Table Of ContentWEYL, DEMAZURE AND FUSION MODULES
FOR THE CURRENT ALGEBRA OF sl
7 r+1
0
0
2 VYJAYANTHICHARIANDSERGEILOKTEV
n
a Abstract. We construct a Poincare-Birkhoff–Witt type basis for the Weyl modules [CP1] of the
J currentalgebraofslr+1. Asacorollaryweprovetheconjecturemadein[CP1],[CP2]onthedimension
3 oftheWeylmodulesinthiscase. Further,werelatetheWeylmodulestothefusionmodulesdefinedin
1 [FL]ofthecurrentalgebraandtheDemazuremodulesinlevelonerepresentationsofthecorresponding
affinealgebra. Inparticular,thisallowsustoestablishsubstantialcasesoftheconjecturesin[FL]on
] thestructureandgradedcharacter ofthefusionmodules.
A
Q
.
h
t Introduction
a
m
Thestudyoffinite–dimensionalrepresentationsofquantumaffinealgebrashasbeenofsomeinterest
[ inrecentyearsandthere areseveraldifferentapproachesto the subject, [CP1], [FR],[FM],[N],[Ka1]for
instance.
2
v An approach that was developed in [CP1] was to study these representations by specializing to the
5 case of classical affine algebras. It was noticed in that paper, that an irreducible finite–dimensional
6 representation of a quantum affine algebra, in general specialized to an indecomposable but reducible
1
representationoftheaffineLiealgebra. ThisbehaviorisseenintherepresentationtheoryofsimpleLie
2
algebras,whenpassingfromthecharacteristiczerosituationtothecaseofnon–zerocharacteristic. The
0
5 modules in non–zerocharacteristicwhichare obtainedfromthe irreducible finite–dimensionalmodules
0 of the simple Lie algebra are called Weyl modules and are given by the same generators and relations
/ as the modules in characteristic zero. This analogy motivated the definition of the Weyl modules in
h
t [CP1], in terms of generators and relations for both the classical and quantum affine Lie algebras. It
a
was also conjectured there, that the Weyl modules of the affine algebra are the classical (q →1) limit
m
of the standard modules of the quantum affine algebra defined in [N]. In particular, a proof of the
:
v conjecture would give generators and relations for these representations, or equivalently, would prove
i thatthestandardmodulesareisomorphictothequantumWeylmodules. Thislatterresultwasproved
X
recently in [CdM] using some deep geometric results of Nakajima.
r
a However, the results of [CP1] and [CP2] showed that the conjecture on the isomorphism between
the quantum Weyl modules and the standard modules would also follow, if a further conjecture on
the dimension ofthe classicalWeyl modules could be proved. This latter conjecture was establishedin
[CP1]forsl . Moreover,itwasshownthatitwasenoughtostudytheanalogousmodulesforthecurrent
2
algebra of a simple Lie algebra, namely the natural parabolic subalgebra of the affine Lie algebra.
In this paper we show that the conjecture on the dimension of the classical Weyl modules is true
for sl , by constructing an explicit basis for the Weyl modules over the current algebra of sl .
r+1 r+1
We use the basis to give a graded fermionic type character formula for the Weyl modules (see also
[HKOTT]). We then make connections with several other interesting problems in the representation
theoryofaffineandcurrentalgebras. Thus,we areabletoestablishasubstantialcaseofconjecturesin
[FL]onthefusionmodulesforacurrentLiealgebras. Thedefinitionofthesemoduleswasmotivatedby
studyingthe spaceofconformalblocks foraffine Lie algebras. The fusionmodules aregivenby a setof
representationsofthesimpleLiealgebraandasetofcomplexpoints,oneforeachrepresentation. Itwas
1
2 VYJAYANTHICHARIANDSERGEILOKTEV
conjecturedthat inthe caseofa simple Lie algebraandwherethe representationswereirreducible and
finite–dimensional the fusion modules are independent of these points. This conjecture is established
in this paper for the fusion of fundamental representation of sl by showing that the fusion module
r+1
is isomorphic to a Weyl modules.
Another well–studied family of modules are the Demazure modules. These modules for the current
algebraareobtainedbytakingthecurrentalgebramodulegeneratedbytheextremalvectorsinmodules
of positive level of affine Lie algebras. The dimension (and, actually, character) of these modules was
computed in [KMOTU], [M]. We use these results to prove that the Weyl modules are isomorphic to
the Demazure modules in the level one representations of the affine Lie algebra of sl . Further, we
r+1
prove that the graded character of the fusion modules can be written in terms of Kostka polynomials
as conjectured in [FL]. Moreover,since the Weyl modules for the current algebracan be regardedas a
pull–backofWeylmodulesfortheaffineLiealgebra,itfollowsthatthesl –structureoftheDemazure
r+1
modules can be extended to a structure of affine Lie algebra modules. This is related to a conjecture
in [FoL].
For an arbitrary simple Lie algebra, the conjecture appears to be harder to establish. This is
not surprising, since the representation theory of the corresponding quantum affine algebra is much
more complicated. In particular the Weyl module associated with a fundamental weight is no longer
irreducibleasamoduleforthesimpleLiealgebra. Butweareconvincedthattheconjectureof[CP1]is
trueforthese modulesandcanbeestablishedina similarway. Notehowever,thatsince the dimension
of fusion modules is known from the definition the isomorphism between the Weyl modules, and the
corresponding fusion modules would follow as in the case of sl once the conjecture on dimension of
r+1
the Weyl modules is proved.
The paper is organizedas follows. In Section 1, we define the Weyl modules and recall some results
from [CP1]. We also recall the definition of the fusion modules from [FL] and the Demazure modules
and prove that these modules are quotients of Weyl modules. We then state the main theorem on the
dimension of the Weyl modules and show that it gives a sufficient condition for an isomorphism to
existbetweenthe Weyl modules, the fusion modules andthe Demazure modules. InSection 2 we start
provingthe conjecture by identifying a basis for the Weyl module. The proofthat it is actually a basis
involves constructing a filtration (which we call a Gelfand–Tsetlin filtration) for the Weyl modules,
studying the associated graded spaces and using an induction on r. Section 3, is devoted to proving
that the associated graded space is actually isomorphic to a sum of Weyl modules for sl . This latter
r
result is the motivation for calling it a Gelfand–Tsetlin filtration. We provide an index of notation at
the end of the paper.
Acknowledgments. We are grateful to B. Feigin, P. Littelmann, T. Miwa, E. Mukhin, and
M. Okado for useful discussions. SL is partially supported by the Grants CRDF RM1-2545-MO-03,
PICS 2094, RFBR - 04-01-00702,RF President Grants MK-3419.2005.1and N.Sh-8004.2006.2. VC is
partially supported by the NSF grant DMS-0500751.
Added in Proof: There has been progress on the various conjectures since our paper was posted
on the web as math.QA/0502165. Thus, in math.RT/0509276 the authors prove that the Demazure
modules, the Weyl modules and the fusion modules are all isomorphic for simply–laced Lie algebras
extending our results for sl . In particular, this establishes the conjecture on the dimension of the
r+1
Weyl modules in [CP1] for the simply laced algebras. For the nonsimply–laced case, they give an
example to show that the Demazure modules are smaller than the Weyl modules and formulate a
natural conjecture for the Demazure modules. They have also have a conjecture for the Weyl modules
whichwould be animmediate consequence of the conjecture in [CP1]. The methods they use are quite
different from ours and in particular they do not have an analog of the fermionic character formula
that we give in Section 2 of this paper.
WEYL, DEMAZURE AND FUSION MODULES FOR THE CURRENT ALGEBRA OF slr+1 3
1. The main theorem and it s applications
1.1. Preliminaries. Throughout the paper we restrict ourselves to the case of the Lie algebra of
(r+1)×(r+1)trace zeromatrices. However,we prefer to use the moregeneralnotationofsimple Lie
algebras since we expect the results to hold in that generality.
1.1.1. LetZdenote the setofintegers,N the setofnon–negativeintegersandN the setofpositive
+
integers. Let g =sl be the Lie algebra of (r+1)×(r+1)–matrices of trace zero, h be the Cartan
r+1
subalgebraofgconsistingofdiagonalmatricesandα ,1≤i≤r,asetofsimplerootsforgwithrespect
i
toh. For1≤i≤j ≤r, letα =(α +···+α )andletx+ (resp. x− )be the (r+1)×(r+1)–matrix
i,j i j i,j i,j
with 1 in the (i,j+1)th (resp. (j+1,i)th)–position and 0 elsewhere. Define subalgebras n± of g by
n± = Cx± .
i,j
1≤i≤j≤r
M
For1≤i≤r+1letH bethediagonalmatrixwith1intheith placeandzeroelsewhere. Theelements
i
h =H −H ,1≤i≤r areabasisofh. Let{ω :1≤i≤r}bethethesetoffundamentalweightsof
i i i+1 i
g, namely the basis of h∗ which is dual to {α :1≤i≤r}. Let P = r Zω , (resp. Q= r Zα )
i i=1 i i=1 i
be the weightlattice (resp. root lattice) of g and set P+ = r Nω , (resp. Q+ = r Nα ). Given
i=1 iP i=1 Pi
a Lie algebra a, we let U(a) denote the universal enveloping algebra a.
P P
1.1.2. LetZ[P]be the integralgroupringofP withbasise(µ),µ∈P. IfV isanyfinite–dimensional
g–module with
V = V , V ={v ∈V :hv =µ(h)v, h∈h},
µ µ
µ∈P
M
let ch (V)= dim(V )e(µ)∈Z[P] be the character of V.
g µ∈P µ
Given λ = r m ω ∈ P+, let V(λ) denote the irreducible finite–dimensional g–module with
P i=1 i i
highest weight λ and highest weight vector v . More precisely, V(λ)=U(g)v with defining relations:
λ λ
P
(1.1) x+i,jvλ =0, (h−λ(h))vλ =0, (x−i,i)mi+1vλ =0,
for all 1≤i≤j ≤r and h∈h.
1.1.3. Let C[t] denote the polynomial ring in an indeterminate t and for any Lie algebra a, set
a[t]=a⊗C[t]. Clearly,a[t]isaLiealgebrawiththeLiebracketbeinggivenby[x⊗f,y⊗g]=[x,y]⊗fg
forallx,y ∈aandf,g ∈C[t]. We regardaasasubalgebraofa[t]bymappingx→x⊗1forx∈a. We
denote the maximal ideal of a[t] generated by elements of the form a⊗tn, a ∈ a, n > 0 by at[t]. The
Lie algebra a[t] is a N–gradedLie algebra,the grading being given by powers of t and hence U(a[t]) is
a N–gradedalgebra.
Given a N–graded g[t]–module M = M . Observe that M , s ∈ N are g–submodules of M.
s∈N s s
In the case when
L
(1.2) M = M , M ={v ∈M :hv =µ(h)v, h∈h},
µ,s µ,s s
(µ,sM)∈P×N
recall, that the graded character of M to be the element of Z[P][t] is defined as,
ch (M)= dim(M )tse(µ)= ch (M )ts.
t µ,s g s
(µ,sX)∈P×N sX∈N
1.2. Weyl modules.
4 VYJAYANTHICHARIANDSERGEILOKTEV
1.2.1. We now recall the definition of the Weyl modules.
Definition. Given λ = r m ω ∈ P+, the Weyl module W(λ) is the g[t]–module generated by an
i=1 i i
element w with defining relations:
λ
P
(1.3) n+[t]wλ =0, ht[t]wλ =0, hwλ =λ(h)wλ, (x−i,i⊗1)mi+1wλ =0,
for all h∈h, 1≤i≤r. (cid:3)
Remark. The modules W(λ) were defined and initially studied in [CP1] for the affine Lie algebras.
All the results of that paper go over with no difficulty to the case of the current algebras, in fact
an inspection of the proofs given there show that the results for the affine algebras were obtained by
considering the action of the current algebras g⊗C[t]. (See Subsection 1.5.3 for further details.) The
notationused in this paper is different from that in [CP1], [CP2]. In those papers, the modules for the
affine Lie algebra were denoted as W(π) where π is an r–tuple of polynomials in an indeterminate u
with constant term one. For λ= r m ω ∈P+ and a∈C×, let
i=1 i i
πPλ =((1−au)m1,··· ,(1−au)mr).
The module W(λ) is obtained from W(π ) by pulling back W(π ) by the Lie algebra homomorphism
λ λ
g[t]→g[t] which maps x⊗ts →x⊗(t−a)s for all x∈g, s≥0.
1.2.2. The following lemma is an elementary consequence of the fact that the relations which define
the Weyl modules are graded.
Lemma.
(i) The modules W(λ) admit a N–grading induced by the grading on g[t],
W(λ)= W(λ) .
s
s∈N
M
(ii) For all s≥0, the subspaces W(λ) are finite–dimensional g–submodules, and we have,
s
W(λ)= W(λ) ,
µ,s
(µ,sM)∈P×N
where W(λ) ={w∈W(λ) :hw =µ(h)w ∀ h∈h.}.
µ,s s
(cid:3)
For µ∈P we set
W(λ) = W(λ) ={w∈W(λ):hw =µ(h)w ∀ h∈h.}.
µ µ,s
s∈N
M
Theorem. [CP1] For all λ ∈ P+, the modules W(λ) are finite–dimensional. Moreover, any finite–
dimensional g[t]–module V generated by an element v ∈V satisfying the relations
(1.4) n+[t]v =0, ht[t]v =0, hv =λ(h)v,
is a quotient of W(λ). (cid:3)
1.3. Fusion modules.
1.3.1. The definition of the fusion product of g[t]–modules was given in [FL] and we recall the defi-
nition in the case of interest to this paper. Given a ∈ C, let ev : g[t] → g be the homomorphism of
a
Lie algebraswhichmaps x⊗ts to asx, x∈g, s∈N. LetV (λ) be the g[t]–module obtainedby pulling
a
back V(λ) through ev . The following proposition is well–known, [CP], [FL] for instance.
a
Proposition. Let k ∈ N , λ ∈ P+, a ∈ C, 1 ≤ s ≤ k and assume that a 6= a if s 6= s′. Then,
+ s s s s′
V (λ )⊗···⊗V (λ ) is an irreducible g[t]–module. (cid:3)
a1 1 ak k
WEYL, DEMAZURE AND FUSION MODULES FOR THE CURRENT ALGEBRA OF slr+1 5
1.3.2. Assume that the hypotheses of Proposition1.3.1are satisfied and set
Va(λ)=Va1(λ1)⊗···⊗Vak(λk), v=vλ1 ⊗···⊗vλk.
ThemoduleVa(λ)isquiteclearlynotaN–gradedg[t]–module. However,theN–gradingong[t]induces
ag–equivariantfiltrationonVa(λ) asfollows: letVan(λ)be the subspaceofVa(λ)spannedby elements
of the form gv, where g ∈U(g[t]) has grade at most n.
Set Va−1(λ)=0, and set
V (λ )∗···∗V (λ )= Vn(λ),/Vn−1(λ).
a1 1 ak k a a
n∈N
M
For v′ ∈Va(λ) let v′ denote its image in Va1(λ1)∗···∗Vak(λk).
Lemma. The following formula defines a N–graded g[t]–module structure on V (λ )∗···∗V (λ ):
a1 1 ak k
(x⊗ts)v′ =(x⊗ts)v′,
for all s∈N, x∈g, v′ ∈V (λ )∗···∗V (λ ).
a1 1 ak k
Proof. It suffices to prove that the action is well–defined. But this follows, since (x ⊗ts)Van(λ) ⊂
Vn+s(λ). (cid:3)
a
Remark. The resulting g[t]–module is called the fusion product of the modules V (λ ), 1≤i≤k.
ai i
1.3.3. The next lemma is an immediate consequence of Proposition 1.3.1.
Lemma. We have,
V (λ )∗···∗V (λ )=U(g[t])v.
a1 1 ak k
(cid:3)
In this paper we prove a significant case of the following conjecture.
Conjecture. [FL] Let k ∈ N , λ ∈ P+, 1 ≤ s ≤ k. Assume that a ,b ∈ C, 1 ≤ s ≤ k and that
+ s s s
a 6=a , b 6=b if 1≤s6=s′ ≤k. Then,
s s′ s s′
Va1(λ1)∗···∗Vak(λk)∼=Vb1(λ1)∗···∗Vbk(λk)
as g[t]–modules.
(cid:3)
1.3.4. We now prove,
Proposition. Suppose that k ∈ N , λ ∈ P+, 1 ≤ i ≤ k and that a ,··· ,a ∈ C are distinct. The
+ i 1 k
fusion product V (λ )∗···∗V (λ ) is a quotient of W(λ +···+λ ).
a1 1 ak k 1 k
k
Proof. ItsufficesinviewofLemma1.3.3toshowthatvsatisfiestherelationsin(1.4). Setλ= λ .
i=1 i
Since, (x+ ⊗ts)v = 0 for all 1 ≤ i ≤ j ≤ r and all s ≥ 0, it follows that (x+ ⊗ts)v = 0. Further,
i,j i,j P
hv=λ(h)v. Finally, we have
k
(h⊗ts+1)v= λ (h)as+1 v,
j j
j=1
X
which implies that (h ⊗ts+1)v=0, for all s≥0, thus proving the proposition.
i
(cid:3)
6 VYJAYANTHICHARIANDSERGEILOKTEV
r
Corollary. For λ= m ω we have
i=1 i i
P r r+1 mi
dimW(λ)≥ .
i
i=1(cid:18) (cid:19)
Y
r
Proof. The proofis immediate fromProposition1.3.4by taking k = m , λ =ω , 1≤s≤k and
i=1 i s is
we assume that each ω occurs m times. (cid:3)
i i
P
The corollary was proved originally in [CP1] by passing to the quantum case and the proof given
there was much more complicated.
1.4. Demazure modules.
1.4.1. We now show that the Demazure modules in the highest weight integrable representations
of the affine algebra associated to sl are quotients of Weyl modules. We begin by recalling the
r+1
definition of the affine Kac–Moody algebra. Thus, let C[t,t−1] be the ring of Laurent polynomials in
an indeterminate t. Let g be the affine Lie algebra defined by
g=g⊗C[t,t−1]⊕Cc⊕Cd,
b
where c is central and
b
[x⊗ts,y⊗tk]=[x,y]⊗ts+k+sδ hx,yic, [d,x⊗ts]=sx⊗ts,
s+k,0
for all x,y ∈g, s,k ∈Z and where h , i is the Killing form of g. Set
h=h⊕Cc⊕Cd,
and regard h∗ as a subspace of h∗ by setting λ(c)=λ(d)=0 for λ∈h∗.
b
For 0≤i≤r, define elements Λ ∈h∗, by
i
b
Λ (h )=δ , Λ (d)=0, Λ (c)=1, 1≤j ≤r,
i j i,j i i
b
and let P+ ⊂ h∗ be the non–negative integer linear span of Λ , 0 ≤ i ≤ r. Let δ ∈ (h)∗ be defined by
i
setting
b b δ|h =0, δ(d)=1, δ(c)=0. b
The simple roots for g with respect to h are α , 0≤i≤r, where α =δ−(α +···+α ). Let Q+ be
i 0 1 r
the subset of P+ spanned by the elements α , 0 ≤ i ≤ r. Let W be the affine Weyl group and (,) be
i
the W–invariant formbon h∗ obtained bby requiring (Λ ,α )=δ for all 0≤i,j ≤r and (δ,α )=b0 for
i j ij i
all 1≤i≤r. Tbhe following Lemma is well–known, see [FoL] focr instance.
b
Lemma. Let µ∈P+. There exists 0≤i ≤r and w(µ)∈W such that w(µ)Λ | =µ. (cid:3)
µ iµ h
1.4.2. Given Λ = r m Λ ∈ P+, let L(Λ) be the g–module generated by an element v with
i=0 i i c Λ
relations:
P gt[t]vΛ =0,b (n+⊗1)vΛ =0, b(h⊗1)vΛ =Λ(h)vΛ,
(x− ⊗1)mi+1v =0, (x+ ⊗t−1)m0+1v =0,
i,i Λ 1,r Λ
for 1 ≤ i ≤ r. This module is known to be irreducible and integrable (see [K]). The next proposition
also can be found in [K].
Proposition. Let Λ∈P+.
(i) We have
b
L(Λ)= L(Λ)Λ′, where L(Λ)Λ′ ={v ∈L(Λ):hv =Λ′(h)v, h∈h}.
ΛM′∈Pb
b
Moreover, dim(L(Λ)Λ′)<∞, dim(L(Λ))Λ =1, and L(Λ)Λ′ =0 if Λ−Λ′ ∈/ Q+.
b
WEYL, DEMAZURE AND FUSION MODULES FOR THE CURRENT ALGEBRA OF slr+1 7
(ii) The set
wt(L(Λ))={Λ′ ∈P :L(Λ) 6=0}⊂Λ−Q+.
Λ′
is preserved by W and
b b
dim(L(Λ) )=dim(L(Λ) ) ∀ w ∈W, Λ′ ∈P.
c wΛ′ Λ′
In particular, dim(L(Λ) )=1 for all w ∈W.
wΛ c b
(cid:3)
c
1.4.3. Given w ∈W, let v be a non–zero element in L(Λ) and let
wΛ wΛ
D(wΛ)=U(g[t])v ⊂L(Λ).
c wΛ
The modules D(wΛ), w ∈ W are called the Demazure modules. Note that if w,w′ ∈ W are such
that w−1w′ ∈ W ⊂ W then D(wΛ) = D(w′Λ). It is clear from Proposition 1.4.2 that D(wΛ) is
finite–dimensional for all w ∈cW. We can now prove, c
c
Proposition. Let Λ ∈ P+ and w ∈ W be such that wΛ| ∈ P+. Then, D(wΛ) is a quotient of
c h
W(wΛ| ).
h
b c
Proof. To prove the proposition it suffices to show that v satisfies (1.4). Suppose that
wΛ
(x+ ⊗ts)v 6=0,
i,j wΛ
for some 1≤i≤j ≤r and s∈N. It follows that wΛ+α +sδ ∈wt(L(Λ)). By Proposition1.4.2 (ii),
ij
we get Λ+w−1(α )+sδ ∈ wt(L(Λ)) and so −w−1α −sδ ∈ Q+. This implies in particular, that
i,j i,j
−w−1α ∈Q+, and we get
i,j
b
(wΛ,α )=(Λ,w−1α )≤(Λ,w−1α +sδ)≤0
b i,j i,j i,j
which contradicts wΛ| ∈P+ unless (Λ,w−1α )=(Λ,w−1α +sδ)=0. In this case, it follows that
h i,j i,j
the reflection s ∈W fixes Λ, and so we get
w−1αi,j+sδ
Λ−(w−1α +sδ)=s (Λ+w−1α +sδ)∈wt(L(Λ)),
ci,j w−1αi,j+sδ i,j
which implies w−1α +sδ ∈Q+, contradicting −w−1α −sδ ∈Q+. Hence
i,j i,j
(x+ ⊗ts)v =0, ∀ 1≤i≤j ≤r, s∈N.
bi,j wΛ b
If(h⊗ts)v 6=0,thenbyasimilarargument,oneseesthatΛ+sδ ∈wt(L(Λ))whichisimpossibleif
wΛ
s>0since δ ∈Q+. Since D(wΛ)is afinite–dimensionalg[t]–modulethe propositionisnowimmediate
from (1.4). (cid:3)
b
1.4.4. The following is a special case of a result that can be found in [KMOTU],[M],[FoL].
r
Theorem. Assume that Λ=Λ , 0≤i ≤r and that w ∈W is such that wΛ| = m ω , where
i0 0 h i=1 i i
m ∈N. Then, we have an isomorphism of g–modules,
i
P
c
D(wΛi0)∼=V(ω1)m1 ⊗···⊗V(ωk)mk.
(cid:3)
1.5. Main theorem.
8 VYJAYANTHICHARIANDSERGEILOKTEV
1.5.1. Thenextresultwhichwasconjecturedin[CP1]andmadeprecisein[CP2]isthemaintheorem
of this paper.
Theorem. Let λ= r m ω ∈P+. We have,
i=1 i i
P r r+1 mi
(1.5) dim(W(λ))= .
i
i=1(cid:18) (cid:19)
Y
WeproveTheorem1.5.1inthefollowingsections. Weconcludethissectionbynotingsomecorollaries
of the theorem. Notice that the next corollaryestablishes the Conjecture 1.3.3 for a substantialfamily
of modules.
Corollary. Given i ,··· ,i ∈ {1,··· ,r}, set λ = k ω . Let w(λ) ∈ W and 0 ≤ i ≤ r be as in
1 k s=1 is λ
Lemma 1.4.1, so that w(λ)Λ | =λ. Let a ,...,a ∈C be such that a 6=a , 1≤i6=j ≤k. Then we
iλ h 1 kP i j
have an isomorphism of g[t]–modules c
Va1(ωi1)∗···∗Vak(ωik)∼=W(λ)∼=D(w(λ)Λiλ).
Proof. The corollary is immediate from the Theorem 1.5.1 together with Proposition 1.3.4, Proposi-
tion 1.4.3 and Theorem 1.4.4. (cid:3)
Remark. To prove the conjecture of [FL] for arbitrary irreducible representations by identifying it
with a g[t]-module which is obviously independent of the points, would involve identifying a suitable
quotient of the Weyl modules. A possible approachis proposed in [FKL].
r
1.5.2. Forapartitionξ =(ξ ≥ξ ≥···≥ξ )ofnon–negativeintegers,setλ = (ξ −ξ )ω .
1 2 r+1 ξ i=1 i i+1 i
Given a partition ξ and regarding µ = r n ω ∈ P+ as the vector (n ,...,n ), let K (t) be the
i=1 i i 1 r P µ,ξ
corresponding Kostka polynomial (see [Mac]). Note that K (t)=0 unless ξ =0.
µ,ξ r+2
P
Corollary. Retain the notation of Corollary 1.5.1. Then,
ch (V(ω ) ∗···∗V(ω ) )=ch W(λ)= K (t)·ch V(λ ),
t i1 a1 ik ak t λ,ξtr g ξ
ξ
X
where the sum is over all partitions ξ satisfying ξ =0, and ξtr is the transposed partition.
r+2
Proof. The second equality follows by first using the identification between the Weyl modules and
Demazure modules given in Corollary 1.5.1 and then using Theorem 5.2 in [KMOTU] which gives the
formula for ch (D(wΛ)) . The first equality follows from the isomorphism between the Weyl modules
t
and the fusion product. (cid:3)
Remark. This corollary establishes a particular case of a conjecture stated in [FL]. In [Ke] another
case of the conjecture was proved, namely it was shown that the graded character of module
V(n ω )∗···∗V(n ω ), n ∈N, 1≤j ≤k,
1 1 k 1 j
is given by a similar formula based on Kostka polynomials.
In the next section we shall give a fermionic character formula for these modules.
1.5.3. We conclude this section by showing that Proposition 1.4.3 and Theorem 1.5.1 are related to
Conjecture 1 in [FoL]. Namely, we shall prove that the Demazure module is isomorphic as a g–module
to a cyclic finite–dimensionalmodule for g. For a∈C×, λ∈P+, define the module W (λ) for g as the
a
module with generator w and relations
a,λ
(1.6) (n+⊗C[t,t−1])wa,λ =0, (hb⊗ts)wa,λ =asλ(h)wa,λ, (x−i,i⊗1)mi+1wa,λ =0, b
for all h∈h, 1≤i≤r, s∈Z. It was proved in [CP1] that as g[t]–modules,
Wa(λ)∼=U(g[t])wa,λ,
WEYL, DEMAZURE AND FUSION MODULES FOR THE CURRENT ALGEBRA OF slr+1 9
and that
dimW (λ)=dimW (λ), ∀ a,b∈C×.
a b
In fact, it was proved there, that the action of x⊗t−s with s∈N on W (λ) was a linear combination
a
of elements from U(g[t]).
Moreoverif we pull back W (λ) by the automorphismτ :g[t]→g[t] mapping x⊗ts to x⊗(t+a)s,
a a
we find that τ∗(W (λ)) is a quotient of W(λ). Hence by Theorem 1.5.1 and the lower bound for the
a a
dimension of W (λ) obtained in [CP1] we have,
a
τa∗(Wa(λ))∼=W(λ).
LetN(λ)betheg[t]–submoduleofW(λ)suchthatthequotientisaDemazuremoduleandletN (λ)=
a
τ∗ (N(λ)) bethe correspondingsubmodule inW (λ). ItsufficestoprovethatN (λ) isag⊗C[t,t−1]–
−a a a
submodule, but this is now clear, since the action of x ⊗ t−s with s ∈ N on W (λ) was a linear
a
combination of elements from U(g[t]).
2. A basis of W(λ)
We proveTheorem1.5.1by givinganexplicit basisforW(λ). Inthis sectionwedefine the proposed
basis of W(λ) by identifying a suitable subset of the Poincare–Birkhoff–Witt basis of U(n−[t]) and
prove that it has the desired cardinality. We also deduce the fermionic version of a character formula
for W(λ).
2.1. The set Br(λ).
2.1.1. Let F denote the set of all pairs (ℓ,s) which satisfy:
ℓ∈N, s=(s(1)≤···≤s(ℓ))∈Nℓ,
including the pair (0,∅), where ∅ is the empty partition. Given (ℓ,s) ∈ F and 1 ≤ i ≤ j ≤ r, let
x± (ℓ,s) be the element of U(n±[t]) defined by,
i,j
x± (ℓ,s)=(x± ⊗ts(1))···(x± ⊗ts(ℓ)),
i,j i,j i,j
if ℓ > 0 and x± (0,∅) = 1. From now on, given an element ((ℓ ,s ),··· ,(ℓ ,s )) of Fj, j > 0, we let
i,j 1 1 j j
(ℓ,s), be the pair of j–tuples of integers, ℓ=(ℓ ,··· ,ℓ ) and s=(s ,··· ,s ). Given (ℓ,s)∈Fj, set,
1 j 1 j
x±(ℓ,s)=x± (ℓ ,s )···x± (ℓ ,s ).
j 1,j 1 1 j,j j j
Definition. Let Br be the subset of U(n−[t]) consisting of elements of the form
(2.1) x−(ℓ ,s )...x−(ℓ ,s )
1 1 1 r r r
where (ℓ ,s )∈Fj for 1≤j ≤r. (cid:3)
j j
In other words, writing (ℓ ,s ) as ((ℓ ,s ),...,(ℓ ,s )), the set Br consists of the elements
j j 1,j 1,j j,j j,j
(2.2) x− (ℓ ,s )x− (ℓ ,s )x− (ℓ ,s )x− (ℓ ,s )···x− (ℓ ,s ),
1,1 1,1 1,1 1,2 1,2 1,2 2,2 2,2 2,2 1,3 1,3 1,3 r,r r,r r,r
in U(n−[t]) with (ℓ ,s )∈F, 1≤i≤j ≤r.
i,j i,j
Proposition. The set Br ⊂U(n−[t]) is a basis of U(n−[t]).
Proof. Fix anorderedbasis of n−[t] as follows: the elements ofthe basis are x− ⊗ts for 1≤i≤j ≤r,
i,j
s ∈ N and the order is given by, x− ⊗ts precedes x− ⊗ts′ when either j < j′ or j = j′, i < i′ or
i,j i′,j′
j =j′, i=i′, s<s′. The proposition follows from the Poincare–Birkhoff–Witttheorem. (cid:3)
10 VYJAYANTHICHARIANDSERGEILOKTEV
2.1.2. For m∈N, let
(2.3) F(m)={(ℓ,s):ℓ>0, 0≤s(i)≤m−ℓ, 1≤i≤ℓ}∪{(0,∅)}.
If m<0 then we set F(m)=∅. For λ= r m ω ∈P+ and 1≤j ≤r, let
i=1 i i
(2.4) Fj(λ)= (ℓ,s)∈PFj :(ℓi,si)∈F(mi), 1≤i≤j .
Given ℓ∈Nj, set η (ℓ)= j (cid:8)ℓ α ∈Q+. For any h∈h we have (cid:9)
j i=1 i i,j
P [h,x±j (ℓ,s)]=±ηj(ℓ)(h)·x±j (ℓ,s).
Definition. For λ ∈ P+, let Br(λ) be the subset of Br consisting of those elements in (2.1) which
satisfy
r
(2.5) (ℓ ,s )∈Fj λ− η (ℓ ) , 1≤j ≤r.
j j s s
s=j+1
X
(cid:3)
Write ℓ =(ℓ ,···ℓ )∈Nj, 1≤j ≤r. Observe that if λ= r m ω ∈P+, then
j 1,j j,j i=1 i i
r j r P r
η (ℓ )− (ℓ −ℓ ) ω ∈ Zω .
s s i,s i+1,s i i
s=j+1 i=1 s=j+1 i=j+1
X X X X
It is now easy to check that (2.5) is equivalent to
r
(ℓ ,s )∈F m + (ℓ −ℓ ) ,
i,j i,j i i+1,s i,s
s=j+1
X
or even more explicitly to: either ℓ =0, or ℓ >0 and
i,j i,j
r r
(2.6) s (ℓ )≤m + ℓ − ℓ ,
i,j i,j i i+1,s i,s
s=j+1 s=j
X X
for all 1≤i≤j ≤r.
Note that for λ∈P+, and ℓ ∈Nr, we have
r
Br−1(λ−η (ℓ))⊂Br(λ).
r
Proposition. (i) We have
Br(λ)= Br−1(λ−η (ℓ))x−(ℓ,s),
r r
(ℓ,s)[∈Fr(λ)
(ii) We have
r r+1 mi
|Br(λ)|= .
i
i=1(cid:18) (cid:19)
Y
Proof. Part (i) is clear. For (ii), recall that for a fixed pair ℓ, n of integers, the number of elements s
of the form, (s=(s(1)≤···≤s(ℓ)≤n) is equal to n+ℓ , and so we have
ℓ
m (cid:0) (cid:1) m m
(2.7) |{s:(ℓ,s)∈F(m)}|= , |F(m)|= =2m.
ℓ ℓ
(cid:18) (cid:19) ℓ=0(cid:18) (cid:19)
X
