (after [BL, Section 13]) Let X be an H-space andlet : X {pt} be the map to a point. It induces a direct image functor

: D+H(X, k) D

+H(pt, k).

The category D+

H(pt, k) is a full subcategory of D+

(BH,k), where BH =EH/H is the classifying space of H. We denote by

AH := H(BH,k)

its ordinary cohomology with coefficients in k.

Definition 3.25. Let F D+H(X, k). The equivariant hypercohomologyof F is the graded AH-module

HH(F) := H(F),

where H(F) denotes the ordinary cohomology of the sheaf F on BH.

Suppose that i : Y X is the inclusion of a T-stable subvariety. For anysheaf F D+T(Y, k) we denote by

FY := iF

the stalk of F on Y. Note that it is a sheaf on Y. One part of the Gysinsequence gives us a natural homomorphism

F iiF = iFY

between sheaves on X. Applying the hypercohomology functor HH yields therestriction morphism

HH(F) HH(FY)

between AH-modules.

In the case of a torus T we have the following: Note that BT = (C \{0})r = P, so AT can be identified with the symmetric algebra S = S(X

(T)Zk) (here X(T) = Hom(T,C) is the character lattice of T).

From now on we restrict ourselves to the case that the group H is a com-plex torus T. One of many possible methods to calculate the hypercohomologyHT(F) of a T-equivariant sheafF is the localization method of Goresky, Kot-twitz and MacPherson. In order to describe this, we introduce thenotion ofmoment graphs.


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MOMENT GRAPHS 17

The moment graph associated to a T-space X. Each X(T)defines an action of T on the affine variety C by t.x = (t)x for t T,x C. We denote the resulting T-variety by C .

In order to associate a moment graph to a T-variety X we now assume thefollowing.

(1) There are only finitely many 0- and 1-dimensional T-orbits in X.(2) Each fixpoint is attractive (recall that a fixed point x X is called

attractive if all weights of T on the tangent space TxX lie in an openhalf space of (Lie T)).

(3) The closure of a 1-dimensional orbit in X is smooth.(4) For each 1-dimensional orbit E in X there is a character X(T)

such that E = C as a T-space.

Note that the character E in part (3) is only well-defined up to a sign. Foreach 1-dimensional orbit E we now fix E. Nothing that follows depends onthis choice.

Let us denote by XT the points ofX fixed by T and by X1 X/T the setof one dimensional T-orbits. The moment graph GX associated to a T-varietyX that satisfies the assumptions above is the following. Its set of vertices isthe set XT of fixed points of X. The vertices x, y XT, x = y are connectedby an edge if there is a 1-dimensional orbit E X1 such that E = E {x, y}.We denote this edge by E as well and we set (E) := E.

Our next step is to associate a k-sheafW(F) on GX to any F D+T(X, k).

For a vertex x XT we set

W(F)x := HT(Fx),

and for a one dimensional orbit E we set

W(F)E := HT(FE).

For the construction of the homomorphisms x,E we need a little lemma.

Fixed points in the closure of a one-dimensional orbit. Let x XT

be an attractive fixed point. We denote by ix : {x} X the inclusion andby x : X {x} the projection. If we apply the functor to the adjunctionid ix,ix then we get a natural morphism

x, ix

(since ix is the identity map).

Lemma 3.26. Suppose that X is connected and affine and let x XT

bean attractive fixed point. Then the morphism x, i

x constructed above is

an isomorphism.

Now let F be an equivariant sheaf on X, let E X1 be a 1-dimensionalorbit and choose a fixpoint x in the closure of E. From the above lemma weobtain an isomorphism

x,FEx

ixFEx = Fx

of sheaves on the point {x} and after taking hypercohomology we get an iso-morphism

HT(FEx)

HT(Fx).


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18 MOMENT GRAPHS

We can now define x,E: HT(Fx) H

T(FE) as the composition of the invers

of the above morphism with the restriction morphism:

x,E: HT(Fx)

HT(FEx) H

T(FE).

Hence we constructed the remaining ingredient for a moment graph sheaf.

The constructions above are clearly functorial, so we now have a functor

W : D+T(X, k) GX-modk.

The localization theorem. The next result shows that one can recoverthe global hypercohomology HT(F) of certain equivariant sheaves F from thelocal hypercohomologies on fixed points and one dimensional orbits. In thecase char k = 0, this is a result of Goresky, Kottwitz and MacPherson. Withsome additional care one can make their arguments work for almost arbitrarycharacteristic (cf. [FW]). Here is the only restriction that we impose:

Definition 3.27. We say that(X, k) satisfies the Goresky-Kottwitz-MacPherson

property (orGKM-property for short), if the following conditions are satisfied:

(1) The characteristic of k is different from 2.(2) If E is a one dimensional orbit in X and n Z is such that E is

invertible by n in X(T), then n is invertible in k.(3) If E and E are one dimensional orbits in X such that E E = ,

then E kE (this is to be read in X(T) Z k).

Theorem 3.28. Suppose that (X, k) satisfies the GKM-assumption andsuppose that F D+T(X, k) is such thatH

T(F) is a free Sk-module. Then we

have an isomorphism

H

T(F) = (W(F)).

The idea of the proof. The direct sum of the restriction maps HT(F) HT(Fx) to fixed points yields the localization map

HT(F) xXT

HT(Fx) =xXT

W(F)x.

The theorem is shown by proving that the localization map is injective andthat its image coincides with the space of global sections (W(F)) ofF.

In a first step we show that it suffices to consider K-equivariant cohomologyinstead ofT-equivariant cohomology, where K = (S1)r is the compact subtorusof T, i.e.

HK(G) = HT(G)

for all G D+T(X, k).

Let T be a closed subgroup and let U X be an open, K-stableneighbourhood of its set of fixed points X in X. We denote by j : U Xthe inclusion and by i : X\ U X its closed complement. We obtain a Gysinsequence

i!i!F F jj

F[1]

As i = i! we obtain a long exact sequence of hypercohomology groups

HnK(i!F) HnK(F) H

nK(FU) . . .


19/21

MOMENT GRAPHS 19

Now i!F is a sheaf on a topological space without -fixed points. One showsthat its hypercohomology is then annihilated by

s =

E,

where the product runs over all one dimensional T-orbits E in X \ X. Inparticular, it is a torsion Sk-module. As HK(F) = HT(F) is assumed to befree, the restriction map HK(F) H

K(FU) is injective and an isomorphism

after inverting s.

Finally one shows that

HK(FX) = limHT(FU),

which yields the injectivity of the map HK(F) HK(FX) for any closed

subgroup of T.

Now we look at the special case of the connected component of ker for some X(T). Then X is a disjoint union of P1s and points, by

our assumptions. One shows in this case explicitely, that H

T(FX ) is givenby the sections ofW(F) obtained by deleting all edges that are not labelledby . We obtain (W(F)) by intersecting these spaces of local sections inside

xXTW(F)x. As the restriction map HT(F) H

T(FXT) factors over the

restriction map HT(FX) HT(FXT) for any , we deduce that H

T(F) is

contained in that intersection as well. So it remains to verify the oppositeinclusion: If f HT(FXT) is contained in H

T(FX ) for any X

(T), thenf HT(F).

Now we use again the assumption that HT(F) is a free Sk-module. Let{e1, . . . , en} be a basis ofHT(F) and denote by e

1, . . . , e

n HomSk(H

T(F), Sk)

the dual basis. Let s be the product of all E associated to one dimensional

orbits E. As we have shown above, the inclusionHT(F) H

T(FXT)

becomes an isomorphism after inverting s, i.e. after applying the functorSkSk[1/s]. Now we can characterize H

T(F) as the subspace inside H

T(F)Sk

Sk[1/s] = HT(FXT)Sk Sk[1/s] that consists of all elements f such that e

i (f)

Sk for i = 1, . . . , n. But f is contained in HT(FX), then e

i (f) Sk[1/s].

From

Sk =

X(T)

Sk[1/s]

we obtain our claim.

Of course, now we should be looking for equivariant sheaves on X thatcorrespond to the Braden-MacPherson sheaves on G. The answer may comeas a surprise: these are not the intersection cohomology sheaves, but the paritysheaves.

Stratified varieties. For the definition of parity sheaves we need yet an-other piece of data, namely a stratification of the variety X. Recall that astratification of X is a decomposition

X =

X


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20 MOMENT GRAPHS

by locally closed subvarieties X X such that the closure of each stratum isa union of strata. We furthermore impose the following assumptions:

Each stratum is T-stable and there is a T-equivariant isomorphismX Cn, where Cn carries a linear T-action.

There are only finitely many 0- and 1-dimensional orbits in X. The first two assumptions imply that there is a unique fixpoint x inX for all . We assume that x is attractive.

We assume that the stratification is a Whitney-stratification.

In the case of our main example, i.e. the flag manifold X = G/B, weconsider the stratificationis given by B-orbits. By the Bruhat decompositionwe can identfify the set of orbits with the Weyl group W.

Parity sheaves. For any we denote by i : X X the inclusionof the stratum X.

Definition 3.29. LetP DT(X, k). P is called even, if for all the sheaves iP and i

!P are isomor-

phic to a direct sum of constant sheaves shifted by even degrees. P is called odd, if P[1] is even. P is called parity, if it is either even or odd.

Helas, parity sheaves need not exist on a stratified variety. But we havethe following uniqueness result.

Lemma 3.30. For each there exists at most one indecomposable paritysheaf P() with supp P() X and P() = CX.

Parity sheaves on flag varieties. The situation on flag varieties is par-ticularly nice:

Proposition 3.31. On the flag variety X = G/B the indecomposableparity sheaf P(w) exists for each w W.

Now we can link the topology of flag varieties to the Braden-MacPhersonsheaves defined before:

Theorem 3.32. For each w W there is an isomorphism

W(P(w)) = B(w).

In particular, for each pair x, w W we have an isomorphism

HT(P(w)x)= B(w)x.

Hence, using Theorem 2.23 we get the following link between parity sheaveson the flag variety and the JordanHolder multiplicities of Verma modules:

Theorem 3.33. For each x, w W we have

((x 0) : L(x 0)) = rkHT(P(w)x).

We need one last step to prove the KazdhanLusztig conjecture.


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MOMENT GRAPHS 21

Parity sheaves and intersection cohomology. The decomposition the-oremof Beilinson, Bernstein, Deligne and Gabber (cf. [BBD]) is used to provethe following:

Theorem 3.34. Suppose that char k = 0. Then Pk(w) is the intersection

cohomology complex IC(BwB/B,k) on the Schubert variety BwB/B.Kazhdan and Lusztig managed to calculate the ranks of the stalks of the

intersection cohomology complexes (cf. [KL1, KL2]):

Theorem 3.35. Letx, w W. Then

rkHT(IC(BwB/B), k)x) = hx,w(1).

Taking Theorems 3.35 and 3.33 together we obtain Conjecture 2.13, hencealso the Kazhdan-Lusztig conjecture 2.11.

References

[BB] A. Beilinson and J. Bernstein, Localisation de g-modules, c. R. Acad. Sci. , Paris,Ser. I 292, 1518 (1981).

[BBD] A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and topologyon singular spaces, I (Luminy, 1981), Asterisque, vol. 100, Soc. Math. France, Paris,1982, pp. 5171.

[BL] J. Bernstein and V. Lunts, Equivariant sheaves and functors, Lecture Notes inMathematics 1578 (1994), Springer Verlag, 139p.

[D] A. Dold, Partitions of unity in the theory of fibrations Ann. Math. (2) 78 (1963),223-255.

[BMP] T. Braden and R. MacPherson, From moment graphs to intersection cohomology,Math. Ann. 321 (2001), no. 3, 533551.

[F1] P. Fiebig, Centers and translation functors for category O over symmetrizable

Kac-Moody algebras, Math. Z.243

(2003), No. 4, 689-717.[F2] , The combinatorics of categoryO over symmetrizable KacMoody algebras,Transformation Groups 11, no. 1 (2006), 2949.

[F3] , The combinatorics of Coxeter categories, Trans. Amer. Math. Soc. 360(2008), 4211-4233.

[F4] , Sheaves on moment graphs and a localization of Verma flags , Adv. Math.217 (2008), 683712.

[F5] , Sheaves on affine Schubert varieties, modular representations andLusztigs conjecture, preprint 2007, arXiv:0711.0871.

[F6] , Lusztigs conjecture as a moment graph problem, preprint 2007,arXiv:0712.3909.

[FW] P. Fiebig and G. Williamson, On the p-smooth locus of Schubert varieties, in prepa-ration.

[GKM] M. Goresky, R. Kottwitz, and R. MacPherson, Equivariant cohomology, Koszulduality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 2583.

[JMW] D. Juteau, K. Mautner and G. Williamson, Parity Sheaves, preprint 2009,arXiv:0906.2994v1.

[KL1] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,Invent. Math. 53 (1979), no. 2, 165184.

[KL2] , Schubert varieties and Poincare duality, Geometry of the Laplace operator(Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos.Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185203.

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