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]).