Bases of representations of type A affine Lie algebras via q(3)

(v)=0.Now,v′∈Si′forsomei′∈IbythestabilityofSv′=xh′x′...xh′

1h2N 1(henceji′(v′)=0)andv′∈kerxi′→i′ 1∩kerxi′+1→i′byourchoiceofN.Thiscontradictsthefactthatxisanelementoftherighthandside.

n+1,wede nethevarietiesΛ( v,w),Λ( v,w)standL (v,w)byInthecaseofgl

Vintheabove.replacingΛVbyΛ

3.TheLieAlgebraAction

Wesummarizeheresomeresultsfrom[N1]thatwillbeneededinthesequel.Seethisreferenceformoredetails,includingproofs.WekeepthenotationofSections1and2(withgarbitrary).′′′Letw,v,v′,v′′∈ZI≥0besuchthatv=v+v.Considerthemaps

(3.0.1)p1p2p3 (v,w;v′′)→Λ(v′′,0)×Λ(v′,w)←FF(v,w;v′′)→Λ(v,w),

wherethenotationisasfollows.ApointofF(v,w;v′′)isapoint(x,j)∈Λ(v,w)togetherwithanI-graded,x-stablesubspaceSofVsuchthatdimS=v′=v v′′. (v,w;v′′)isapoint(x,j,S)ofF(v,w;v′′)togetherwithacollectionApointofF

′′′′～′′～:ViofisomorphismsRi:Vi=Vi/Siforeachi∈I.Thenwede ne=SiandRi

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a

AFFINELIEALGEBRAS,QUIVERVARIETIESANDSTATISTICALMECHANICS7p2(x,j,S,R′,R′′)=(x,j,S),p3(x,j,S)=(x,j)andp1(x,j,S,R′,R′′)=(x′′,x′,j′)wherex′′,x′,j′aredeterminedby

′′′′Rin(h)xh=xhRout(h):Vout(h)→Sin(h),

′′′:Vi→Wiji=jiRi

′′′′′′′′Rin(h)xh=xhRout(h):Vout(h)→Vin(h)/Sin(h).

Itfollowsthatx′andx′′arenilpotent.

Lemma3.0.1([N1,Lemma10.3]).Onehas 1(p3 p2) 1(Λ(v,w)st) p1(Λ(v′′,0)×Λ(v′,w)st).

Thus,wecanrestrict(3.0.1)toΛst,forgettheΛ(v′′,0)-factorandconsiderthequotientbyGV,GV′.Thisyieldsthediagram

(3.0.2)

where

F(v,w,v v′)={(x,j,S)∈F(v,w;v v′)|(x,j)∈Λ(v,w)st}/GV.

LetM(L(v,w))bethevectorspaceofallconstructiblefunctionsonL(v,w).ForasubvarietyYofavarietyA,let1YdenotethefunctiononAwhichtakesthevalue1onYand0elsewhere.Letχ(Y)denotetheEulercharacteristicofthealgebraicvarietyY.ThenforamapπbetweenalgebraicvarietiesAandB,letπ!denotethemapbetweentheabeliangroupsofconstructiblefunctionsonAandBgivenby

π!(1Y)(y)=χ(π 1(y)∩Y),Y A

andletπ bethepullbackmapfromfunctionsonBtofunctionsonAactingasπ f(y)=f(π(y)).Thende ne

Hi:M(L(v,w))→M(L(v,w));Hif=uif,

Eif=(π1)!(π2f),

Fig=(π2)!(π1g).def21L(v,w),F(v,w;v v′)→L(v′,w)←ππEi:M(L(v,w))→M(L(v ei,w));Fi:M(L(v ei,w))→M(L(v,w));

Here

u=t(u0,...,un)=w Cv

whereCistheCartanmatrixofgandweareusingdiagram(3.0.2)withv′=v eiwhereeiisthevectorwhosecomponentsaregivenbyeii′=δii′.

Nowlet betheconstantfunctiononL(0,w)withvalue1.LetL(w)bethevectorspaceoffunctionsgeneratedbyactingon withallpossiblecombinationsoftheoperatorsFi.ThenletL(v,w)=M(L(v,w))∩L(w).

Proposition3.0.2([N1,Thm10.14]).TheoperatorsEi,Fi,HionL(w)providethestructureoftheirreduciblehighestweightintegrablerepresentationofgwith highestweightw.EachsummandofthedecompositionL(w)=vL(v,w)isaweightspacewithweightw Cv.

LetX∈IrrL(v,w)andde nealinearmapTX:L(v,w)→Casin[L2,3.8].ThemapTXassociatestoaconstructiblefunctionf∈L(v,w)the(constant)valueoffonasuitableopendensesubsetofX.ThefactthatL(v,w)is nite-dimensionalallowsustotakesuchanopensetonwhichanyf∈L(v,w)isconstant.Sowehavealinearmap

Φ:L(v,w)→CIrrL(v,w).

Thefollowingpropositionisprovedin[L2,4.16](slightlygeneralizedin[N1,Propo-sition10.15]).

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a

8IGORB.FRENKELANDALISTAIRSAVAGE

Proposition3.0.3.ThemapΦisanisomorphism;foranyX∈IrrL(v,w),thereisauniquefunctiongX∈L(v,w)suchthatforsomeopendensesubsetOofXwehavegX|O=1andforsomeclosedGV-invariantsubsetK L(v,w)ofdimension<dimL(v,w)wehavegX=0outsideX∪K.ThefunctionsgXforX∈IrrΛ(v,w)formabasisofL(v,w).

4.LevelOneRepresentations

WenowseektodescribetheirreduciblecomponentsofNakajima’squivervari-ety.BythecommentmadeinSection2,itsu cestodeterminewhichirreduciblecomponentsofΛ(v,w)arenotkilledbythestabilitycondition.ByDe nition2.0.1andLemma2.0.4,thesearepreciselythoseirreduciblecomponentswhichcontainpointsxsuchthat

(4.0.3)dim(kerxi→i 1∩kerxi+1←i)≤wi i.

We rstconsiderthebasicrepresentationofhighestweightΛ0whereΛ0(αi)=δ0i.Thiscorrespondstow=w0,thevectorwithzerocomponent1andallothercomponentsequaltozero.

4.1.TypeA∞.ConsiderthecasewheregisoftypeA∞.LetYbethesetofallYoungdiagrams,thatis,thesetofallweaklydecreasingsequences[l1,...,ls]ofnon-negativeintegers(lj=0forj>s).ForY=[l1,...,ls]∈Y,letAYbetheset{(1 i,li i)|1≤i≤s}.

Theorem4.1.1.TheirreduciblecomponentsofL(v,w0)arepreciselythoseXf ∞suchthatwheref∈ZV

{(k′,k)|f(k′,k)=1}=AY

forsomeY∈Yandf(k′,k)=0for(k′,k)∈AY.Denotethecomponentcor-respondingtosuchanfbyXY.Thus,Y XYisanatural1-1correspondencebetweenthesetYandtheirreduciblecomponentsof∪vL(v,w0).

′′′′Proof.Considerthetworepresentations(V∞(k1,k1),x∞(k1,k1))and(V∞(k2,k2),x∞(k2,k2))′ofourorientedgraphasdescribedinSection1wherethebasisofV∞(ki,ki)is′{eir|ki≤r≤ki}.LetWbetheconormalbundletotheGV-orbitthroughthe

point

′′′,k)⊕V(k′,k), .x =(xh)h∈ =x∞(k1,k1)⊕x∞(k2,k2)∈EV∞(k11∞22

ByProposition1.2.3,x=(x ,x ¯)=(xh)h∈HisinWifandonlyif

xi+1→ixi+1←i=xi←i 1xi→i 1

foralli.1′2Leteir=0forr<kiorr>ki.Now,xr+1←r(er)=crer+1forsomecr∈Csince

2′xr+1←r(e2r)canhavenoer+1-componentbynilpotency.Supposethatk1≤r+1≤

′k1andcr=0(thatis,xr+1←r(e2r)=0).Thenifr+1>k1,

221xr←r 1(e2r 1)=xr←r 1xr→r 1(er)=xr+1→rxr+1←r(er)=crer=0.

′Inparticular,e2r 1=0andsor 1≥k2.Continuinginthismanner,weseethat

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