Modular intersection cohomology complexes on flag varieties

0.28 MB·English

by  Geordie Williamson

INTERSECTION COHOMOLOGY COMPLEXES ON LOW
RANK FLAG VARIETIES
0
1 GEORDIEWILLIAMSON
0 WITHANAPPENDIXBYTOMBRADEN
2

b
e ABSTRACT. We presentacombinatorial procedure,based onthe
F W-graphof the Coxetergroup, which shows thatthe gradeddi-
mension of the stalks of intersection cohomology complexes of
0
certain Schubert varieties is independent of the characteristic of
2
the coefficient field. Our procedure exploits the existence and

] uniqueness of parity sheaves. In particular we are able to show
T
thatthecharactersofallintersectioncohomologycomplexeswith
R
coefficientsinafieldontheflagvarietyG/B oftypeAnforn<7
.
h are given by Kazhdan-Lusztig basis elements. By results of So-
t ergel, this implies a part of Lusztig’s conjecture for SL(n) with
a
m n≤7. Wealsogiveexampleswhereourtechniquesfail.
In the appendix by Tom Braden examples are given of inter-
[
section cohomology complexes on the flag varities of SL(8) and

3 SO(8) whose stalks have different graded dimension in charac-
v
teristic2.
7
0
2
0
.
9 1. INTRODUCTION
0
7 Letkbeafieldofcharacteristicp ≥ 0. LetGbeareducivealgebraic
0
: group over C, B ⊂ G a Borel subgroup and (W,S) the Weyl group
v
anditssimplereflections. DenotebyDb(G/B)theboundedderived
i Λ
X
categoryofsheavesofk-vectorspacesonG/BconstructiblealongB-
r orbits. In Db(G/B) there exist the intersection cohomology sheaves
a Λ
IC(w). Thesesheavesareacertainextension oftheconstantsheafin
degree−ℓ(w) on the Bruhat cell BwB/B to its closure.
Let H be the Hecke algebra of (W,S) over Z[v,v−1] normalised so
astosatisfy
H ifsw > w
H H = sw
s w (cid:26) (v −v−1)Hw +Hsw ifsw < w
andlet{H | w ∈ W}bethe Kazhdan-Lusztig basisofH. Itsatisfies
w
H ∈ H + ⊕ v−1N[v−1]H . Given a finite dimensional graded
w w x<w x
vector space V = ⊕V let P(V) = (dimV )vi be its Poincare´ poly-
i i
nomial. P
1
2 GEORDIEWILLIAMSON WITHANAPPENDIXBYTOMBRADEN
The character of a sheaf F ∈ Db(G/B) is the element of H given
Λ
by
ch(F) = P(H∗(F ))vℓ(w)H .
w w
wX∈W
Ifkisofcharacteristiczero,atheoremofKazhdanandLusztig[KL80]
says that ch(IC(w)) = H . Thus the Poincare´ polynomials of the
w
stalksoftheintersectioncohomologysheavesaregivenbyKazhdan-
Lusztig polynomials. It then follows that the same is true in almost
allcharacteristics, howeverfor anygiven characteristic almostnoth-
ingisknown.
It is a difficult question to determine over which fields one has
ch(IC(w)) = H and, if not, what these characters are. It has been
w
knownsincetheoriginalpapersofKazhdanandLusztig([KL79]and
[KL80]) that in non-simply laced cases the intersection cohomology
complexes may have a different character in characteristic 2. (This
happens, for example, in the only non-smooth Schubert variety in
the flag variety of SP(4).) In 2002 Braden discovered examples of
Schubert varieties in simply laced types A and D where the char-
7 4
acter of the interesection cohomology sheaf in characteristic 2 is dif-
ferentto all othercharacteristics (seethe appendix).
In this article we define combinatorially a certain subset σ(W) ⊂
W of separatedelementsandshow:
Theorem 1.1. Suppose that x ∈ σ(W), then ch(IC(w)) = H for any
w
fieldk.
The determination of the characters of IC(w) is closely related to
the decomposition theorem in positive characteristic. Given a sim-
ple reflection s ∈ S let P be the corresponding standard minimal
s
parabolicsubgroup andconsider the quotient map
G/B →πs G/P .
s
Ifk is of characteristic zero, the decomposition theorem implies that
π IC(w)isadirectsumofshiftsofintersectioncohomologysheaves.
s∗
This need not be true if k is of positive characteristic. Given w ∈ W
and s ∈ S let {w ,...,w } be all the parameters of Kazhdan-Lusztig
1 i
basiselementsthatappearin the product H H . Then we have:
w s
Theorem 1.2. Suppose that w and w ,...,w lie in σ(W). Then the de-
1 n
compositiontheoremholdsfor π IC(w).
s∗
Whilst being of considerable intrinsic interest, these questions are
alsoimportantinrepresentationtheory. Assumethatkisalgebraically
closed, that chark = p is greater than the Coxeter number of W and
INTERSECTIONCOHOMOLOGYCOMPLEXES 3
let G∨ be a split semi-simple and simply connected algebraic group
over k whose root system is dual to that of G, and let B∨ ⊃ T∨
be a Borel subgroup and maximal torus in G∨. Fix positive roots
R+ ⊂ X(T∨) so that the roots corresponding to B are those lying in
−R+. Toeachweightλ ∈ X(T∨)one mayassociate a standard mod-
ule H0(λ) which is non-zero if and only if λ is dominant, in which
case itcontains a uniquesimple submodule L(λ).
AconjectureofLusztig[Lus80]expressesthecharactersofthesim-
ple modules L(λ) in terms of the (known) characters of standard
modules. A particular case of the conjecture is the following: let
ρ ∈ X(T∨)denotethehalf-sumofthepositiveroots,andst = (p−1)ρ
the Steinberg weight, then it isconjectured that, for allx,y ∈ W,
(1) [H0(st+xρ) : L(st+yρ)] = h (1)
x,y
where h ∈ Z[v−1] is the Kazhdan-Lusztig polynomial indexed by
x,y
x,y ∈ W. A theorem of Soergel [Soe00] says that (1) is equivalent to
the semi-simplicity ofπ IC(x) for allx ∈ W and s ∈ S.
s∗
Of course, in order to apply Theorems 1.1 and 1.2 it is necessary
to know the set σ(W). The essential ingredient in the calculation of
σ(W) is the W-graph of the Coxeter system (W,S). Unfortunately,
even in simple situations the W-graph can be very complicated and
no general description is known. However, using Fokko du Cloux’s
program Coxeter [dC] it is possible to use a computer to determine
the set σ(W) for low rank Weyl groups. The simplest situation is
whenσ(W) = W. This onlyoccurs in type Ain low rank:
Theorem 1.3. If G is of type A for n ≤ 6 then σ(W) = W. Hence, in
n
all characteristics the intersection cohomology complexes have characters
given by Kazhdan-Lusztig basis elements and the decomposition theorem
holdsforπ IC(x) forall s ∈ S and x ∈ W.
s∗
It also follows that (1) holds for G∨ = SL (k) with n ≤ 7 if k has
n
characteristic atleastn+1. Usingresults ofBradeninthe appendix,
weare also ableto extend this theorem to cover G∨ of type D .
4
In other types and type A for n ≥ 7 our techniques are not as
n
effective. In most examples that we have computed σ(W) is not the
entire Weyl group. However, we are able to confirm the characters
anddecompositiontheoremformanyintersectioncohomologycom-
plexes in ranks ≤ 6. It also seems that the elements x ∈/ σ(W) for
which our methods fail will provide an interesting source of future
research.
IndeedintheappendixBradenshowsthat,bothintypeD andA ,
4 7
ifk isofcharacteristic 2,thenforminimalelementsinW \σ(W)one
4 GEORDIEWILLIAMSON WITHANAPPENDIXBYTOMBRADEN
has ch(IC(w)) 6= H . These two examples, combined with the case
w
ofdihedralgroups,leadsonetosuspectacloserelationshipbetween
W \ σ(W) and those intersection cohomology complexes for which
one hasch(IC(w)) 6= H for some fieldof coefficients k.
w
Let us briefly mention that, in [BM01] Braden and MacPherson
give an algorithm for the calculation of the stalks of the intersec-
tion cohomology complexes, using only the fixed points and one-
dimensional orbits of a maximal torus on the flagvariety. (Thisdata
isencoded in the so-called “momentgraph” of the flagvariety.) The
forthcoming paper [FW] of Fiebig and the author extends this re-
sult, showing that the moment graph of the flag variety can be used
to calculate the characters of parity sheaves, and hence determine
those intersection cohomology complexes for which ch(IC(w)) 6=
H . Thusthe results ofthis papercould (at least in principle)be de-
w
duced from the moment graph. In fact, the computations of torsion
intheappendixtranslateeasilyintothemomentgraphlanguageand
giveaproofthatthemomentgraphsheavescorrespondingtocertain
Bott-Samelsonvarietiesdonotsplitasexpectedunless2isinvertible
inthe coefficient ring
Lastly, in this article we only consider field coefficients. However
versions of the intersection cohomology complex exist with coeffi-
cientsinZandthestatementthatch(IC(w)) = H forafieldofchar-
w
acteristic p is equivalent to the absence of p-torsion in the stalks and
costalks of an intersection cohomology complex over Z. The reader
isreferred to [Jut09]for more details.
Thestructure ofthepaperisasfollows. InSection2wereviewthe
Hecke algebra and Kazhdan-Lusztig basis in more detail and recall
the W-graph associated to (W,S). In Section 3 we discuss the parity
sheaves,introduced in [JMW09], which are our main theoretical tool.
In Section 4 we define the subset σ(W) ⊂ W and prove Theorems
1.1 and 1.2. In Section 5 we discuss the calculation of the sets σ(W)
viacomputer andgive some examplesofthe setsσ(W)for lowrank
Weylgroups.
1.1. Acknowledgments. I would like to thank Tom Braden for use-
ful correspondence, pointing out errors in previous versions, and
contributing the appendix. I would also like to thank Daniel Juteau
andCarlMautnerforcommentson aprevious version ofthispaper.
2. THE HECKE ALGEBRA AND W-GRAPHS
In this section we recall the Hecke algebra and Kazhdan-Lusztig
basis in slightly more detail. Let (W,S) be a Coxeter system with
INTERSECTIONCOHOMOLOGYCOMPLEXES 5
Bruhat order ≤ and length function ℓ : W → N. Given w ∈ W we
definethe leftand rightdescentsetto be
L(w) = {s ∈ S | sw < w} and R(w) = {s ∈ S | ws < w}.
Recall that the Hecke algebra is the free Z[v,v−1]-module with mul-
tiplication given by
H ifsw > w,
H H = sw
s w (cid:26) (v−v−1)Hw +Hsw ifsw < w.
The elements H are invertible and there is an involution h 7→ h on
w
H which sends H to H−1 and v to v−1. We will call elements fixed
w w−1
bythis involution self-dual.
There exists a basis {H } of H called the Kazhdan-Lusztig basis
w
which isuniquelydeterminedby requiring:
(1) the H are self-dual;
w
(2) H = h H where h = 1 and h ∈ v−1Z[v−1] for
w x≤w x,w x w,w x,w
x 6= w.P
Thepolynomialsh are(uptoarenormalisation)theKazhdan-Lusztig
x,w
polynomials. One maycheck, for example,that H = H +v−1H .
s s id
The action of H for s ∈ S on the Kazhdan-Lusztig basis has a
s
particularly simple form. We denote by µ(x,w)the coefficient of v−1
inh . Then:
x,w
(v +v−1)H ifws < w,
(2) H H = sw
w s (cid:26) H + µ(x,w)H ifws > w.
sw x<w;s∈R(x) x
P
(Asimilarformula describesmultiplication byH on the left).
s
ThusalltheinformationabouttheactionofH ontheleftandright
s
on the Kazhdan-Lusztig basis may be encoded in a labelled graph,
known as the W-graph. The vertices correspond to the elements of
W and are labelled with the left and right descent sets. There is a
directed edge between x and y ∈ W if µ(x,y) 6= 0, in which case
the edge is labelled with the value of µ(x,y). For more details on
the Kazhdan-Lusztig basis and W-graphs the reader is referred to
[KL79], [Hum90]or [Soe97].
3. PARITY SHEAVES
In this section we briefly recall some basic properties of “parity
sheaves” introduced in [JMW09] and motivated by [Soe00]. These
are ourmain technical tool.
We recall briefly the setting of [JMW09]. We fix a field of coef-
ficients k. All spaces will be complex algebraic H-varieties, for H
a complex linear algebraic group. Given an H-space X, we write
6 GEORDIEWILLIAMSON WITHANAPPENDIXBYTOMBRADEN
Db(X) for the bounded derived category of constructible k-sheaves
c
on X and Db (X) for the bounded H-equivariant derived category
H
of constructible sheaves of k-vector spaces on X (see [BL94]). By
abuse of language, we call objects in Db (X) complexes. We denote
H
by For : Db (X) → Db(X) the forgetful functor (see [BL94]). If H
H c
hasfinitelymanyorbits onX thentheimageoftheforgetful functor
is contained in Db(X), the full subcategory of Db(X) consisting of
Λ c
sheaves whose cohomology is locally constant along H-orbits. The
endomorphismringofanyindecomposableobjectinDb (X)islocal,
H
andhence the Krull-Schmidt theorem holds in Db (X).
H
All maps will be equivariant morphisms of complex algebraic va-
rieties. Given a map f : X → Y we have functors f , f from Db (X)
∗ ! H
to Db (Y) and f∗, f! from Db (Y) to Db (X). Similar functors exist
H H H
betweenDb(X)andDb(Y). OnthecategoriesDb (X),Db (Y),Db(X)
c c H H c
andDb(Y) we have the Verdierdualityfunctor, which we denote by
c
D. Wehave isomorphisms of functors Df ∼= fD and Df∗ ∼= f!D. All
∗ !
functors f∗,f!,f,f ,Dcommute with the forgetful functor.
! ∗
NowletGdenoteareductivecomplexalgebraicgroupandB ⊃ T
a Borel subgroup and maximal torus. Let W denote the Weyl group
and S ⊂ W the set of simple reflections. Throughout X = G/P,
where P is either B or a minimal standard parabolic subgroup P
s
corresponding to s ∈ S (i.e. P := BsB). We regard X as a B-
s
variety. Each B-orbit is isomorphic to an affine space and the strata
are classified by W if P = B and W/hsi if P = P . Given w ∈ W
s
(resp. w ∈ W/hsi) we denote by X (resp. X ) the stratum BwB/B
w w
(resp. BwP /P ), by i : X ֒→ G/B (resp. i : X ֒→ G/P ) its
s s w w w w s
inclusion and by k (resp. k ) in the B-equivariant constant sheaf
w w
on X (resp. X ).
w w
Forbrevity,? ∈ {∗,!}. AcomplexF ∈ Db (X)is?-evenifi?F (resp.
B w
i?F) is isomorphic to a direct sum of constant sheaves concentrated
w
in even degrees, for all strata X ⊂ G/B (resp. X ⊂ G/P ). A
w w s
complex is even if it is both ∗- and !-even. A complex F is (?-) odd if
F[1] is (?-) even. A complex F ∈ Db (X) is (?)-parity if we have an
B
∼
isomorphism F = F ⊕ F with F (?)-even and F (?)-odd. Note
0 1 0 1
thatdirect sumsand summandsof (?)-parity sheavesare (?)-parity.
An indecomposable parity complex is called a parity sheaf. The
followingtheoremshowsthatonemayclassifyparitysheavesonthe
flagvariety in a similarwayto intersection cohomology complexes:
Theorem 3.1 ([JMW09, 4.4]). There exists (up to isomorphism)a unique
parity sheaf E(w) ∈ Db (X) with support contained in X and i∗E(w) ∼=
B w w
INTERSECTIONCOHOMOLOGYCOMPLEXES 7
k [dimX ]. Each E(w) is self-dual and any indecomposable parity com-
w w
plexisisomorphictoE(w)[m]for somew ∈ W and m ∈ N.
Given a∗-paritycomplexE ∈ Db (G/B)andw ∈ W wemaywrite
B
i∗E ∼= V(w)⊗k for some finite dimensional graded k-vector space
w w
V(w) = ⊕V(w) . We define the character of E in the Hecke algebra
i
tobe:
ch(E) = (dimV(w) )vℓ(w)+iH .
i w
i∈ZX,w∈W
Remark 1. Given a ∗-parity complex E it is easily seen that ch(E)
agrees with the character of For(E) ∈ Db(G/B) as defined in the
Λ
introduction.
AsimilarcharactermapexistsforDb (G/P )forasimplereflection
B s
s ∈ S. LetH denotetheleftidealHH inH. ThenH isfreewithba-
s s s
sisH H forw ∈ Ws,whereWs ⊂ W denotesthesubsetofelements
w s
w ∈ W such that s ∈/ R(w). Given a ∗-parity complex E ∈ Db (G/P )
B s
andw ∈ Ws wecanwritei∗E ∼= V(w)⊗k forsomegradedk-vector
w w
spaceV(w) = ⊕V(w) . Wedefinethe character of E to be:
i
ch(E) = (dimV(w) )vℓ(w)+iH H .
i w s
i∈ZX,w∈Ws
For anys ∈ S wehaveobvious mapsgiven byinclusion andmul-
tiplication:
·H
s
%%
H H
dd s
inc
Thequotient map π : G/B → G/P inducesfunctors:
s s
πs∗
%%
Db (G/B) Db (G/P )
B ee B s
π∗
s
Thefollowing lemmaiswell-known (see [Spr82], Lemme2.6):
Lemma3.2. (1) If E ∈ Db (G/B) is∗-parity,then sois π E and
B s∗
ch(π E) = ch(E)H .
s∗ s
(2) IfE ∈ Db (G/P ) is∗-parity, thensois π∗E and
B s s
ch(π∗E[1]) = inc(ch(E)).
s
8 GEORDIEWILLIAMSON WITHANAPPENDIXBYTOMBRADEN
(3) IfE ∈ Db (G/B) or Db (G/P ) is∗-paritythen
B B s
ch(E[1]) ∼= v−1ch(E).
Proof. Wefirstshowthecharacterrelations. Statement(3)isastraight-
forward consequenceofthedefinitionsand(2)followsfrom thedef-
initions andthe fact that π−1(X ) = X ⊔X for w ∈ W. Itremains
s w w ws
toshow (1).
We prove (1) by induction on the number of w ∈ W for which
i∗E =6 0. If this number is one, then (by definition of ∗-parity) E is
w
necessarily a direct sum of shifts of i k , for some w ∈ W. We may
w! w
∼
assume that E = i k [ℓ(w)]. Let us write w for the image of w in
w! w
W/hsi. If ws > w then π restricts to an isomorphism X → X .
s w w
Hence
∼
π E = i k [ℓ(w)].
s∗ w! w
Ifws < w thentherestriction ofπ toX inducesa(trivial)C-bundle
s w
overX , hence
w
∼
π E = i k [ℓ(w)−2].
s∗ w! w
A simple calculation in the Hecke algebra then shows that in both
cases
ch(π E) = ch(E)H
s∗ s
asclaimed.
We now turn to the general case. We may assume without loss
of generality that E is ∗-even. Choose w ∈ W so that X is open in
w
the support of E and let i : suppE \ X ֒→ G/B denote the inclu-
w
sion. Theni! E ∼= i∗E andwehaveadistinguishedtriangleof∗-even
w w
sheaves
i i∗E → E → i i∗E →[1]
w! w ∗
By induction π applied to the first or third terms is ∗-parity and
s∗
(1) holds. It follows that the same is true of E because ch(E) =
ch(i i∗E)+ch(i i∗E)and ch(π E) = ch(π i i∗E)+ch(π i i∗E)
w! w ∗ s∗ s∗ w! w s∗ ∗
Itremainstoseethatπ andπ∗preservetheclassesofparitysheaves.
∗
∼
Howeverthisfollowsimmediatelybecauseπ D = Dπ (asπisproper)
∗ ∗
and (π∗[1])D ∼= D(π∗[1]) (because π is a smooth fibration with fibres
(cid:3)
ofcomplex dimension 1).
ConsiderGasaB×B-spacevia(b ,b )·g := b gb−1. Asthesecond
1 2 1 2
copy of B-acts freely on G, the quotient equivalence ([BL94, 2.6.2])
yieldsanequivalence oftriangulated categories:
Q∗ : Db (G/B) →∼ Db (G).
B B×B
INTERSECTIONCOHOMOLOGYCOMPLEXES 9
Considertheinversionmapi : G → G. ThenthisisB×B-equivariant
with respect to the swap map B × B → B × B : (b ,b ) 7→ (b ,b ).
1 2 2 1
Thisinducesan equivalence
i∗ : Db (G) → Db (G)
B×B B×B
Consider the functor ι := (Q∗)−1i∗Q∗ : Db (G/B) → Db (G/B). Then
B B
ι commutes with D. It is easy to see ι preserves parity complex and
that, for a parity complexE ∈ Db (G/B),
B
(3) ch(ι(E)) = j(ch(E))
where j : H → H isthe anti-involution Hw 7→ Hw−1.
Define endofunctors on Db (G/B)by
B
(−)ϑ := π∗π (−)[1] and ϑ (−) := ι∗π∗π ι∗(−)[1].
s s s∗ s s s∗
Then the functors (−)ϑ and ϑ (−) preserve parity sheaves; the shift
s s
∼ ∼
is chosen so that D(ϑE) = ϑ(DE) and D(Eϑ ) = (DE)ϑ . By (3) and
s s
the above lemma,
ch(Eϑ ) = ch(E)H and ch(ϑ E) = H ch(E)
s s s s
forparity sheavesE ∈ Db (G/B).
B
Thefirstresultaboutthecharactersofparitysheavesisthefollow-
ing:
Proposition 3.3. For all w ∈ W,ch(E(w)) ∈ H is self-dual.
Proof. We proceed via induction on ℓ(w) with the base case being
trivial. Let us fix w and choose s ∈ S with sw < w. By Theorem 3.1
wemaywrite
∼
ϑ E(sw) = E(w)⊕G
s
where
G ∼= E(x)[η]⊕mx,η.
Mx<w
η∈Z
TheVerdierself-duality ofϑ E(sw)andeach E(x)forx < w together
s
with Krull-Schmidt implies
m = m .
x,−η x,η
By induction, the ch(E(x)) for x < w are self-dual. Hence both ch(G)
and ch(ϑ E(sw)) = H ch(E(sw)) are self-dual. Thus so is ch(E(w)).
s s
(cid:3)
Lets ∈ S beasimplereflection. Thenextpropositionrelatesparity
sheaveson G/P to those on G/B:
s
10 GEORDIEWILLIAMSON WITHANAPPENDIXBYTOMBRADEN
Proposition 3.4. Let w ∈ W be such that ws < w and denote by w the
imageofw inW/hsi. We have isomorphisms
π∗E(w)[1] ∼= E(w)
s
and
∼
π E(w) = E(w)[−1]⊕E(w)[1].
s∗
Proof. AsE(w)isa direct summand ofπ∗E(w)[1] we have
s
ch(E(w)) = H +v−1H +....
w ws
Itfollows (byconsidering i∗π E(w))that
w s∗
∼
(4) π E(w) = E(w)[1]⊕E(w)[−1]⊕G
s∗
forsome paritycomplex G. Wemayalsodecompose
π∗E(w)[1] ∼= E(w)⊕G′.
s
Hence
π π∗E(w)[1] ∼= E(w)[1]⊕E(w)[−1]⊕G ⊕π G′.
s∗ s s∗
However,by Lemma3.2,
ch(π π∗E(w)[1]) = (v +v−1)ch(E(w))
s∗ s
andso ch(G) = ch(π G′) = 0. Hence G andG′ are zero. (cid:3)
s∗
InDb(G/B)thereexisttheintersectioncohomologysheavesIC(w)
Λ
which may be defined as (i ) k [ℓ(w)] (see [BBD82]). The intersec-
w !∗ w
tioncohomologycomplexesadmitequivariantliftsIC (w) ∈ Db (G/B)
B B
which are uniquely determined (up to isomorphism) by requiring
For(IC (w)) ∼= IC(w)(see [BL94, 5.2]).
B
Wewill needthe following in the nextsection:
Proposition 3.5. For w ∈ W, E(w) ∼= IC (w) ifand only ifch(E(w)) =
B
H . Henceif ch(E(w)) = H thenFor(E(w)) ∼= IC(w).
w w
Proof. RecallthatIC(w)istheuniquecomplexinDb(G/B)satisfying
Λ
the four conditions:
(1) IC(w) isVerdierself-dual;
(2) i∗IC(w) = 0for y (cid:2) w;
y
(3) i∗IC(w) ∼= k [ℓ(w)];
w w
(4) i∗IC(w)isconcentrated in degrees< −ℓ(x) for x < w.
x
Note that 1),2)and 3)are alwayssatisfied by For(E(w)).
∼
Now,ifch(E(w)) = H then(4)isalsosatisfiedandsoFor(E(w)) =
w
IC(w). On the other hand, if ch(E(w)) 6= H then ch(E(w)) is a self-
w
dual and positive combination of Kazhdan-Lusztig basis elements