箙多様体と

(1)

(Quiver varieties and quantum ane algebras)

中島啓^(Hiraku^Nakajima)

Abstract. この論文は,math.QA/0009231(t-analogueoftheq -characters ofnitedimensionalrepresentationsofquantumane algebras )に

App endixとして箙多様体の定義を付け加えたものである.

もともとの^abstract: Frenkel-Reshetikhinintroducedq -characterofnite dimensionalrepresentationsofquantumanealgebras[ 6]. Wegiveacom-

binatorialalgorithm to compute them for all simple mo dules. Our to ol is

t-analogueoftheq -characters,whichissimilartoKazhdan-Lusztigp olynomi-

als,andouralgorithmhasaresemblancewiththeirdenition.

Weneed thetheoryofquivervarietiesforthedenition of t-analogues

andthepro of. Butitapp earonlyinthelastsection.Therestofthepap eris

devotedtoanexplanationofthealgorithm,whichonecanreadwithoutthe

knowledgeab outquivervarieties. Apro ofisgivenonlyinpart. Afullpro of

willapp earelsewhere.

1. The quantumloop algebra

Letgb easimpleLiealgebraoftypeADEoverC,Lg=gC[z;z 01

]b eitslo op

algebra,andU

q

(Lg)b e itsquantumuniversalenvelopingalgebra,orthequantum

lo opalgebraforshort. Itisasub quotientofthequantumanealgebraU

q (b

g),i.e.,

withoutcentralextensionanddegreeop erator. LetI b ethesetofsimplero ots,P

b e theweightlattice, andP 3

b e its dual lattice (all for g). The algebra has the

so-called Drinfeld's newrealization: It is a C(q )-algebra with generators q h

, e

k ;r ,

f

k ;r , h

k ;n

(h2 P 3

, k2I, r2Z, n 2Znf0g) withcertain relations(see e.g.,[ 1,

12.2]).

The algebra U

q

(Lg) is a Hopf algebra, where the coproduct is dened using

the Drinfeld-Jimbo realization of U

q

(Lg). So a tensor pro duct M

C(q ) M

0

of

U

q

(Lg)-mo dulesM,M 0

hasa structureofaU

q

(Lg)-module.

Let U

"

(Lg) b e its sp ecialization at q = " 2 C 3

. For precise denition of

the sp ecialization,we rstintroduce anintegral form U Z

q

(Lg) ofU

q

(Lg) and set

U

"

(Lg)=U Z

q (Lg)

Z[q ;q 01

]

C,where Z[q ;q 01

]!Cisgivenbyq 61

7!"

61

. See[3]

fordetail. Butweassume"isnot a ro otofunityinthispap er. Sowejustreplace

qby"inthedenitionof U

q (Lg).

SupportedbytheGrant-in-aidforScienticResearch(No.11740011),theMinistry ofEdu-

cation,Japan.

(2)

2 中島啓(HirakuNakajima)

The quantumlo op algebra U

q

(Lg) contains the quantumenveloping algebra

U

q

(g)for thenitedimensional Liealgebra gas asubalgebra. Thesp ecialization

U

"

(Lg)containsthesp ecializationU

"

(g)ofU

q (g).

1.1. Finite dimensionalrepresentationsofU

"

(Lg). ThealgebraU

"

(Lg)

contains a commutative subalgebra generated by q h

, h

k;n

(h 2 P 3

, k 2 I, n 2

Znf0g). Letusintroducegenerating functions 6

k

(z)(k2I)by

6

k (z)

def:

= q 6hk

exp 6(q0q 01

) 1

X

m=1 h

k ;6m z

7m

!

:

AU

"

(Lg)-mo duleM iscalledoftype1ifM hasaweightspacedecomposition

as aU

"

(g)-mo dule:

M= M

2P

M(); M()=

n

m2M

q

h

3m="

hh;i

m o

:

Wewillonlyconsidertype1modulesinthispap er.

A type 1 mo dule M is an l-highest weight module ('l' stands for the lo op)if

thereexistsa vectorm

0

2M suchthat

e

k ;r 3m

0

=0; U

"

(Lg) 0

3m

0

=M;

6

k

(z)3m

0

=9 6

k (z )m

0

fork2I

for some 9 6

k

(z) 2 C[[z 7

]]. The pair of the I-tuple (9 +

(z);9 0

(z)) = (9 +

k (z);

9 0

k (z))

k2I

2(C[[z 7

]]

I

) 2

is called the l-highest weight of M, and m

0

is called the

l-highestweightvector.

Theorem 1.1.1(Chari-Pressley[2]). (1) Every nite-dimensional simple

U

"

(Lg)-module of type 1 is an l-highest weight module, and its l-highest weight

isgivenby

9 6

k

(z)="

degP

k

P

k ("

01

=z)

P

k ("=z )

6

(1.1.2)

for some polynomials P

k

(u) 2 C[u] with P

k

(0) =1. Here( ) 6

2 C[[z 7

]] denotes

the expansionatz=1and0 respectively.

(2)Conversely,forgivenP

k

(u)asabove,thereexistsanite-dimensional sim-

ple l-highest weight U

"

(Lg)-module M of type 1 such that the l-highest weight is

givenbythe above formula.

AssigningtoM theI-tupleP =(P

k )

k2I 2C[u]

I

(P

k

(0)=1)denesabijection

betweenthe setof allP'sandthe set ofisomorphism classesof nite-dimensional

simple U

"

(Lg)-modulesoftype1.

Wedenote by L

P

the simpleU

"

(Lg)-module asso ciated to P. Wecall P the

Drinfeld polynomial . Fortheabuseof terminology,we alsosay `P is thel-highest

weightofL

P '.

SinceChq h

,h

k ;n

iisacommutativesubalgebraofU

"

(Lg),anyU

"

(Lg)-mo dule

M decomp oses intoa direct sumM = L

M(9 +

;9 0

)of generalized eigenspaces,

where

M(9 +

;9 0

)

def:

= 8

m2M

( 6

(z)09 6

(z )Id) N

3m=0 fork2I andsucientlylargeN 9

;

(3)

for9 6

k

(z)2C[[z 7

]]. ThepairoftheI-tuple (9 +

;9 0

)=(9 +

k

;9 0

k )

k 2I

is calledan

l-weight,andM(9 +

;9 0

)iscalledanl-weight space ofM ifM(9 +

;9 0

)6=0.

Theorem 1.1.3(Frenkel-Reshetikhin[6]). Any l-weight of any nite dimen-

sional U

"

(Lg)-module M of type1 has thefollowing form

9 6

k (z)="

degQ

k 0degR

k

Q

k ("

01

=z )R

k ("=z )

Q

k ("=z )R

k ("

01

=z)

6

(1.1.4)

forsomepolynomials

Q

k (u)=

sk

Y

i=1 (10a

ki u); R

k (u)=

rk

Y

j=1 (10b

k j u):

Againfortheabuseofterminology,wealsosay`Q=Risanl-weightofM'. We

denotethel-weightspaceM(9 +

;9 0

)byM(Q=R).

Frenkel-Reshetikhin[ 6]denedtheq-characterofM by

q (M)

def:

= X

Q=R

dimM(Q=R) Y

k 2I sk

Y

i=1 rk

Y

j =1 Y

k ;a

k i Y

01

k ;b

kj :

Theorem 1.1.5(Frenkel-Reshetikhin[6]). (1)

q

denesaninjectiveringho-

momorphismfromtheGrothendieckringRepU

"

(Lg)ofnitedimensionalU

"

(Lg)-

modules of type 1 to Z[Y 6

k ;a ]

k 2I;a2C 3

(a ring of Laurent polynomials in innitely

many variables).

(2) Ifwe composeamap Y 6

k ;a 7!y

6

k

(forgetting`spectralparameters'), itgives

the usualcharacterof the restrictionof M toaU

"

(g)-module.

Definition1.1.6. Amonomial Y

k 2I sk

Y

i=1 rk

Y

j =1 Y

k ;ak i Y

01

k;bkj

app earingintheq-char-

acter

q

is called l-dominant ifr

k

=0 forallk , i.e., a pro duct of p ositivepowers

ofY

k ;c 'sor1.

If L

P

is the simple U

"

(Lg)-mo dule with l-highest weight P, its q -character

containsanl-dominantmonomialcorrespondingtothel-highestweight. Wedenote

itbym

P

. Itscoecientin

q (L

P )is1.

SincefL

P g

P

formsa basisofRepU

"

(Lg),wehavethefollowingusefulcondi-

tionforthesimplicityof anitedimensionalU

"

(Lg)-moduleM of typ e1:

If

q

(M)containsonlyonel-dominantterm,thenM issimple.

(1.1.7)

1.2. Example. Wegiveexamples ofq -characters.

Ifg=A

n

,we haveanevaluationhomomorphismev

a :U

"

(Lg)!U

"

(g) corre-

sp ondingtoLg!g;z7!a(Jimbo). HencepullbacksofsimpleU

"

(g)-mo dulesare

simpleU

"

(Lg)-mo dules.

Example1.2.1. Letg=A

1

=sl

2

andV b ethe2-dimensionalsimpleU

"

(Lg)-

mo dule. Thentheq-characterofM

a

=ev

a

(M)isgivenby 1

q (M

a )=Y

1;a +Y

01

1;a"

2 :

1

(4)

Since

q

isaringhomomorphism,wehave

q (M

a

M

b )=

Y

1;a +Y

01

1;a"

2

Y

1;b +Y

01

1;b"

2

=Y

1;a Y

1;b +Y

01

1;a"

2 Y

1;b +Y

1;a Y

01

1;b"

2 +Y

01

1;a"

2 Y

01

1;b"

2 :

Ifb6=a"

2

;a"

02

,thenM

a

M

b

issimplebythecriterion(1.1.7).

If b = a"

2

or a"

02

, then the second or third term b ecomes 1. In fact, it is

knownthatM

a

M

a"

2 decomp oses(inRepU

"

(Lg))toasumM 0

a 8M

00

,whereM 0

is the3-dimensional simpleU

"

(Lg)-mo dule, and M 00

is the trivial module. Thus

wehave

q (M

a

M

a"

2)

=

q (M

0

a )+

q (M

00

)

=Y

1;a Y

1;a"

2

+Y

1;a Y

01

1;a"

4 +Y

01

1;a"

2 Y

01

1;a"

4 +1:

SeealsoExamples4.1.6,6.1.3,7.2.1.

2. Standardmo dules

2.1. In[15]wedenedafamilyofnitedimensionalU

"

(Lg)-mo dulesoftyp e

1 andcalled them standardmodules. Theyare parametrizedbytheI-tuples P =

(P

k )

k 2I

2 C[u]

I

exactly as simplemo dules. Wedenote by M

P

asso ciated to P.

Thedenitionwillberecalledin x8,but wegiveheretheiralgebraicidentication

duetoVaragnolo-Vasserot[16].

Definition2.1.1. WesayL

P

anl-fundamentalrepresentation if

P

k (u)=

(

10su ifk=k

0 ,

1 otherwise;

forsomes2C 3

andk

0

2I. WedenoteL

P

byL(3

k

0 )

s . (3

k

isthek-thfundamental

weightofg.)

For s 2 C 3

and a nite sequence (k

)

= (k

1

;k

2

;:::) in I and a sequence

(n

)

=(n

1 n

2

:::)ofintegers,weset

M(s;(k

)

;(n

)

def:

= L(3

k1 )

"

n

1s L(3

k2 )

"

n

2

s 111:

Note that U

"

(Lg) is not co commutativeHopf algebra,so thetensor pro duct de-

p endsontheorderingoffactors.

Theorem 2.1.2(Varagnolo-Vasserot[16]). (1) A standard module M is iso-

morphictoamodule ofthe form

O

i M(s

i

;(k i

i )

i

;(n i

i )

=M(s 1

;(k 1

1 )

1

;(n 1

1 )

1

)M(s 2

;(k 2

2 )

2

;(n 2

2 )

2

)111 (nite tensor product)

suchthats i

=s j

= 2"

Z

fori6=j andn i

1 n

i

2

::: foreachi.

(2) The above tensor product is independent of the ordering of the factors

M(s i

;(k i

i )

i

;(n i

i )

i ).

(3)TheI-tupleofpolynomialsP correspondingtoM istheproductofDrinfeld

(5)

NotethatifP isgiven,wecandeneamo duleMoftheab oveformbydecom-

p osingP intoaproductofDrinfeldp olynomials ofl-fundamentalrepresentations.

Thuswemaydenote theab ovemo dulebyM

P .

Thefollowingprop ertiesofM

P

wereshownin [15]:

(1) fM

P

g isa basisofRepU

"

(Lg).

(2) M

P

isan l-highest weightmo dule with l-highestweightP (i.e.,given by

(1.1.2)).

(3) L

P

istheuniquesimplequotientofM

P .

(4) M

P

dep ends`continuously'onP ina certainsense. For example,dimM

P

isindep endentofP.

(5) fora genericP,M

P

= L

P .

Conjecturally M

P

isisomorphic to thesp ecializationof themo dule V max

(),

introducedbyKashiwara[8],andfurther studiedbyChari-Pressley[4].

3. t-analoguesof q -characters

Amainto ol inthispap eris at-analogueoftheq-character:

q ;t

: RepU

"

(Lg)!Z[t;t 01

][Y 6

k;a ]

k 2I;a2C 3:

This is a homomorphism of additive groups, not of rings, and has the prop erty

q ;t=1

=

q

. Wedene

q ;t

for allstandardmo dulesM

P

. SincefM

P g

P

isabasis

ofRepU

"

(Lg),wecanextenditlinearlytoanynitedimensionalU

"

(Lg)-mo dules.

Forthedenitionweneedgeometricconstructionsofstandardmo dules,sowe

will p ostp one it to x8.3. Wegive an alternativedenition,which is conjecturally

thesameasthegeometricdenition.

3.1. Aconjecturaldenition. LetM=M

P

b eastandardmo dule,Q=Rb e

anl-weightofM,M(Q=R)b ethecorresp ondingl-weightspace. Denealtration

onM(Q=R)by

0=M 01

(Q=R)M 0

(Q=R)M 1

(Q=R)111

M n

(Q=R ) def:

=

\

k Ker(

6

k

(z)09 6

k (z)id)

n+1

:

Conjecture 3.1.1. Thet-analogue

q ;t (M

P

),denedgeometrical lyinx8.3,is

equalto

q ;t (M

P )=

X

Q=R X

n t

2n0d(Q=R ;P)

dim 0

M n

(Q=R)=M n01

(Q=R) 1

m

Q=R

;

where d(Q=R ;P) is an integer (determinedexplicitly fromQ=R , P by (5.1.2) be-

low),andm

Q=R

isamonomialinY 6

k ;a

correspondingtothel-weightspaceM(Q=R ).

Thisdenitionmakessenseforanynitedimensionalmodules,butisnot well-

denedontheGrothendiek group RepU

"

(Lg). Thusthe ab ovedo es notholdfor

simplemo dules.

3.2. Amain resultofthis pap erisa combinatorialalgorithmforcomputing

q ;t (M

P

)and[M

P :L

Q

]. It isdividedintothreesteps:

Step 1: Compute

q ;t

foralll-fundamentalrepsentations.

Step 2: Compute

q ;t (M

P

)forallstandard mo dulesM

P .

Step 3: Expressthemultiplicity[ M

P :L

Q

]intermsof

q ;t (M

R

)forvarious

(6)

Step1isamo dicationofFrenkel-Mukhin'salgorithm[5]forcomputing

q ofl-

fundamentalrepresentations. Step2isnothingbutastudyof

q ;t

oftensorproducts

ofl-fundamentalrepresentations. Although

q ;t

isnotaringhomomorphism,

q ;t

of tensorpro ducts isgiven bya simplymo died multiplication. Forthe pro ofwe

useanideain [13]. Step3wasessentiallydonein [15].

4. Step 3

Westart withStep 3. The algorithm issimilar to thedenition of Kazhdan-

Lusztig p olynomials [9]. It is also similar to the algorithm for computing the

transition matrix between the canonical basis and the PBW basis of type ADE

[10].

4.1. Let

A

k;a def:

= Y

k;a"

Y

k;a"

01 Y

l:l 6=k Y

ckl

l;a

;

wherec

kl

isthe(k;l)-entryoftheCartanmatrix.

Definition4.1.1. (1) Let m, m 0

b e monomialsin Y 6

k ;a

(k2I, a2 C 3

). We

deneanorderingamongmonomialsby

mm 0

() m

0

m

isa monimialinA 01

k ;a

(k2I,a2C 3

):

Here amonomial in A 01

k ;a

meansa pro ductof nonnegativep owersofA 01

k;a

. Itdo es

notcontainanyfactorsA

k ;a .

(2)If9 6

;9 06

arel-weightsofnitedimensionalU

"

(Lg)-mo dules,orQ=R ;Q 0

=R 0

arerelatedtol-weightsby(1.1.4),wewrite 9 6

9

06

,Q=RQ 0

=R 0

ifthecorre-

sp ondingmonomialsm,m 0

satisfymm 0

.

Recall that

q (L

P

) contains an l-dominant monomial m

P

corresp onding to

the highest weight vector. It is known that any monomialm app earing

q (L

P ),

q (M

P

)satisesmm

P

([5,4.1],[15, 13.5.2]).

Let

c

QP (t)

def:

= thecoecientofm

Q in

q ;t (M

P ):

Then (c

QP (t))

P ;Q

is upp er-triangular and c

PP

(t) = 1 by the ab ove mentioned

result.

Let( c QP

(t))b e theinversematrix( c

QP (t))

01

. Let

u

RP (t)

def:

= X

Q c

RQ

(t 01

)c

QP (t):

Let b etheinvolutiononZ[t;t 01

]givenbyt 61

7!t 71

.

Lemma 4.1.2(Lusztig[10, 7.10]). There exists a unique solution Z

QP (t) 2

Z[t 01

](QP)of

Z

RP (t)=

X

Q:RQP Z

RQ (t)u

QP (t);

(4.1.3)

Z

PP

(t)=1; Z

QP (t)2t

01

Z[t 01

]forQ<P: (4.1.4)

Thislemmaisprovedbyinduction,andholdsina generalsetting. Lusztighas

(7)

Theorem 4.1.5. Themultiplicity[M

P :L

Q

]ofasimple moduleL

Q

inastan-

dardmodule M

P

isequaltoZ

QP (1).

Theproofwillb egivenin x8.4.

Example4.1.6. Let g = A

1 , M

a

= ev 3

a

(M) where M is the 2-dimensional

simpleU

"

(g)-mo duleasbefore. By steps1,2explainedbelow 2

wehave

q ;t (M

a"

2M

a )=Y

1;a Y

1;a"

2+Y

1;a Y

01

1;a"

4 +Y

01

1;a"

2 Y

01

1;a"

4 +t

01

1:

(4.1.7)

LetP(u)=(10au)(10a"

2

u)(i.e.,M

P

=M

a"

2M

a

),Q(u)=1(i.e. M

Q

=trivial

mo dule). Thentheab ovealgorithmgivesusZ

QP (t)=t

01

:

5. Step 1

5.1. Somedenitions. LetM

P

b eastandardmo dule. Letm

P

bethemono-

mial corresp onding to the l-highest weightvector. Let M

P

(Q=R) b e anl-weight

space as b efore. We denote by m

Q=R

the corresp onding monomial. We dene

w

k;a (P);v

k ;a

(Q=R ;P)2Z

0

;u

k ;a

(Q=R )2Zby

m

P

= Y

k2I;a2C 3

Y w

k;a (P)

k ;a

;

m

Q=R

=m

P Y

k 2I;a2C 3

A

0vk;a(Q=R ;P)

k ;a

= Y

k2I;a2C 3

Y

uk;a(Q=R )

k ;a :

Supp osetwostandardmo dulesM

P 1

,M

P 2

andl-weightspacesM

P 1(Q

1

=R 1

)

M

P 1,

M

P 2

(Q 2

=R 2

)M

P 2

aregiven. Wedene

d(Q 1

=R 1

;P 1

;Q 2

=R 2

;P 2

)

def:

= X

k;a 0

v

k;a (Q

1

=R 1

;P 1

)u

k ;a"

01(Q 2

=R 2

)+w

k ;a"

(P 1

)v

k;a (Q

2

=R 2

;P 2

) 1

: (5.1.1)

Wealsodene

d(Q=R ;P) def:

= d(Q=R;P;Q=R;P):

(5.1.2)

Wedenoted(Q 1

=R 1

;P 1

;Q 2

=R 2

;P 2

)alsobyd(m

Q 1

=R 1

;m

P 1;

m

Q 2

=R 2;

m

P 2

).

We needthe followingmodicationof

q ;t

. Write

q ;t (M

P )=

P

m a

m (t) m,

wheremisa monomialanda

m

(t)isitsco ecient. Let

g

q ;t (M

P )

def:

= X

m t

d(m;m

P )

a

m (t)m;

(5.1.3)

whered(m;m

P

)isdenedin(5.1.2).

3

5.2. Frenkel-Mukhin[5,5.1,5.2]provedthattheimageof theq -character

q

iscontainedin

\

k2I

Z[Y 6

l ;a ]

l6=k;a2C 3Z[Y

k;b (1+A

01

k ;b"

)]

b2C 3

:

2

ordirectcalculationforthedenition(8.3.1)

3

Infact,d(m;m

P

)isdeterminedfromam(t)sothatt d(m;m

P )

am(t)isap olynomialintwith

(8)

Wehavethet-analogueof thisresult,replacing(1+A 01

k ;b"

) n

by

1+A 01

k ;b"

n

t def:

= n

X

r =0 t

r (n0r)

n

r

t A

0r

k ;b"

;

where[ n

r ]

t

isthet-binomialco ecient. Moreprecisely,wehave

Theorem 5.2.1. (1)Foreachk2I,g

q ;t (M

P

)isexpressedasalinearcombi-

nationof

Y

i Y

ni

k ;bi

1+A 01

k;bi"

ni

t

=Y n1

k ;b1

1+A 01

k ;b1"

n1

t Y

n2

k ;b2

1+A 01

k;b2"

n2

t 111

with coecients in Z[t][Y 6

l;a ]

l 6=k;a2C 3,

where b

i 2 C

3

, n

i 2 Z

>0 with b

i 6= b

j for

i6=j.

(2) If L

P

is an l-fundamental representation (and hence M

P

= L

P ), then

q ;t (M

P

)containsnol-dominantmonomialsotherthanm

P

andtheconditionabove

uniquelydetermines

q ;t (M

P ).

Remark 5.2.2. Thestatement(1)fort=1wasprovedbyFrenkel-Mukhin[5].

And thepro of of (2) is thesame for t =1 andthe generalcase, as illustrated in

thefollowingexamples. Inthissense,(2)should alsob ecredittedtothem.

5.3. Graph. Wegive few examples of

q ;t

of l-fundamentalrepresentations

determinedbytheab ovetheorem.

We attach to each standard mo dule M

P

, an oriented colored graph 0

P . (It

is a slight mo dication of the graph in [6, 5.3].) The vertices are monomials in

q ;t (M

P

). Wedraw ancolorededge k ;a

00!fromm

1 tom

2 ifm

2

=m

1 A

01

k ;a

. Wealso

writethemultiplicityofthemonomialsin

q ;t (M

P ).

3

=sl

4 andM

P

=L(3

2 )

1

. Thenthecorresponding

graph 0

P is

Y

2;1 2;"

0000! Y

1;"

Y 01

2;"

2 Y

3;"

1;"

2

0000! Y 01

1;"

3 Y

3;"

2

?

y

?

y3;"

2

Y

1;"

Y 01

3;"

3 1;"

2

0000! Y 01

1;"

3 Y

2;"

2Y 01

3;"

3 2;"

3

0000! Y 01

2;"

4 :

Let us explain how we determine this graph inductively. We start with the l-

highestweightY

2;1

. Weknowthat itsco ecientis1. ApplyingTheorem5.2.1(1)

withk=2,wegetY

1;"

Y 01

2;"

2 Y

3;"

withco ecient1. ThenweapplyTheorem5.2.1(1)

withk=1 togetY 01

1;"

3 Y

3;"

. Andso on. All multiplicitiesare1 inthiscase.

Forg=A

n

, itisknownthat theco ecientsof

q ;t (L(3

k )

a

)areall 1.

4

Thus

q ;t (L(3

k )

a )=

q ;t=1 (L(3

k )

a ).

Example5.3.2. Letg=D

4 andM

P

=L(3

2 )

1

. Thegraph0

P

isFigure1. Itis

knownthattherestrictionofM

P toaU

"

(g)-mo duleisadirectsumoftheadjoint

representation and the trivial representation. This fact is reected in

q ;t (M

P )

whereY

2;"

2Y 01

2;"

4

hastheco ecient[2]

t

andallothershas1. Notethatthenumber

4

Moregenerally,iftheco ecientsof

k

inthehighestro otis1,thenthesameholds. This

resulteasilyfollowsfromthetheoryofquivervarieties. Exercise: Checkthisusingtheab ove

(9)

Y

2;1

Y

1;"

Y 01

2;"

2 Y

3;"

Y

4;"

Y

1;"

Y 01

3;"

3 Y

4;"

Y 01

1;"

3 Y

3;"

Y

4;"

Y

1;"

Y

3;"

Y 01

4;"

3

Y 01

1;"

3 Y

2;"

2Y 01

3;"

3 Y

4;"

Y

1;"

Y

2;"

2Y 01

3;"

3 Y

01

4;"

3 Y

01

1;"

3 Y

2;"

2Y

3;"

Y 01

4;"

3

Y 01

2;"

4 Y

4;"

Y

4;"

3 Y

1;"

Y

1;"

3Y 01

2;"

4 Y

01

1;"

3 Y

2

2;"

2Y 01

3;"

3 Y

01

4;"

3 Y

01

2;"

4 Y

3;"

Y

3;"

3

Y

4;"

Y 01

4;"

5

Y

1;"

Y 01

1;"

5

[2]

t Y

2;"

2Y 01

2;"

4

Y

3;"

Y 01

3;"

5

Y

2;"

2Y 01

4;"

3 Y

01

4;"

5

Y 01

1;"

3 Y

01

1;"

5 Y

2;"

2 Y

1;"

3Y 02

2;"

4 Y

3;"

3Y

4;"

3 Y

2;"

2Y 01

3;"

3 Y

01

3;"

5

Y

1;"

3Y 01

2;"

4 Y

3;"

3Y 01

4;"

5 Y

01

1;"

5 Y

01

2;"

4 Y

3;"

3Y

4;"

3 Y

1;"

3Y 01

2;"

4 Y

01

3;"

5 Y

4;"

3

Y 01

1;"

5 Y

3;"

3Y 01

4;"

5

Y

1;"

3Y 01

3;"

5 Y

01

4;"

5

Y 01

1;"

5 Y

01

3;"

5 Y

4;"

3

Y 01

1;"

5 Y

2;"

4Y 01

3;"

5 Y

01

4;"

5

Y 01

2;"

6

? 2;"

3;"

2

? 1;"

2

j 4;"

2

? 1;"

2

j 4;"

2

3;"

2

j 4;"

2

3;"

2

? 1;"

2

2;"

3

j 4;"

2

2;"

3

? 1;"

2

3;"

2

? 2;"

3

? 4;"

4

z 4;"

2

? 1;"

4

j 1;"

2

? 2;"

3

3;"

2

? 3;"

4

? 4;"

2

? 1;"

2

? 2;"

3

? 3;"

2

j 2;"

3

j 2;"

3

4;"

4

? 1;"

4

j 3;"

4

? 2;"

3

? 1;"

4

j 3;"

4

4;"

4

j 3;"

4

4;"

4

? 1;"

4

j 3;"

4

? 1;"

4

4;"

4

? 2;"

5

Figure 1. ThegraphforL(3

2 )

1

(10)

10 中島啓(HirakuNaka jima)

ofmonomialsis 28,whichis thedimensionoftheadjointrepresentation. Seealso

Example7.2.3below.

Letusgiveamore complicatedexample.

Example5.3.3. Let g=A

2

and M

P

=L(3

2 )

2

"

L(3

1 )

1

. Although this is

notanl-fundamentalrepresentation,

q ;t (M

P

)hasnol-dominanttermsotherthan

m

P

,so theconditionTheorem5.2.1(1)givesus

q ;t

. Thegraphis Figure2.

Y

1;1 Y

2

2;"

Y 01

1;"

2 Y

3

2;"

[2]

t Y

1;1 Y

1;"

2Y

2;"

Y 01

2;"

3

[2]

t Y

1;1 Y

01

1;"

4 Y

2;"

[3]

t Y

2

2;"

Y 01

2;"

3

Y

1;1 Y

2

1;"

2 Y

02

2;"

3

[2]

t Y

01

1;"

2 Y

01

1;"

4 Y

2

2;"

[3]

t Y

1;"

2

Y

2;"

Y 02

2;"

3

[2]

t Y

1;1 Y

1;"

2

Y 01

1;"

4 Y

01

2;"

3

Y 2

1;"

2Y 03

2;"

3

([3]

t +1)Y

01

1;"

4 Y

2;"

Y 01

2;"

3

Y

1;1 Y

02

1;"

4

[2]

t Y

1;"

2Y 01

1;"

4 Y

02

2;"

3

Y 01

1;"

2 Y

02

1;"

4 Y

2;"

Y 02

1;"

4 Y

01

2;"

3

1;"

j 2;"

2

j 2;"

2

9

1;"

3

1;"

? 2;"

2

? 2;"

2

? 2;"

2

1;"

? 1;"

3

j 2;"

2

2;"

2

? 1;"

3

1;"

? 1;"

3

? 2;"

2

2;"

2

? 1;"

j 1;"

3

2;"

2

Figure 2. ThegraphforL(3

2 )

2

"

L(3

1 )

1

Remark 5.3.4. Aswe cansee inab oveexamples,thecrystal graphsare sub-

graphs of0

P

. Thesetof verticesisthesame,but thesetof arrowsissmaller. We

wouldliketodiscussthis furtherelsewhere.

6. Step 2

6.1. LetM

P

=M(s 1

;(k 1

1 );(n

1

))M(s 2

;(k 2

2 );(n

2

))111 b eastandard

i j Z

(11)

Proposition6.1.1. Wehave

q ;t (M

P )=

q ;t (M(s

1

;(k 1

1 )

1

;(n 1

1 )

1 ))

q ;t (M(s

2

;(k 2

2 )

2

;(n 2

2 )

2 ))111

if s i

=s j

= 2"

Z

fori6=j.

Thusitisenoughto study

q ;t

(M(s;(k

)

;(n

)

))=

q ;t (L(3

k

1 )

"

n

1s L(3

k

2 )

"

n

2

s 111):

Let

q ;t (L(3

k

)

"

n

s ))=

X

r a

m

;r

(t)m

;r

;

wherem

;r

is amonomialinY 6

k ;a anda

m;r

(t)2Z[t;t 01

]isitsco ecient.

Ift=1,

q ;1

isa ringhomomorphism,hencewehave

q ;1

(M(a;(k

)

;(n

)

))=

X

r

1

;r

2

;:::

Y

a

m

r

;r

(1)m

;r

:

Theorem 6.1.2. LetP

be the Drinfeld polynomial of L(3

k )

"

n

s

. Then we

have

q ;t

(M(a;(k

);(n

)))=

X

r

1

;r

2

;:::

t P

; 6d(m;r

;m

P

;m ;r

;m

P

) Y

a

m

r

;r

(t)m

;r

;

wherethe signford(m

;r

;m

P

;

m

;r

;m

P

)is0if and+otherwise.

Example6.1.3. Forg=A

1

, wehave

d(Y 01

1;a"

2

;Y

1;a

;Y

1;a

;Y

1;a

)=1; d(Y

1;a

;Y

1;a

;Y 01

1;a

;Y

1;a"

02)=1

andallothersare0. Thenweget(4.1.7).

IfP =(10au) n

, weget

q ;t (M

P )=

n

X

r =0

n

r

t Y

n0r

1;a Y

0r

1;a"

2

from

q ;t (L(3

1 )

a ) = Y

1;a +Y

01

1;a"

2

. This also follows directly from the deni-

tion (8.3.1) b elow. The t-binomial co ecients app ear as Poincare p olynomials

ofGrassmannmanifolds.

7. Restrition to U

"

(g)

FinitedimensionalsimpleU

"

(g)-mo dulesareclassiedbyhighestweights. Let

ResM

P

b e the restriction of a standard mo dule M

P

to a U

"

(g)-mo dule. It de-

comp oses into a sumof various simplemo dules. Once

q (M

P

) is computed, the

characterofResM

P

isgivenbyreplacingY 6

k ;a byy

6

k

(Theorem1.1.5(2)). Combin-

ing with theknowledgeof charactersofsimplenite dimensional U

"

(g)-mo dules,

wecandeterminethemultiplicityofsimplemo dulesin ResM

P .

CharactersofsimplenitedimensionalU

"

(g)-mo dulesarethesame asthatof

simpleg-mo dules,henceareknown. However,weexpress themintermsof

q ;t in

(12)

7.1. Foradominantweightw= P

w

k 3

k

wedenotebyL

w

thesimplehighest

weightU

"

(g)-mo dulewiththehighestweightw.

Weconsidera standard mo duleM

P

withdegP

k

=w

k

. By the`continuity'of

M

P

onP,ResM

P

dep endsonlyonw

k

=degP

k

,andnotonP itself. Letusdenote

themultiplicityofL

w

0 inResM

P byZ

w 0

;w ,i.e.,

ResM

P

= M

w 0

L 8Z

w 0

;w

w 0

:

We will give a formula expressing Z

w 0

;w

in termsof

q ;t (M

P

). Although we

can give algorithm forarbitary P in principle, the followingchoicewill make the

formulasimple.

Cho oseandxorientationsofedgesin theDynkindiagram. Wedeneinteger

m(k )foreachvertexksothatm(k )0m(l )=1ifwehaveanorientededgefromk

tol,i.e.,k!l . Thenwedene P by

P

k

(u)=(10u"

m(k)

) w

k

:

Let g

q ;t (M

P

)as in (5.1.3). Let e

t (M

P

)2 Z[t]Z[t 6

k ]

k 2I

bea t-analogue of

theordinarycharacterwhichisobtained fromg

q ;t (M

P

)bysendingY 6

k ;a toy

6

k .

Foranotherdominantweightw 0

= P

w 0

k 3

k ,let

c

w 0

;w (t)

def:

= theco ecientof Y

y w

0

k

k in e

t (M

P ):

The matrix(c

w 0

;w (t))

w 0

;w

is upper-triangular with resp ect to the usual order on

weights,anddiagonalentriesareall1.

Theorem 7.1.1. c

w 0

;w

(0)isthe weightmultiplicity ofw 0

inthehighestweight

moduleL

w

withthe highestweightw .

Thisisjustasimplerephrasingofa mainresultin [12,14]. Theproofwillb e

giveninx8.5.

Notethat c

w 0

;w

(1)givestheweightmultiplicityof w 0

in ResM

P

sinceg

t=1 is

theordinarycharacter. Thuswehave

c

w 00

;w (1)=

X

w 0

c

w 00

;w 0

(0)Z

w 0

;w :

ThisequationdeterminesthemultiplicityZ

w 0

;w

onlyfromtheknowledgeof

q ;t .

AccordingtoaconjectureofLusztig[11]togetherwithaformula(8.5.1)b elow,

c

w 0

;w

(t)shouldb ewrittenbyferminonicformofHatayamaelal.[7]. Moreprecisely,

we should have P

w 0

c w

00

;w 0

(0)c

w 0

;w

(t) = M(w ;w 00

;t 2

); where (c w

00

;w 0

(0)) is the

inversematrixof(c

w 00

;w

0(0)). See[11]for thedentionofM(w ;w 0

;q). Although

thisformulacanb echeckedinmanyexamples,thecomplexityofthecombinatorics

preventus from provingit in full generality. Conjecturally M(w ;w 0

;q=1)gives

us the multiplicities of the restriction of M

P

(Kirillov-Reshetikhin 5

). Thus the

conjectureiscompatiblewithourresultinthissection.

7.2.

1

andw=23

1

. WetakeP =(10u) 2

bytheab ove

choice. By Example6.1.3,wehave

e

t (M

P )=y

2

1

+(1+t 2

)+y 02

1 :

5

(13)

Thus Z

0;w

= 1. Since Res(M

P ) = L

31 L

31

= L

231 8L

0

, this is the correct

answer!

3

andw=3

2

. ByExample5.3.1alltheco ecients

of

q ;t (L(3

2 )

1

)are1. HenceResM

P

=ResL(3

2 )

1

issimpleas aU

"

(g)-module.

Example7.2.3. Let g = D

4

, w = 3

2 , w

0

= 0. By Example 5.3.2 wehave

c

w 0

;w

(t)=4+t 2

. ThusZ

w 0

;w

=1,i.e. Res(L(3

2 )

1 )=L

32 8L

0 .

8. Quiver varieties

Inthissection, wegivethe denitionof

q ;t

and proveTheorems 4.1.5,7.1.1.

Aswementioned,thosepro ofsareessentiallygivenin[12,14]and[15]resp ectively.

Theonlythingswedoherearetranslationofresultsintothelanguageof

q ;t . We

b elievethat thissectiongivesgo o d introductionsto [12, 14,15].

8.1. Let w = P

w

k 3

k (w

k 2 Z

0

) b e a dominant weight of the nite di-

mensional Lie algebra g. In [12, 14, 15], we have attached to each w , a map

:M(w)!M

0

(1;w)withthefollowingproperties:

(1) M(w ) isa nitedisjointunionof nonsingularquasi-projectivevarietiesof

variousdimensions.

(2) M

0

(1;w )isananealgebraicvariety.

(3) isa projectivemorphism.

(4) There exist actionsof G

w 2C

3

on M(w ) and M

0

(1;w ) such that is

equivariant.

(5) M

0

(1;w )isacone,andthevertex(denotedby0)istheuniquexedp oint

oftheC 3

-action(restrictionofG

w 2C

3

-actiontothesecondfactor).

HereG

w

= Q

k2I GL(w

k

;C).

Weconsidertheb erpro duct

Z(w ) def:

= M(w)2

M0(1;w ) M(w ):

The convolutionpro duct makes the (Borel-Mo ore) homology group H

3

(Z(w);C)

into an associative (noncommutative) algebra. One of main results in [14] is a

constructionofasurjectivealgebra homomorphism

U(g)!H

top

(Z(w );C);

whereU(g)istheuniversalenvelopingalgebraofg(NB:nota`quantum'version).

Here H

top

( ) meansthe degree =dim

R

Z(w) part ofthe homology group. More

precisely,wetakedegree=dimensionpartoneachconnectedcomp onentsofZ(w),

and then make the direct sum. Note that the the dimension diers on various

comp onents.

LetL(w )= 01

(0). ItisknownthatM(w )hasaholomorphicsymplecticform

suchthat L(w ) is alagrangian subvariety. TheconvolutionmakesH

top

(L(w );C)

(thetopdegreepartoftheBorel-Mo orehomologygroup,inthesamesenseasabove)

intoanH

top

(Z(w);C)-mo dule. Itis aU(g)-mo dulebytheab ovehomomorphism.

By [14, 10.2] it is the simple nite dimensional U(g)-module L

w

with highest

weight w . And connected components M(v;w ) of M(w) are parametrized by

vectorsv= P

v

k

k (

k

is thek th simplero otofg)so that

H

top

(L(w );C)= M

H

top

(M(v ;w)\L(w );C)

(14)

istheweightspacedecomp osition ofthesimplehighestweightmo duleL

w , where

H

top

(M(v ;w)\L(w );C) has weightw0v . In particular, v =0 corresp onds to

thehighestweightvector. Infact,M(0;w )isconsistingofasinglep oint.

ThespaceM

0

(1;w )hasa stratication

M

0

(1;w)= [

M reg

0

(v ;w);

where v runs over theset of vectors such that w0v is a weightof L

w

which is

dominant [12,x3].

8.2. LetusgivetheU

q

(Lg)-versionof theconstructionoftheprevioussub-

section.

Weuse the followingnotation: Let R(G) denote therepresentation ring ofa

linearalgebraic group G. If Gactsa quasi-projectivevarietyX, K G

(X) denotes

theGrothendieckgroupofG-equivariantcoherentsheavesonX.

The representation ring R(G

w 2C

3

) of G

w 2C

3

is isomorphicto the tensor

pro duct R(G

w )

Z R (C

3

). Moreover,R(C 3

)is isomorphicto Z[q;q 01

], whereq is

thecanonical1-dimensionalrepresentationofC 3

.

TheconvolutionmakestheGrothendieckgroupK G

w 2C

3

(Z(w ))intoaR(G

w 2

C 3

)=R (G

w )[q;q

01

]-algebra. Oneof main resultsin [15] is a constructionof an

algebra homomorphism

U Z

q (Lg)

Z R(G

w )!K

G

w 2C

3

(Z(w ))=torsion:

By the equivariance of , L(w) = 01

(0) is invariant under G

w 2C

3

. The

convolutionmakesK G

w 2C

3

(L(w))intoaK G

w 2C

3

(Z(w ))-mo dule. Moreover,itis

freeofniterankoverR(G

w 2C

3

)[15, x7]. Itis aU Z

q (Lg)

Z R (G

w

)-moduleby

theab ovehomomorphism. By[15,x13],itcontainsavectorm

0

suchthat

e

k ;r 3m

0

=0;

0

U Z

q (Lg)

0

Z R(G

w )

1

3m

0

=K Gw2C

3

(L(w ));

6

k

(z)3m

0

=q wk

V

01=q z q

01

W

k

V

0q =z q

01

W

k

!

6

3m

0

fork2I:

(8.2.1)

The right hand side ofthe third equation needsan explanation: First W

k is the

vectorrepresentationofGL(w

k

;C),consideredasaG

w 2C

3

-module. Then V

u V =

P

u i

V

i

V. Since V

0q =z q

01

W

k

is 10(1=z)W

k

+::: (1 is the trivial mo dule), we

can dene

V

0q =z q

01

W

k

01

as a formal p ower series in 1=z . This gives us the

case ( ) +

of the above formula. In the case ( ) 0

, we expand as V

0q =z q

01

W

k

=

(01=z) wk

V

wk

W

k 0z

V

wk01

W

k +111

. Then V

wk

W

k

is an invertible element,

wecanalsodene

V

0q =z q

01

W

k

01

. Thevectorm

0

isthecanonicalgeneratorof

K Gw2C

3

(M(0;w )). (RecallM(0;w )isa p oint.)

The mo dule K G

w 2C

3

(L(w )) should b e considered as a `universal' standard

mo dulesincestandardmo dulesareobtainedfromitbysp ecializationsasweexplain

now.

Leta=(s;")2G

w 2C

3

beasemisimpleelement. Itdenesahomomorphism

a :R (G

w 2C

3

)!Cbysendinga representationtothevalueofthecharacterat

a. Then

K Gw2C

3

(L(w))

R (G 2C 3

) C (8.2.2)

(15)

isa mo duleoverU

"

(Lg)=U Z

q (Lg)

Z[q ;q 01

]

C. By(8.2.1)itisa nite-dimensional

l-highestweightmo dule. Thisisthestandardmodule M

P

,whereP

k (u)=

a (

V

0u

q 01

W

k

). Notethatthesetofconjugacyclassesofa=(s;")bijectivelycorresp onds

tothesetofI-tupleofp olynomialsP withdegP

k

=w

k .

8.3. LetAb etheZariskiclosureofa Z

inG

w 2C

3

. Itisanab elianreductive

group. WehaveK G

w 2C

3

(L(w ))

R(G

w 2C

3

) R (A)

= K

A

(L(w ))[15,x7]. Since

a

factorsthroughR (A),thestandardmoduleM

P

isisomorphictoK A

(L(w ))

R(A) C.

By Thomason's lo calization theorem, it is isomorphic to K(L(w) A

)

Z

C, where

L(w) A

is thexed p oint set. Furthermore, it is isomorphicto H

3 (L(w )

A

;C)via

theCherncharacterhomomorphism[15,x7].

Let A

: M(w) A

! M

0 (1;w)

A

be the restriction of the map : M(w ) !

M

0

(1;w ) to the xed point set. Let M(w ) A

= F

M() be thedecomp osition

intoconnected comp onents. EachM() is a nonsingular quasi-projective variety.

Thenwehavethedirect sumdecomposition

M

P

= H

3 (L(w )

A

;C)

= M

H

3

(M()\L(w );C):

In [15, x13, x14]we haveshownthat this is the l-weight spacedecomposition of

M

P

. Inparticular, theindex can b econsidered as anl-weightofM

P

. Thuswe

havearrivedat ageometricinterpretationof

q :

q (M

P )=

X

dimH

3

(M()\L(w );C)m

;

wherem

isthemonomialcorresp ondingtothel-weight.

Nowwedenethet-analogue

q ;t by

q ;t (M

P )

def:

= X

X

k dimH

k

(M()\L(w );C)t

k0dimCM()

m

: (8.3.1)

By[15, x14]wehaveastratication

M

0 (1;w )

A

= [

M

reg

0 ();

consisting ofnonsingularlocally closedsubvarieties. Heretheindex set fgisthe

subsetoftheab oveindexset consistingofl-dominant l-weights.

8.4. Proof of Theorem 4.1.5. Thel-highest weightP is xed throughout

the pro of. Thus thedominant weightvector w and the element a=(s;")2 G

w

arexed.

Wechangethenotationnow. Ifcorresp ondstoanl-weightspaceM

P (Q=R),

we denote ab oveM() by M(Q=R;P). Wealso denote by M reg

0

(Q;P) for above

M reg

0

() ifcorresp onds toanl-dominantl-weightQ. Thuswehave

M(w ) A

= G

Q=R

M(Q=R;P); M

0 (1;w)

A

= [

Q M

reg

0

(Q;P):

InthisnotationH

3

(M(P ;P) \L(w );C)isthel-highestweightspace. SinceM(0;w )

is a single p oint as we explained, we have M(P ;P) = M(0;w ). We also have

M reg

(P ;P)=f0g.

(16)

Lemma 8.4.1. (1)dim

C

M(Q=R ;P)=d(Q=R ;P). dim

C M

reg

0

(Q;P)=d(Q;P).

(2)If M reg

0

(Q;P)M reg

0

(R;P), thenRQ.

(3) Choose x 2 M reg

0

(Q;P). Then ( A

) 01

(x)\M(S=T;P) is isomorphic to

M(S=T;Q)\L(w).

Proof. (1)Therstequationisthedimensionformula[15,4.1.6]. Thesecond

equation followsfrom dim

C M

reg

0

(Q;P) =dim

C

M(Q;P), whichis clear from the

denition [15,x4].

(2),(3) The resultsare known or trivial for Q =P. Now usethe transversal

slice atx2M reg

0

(Q;P)[15,x3] toreducea generalcasetothiscase.

LetD b

(M

0 (1;w )

A

)b etheb oundedderivedcategory ofcomplexesofsheaves

such that cohomology sheaves are constantalong each stratumM reg

0

(Q;P). Let

IC(M reg

0

(Q;P)) b e the intersection homology complex asso ciated with the con-

stant lo cal systemC

M reg

0 (Q;P)

onM reg

0

(Q;P). By using thetransversal slice [15,

x3], one can check that it is an object in D b

(M

0 (1;w )

A

). Let C

M(Q=R;P) be

the constant lo cal system on M(Q=R ;P). Then A

3 (C

M(Q=R;P)

) is an object of

D b

(M

0 (1;w )

A

) again by the transversal slice argument. Using the decomp osi-

tion theorem of Beilinson-Bernstein-Deligne, we have shownthat there exists an

isomorphism inD b

(M

0 (1;w )

A

):

A

3 (C

M(R;P) [dim

C

M(R;P)])

= M

Q;k L

Q;k

(R;P)IC(M reg

0

(Q;P))[k ] (8.4.2)

for some vector space L

Q;k

(R ;P) [15, 14.3.2]. Since A

(M(R ;P)) M reg

0

(R;P)

bydenition[15, x4],thesummationruns overQRbyLemma 8.4.1. Let

L

RQ (t)

def:

= X

k dimL

Q;k (R;P)t

0k

:

Applying theVerdierduality to the b oth hand side of (8.4.2) and using the self-

duality of A

3 (C

M(R;P) [dim

C

M(R;P)]) and IC(M reg

0

(Q;P)), we nd L

RQ (t) =

L

RQ (t).

Cho ose a p oint x

Q

from M reg

0

(Q;P) for each stratum. Let i

xQ : fx

Q g !

M

0 (1;w )

A

denotetheinclusion. Consider

H k

(i

!

x

Q

A

3 C

M(R;P)

[dimM(R;P)])=H

dim

C

M(R;P)0k ((

A

) 01

(x

Q

)\M(R;P);C):

ByLemma8.4.1(3)thisisisomorphictoH

dimCM(R;P)0k

(M(R;Q)\L(w);C).There-

forewehave

X

k dimH

k

(i

!

xQ

A

3 C

M(R;P)

[dimM(R ;P)])t dim

C

M(Q;P)0k

= X

d dimH

d

(M(R;Q)\L(w );C)t d+dim

C

M(Q;P)0dim

C M(R;P)

=c

R Q (t);

(8.4.3)

whereweuseddim

C

M(R;P)0dim

C

M(Q;P)=dim

C

M(R ;Q)inthelastequality.

By[15, 14.3.10],wehave

[M

Q :L

R

]=dimH 3

(i

!

xQ IC(M

reg

0

(R;P))):

(Infact, we denedthestandard mo duleM

Q as H

3 ((

A

) 01

(x

Q

);C)in [15, x13],

(17)

However,byusingthetransversalslice,wecanshowthattherighthand sideisthe

same forbothdenitions. cf.Lemma 8.4.1.)

Let

Z

RQ (t)

def:

= X

k dimH

k

(i

!

x

Q IC(M

reg

0

(R ;P)))t dimM

reg

0

(Q;P)0k

:

Wehave[M

Q :L

R ]=Z

R Q

(1). By thedeningpropertyoftheintersectionhomol-

ogy,Z

R Q

(t)satises(4.1.4).

Substituting(8.4.2)into(8.4.3), weget

c

SQ (t)=

X

R L

SR (t)Z

RQ (t):

NowL

S R (t)=L

S R

(t)implies(4.1.3). Thiscompletesthepro ofofTheorem 4.1.5.

8.5. Proof of Theorem 7.1.1. By theresult explainedin x8.1, the weight

multiplicityofw 0

inL

w

isequal to

dimH

top

(M(w0w 0

;w )\L(w );C):

Theassertionfollowsfrom moregeneralformula

e

t (M

P )=

X

w 0

X

d H

d

(M(w0w 0

;w)\L(w);C)t dim

C M(w0w

0

;w)0d Y

k y

w 0

k

k : (8.5.1)

NotethatM(w0w 0

;w )\L(w )isalagrangiansubvarietyinM(w0w 0

;w),sowe

havetop=dim

C

M(w0w 0

;w ).

Inorder toprove(8.5.1), we use[12, 5.7],where theBetti numbersare given

in termsof those of xed p oint comp onents. It lo oks almost thesame as ab ove.

However,there is one signicantdierence. TheC 3

-actionused there is dierent

fromourC 3

-actionusedhere,denedin[15,x2]. Thisisthereasonwhywechoose

P and corresp ondinga=(s;")as explainedin x7. Then A=a Z

is isomorphicto

C 3

andtheactionisthesame astheC 3

-actionconsideredin [12,x5].

We decomp ose M(w ) A

= F

M() intoconnected components as b efore. By

[12, 5.7]wehave 6

dimH

d

(M(w0w 0

;w )\L(w );C)= X

dimH

dim

C M(w 0w

0

;w )0d

(M();C);

wherethesummationruns overthesetofsuchthatthecorresp ondingmonomial

m

issentto Q

k y

w 0

k

after Y

k ;a

! y

k

. TheC 3

-action makesM

0 (1;w)

C 3

=f0g,

soM()=M()\L(w ). Hencetheaboveexpressioncoincideswiththedenition

ofthecoecientse

t .

Acknowledgement. Wewould liketo thank E. Frenkel and E. Mukhin for

explanationsoftheiralgorithmcomputingq -charactersoffundamentalrepresenta-

tions.

6

Infact, thisformulaevenholdsforgeneralP if wereplacedim

C

M(w0w 0

;w)0dbya

suitabledegree. However,thisdegreeshiftisgivenbyacomplicatedexpressionin.Soourchoice

(18)

App endix A. 箙多様体

A.1. 箙^(quiver). ^quiver^Qは^,頂点の有限集合^Iと⁽向きのついた⁾辺の集合

であって^,辺の始点と終点を対応させる写像ôut:^!Î,ⁱⁿ^:^!Îが与えられているもののことである^. すなわち^,有限グラフの辺に向きを入れたものに他ならない^.

(図³参照⁾

h

in(h) out(h)

Figure 3. quiverの例このとき道^(path)とは^, 辺の有限列^(h

1

;h

2

;:::;h

N

)で^in(h

1

) = out(h

2 ), :::,

in(h

N01

)=out(h

N

)が成り立つもののことを言う^. ^Nを道の長さという^. さらに長さ^Nが⁰の道^,すなわちひとつの頂点^k²^Iだけからなるものも道であると約束する^.

Fを体とするとき^,道代数^(path^algebra) ^FQを道を基底とするベクトル空間に道の合成によって積を入れたものとする^. もしも二つの道が合成できないとき^,すなわちそれぞれの終点と始点とが一致しないときは^,積は⁰と定める^. 例えば下の図⁴ の定める道代数は^,一変数多項式環^F[x]に他ならない^. 一般には^,道代数は非可換の環になる^.

Figure4. F[x]に対応する^quiver

また^Iを^FQの両側イデアルとするとき^, 道代数を関係式^Iで割った環^FQ=Iもよく考える^. 例えば^,上の例で^m回まわるループ^h^mを⁰とする関係式を考えれば^, 対応する環は^F[x]=x^mに他ならない^.

quiverの表現とは^,道代数^FQの左加群のことを言う^. 言い直せば^,各頂点^k²^Iに対して^F-ベクトル空間^Vkが与えられ^,各辺^h²に対して線型写像^Bh

:V

out(h)

!

V

in(h)

が与えられているもののことである^. また^FQ=Iの表現は^, 上のような^Bhに

Iが定める条件を課したものに他ならない^.

したがって^, ^quiverの表現の同型類⁽ただし次元が同じものだけを考える⁾の集合

(すなわち^, 表現のモジュライ空間⁾は^,

M

h2 Hom(V

out(h)

;V

in(h) )

,

Y

k2I GL(V

k ) (A.1.1)

に他ならない^. ベクトル空間を^E (V),

Q

k 2I GL(V

k

)を^GV で表わす^.

表現のモジュライ空間^(A.1.1)は^,^Riemann面上の正則ベクトル束のモジュライ空間と類似点がある^. 実際後者は^,⁽無限次元の⁾アファイン空間を⁽やはり無限次元の^)Lie 群で割った空間になっている^.

(19)

relationの例であとの箙多様体と深く関係するものがあるので説明する^. ^Q^#を

Q=(I;)のダブルとする^. すなわち^,^Iは変えず^, の辺^hにその向きを逆にしたもの^hを付け加えたものである^. このとき^,^Q^#の辺 ^hに対し^,^"(h)を^h²のときに^1,^h²のときに⁰¹と定める^. そこで^,

X

in(h)=k

"(h)hh=0; (k2I)

で生成される両側イデアルを考え^Iとおく^. ^FQ^#^=Iがpreprojectivealgebraと呼ばれるものである^. その表現の同型類は^,

8

<

: (B

h )2E

[ (V)

X

in(h)=k

"(h)B

h B

h

=0 9

=

; ,

G

V (A.1.2)

で与えられる^. ^E

[

(V)=E

(V)8E

(V)はシンプレクティックベクトル空間⁷で^,

P

"(h)B

h B

h

は ^GV の作用に関するモーメント写像である^. シンプレクティック形式やモーメント写像については^, 詳しい一般論がこの論説で必要になるわけではないが^,あとに出てくる幾何学的不変式論や^hyper-K^ahler構造との関連も含めて^,^[19, 第³章^]を参照されればありがたい^. よって上の空間は^, もしも商空間が多様体の構造を持てばいわゆるシンプレクティック商と捉えることが出来る^. またシンプレクティック商の一般論によれば^,^(A.1.1)の^E

(V)=G

V が多様体になれば^,その余接束

T 3

(E

(V)=G

V

)と同型になる^. ところが^, ^GV のような代数群に関する商空間は^,一般には位相空間として^Hausdorにならない^. この困難を解消するためには幾何学的不変式論が必要で^,のちに述べる箙多様体もその様に定義される^. その様な技術的な相違点があるにせよ^, preprojectivealgebraの幾何学的な意味は^`余接束^'である^.

ALE空間⁽より正確にはその下部構造の複素曲面⁾や箙多様体が^(A.1.2)もしくはそのmo dicationとして定義される^. その意味で^,^(A.1.2)は有益なシンプレクティック多様体を構成するレシピと思うことが出来る^.

A.2. 箙多様体^(quiver^variety). 箙多様体の定義をする^. まず^(A.1.1)の^E (V)

とその双対空間の直和を考える^:

E

(V)8E

(V)

3

= M

h2 Hom(V

out(h)

;V

in(h)

)8Hom(V

in(h)

;V

out(h) )

これは^,Êのダブル^,すなわちÊの辺に逆向きの辺も付け加えてできた^quiver⁽これを^Q^#で表わす⁾に対応するベクトル空間と思える^. Êのダブルを^Hで表わす^. さらに^, 各頂点ごとにベクトル空間^Wkを置いて^,

M

h2H Hom(V

out(h)

;V

in(h) )

!

8 M

k2I

Hom(W

k

;V

k

)8Hom(V

k

;W

k )

!

(A.2.1)

というベクトル空間を考える^. このベクトル空間を^M(V;^W⁾と書こう^. ^M(V;^W⁾には自然に^G

V

= Q

k2I GL(V

k

)が作用する^.

さて^,^(A.2.1)の点の意味するところは何であろうか？第一の成分は^,^quiver^Hの表現であるから意味は明らかであろう^. 一方^,第二成分は^[17]で現れたものの^,^quiver の表現論で扱われたことはなかったようである^. 前節のように表現論との関連から^,第二成分の最高ウェイトベクトルとしての意味がはっきりしてくるのではあるが^,ここで^[17]での意味を説明しよう^. 箙多様体を^ALE空間上の正則ベクトル束のモジュライ空間として見るとき^,^(W

k )

k 2I

は^,^framing, すなわち無限遠でのベクトル束の自明

7この論説では,シンプレクティック形式はすべて複素数体上で考える.他の文献では,実のシンプレクティック形式と区別するために正則シンプレクティック形式と呼ぶ場合も多いので注意.

(20)

化に対応する^. これを^quiverの言葉に無理矢理に翻訳すると次のようになる^. ^quiver

Q

#から一切の辺を取り除いた^quiverを^Q⁰とする^. このとき^,^Q⁰の表現とは^,頂点の上にベクトル空間を置いたもの^,そしてベクトル空間の間の線型写像は一切考えないものに他ならない^. ^Q^#の表現^(V

k )

k 2I , (B

h )

h2H

に対し^,^B

h

を忘れると^, ^Q⁰の表現が与えられる^. これは^,表現の間の関手である^. ⁽より一般に^,^Q^#の部分グラフを考えれば^,同様の関手が定義される^.) この関手を制限と呼ぼう^. さて^,上の式で^,^(W

k )

k 2I

は^, ^Q⁰の表現と考えることができる^. すると

M

k 2I

Hom(W

k

;V

k )

は^, ^(Wk )

k 2Iから^(Vk )

k2i

;(B

h )

h2Iの制限への^quiver^Q⁰での準同型の全体である^.

M

k 2I Hom(V

k

;W

k )

はもちろん^,^(Vk )

k 2I

;(B

h )

h2Hの制限から^(Wk )

k 2iへの^quiver^Q⁰での準同型の全体である^.

前に述べたような^,^quiverの表現とベクトル束の類似のもとでは^,次の対応があることは納得が行くであろう^.

(1) 上の関手は^,ベクトル束を部分多様体へ制限することに対応する^.

(2) (A.2.1)の第二成分は^,部分多様体に与えられたベクトル束との間の準同型に

対応する^.

さて^,^(A.2.1)の各成分を^Bh ,i

k ,j

k と書こう^. このとき^, 次の写像を考える^.

:M! M

k 2I End(V

k ); (B

h

;i

k

;j

k )

h2H;k 2I 70!

0

@ X

h2H:in(h)=k

"(h)B

h B

h +i

k j

k 1

A

k2I :

M(V;W)は^,その^`半分^'

M

h2 Hom(V

out(h)

;V

in(h) )8

M

k2I

Hom(W

k

;V

k ) (A.2.2)

の余接束と見ることができ^, 特に自然なシンプレクティック形式を持つ^. 上のベクトル空間を^M

(V;W)と書こう^. ^GV の作用は^,^M(V;^W⁾のシンプレクティック形式を保つ^. 上の写像は^,運動量写像である^. このあたりは^,preprojectivealgebraの説明をしたときと同様である^.

さて^,シンプレクティック商

01

(0)=G

V

を考えたい^. もしも商空間^M

(V;W)=G

V が多様体であれば^, これはその余接束と同型である^. しかし^,集合論的な商空間は一般には代数多様体になることが期待できない^. 幾何学的不変式論^[18]の教えは^,商空間を考える代わりに商空間上の関数が何かを考えよ^,と言うことである^. 今の場合は^, 二つの候補がある^. まず第一のものは^,

01

(0)をアファイン代数多様体と思って^, 座標環の^G

V

不変な部分環に対応するアファイン代数多様体を考えるものである^. これは⁰¹⁽⁰⁾⁼^=G

V

と書かれるもので^,その点は⁰¹⁽⁰⁾の閉軌道に他ならない^.

もう一つは^,⁰¹⁽⁰⁾上の直線束⁽今の場合^,自明なものを取る⁾に^,^G

V

の作用を持ち上げ^,直線束のテンソル積の^GV 不変な切断のなす次数つき環に対応する準射影多様体を考えるものである^. 自明な直線束への作用の持ち上げを^, ^GL(Vk

)の行列式の積と取ると^,以下のように定義される^`安定^(stable)'な点の全体の商空間になる^.