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
6IGORB.FRENKELANDALISTAIRSAVAGE
Now,supposethatSisanI-gradedsubspaceofV.Forx∈ΛVwesaythatSisx-stableifx(S) S.
De nition2.0.2([N1]).LetΛst=Λ(v,w)stbethesetofall(x,j)∈Λ(v,w)satisfyingthefollowingcondition:IfS=(Si)withSi Viisx-stableandji(Si)=0fori∈I,thenSi=0fori∈I.
ThegroupGVactsonΛ(v,w)via
1 1(g,(x,j))=((gi),((xh),(ji)))→((gin(h)xhgout(h)),jigi).
andthestabilizerofanypointofΛ(v,w)stinGVistrivial(see[N3,Lemma3.10]).Wethenmakethefollowingde nition.
De nition2.0.3([N1]).LetL≡L(v,w)=Λ(v,w)st/GV.
LetIrrL(v,w)(resp.IrrΛ(v,w))bethesetofirreduciblecomponentsofL(v,w)(resp.Λ(v,w)).ThenIrrL(v,w)canbeidenti edwith
{Y∈IrrΛ(v,w)|Y∩Λ(v,w)st= }.
Speci cally,theirreduciblecomponentsofIrrL(v,w)arepreciselythose def¯f×Xf=CHom(Vi,Wi)∩Λ(v,w)st/GV
i∈I
whicharenonempty.
Thefollowingwillbeusedinthesequel.
Lemma2.0.4.Onehas
Λst={x∈Λ|kerxi→i 1∩kerxi+1←i∩kerji=0 i}
.
Proof.Sinceeachkerxi→i 1∩kerxi+1←iisx-stable,thelefthandsideisobviouslycontainedintherighthandside.Nowsupposexisanelementoftherighthandside.LetS=(Si)withSi Vibex-stableandji(Si)=0fori∈I.AssumethatS=0.SinceallelementsofΛarenilpotent,wecan ndaminimalvalueofNsuchthattheconditioninDe nition1.0.1issatis ed.Thenwecan nd′′av∈Siforsomeiandasequenceh′1,h2,...,hN 1(emptyifN=1)inH′′′′′suchthatout(h′1)=in(h2),out(h2)=in(h3),...,out(hN 2)=in(hN 1)and
(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