SHEAVES ON AFFINE SCHUBERT VARIETIES, MODULAR REPRESENTATIONS, AND ...

49 Pages·2010·0.95 MB·English

JOURNALOFTHE
AMERICANMATHEMATICALSOCIETY
Volume24,Number1,January2011,Pages133–181
S0894-0347(2010)00679-0
ArticleelectronicallypublishedonSeptember23,2010
SHEAVES ON AFFINE SCHUBERT VARIETIES, MODULAR
REPRESENTATIONS, AND LUSZTIG’S CONJECTURE
PETERFIEBIG
1. Introduction
One of the fundamental problems in representation theory is the calculation of
the simple charactersof agiven group. This problem oftenturns out tobedifficult
and there is an abundance of situations in which a solution is out of reach. In the
case of algebraic groups over fields of positive characteristic we have a partial, but
not yet a full answer.
In 1979, George Lusztig conjectured a formula for the simple characters of a
reductive algebraic group defined over a field of characteristic greater than the
associated Coxeter number; cf. [Lus80b]. Lusztig outlined in 1990 a program that
led, in a combined effort of several authors, to a proof of the conjecture for almost
allcharacteristics. ThismeansthatforagivenrootsystemRthereexistsanumber
N =N(R) such that the conjecture holds for all algebraic groups associated tothe
root system R if the underlying field is of characteristic greater than N. This
number, however, is unknown in all but low rank cases.
One of the essential steps in Lusztig’s program was the construction of a func-
tor between the category of intersection cohomology sheaves with complex coeffi-
cients on an affine flag manifold and the category of representations of a quantum
group(thiscombinesresultsofKashiwara–Tanisaki,[KT95],andKazhdan–Lusztig,
[KL93]). This led to a proof of the quantum (i.e. characteristic 0) analog of the
conjecture. Andersen,JantzenandSoergelthenshowedthatthecharacteristiczero
case implies the characteristic p case for almost all p (cf. [AJS94]).
OneoftheprincipalfunctorsutilizedinLusztig’sprogramwastheaffineversion
of the Beilinson–Bernstein localization functor. It amounts to realizing an affine
Kac–Moody algebra inside the space of global differential operators on an affine
flagmanifold. Acharacteristicpversionofthisfunctorisafundamental ingredient
in Bezrukavnikov’s program for modular representation theory (cf. [BMR08]), and
recently Frenkel and Gaitsgory used the Beilinson–Bernstein localization idea in
order to study the critical level representations of an affine Kac–Moody algebra
(cf. [FG06]).
There is, however, an alternative approach that links the geometry of an al-
gebraic variety to representation theory. It was originally developed in the case
of finite-dimensional complex simple Lie algebras by Soergel (cf. [Soe90]). The
idea was to give a “combinatorial description” of both the topological and the
Receivedby theeditorsJune 24,2008and,inrevisedform,November26,2009,andJuly16,
2010.
2010 MathematicsSubjectClassification. Primary20C20;Secondary55N30.
(cid:2)c2010AmericanMathematicalSociety
Reverts to public domain 28 years from publication
133
134 PETERFIEBIG
representation-theoretic categories in terms of the underlying root system using
Jantzen’s translation functors. This approach gives a new proof of the Kazhdan–
Lusztig conjecture, but it is also important in its own right: when taken together
with the Beilinson–Bernstein localizationit establishes the celebratedKoszul dual-
ity for simple finite-dimensional complex Lie algebras (cf. [Soe90, BGS96]).
In this paper we develop the combinatorial approach for quantum and modular
representations. We relate a certain category of sheaves of k-vector spaces on an
affine flag manifold to representations of the k-Lie algebra or the quantum group
associated to Langlands’ dual root datum (the occurrence of Langlands’ duality is
typicalforthistypeofapproach). AsacorollaryweobtainLusztig’sconjecturefor
quantum groups and for modular representations for large enough characteristics.
The main tool that we use is the theory of sheaves on moment graphs, which
originally appeared in the work on the localizationtheorem for equivariant sheaves
on topological spaces by Goresky, Kottwitz and MacPherson (cf. [GKM98]) and
BradenandMacPherson(cf.[BM01]). Inparticular, westateaconjectureinterms
of moment graphs that implies Lusztig’s quantum and modular conjectures for all
relevant characteristics.
Although there is no general proof of this moment graph conjecture yet, some
importantinstancesareknown: Thesmoothlocusofamomentgraphisdetermined
in [Fie06], which yields the multiplicity-one case of Lusztig’s conjecture in full
generality. Moreover, by developing a Lefschetz theory on a moment graph we
obtain in [Fie08c] an upper bound on the exceptional primes, i.e. an upper bound
for the number N referred to above. Although this bound is huge (in particular,
muchgreaterthantheCoxeternumber),itcanbecalculatedbyanexplicitformula
in terms of the underlying root system.
In theremainder ofthis introductionwe explainour approach andthe resultsin
more detail.
1.1. The basic data. Let G = GC be a connected, simply connected complex
algebraic group. We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ B.
Let X = Hom(T,C×) be the character lattice of T and R+ ⊂ R ⊂ X the sets of
roots of B and of G. We denote by X∨ = Hom(C×,T) the cocharacter lattice,
and by R∨ ⊂ X∨ the dual root system. We let W ⊂ GL(X∨) be the Weyl group,
W(cid:2)=W(cid:2)ZR∨ ⊂Aff(X∨) the affine Weyl group and h the Coxeter number of our
data, i.e. the height of the highest root +1. By S(cid:3)⊂W(cid:2)we denote the set of simple
affine reflections.
We fix an algebraically closed fieldk of characteristic p>h andlet G∨ =G∨ be
k
the connnected simple algebraic group over k that is the Langlands dual of G. We
let T∨ ⊂ G∨ be a maximal torus and identify Hom(T∨,C×) with the cocharacter
lattice X∨. We let g∨ and h∨ be the Lie algebras of G∨ and T∨.
1.2. Multiplicities of modular representations. It is known how to calcu-
late the characters of the simple rational representations of the reductive algebraic
groupswithrootsystemR∨ fromthecharactersofsimpleobjectsinacertaincate-
goryC ofrestrictedrepresentationsofg∨ thatcarryanadditionalactionofT∨ (i.e.,
an X∨-grading), such that h∨ ⊂g∨ acts via the differential of the T∨-action. The
simple objects in C are parametrized by their highest weights, so for λ ∈ X∨ de-
note by L(λ) the corresponding simple object. It can be constructed as the unique
SHEAVES ON AFFINE SCHUBERT VARIETIES AND LUSZTIG’S CONJECTURE 135
simple quotient of the standard (or baby Verma) module Z(λ) with highest weight
λ.
Thecharactersofthestandardmodulesareeasytocompute,andinordertoget
thecharactersofthesimplemodulesitisenoughtoknowthenumber[Z(λ):L(μ)]
ofoccurrencesofL(μ)asasubquotientinaJordan-H¨older filtrationofZ(λ)forall
λ,μ∈X∨. Fromthelinkageandtranslationprinciplesitfollowsthatitissufficient
to consider the simple and standard modules in the principal block of C, i.e. those
objects which correspond to weights of the form λ = x · 0 and μ = y · 0 for
p p
x,y ∈W(cid:2)(here “· ” denotes a shifted and elongated action of W(cid:2)on X∨).
p
We identify the set W(cid:2) with the set A of alcoves for the affine action of W(cid:2) on
X∨⊗ZR by fixing a base alcove Ae (so that w corresponds to the alcove w.Ae). In
[Soe97]apolynomialp isassociatedtoeachpairofalcoves(following[Lus80a]).
A,B
For x,y ∈ W(cid:2) we write p for p . Let us denote by w ∈ W the longest
x,y x.Ae,y.Ae 0
element in the finite Weyl group.
Conjecture. Suppose that chark >h. For x,y ∈W(cid:2) we have
[Z(x· 0):L(y· 0)]=p (1).
p p w0x,w0y
The above conjecture is known as the generic Lusztig conjecture. In [Fie07] we
prove the (well-known) fact that it is equivalent to Lusztig’s original conjecture on
the characters of simple rational representations (cf. [Lus80b]).
Usinganinherentsymmetryonecanshowthatitisenoughtoverifythisconjec-
ture in the case that y is an element in the “anti-fundamental box”, i.e. in the case
that y satisfies −p<(cid:5)α,y· 0(cid:6)<0 for each simple root α. Denote by W(cid:2)res,− ⊂W(cid:2)
p
the set of such elements.
1.3. The quantum analog. Letusjustquicklystate,withoutgivingdetails,that
there is a characteristic 0 analog of the representation theory described above
(cf. the overview articles [And95] and [Soe95]). Lusztig associated to the root
system R, its set of positive roots R+ ⊂R, an integer l prime to all entries of the
Cartan matrix, and a primitive l-th root of unity ζ the restricted quantum group
u, which is a finite-dimensional ZR-graded algebra over k =Q(ζ). For each λ∈X
one constructs a standard object Z (λ) and a simple object L (λ) in the category
q q
of ZR-graded u-modules and one can state a conjecture in complete analogy to
the conjecture above. Let us refer to these two conjectures as Lusztig’s modular
conjecture and Lusztig’s quantum conjecture.
1.4. Results. The main application of the relations that we construct between
sheaves on affine Schubert varieties, sheaves on the underlying moment graph and
representation theory is a proof of the following theorem.
Theorem 1.1. (1) Lusztig’s quantum conjecture holds.
(2) The multiplicity-one case of Lusztig’s modular conjecture holds, i.e. for all
fields k with chark =p>h we have [Z(x· 0):L(y· 0)]=1 if and only if
p p
p (1)=1.
w0x,w0y
(3) ThereisanexplicitnumberU(w(cid:3) ),definedintermsoftherootsystem,such
0
that Lusztig’s modular conjecture holds for all fields k with chark >U(w(cid:3) ).
0
Part (1) of the above theorem has already been proven by combining results of
Kashiwara–TanisakiandKazhdan–LusztigwhichformmajorstepsinLusztig’spro-
gram. Our proof is independent and uses only the combinatorial description of the
136 PETERFIEBIG
quantum category given by Andersen, Jantzen and Soergel. Part (2) is an applica-
tion of the result on the “smooth locus” of a moment graph given in [Fie06]. Part
(3) finally uses the main result in [Fie08c], where we develop a Lefschetz theory on
a moment graph in order to calculate the number U(w(cid:3) ).
0
In the remainder of the introduction we want to describe the main ideas in the
proofs of the above results.
1.5. Deformed representation theory. Each simple object L(μ) admits a pro-
jectivecoverP(μ)inC. Moreover,eachP(μ)hasafinitefiltrationwithsubquotients
that are isomorphic to various Z(λ). The corresponding multiplicity, denoted by
(P(μ):Z(λ)), is independent of the particular filtration, andthe following Brauer-
type reciprocity formula was shown by Humphreys (cf. [Hum71]):
[Z(λ):L(μ)]=(P(μ):Z(λ)).
Let S = S(h∨) be the symmetric algebra of h∨, and denote by S˜ the comple-
tion of S at the maximal ideal generated by h∨. In order to compute the above
multiplicities, Andersen, Jantzen and Soergel construct in [AJS94] deformed ver-
sions P(cid:4)(μ) and Z(cid:4)(λ) of P(μ) and Z(λ), which appear in a deformed version C(cid:4)of
C. Again, each P(cid:4)(μ) has a finite filtration with subquotients isomorphic to various
(cid:4)
Z(λ), and for the multiplicities we have
(cid:4) (cid:4)
(P(μ):Z(λ))=(P(μ):Z(λ)).
TheobjectsP(cid:4)(y· 0)canbeconstructedusingtranslationfunctors: toeachsimple
p
affine reflections∈S(cid:3)one associates atranslation functor θs on the principal block
of C(cid:4). We show that for each y ∈ W(cid:2)res,− the object P(cid:4)(y· 0) appears as a direct
p
summand in θt ◦···◦θsZ(cid:4)(0) for a suitable sequence s,...,t ∈ S(cid:3). This serves as
a motivation to define R as the smallest full subcategory of C(cid:4)that is stable under
(cid:4)
taking direct sums and direct summands, contains Z(0) and with each object M
and each s∈S(cid:3)the object θsM.
Let us just mention that in the quantum case we find analogous structures and
resultsand,inparticular,wehaveacategoryRoverthefieldk =Q(ζ)thatencodes
the structure of the quantum category.
In the following we relate the category R to equivariant sheaves of k-vector
spaces on a complex affine flag variety.
1.6. Equivariant sheaves. DenotebyG((t))theloopgroupassociatedtoG, and
by I ⊂ G((t)) the Iwahori subgroup corresponding to B. The quotient F(cid:2)l =
G((t))/I carriesanaturalcomplexind-varietystructureandiscalledtheaffineflag
variety. Denote by T(cid:3) = T ×C× the extended torus and let it act on G((t)) such
thatthefirstfactoractsbyleftmultiplicationandthesecondbyrotatingtheloops.
Then T(cid:3) also acts on F(cid:2)l.
Now let k be a field of characteristic (cid:8)= 2 and let D(cid:2)(F(cid:2)l,k) be the bounded
T
equivariant derived category of sheaves of k-vector spaces on F(cid:2)l. We define the
category I of special equivariant sheaves on F(cid:2)l following [Soe00]. It is the smallest
full subcategory of D(cid:2)(F(cid:2)l,k) that is stable under taking direct sums and direct
T
summands and under shifting, contains the constant equivariant sheaf F of rank
e
one on the base point of F(cid:2)l, and with each F and each s ∈ S(cid:3) the sheaf πs∗πs∗F
in DT(cid:2)(F(cid:2)l,k) (here πs: F(cid:2)l → F(cid:2)ls is the canonical map onto the partial affine flag
SHEAVES ON AFFINE SCHUBERT VARIETIES AND LUSZTIG’S CONJECTURE 137
variety corresponding to s). In [FW] we give another, more intrinsic definition of
the category I as the category of certain equivariant parity sheaves on F(cid:2)l.
The set of T(cid:3)-fixed points in F(cid:2)l is discrete and can canonically be identified with
the set W(cid:2). So for x∈W(cid:2)we denote by i : {pt}→F(cid:2)l the corresponding inclusion.
x
The following result is the most important step in the proof of Theorem 1.1.
Theorem 1.2. Suppose k =Q(ζ) or that k is of characteristic p>h.
(1) There exists an additive functor
Φ: I →R
such that Φ(Fe) ∼= Z(cid:4)(0) and such that Φ(πs∗πs∗F) ∼= θsΦ(F) for all s ∈ S(cid:3)
and F ∈I.
(2) Each Φ(F) has a filtration by deformed standard modules, and the ranks of
(cid:3)
the local equivariant hypercohomologies of F on T-fixed points yield multi-
plicities for Φ(F); i.e., we have
(cid:5) (cid:6)
rkH∗(i∗F)= Φ(F):Z(cid:4)(x· 0)
T(cid:2) x p
for all x∈W(cid:2).
(Here, “rk” refers to the rank of a free H∗(pt,k)-module.) Now we explain how
(cid:2)
T
to derive part (1) of Theorem 1.1 from the statement above.
1.7. Intersection cohomology sheaves. We identify the set of I-orbits in F(cid:2)l
with the affine Weyl group, so we denote by O ⊂ F(cid:2)l the orbit corresponding to
y
y ∈ W(cid:2) and let IC(cid:2) ∈ D(cid:2)(F(cid:2)l,k) be the T(cid:3)-equivariant intersection cohomology
T,y T
complex on the Schubert variety O with coefficients in k.
y
Ifkisafieldofcharacteristiczero,thenthedecompositiontheoremandsomeor-
bitcombinatoricsshowthatI isthecategoryofdirectsumsofshiftedT(cid:3)-equivariant
intersection cohomology sheaves on Schubert varieties in F(cid:2)l. The equivariant ana-
logueofaresultofKazhdanandLusztigin[KL79]isthatwehaverkH∗T(cid:2)(i∗xICT(cid:2),y)=
h (1) for all x,y ∈ W(cid:2), where h is an affine Kazhdan-Lusztig polynomial
x,y x,y
(cf. Theorem 4.4).
Now fix an arbitrary sequence s,...,t ∈ S(cid:3). The object πt∗πt∗···πs∗πs∗Fe can
be defined over Z and it decomposes in almost all characteristics as it does in
characteristic zero. We deduce that πt∗πt∗···πs∗πs∗Fe is a direct sum of shifted
intersection cohomology sheaves on Schubert varieties for all fields k of big enough
characteristic (the notion of big enough now depends on the chosen sequence).
Denotebyw(cid:3) ∈W(cid:2)thelargestelementinW(cid:2)res,− (withrespecttotheBruhator-
0
der), and set W(cid:2)◦ ={w ∈W(cid:2)|w ≤w(cid:3) }. Define I◦ ⊂I as the full subcategory that
0
consists of direct sums of shifteddirectsummands ofthe objectsπt∗πt∗···πs∗πs∗Fe,
wheres···tisareducedexpressionofanelementinW(cid:2)◦. Sincefortheconstruction
ofI◦ wehavetoconsideronlyfinitelymanysequences,wecandeducethefollowing:
Theorem 1.3. Suppose that k is a field of characteristic 0 or p(cid:12)0.
(1) I◦ is the full subcategory of D(cid:2)(F(cid:2)l,k) that consists of objects isomorphic to
T
a direct sum of shifted intersection cohomology sheaves IC(cid:2) with y ∈W(cid:2)◦.
T,y
138 PETERFIEBIG
(2) If y ∈W(cid:2)◦ and y =s···t is a reduced expression, then IC(cid:2) occurs as the
T,y
unique indecomposable direct summand in πt∗πt∗···πs∗πs∗Fe that is sup-
ported on O .
y
(3) For y ∈W(cid:2)◦ and x∈W(cid:2) we have rkH∗T(cid:2)(i∗xICT(cid:2),y)=hx,y(1).
Again one hopes that each prime above the Coxeter number is big enough for
the statements of the theorem to hold.
InSection8weshow thatwehaveΦ(ICT(cid:2),y)∼=P(cid:4)(y·p0)undertheassumptionof
Theorem 1.3. Moreover, the following mysterious relation between the Kazhdan–
Lusztig polynomials h and the periodic polynomial p holds. Although h (cid:8)=
x,y
p ingeneral,wehaveh (1)=p (1)forally ∈W(cid:2)res,− andallx∈W(cid:2)
w0x,w0y x,y w0x,w0y
(this phenomenon is explained by the fact that the functor Ψ defined below does
notpreservegradings). Sopart(1)ofTheorem1.1isaconsequenceofTheorem1.2
andTheorem1.3. Inaddition, wecanalreadydeduceLusztig’smodularconjecture
for almost all primes. Let us now discuss the main steps of the definition Φ.
1.8. Categorifications of combinatorial data. Let H be the affine Hecke alge-
bra associated to R and let M be its periodic module (cf. [Soe97]). Denote by A
e
the basis element in M corresponding to the fundamental alcove, and consider the
map ψ: H→M that is obtained by letting H act on A .
e
Foreachfieldk withcharacteristicdifferentfrom2anddifferentfrom3ifRisof
typeG weconstructcategoriesHandMandafunctorΨ: H→Mtogetherwith
2
character maps hH: H (cid:2)(cid:2)(cid:3) H and hM: M (cid:2)(cid:2)(cid:3) M (i.e. maps to the Grothendieck
groups), such that the diagram
H(cid:2) Ψ (cid:3)(cid:3)M(cid:2)
(cid:2) (cid:2)
hH (cid:2) (cid:2)hM
(cid:2)(cid:2) (cid:2)(cid:2)
ψ (cid:3)(cid:3)
H M
commutes.
The category H is a full subcategory of the category of modules over the equi-
variant cohomology of F(cid:2)l (with coefficients in k), and hypercohomology H∗ yields
(cid:2)
T
a functor from I to H. The category M appears in the work of Andersen, Jantzen
and Soergel. There an equivalence V between R and an S˜-linear version M(cid:7) of M
is constructed in the case that k is an algebraically closed field of characteristic
p>h. Our functor is then the composition
Φ: I −H→∗T(cid:2) H−Ψ→M−·⊗−−S→S˜ M(cid:7)−V−→1 R.
We now relate H to yet another category.
1.9. Sheaves on moment graphs. In the papers [Fie06] and [Fie08c] we study
the theory of sheaves on moment graphs, which gives the following alternative
description of the indecomposable objects in H (cf. [Fie08b]).
To the action of T(cid:3) on F(cid:2)l one can associate a moment graph over the lattice
X(cid:3) = Hom(T(cid:3),C×) that plays a prominent role in equivariant algebraic topology
(cf.[GKM98,BM01]). Recall thatan(ordered)moment graphover alatticeY isa
graph together with a partial order on its set of vertices and a map that associates
to each edge a nonzero element in Y.
SHEAVES ON AFFINE SCHUBERT VARIETIES AND LUSZTIG’S CONJECTURE 139
In our situation such a moment graph G(cid:3)is obtained as follows. Its vertices are
theT(cid:3)-fixedpointinF(cid:2)landitsedgesaretheone-dimensionalT(cid:3)-orbits(theclosureof
aone-dimensionalorbitcontainsexactlytwofixedpoints). Eachedgeislabelledby
(cid:3)
the positive affine root associated to the rotation action of T on the corresponding
orbit. The partial order is induced by the closure relation on the Iwahori-orbits in
F(cid:2)l (each such orbit contains a unique T(cid:3)-fixed point). For the applications in this
article, only thefullsubgraph G(cid:3)◦ ⊂G(cid:3)thatconsists oftheverticescorrespondingto
elements in W(cid:2)◦ plays a role. It has the advantage of being finite.
Suppose that k is a field of characteristic (cid:8)=2 and set V(cid:3)k :=X(cid:3) ⊗Zk. To G(cid:3)◦ one
can associate a category of k-sheaves on G(cid:3)◦, and to each vertex y one assigns the
intersection (or canonical) sheaf B . Denote by Bx its stalk at the vertex x. It is
y y
(cid:3) (cid:3)
a graded free module over the symmetric algebra S :=S(V ).
k
Now the restriction on the characteristic of k is as follows. A large part of the
theory of k-sheaves on a moment graph G behaves well only in the case that the
pair(G,k)satisfiestheGKM-property,whichmeansthatthelabelsontwodifferent
edges meeting at a common vertex are linearly independent over k (cf. [Fie08a]).
We will show that for G(cid:3)◦ this is the case if chark is at least the Coxeter number.
We define a certain subcategory H◦ of H that contains all relevant objects.
From the results in [Fie08b] we deduce that for chark ≥ h the category H◦ can
be interpreted as the category of direct sums of the spaces of global sections of
the intersection sheaves B on G(cid:3)◦. This, together with the functor Ψ and the
y
Andersen–Jantzen–Soergel equivalence, gives us a way to link the moment graph
theory to modular representation theory, and in Section 9.3 we state a conjecture
that implies Lusztig’s conjecture. The main results in [Fie06] and [Fie08c] prove
certain instances of this conjecture and yield part (2) and part (3) of Theorem 1.1.
1.10. Contents. InSection2wediscusstheaffinizationofarootsystemanddefine
an algebra Z(cid:3) over the subring Z(cid:6) = Z[1/2] of Q. For each simple affine reflection
we construct a translation functor on the category of Z(cid:3)-modules, and we define
the category H of special Z(cid:3)-modules as the full subcategory generated from a unit
object by repeatedly applying translation functors.
In Section 3 we discuss various localizations of special modules. This is used in
Section 4 to define a functorial filtration on the category H for each partial order
on the set W(cid:2) that is compatible with the involutions given by the simple affine
reflections. To such a filtration corresponds a character map from H to the free
Z[v,v−1]-module withbasis W(cid:2). It turnsout thatthe Bruhatorder naturally yields
characters in the affine Hecke algebra, whereas the generic Bruhat order yields
characters in its periodic module.
In Section 5 we recall the definition of the category M introduced by Andersen,
Jantzen and Soergel. In analogy to the definition of H, also M is generated inside
a certain category K by repeatedly applying translation functors to a unit object.
ThisallowsustodefinethefunctorΨ: H→M. Althoughthefunctoritselfiseasy
to define, it is quite tedious to check that it intertwines the translation functors.
The definition of translation functors on K that we use is different from the
definition of Andersen, Jantzen and Soergel (it is a version “without constants”).
In Section 6 we show that both definitions lead to equivalent categories.
140 PETERFIEBIG
In Section 7we define the category I of special sheaves onthe affine flag variety
and show that the hypercohomology functor can be considered as a functor from I
to H. We use the decomposition theorem to deduce multiplicity formulas for the
objects in H for almost all characteristics.
In Section 8 we recall the main result of [AJS94], which relates the category M
to the category R of representations of a Lie algebra or a quantum group. The
multiplicity formulas that we gained in Section 7 for H, together with the functor
Ψ: H→M, give multiplicity formulas for objects in R.
WeinterpretourmaincategoryHasacategoryofsheavesonmomentgraphsin
Section 9, and state a conjecture on the multiplicities of stalks of the intersection
sheaves on the graph. In [Fie06] and [Fie08c] two instances of this conjecture are
proven. In a last step we apply this to Lusztig’s conjecture.
2. A category associated to a root system
Let V be a finite-dimensional Q-vector space, let V∗ =HomQ(V,Q) be its dual
space and denote by (cid:5)·,·(cid:6): V ×V∗ → Q the canonical pairing. Let R ⊂ V be a
reduced and irreducible root system. For a root α ∈ R denote by α∨ ∈ V∗ its
coroot. Let R∨ ={α∨ |α∈R} be the coroot system.
Denote by s ∈ GL(V∗) the reflection associated to α ∈ R, i.e. the linear map
α
given by s (v) = v − (cid:5)α,v(cid:6)α∨. Let W ⊂ GL(V∗) be the Weyl group, i.e. the
α
subgroup generated by the s with α∈R.
α
2.1. The affine Weyl group. For α∈R and n∈Z define the affine hyperplane
H :={v ∈V∗ |(cid:5)α,v(cid:6)=n}.
α,n
We denote by s the reflection at H , i.e. the affine transformation on V∗ that
α,n α,n
maps v ∈V∗ to
s (v):=v−((cid:5)α,v(cid:6)−n)α∨.
α,n
The affine Weyl group W(cid:2)⊂ Aff(V∗) is the group of affine transformations on V∗
that is generated by the set T(cid:3) of all reflections s with α∈R and n∈Z.
α,n
Sinces =s ,thefiniteWeylgroupW appearsasasubgroupofW(cid:2). LetZR∨ ⊂
α α,0
V∗ be the coroot lattice. To x ∈ ZR∨ we associate the translation t : V∗ → V∗,
x
v (cid:14)→ v+x. For α∨ ∈ R∨ we have tα∨ = sα,1◦sα,0; hence W(cid:2) contains the abelian
group ZR∨. We even have W(cid:2)= W (cid:2)ZR∨ and we denote by ·: W(cid:2)→ W, w (cid:14)→w,
the corresponding quotient map. Then s =s .
α,n α
2.2. A linearization. Set V(cid:3) :=V ⊕Q, so V(cid:3)∗ =V∗⊕Q, and define for α∈R and
n∈Z a linear action of s on V(cid:3)∗ by
α,n
s (v,μ):=(v−((cid:5)α,v(cid:6)−μn)α∨,μ).
α,n
This extends to a linear action of W(cid:2) on V(cid:3)∗ which leaves the level spaces V∗ =
{(v,κ) | v ∈ V∗} ∼= V∗ for κ ∈ Q stable. On V∗ we recover the affine actioκn of
1
W(cid:2) and on V∗ the affine Weyl group W(cid:2) acts via the ordinary action of its finite
0
quotient W.
The hyperplane in V(cid:3)∗ fixed by s is
α,n
H(cid:3) :={(v,μ)∈V(cid:3)∗ |(cid:5)α,v(cid:6)=μn}.
α,n
SHEAVES ON AFFINE SCHUBERT VARIETIES AND LUSZTIG’S CONJECTURE 141
Set δ :=(0,1)∈V(cid:3) =V ⊕Q. Then
α :=(α,0)−nδ ∈V(cid:3)
n
(cid:3)
is an equation of H . We call α a (real) affine root, and we set
α,n n
R(cid:3) :={α |α∈R,n∈Z}=R×Zδ ⊂V(cid:3).
n
Let us choose a system R+ ⊂R of positive (finite) roots. The corresponding set
of positive affine roots is
R(cid:3)+ :={α+nδ |α∈R,n>0}∪{α|α∈R+}.
Then R(cid:3) =R(cid:3)+∪˙ −R(cid:3)+ and for any reflection t∈T(cid:3) there is a unique positive affine
root α =α−nδ such that t=s .
t α,n
Now consider the dual action of W(cid:2) on V(cid:3) that is given by w.φ = φ◦w−1 for
w ∈W(cid:2)and φ∈V(cid:3). More explicitly, it is given by
w(0,ν) = (0,ν),
s (λ,0) = (s (λ),n(cid:5)λ,α∨(cid:6)).
α,n α
Denote by ·: V(cid:3) →V the map (λ,ν)(cid:14)→λ. Then the following is immediate.
Lemma 2.1. For each w ∈W(cid:2) and x∈V(cid:3) we have w(x)=w(x).
RecallthatthesetofpositiverootsR+ determinesasetofsimplerootsΔ⊂R+
and a set of simple reflections S ⊂ W. The corresponding set of simple affine
reflections is
S(cid:3):=S∪{s }⊂W(cid:2),
γ,1
where γ ∈R+ is the highest root. Then (W(cid:2),S(cid:3)) is a Coxeter system and we denote
by l: W(cid:2)→N the associated length function.
2.3. The associated moment graph. Let Y ∼=Zr be a lattice. An (unordered)
moment graphG over Y isgivenby agraph(V,E)withverticesV andedgesE and
a map α: E →Y \{0} which is called the labeling.
Let X := {λ ∈ V | (cid:5)λ,α∨(cid:6) ∈ Z ∀α ∈ R} be the weight lattice and X(cid:3) :=
X⊕Zδ ⊂V(cid:3) theaffine weight lattice. ThelattercontainstheaffinerootlatticeZR(cid:3).
The affine Bruhat graph G(cid:3)=G(cid:3) associated to R is given as follows. Its underlying
R
lattice is X(cid:3). The set of vertices is the affine Weyl group W(cid:2) and x,y ∈ W(cid:2) are
connected by an edge if there is a reflection t ∈ T(cid:3) with tx = y. This edge is
labelled by the positive affine root α ∈X(cid:3) corresponding to t.
t
For a parabolic subgroup W(cid:2) ⊂ W(cid:2) that is given by a subset I of S(cid:3) we obtain
I
another graph G(cid:3)I as follows. Its set of vertices is W(cid:2)I := W(cid:2)/W(cid:2) and x,y ∈ W(cid:2)I
I
are connected by an edge if there exists t∈T(cid:3) such that tx=y. This edge is then
labelled by α .
t
2.4. The structure algebra. We denote by Z(cid:6) the subring Z[2−1] of Q. Let
S = SZ(cid:4) = S(X ⊗Z Z(cid:6)) and S(cid:3) = S(cid:3)Z(cid:4) = S(X(cid:3) ⊗Z Z(cid:6)) be the symmetric algebras
over the free Z(cid:6)-modules associated to the lattices X and X(cid:3). Both S and S(cid:3) are
Z-graded algebras. The grading we choose is determined by setting X ⊗Z Z(cid:6) and
X(cid:3) ⊗Z Z(cid:6) in degree 2 (this will become important only when we consider graded
multiplicities). In this article many objects are considered to be Z-graded. For
142 PETERFIEBIG
n∈Zwe deno(cid:8)teby (·)(cid:5)n(cid:6)thefunctor thatshiftsgradings by n; i.e., for aZ-graded
module M = M we have M(cid:5)n(cid:6) =M .
i∈Z i i i+n
The affine structure algebra associated to the root system R is
⎧ ⎫
(cid:13)
Z(cid:3)=Z(cid:3)Z(cid:4) :=⎨⎩(zw)∈ (cid:12) S(cid:3)(cid:13)(cid:13)(cid:13) for azwll α≡∈zsRα,n+w, nm∈oZd, αwn∈W(cid:2)⎬⎭.
w∈W(cid:3)
(cid:17)
(cid:3)
The infinite product w∈W(cid:3)S appearing in the definition above should be under-
(cid:3)
stoodastheproductinthecategoryofgradedS-modules; i.e.,theproductistaken
degree-wise. As the defining relations are homogeneous, Z(cid:3) is a Z-graded algebra.
It is an S(cid:3)-algebra via the diagonal inclusion S(cid:3)⊂Z(cid:3).
Let k be a ring in which 2 is invertible. For any Z(cid:6)-module M we set M :=
k
M ⊗Z(cid:4) k. In particular, we have Sk = S(X ⊗Z k) and S(cid:3)k = S(X(cid:3) ⊗Z k), and the
structure algebra over k is
Z(cid:3)k =Z(cid:3)⊗Z(cid:4) k. (cid:17)
(cid:3)
Notethatthisdoesnotnecessarilycoincidewiththesubalgebraof w∈W(cid:3)Skdefined
in analogy with Z(cid:3) by congruence relations.
Let us construct some elements in Z(cid:3). For λ ∈ X(cid:3) and x ∈ W(cid:2) define c(λ) :=
x
x(λ). As sα,n(μ) ≡ μ mod αn for all α,n,μ, we have that c(λ) := (c(λ)x)x∈W(cid:3) is
contained in Z(cid:3). This yields a homomorphism
c: X(cid:3) →Z(cid:3)
of abelian groups. Note that the image of c is contained in degree 2.
For β ∈R+ let W(cid:2)β ⊂W(cid:2)be the subgroup generated by all reflections s with
β,n
n∈Z. The next lemma will be used in the proof of Theorem 5.4.
Lemma 2.2. For each β ∈ R+ and w ∈W(cid:2) there is an element z = (zx)x∈W(cid:3) ∈ Z(cid:3)
of degree 2 with the following properties:
(1) z =0,
w
(2) for all x∈W(cid:2)β we have z ∈Z(cid:6)δ if l(x) is even, and z ∈β+Z(cid:6)δ if l(x)
xw xw
is odd.
Proof. For all x ∈ W(cid:2) set z˜x := β −xw−1(β) ∈ X(cid:3) ⊂ S(cid:3). Then z˜ := (z˜x)x∈W(cid:3) =
β ·1Z(cid:2) +c(w−1(β)) is an element in Z(cid:3). Obviously, z˜w = 0. For all x ∈ W(cid:2)β we
calculate (using Lemma 2.1):
z˜ = β−x(β)
xw
≡ β−x(β) mod δ
≡ β−(−1)l(x)β mod δ,
and hence 2−1z˜serves our purpose. (cid:4)
2.5. Invariants. Choose t ∈ T(cid:3) and consider (cid:17)the involution w (cid:14)→ wt on the set
W(cid:2). We denote by σt the algebra involution on w∈W(cid:3)S(cid:3)that is given by s(cid:17)witching
coordinates: σt(zw) = (zw(cid:6) ), where zw(cid:6) = zwt. Then the subalgebra Z(cid:3) ⊂ w∈W(cid:3)S(cid:3)
is σ -stable. Denote by Z(cid:3)t ⊂ Z(cid:3) the subalgebra of σ -invariants. For the ring k we
t t
set Z(cid:3)t = (Z(cid:3)t) . We denote by Z(cid:3)−t ⊂ Z(cid:3) the Z(cid:3)t-module of σ -anti-invariants. As
k k t
we can divide by 2 in Z(cid:3) we have Z(cid:3)=Z(cid:3)t⊕Z(cid:3)−t. Consider the element c(α ) ∈Z(cid:3).
t
Then σ (c(α ))=−c(α ) as t(α )=−α . So c(α )∈Z(cid:3)−t.
t t t t t t