GRADED LEVEL ZERO INTEGRABLE REPRESENTATIONS of AFFINE LIE ALGEBRAS 1. Introduction in This Paper, We Continue the Study ([2, 3

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 6, June 2008, Pages 2923–2940 S 0002-9947(07)04394-2 Article electronically published on December 11, 2007

GRADED LEVEL ZERO INTEGRABLE REPRESENTATIONS OF AFFINE LIE ALGEBRAS

VYJAYANTHI CHARI AND JACOB GREENSTEIN

Abstract. We study the structure of the category of integrable level zero representations with finite dimensional weight spaces of affine Lie algebras. We show that this category possesses a weaker version of the finite length property, namely that an indecomposable object has finitely many simple constituents which are non-trivial as modules over the corresponding loop algebra. Moreover, any object in this category is a direct sum of indecomposables only finitely many of which are non-trivial. We obtain a parametrization of blocks in this category.

1. Introduction In this paper, we continue the study ([2, 3, 7]) of the category Ifin of integrable level zero representations with finite-dimensional weight spaces of affine Lie algebras. The category of integrable representations with finite-dimensional weight spaces and of non-zero level is semi-simple ([16]), and the simple objects are the highest weight modules. In contrast, it is easy to see that the category Ifin is not semi-simple. For instance, the derived Lie algebra of the affine algebra itself provides an example of a non-simple indecomposable object in that category. The simple objects in Ifin were classified in [2, 7] but not much else is known about the structure of the category. The category Ifin can be regarded as the graded version of the category F of finite-dimensional representations of the corresponding loop algebra. The structure of F has been studied extensively in both the quantum and classical cases in recent years([1,5,6,9,14]).ThereisanaturalfunctorL from F to Ifin which we study in this paper (Section 5) and which has many nice properties. But there are important properties that fail, for example in general the functor L only maps irreducible objects to completely reducible objects. In particular, the trivial representation in F is mapped to an infinite direct sum of one-dimensional representations. The latter phenomenon is the main source of difficulty and makes the study of the category Ifin interesting in its own right, since the results and proofs sometimes require substantial modifications from the ones for F. It is immediate from the preceding comments that objects in the category Ifin are not always of finite length. However, in Section 4 of this paper, we show that a weaker version of the finite length property holds, namely that the number of non-trivial constituents of an indecomposable module is finite. Moreover, each indecomposable module admits

Received by the editors February 23, 2006. 2000 Mathematics Subject Classiﬁcation. Primary 17B67. This work was partially supported by the NSF grant DMS-0500751.

c 2007 American Mathematical Society 2923

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an analogue of a composition series which we call a pseudo-Jordan-Hölder series. Such a series is unique up to a natural equivalence. As a result we are able to conclude (Theorem 1) that any object in Ifin is a direct sum of indecomposable modules. Moreover, only finitely many of the indecomposable summands are non- trivial modules for the corresponding loop algebra. This, in particular, implies that Ifin has a block decomposition. The methods used in this part of the paper rely only on facts which remain valid for quantum affine algebras. In Section 6 we obtain a parametrization for the blocks and describe the blocks explicitly (Theorem 2). The blocks in Ifin are parametrized by the orbits for the natural action of the group C× on the set Ξ of finitely supported functions from C× to the quotient group of the weight lattice of the underlying simple finite- dimensional Lie algebra g by its root lattice. It was proved in [5] that Ξ parametrizes blocks in the category F.ToprovetheresultforIfin we use the functor L defined and studied in Section 5. One of the tools used crucially in [5] was the notion of the finite-dimensional Weyl module introduced and studied in [8] (in the quantum case they appear in [12, 13]). One can define in a similar way the notion of the graded Weyl module, which is an object in Ifin and has the usual universal properties. A natural question is then whether the functor L maps Weyl modules in F to graded Weyl modules or at least to direct sums of graded Weyl modules. This question turns out to be rather difficult since it is equivalent to proving the conjecture of [8] on the dimension of the Weyl modules in F. This conjecture was established in [8] for the affine algebra whose underlying simple Lie algebra g is isomorphic to sl2. In other cases, as Hiraku Nakajima has pointed out to us recently, the dimension conjecture can be deduced as follows. The results of [12, 13] imply that the Weyl modules are specializations (the q = 1 limit) of certain finite-dimensional quotients of the extremal modules for the quantum affine algebra. Then it follows from the results in [1] and [14, 15] that these quotients (and hence their specializations) have the correct dimension. Other approaches to proving this conjecture, which do not rely on the quantum case, have been studied recently: in [4] for the case of g isomorphic to slr, r ≥ 3 and in [10] for any simply-laced g.

2. Preliminaries

Throughout the paper, Z (respectively, Z+, N) will denote the set of integers (respectively, non-negative, positive integers).

2.1. Let g be a complex finite-dimensional simple Lie algebra and h a Cartan subalgebra of g.SetI = {1,...,dim h} and let {αi : i ∈ I} (respectively, {i : i ∈ I}) be a set of simple roots (respectively, fundamental weights) of g with respect to h, and R+ (respectively, Q, P ) be the corresponding set of positive roots (respectively + + the root lattice, the weight lattice). Let Q , P be the Z+-span of the simple roots and fundamental weights respectively. It is convenient to set 0 =0.Let≥ be the standard partial order on P defined by: λ ≥ µ if λ − µ ∈ Q+.Letθ ∈ R+ be the highest root and if g is not simply laced denote by θs the highest short root. Denote by W the Weyl group of g. + Given α ∈ R,letgα be the corresponding root space. For α ∈ R , fix non-zero ± ∈ ∨ ∈ elements xα g±α and α h, such that ∨ ± ± ± + − ∨ [α ,xα ]= 2xα , [xα ,xα ]=α

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and write ± n = g±α = Cx±α. α∈R+ α∈R+ It is well-known that Γ = P/Q is a finite abelian group. For any 0 = γ ∈ Γfix + + the unique minimal representative 0 = γ ∈ P , i.e. if λ ∈ P satisfies λ ≤ γ then λ = γ . + Lemma. Let λ ∈ P \{0} and let γ ∈ Γ be such that λ = γ (mod Q).Then + + λ ≥ γ . In addition, if λ ∈ P ∩ Q and λ =0 ,thenλ ≥ β where β = θ if g is simply laced and β = θs otherwise. ∼ Proof. For g = sl+1 one can show by a direct computation that i + j ∈ + i+j (mod +1) + Q ,1≤ i, j ≤ . The statement follows since the minimal repre- ≤ ≤ ∈ + \{ } sentatives of the elements of Γ in this case are i,0 i ,andanyλ P 0 N ≥ ≤ ≤ can be written as λ = r=1 ir for some N 1and1 ir . For all other types the statement is trivially checked. 2.2. For λ ∈ P +,letV (λ) be the irreducible finite-dimensional g-module with highest weight vector vλ, i.e. the cyclic module generated by vλ with defining relations: − ∨ + λ(αi )+1 n vλ =0,hvλ = λ(h)vλ, (xαi ) vλ =0, for all i ∈ I, h ∈ h.Givenanyg-module M and µ ∈ h∗ set M = {v ∈ M : hv = µ(h)v, h ∈ h}. µ If M is finite-dimensional, then M = µ∈h∗ Mµ, and moreover ∼ ⊕mλ(M) M = V (λ) ,mλ(M) ∈ Z+. λ∈P + The following Lemma is standard (see [11] for instance).

+ Lemma. Let λ, µ ∈ P be such that λ ≥ µ.ThenV (λ)µ =0 . 2.3. Given a Lie algebra a,letU(a) denote the universal enveloping algebra of a and let L(a)=a ⊗ C[t, t−1] be the loop algebra of a with the Lie bracket given by

[x ⊗ f,y ⊗ g]=[x, y]a ⊗ fg, for all x, y ∈ a, f,g ∈ C[t±1]. The Lie algebra L(a) and its universal enveloping algebra are Z-graded by the powers of t. We shall identify a with the subalgebra a ⊗ 1ofL(a). Denote by Le(a)=L(a) ⊕ Cd theextendedloopalgebraofa,in which [d, x ⊗ tn]=nx ⊗ tn. ⊕ e ∈ ∗ 2.4. Let he = h Cd, which is an abelian Lie subalgebra of L (g). Define δ he by δ(d)=1,δ(h)=0, ∀ h ∈ h. ∗ ∗ ∈ ∗ We regard elements of h as elements of he by setting λ(d)=0forλ h .In + ∗ particular, we identify P and P with their respective images in he. Obviously, ∗ ∗ ⊕ ⊕ + + ⊕ ⊂{ ∈ ∗ he = h Cδ.SetPe = P Zδ and let Pe = P Zδ.ThenPe λ he : ∨ ∈ ∀ ∈ } λ(αi ) Z i I . Let W be the affine Weyl group associated with W . Its image in the group of ∗ automorphisms of he identifies with the semidirect product of W with an abelian ∈ ∨ − ∈ ∗ group generated by the th, h Z(Wθ )whereth(λ)=λ λ(h)δ for all λ he.

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+ Given λ ∈ P ,letrλ =minh∈Z(Wθ∨){λ(h):λ(h) > 0}∈N. This number exists ∨ ∨ ∈ ∈ since Z(Wθ ) is contained in the Z-span of the αi , i I. Then for all s Z,there exists w ∈ W such that w(λ + sδ)=λ +¯sδ,where0≤ s0

It is easy to see (cf. [8]) that U(L(h)) is the polynomial algebra on Λi,±k, i ∈ I, k ∈ Z, k =0.

3. Elementary properties of the category Ifin 3.1. Recall that an Le(g)-module V is said to be integrable if V = Vµ, ∈ ∗ µ he ± ⊗ s ∈ + ∈ and the elements xα t act locally nilpotently on V for all α R and s Z. I e e { ∈ ∗ Denote by the category of integrable L (g)-modules. Let wt (V )= µ he : Vµ =0 } be the set of weights of V with respect to he. It is well-known that the set wte(V )isW-invariant. The following Lemma follows immediately from Section 2.4 and will be used repeatedly in the rest of the paper. Lemma. Let V ∈ Ob I and assume that µ + sδ ∈ wte(V ) where µ ∈ P +, s ∈ Z. e Then µ + rδ ∈ wt (V ) for some 0 ≤ r

Proof. Part (i) is standard and follows from the representation theory of sl2 applied e { ± ∨} ∈ to the subalgebras of L (g) spanned by the elements xαi ,αi for i I. The ﬁrst two statements in (ii) are straightforward while for the last it is suﬃcient to observe e that the functor V → V ⊗ C−aδ,whereC−aδ is the 1-dimensional L (g)-module on which L(g) acts trivially and d acts by −a, provides an equivalence of categories between I{a¯} and I{0¯}.

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It follows, in particular, that we can restrict ourselves to the subcategory I{0¯} e of I.ObservethatV ∈ Ob I is an object in I{0¯} if and only if wt (V ) ⊂ Pe.Let fin e I be the subcategory of I consisting of modules V such that wt (V ) ⊂ Pe and dim Vµ < ∞ for all µ ∈ Pe.

e ∈ + \ + { ∈ 3.3. Let V be an integrable L (g)-module. For λ Pe Zδ,setVλ = v Vλ : L(n+)v =0} and + + V = Vλ . + λ∈Pe \Zδ :0≤λ(d)

fin Proposition. Let V be an object in I and suppose that V1 V2 ··· is an e e ascending chain of L (g)-submodules of V .Thenwt (Vr/Vr−1) ⊂ Zδ for all but ﬁnitely many r ≥ 1.Inparticular,V + is ﬁnite-dimensional.

Proof. Write V = V [γ], γ∈Γ where V [γ]= Vµ+nδ.

µ=γ (mod Q),n∈Z Since V [γ] is obviously an Le(g)-submodule of V and Γ is a finite group we can assume without loss of generality that V = V [γ]. Suppose for a contradiction that e wt (Vr/Vr−1) ⊂ Zδ for infinitely many r ≥ 1. Since V is integrable, Vr/Vr−1 is a (possibly infinite) direct sum of finite-dimensional g-modules. In particular, + for infinitely many r ≥ 1thereexistsµr ∈ P \{0} and sr ∈ Z such that v ∈ (Vr/Vr−1)µr+sr δ, v = 0 generates a simple highest weight g-submodule isomorphic ≥ to V (µr). By Lemma 2.1, µr γ , hence V (µr)γ = 0 by Lemma 2.2, and we e conclude that γ + srδ ∈ wt (Vr/Vr−1). e Suppose first that γ = 0. Then by Lemma 3.1, γ +¯srδ ∈ wt (Vr/Vr−1) ≤ ≤ ∈ with 0 s¯r

3.4. In the category Ifin the role similar to that of highest weight modules is played by -highest weight modules. ∈ + Deﬁnition. Let V be an integrable module. A non-zero element v Vλ is called an -highest weight vector if U(Le(h))v is an indecomposable U(Le(h))-submodule e of V and if dim(U(L (h))v)λ+rδ≤ 1 for all r ∈ Z.WesaythatV is -highest weight if V = U(Le(g))v where v ∈ V is an -highest weight vector.

It is not hard to see by the usual arguments that an -highest weight module is ∈ + e e indecomposable. Observe also that if v Vλ and U(L (h))v is a simple U(L (h))- e module, then dim(U(L (h))v)λ+rδ ≤ 1andsov is an -highest weight vector.

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3.5. Retain the assumption of Section 3.4 and consider V L(g) = {v ∈ V : L(g)v = 0}. Obviously, V L(g) is an Le(g)-submodule of V . L(g) ∼ mr ∈ Lemma. (i) V = r∈Z Crδ , mr Z+. (ii) Suppose that wte(V ) ⊂ Zδ.ThenV = V L(g). (iii) Suppose that V = V L(g).ThenV/V L(g) does not admit an Le(g)-submodule isomorphic to Crδ, r ∈ Z. L(g) Proof. Parts (i) and (ii) are immediate. For (iii), let V0 = V V and suppose ∈ \ ⊂ that there exists v V V0, such that L(g)v V0.SinceV is a weight module, ∈ ∗ ∈ ∈ we can write, uniquely, v = k vµk , µk he.Sincehv V0 for all h he and e ⊂ ∈ ± ⊗ n wt (V0) Zδ by (i), it follows that µk Zδ.Then(xα t )v =0andsoL(g)v =0. Thus, v ∈ V0, which is a contradiction.

4. Finiteness in the category Ifin The main result of this section is the following theorem. Although we state and prove this result only for the classical affine Lie algebras, it is clear that the proof goes over verbatim to the quantum case. Theorem 1. Let V be an object in Ifin.ThenV is isomorphic to a direct sum of indecomposable modules. More precisely, there exists submodules Uj , j =1, 2,such ⊕ ⊂ L(g) + + that V = U1 U2 with U1 V , U2 = V . Moreover, U2 is isomorphic to a finite direct sum of indecomposable modules. We prove this result in the rest of the section. Remark. Note that there exist indecomposable modules of infinite length in Ifin. For example, let M = g ⊕ C as a g-module with the L(g)-module structure defined k by (x ⊗ t )(y ⊕ a)=[x, y] ⊕ k(x, y)g,where(· , ·)g is the Killing form of g, for all x, y ∈ g, k ∈ Z and a ∈ C.ThenM is an indecomposable L(g)-module. Applying the functor L (cf. Section 5.4) we conclude that L(M)containsL(C)= C ∼ r∈Z rδ as a submodule and L(M)/L(C) = L(g) which is a simple Le(g)-module. Thus, L(M) has infinite length and it is easy to see that L(M) is indecomposable. 4.1. The following proposition was established in [3] (Proposition 3.2) in the quantum case. We provide its proof here for the sake of completeness. Proposition. Let V be an object in Ifin. Then there exists λ ∈ wte(V ) such that ∈ e ∈ + \{ } ∈ + + λ + η/wt (V ) for all η Q 0 .Inparticular,λ Pe and Vλ =0. Proof. By Lemma 3.5, we may assume that wte(V ) ⊂ Zδ. Suppose that for each µ ∈ wte(V )thereexistsν ∈ Q+ such that µ + ν ∈ wte(V ). Fix some µ ∈ wte(V ). Then e there exists an infinite sequence {ηr}r≥1 such that ηr ≤ ηr+1 and µ + ηr ∈ wt (V ) ≥ for all r 1. Set Wr := U(g)Vµ+ηr .ThenWr is an integrable g-module with finite- dimensional weight spaces and hence is isomorphic to a finite direct sum of simple + finite-dimensional g-modules V (µr,s)forsomeµr,s ∈ P .Chooses1 such that

ν1 := µ1,s1 >µ. Such s1 exists since µ + η1 is a weight of W1. Furthermore, let r2 be the smallest positive integer so that there exists s2 with ν2 := µr2,s2 >µ, µ1,s1 =

µr2,s2 .Noticethatr2 always exists since the module W1 is ﬁnite-dimensional and the maximal weights which occur in Wr keep increasing. Repeating this process, we obtain an inﬁnite collection of elements νk >µ, k ≥ 1, such that V (νk)is isomorphic to an irreducible g-submodule W (νk)ofV . By Lemma 2.2 it follows that

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Vµ∩W (νk) =0forall k ≥ 1. Since all the νk are distinct, the sum of W (νk) is direct, e which contradicts the ﬁnite-dimensionality of Vµ.Inparticularλ + αi ∈/ wt (V )for ∈ ∈ + all i I which implies that λ Pe . 4.2. Proposition. Suppose that V is an object in Ifin and assume that wte(V ) is not a subset of Zδ.ThenV contains an -highest weight submodule generated ∈ + ∈ + \{ } ≤ by v Vλ+rδ for some λ P 0 and 0 r0ofV , such that Vr Vr+1. Let vr be an -highest weight vector generating Vr and assume that the weight + + of vr is µr + srδ,whereµr ∈ P ∩ (λ − Q ), sr ∈ Z. As before, we may also ≤ + ∩ − + assume that 0 sr

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This procedure can obviously be repeated to get an ascending chain satisfying the conditions of the above deﬁnition. Proposition 3.3 implies that we must reach a L(g) stage when (V/VN ) = V/VN and hence VN+1 = V , which proves the ﬁniteness condition of the proposition.

fin 4.4. Let V be an object in I and let V = VN ⊇ ··· ⊇ V1 ⊇ V0 =0and ⊇ ··· ⊇ ⊇ e V = VN V1 V0 = 0 be descending chains of its L (g)-submodules. We call these chains equivalent if |{ ≤ e ⊂ }| |{ ≤ e ⊂ }| (i) 0

Proposition. Let V be an indecomposable object in Ifin. Then its pseudo-Jordan- H¨older series is unique up to equivalence deﬁned above.

Proof. It is easy to see that any refinement of a pseudo-Jordan-Hölder series for an indecomposable object V in Ifin is equivalent, in the above sense, to that pseudo- Jordan-Hölder series itself. The statement follows immediately from the Schreier Refinement Theorem.

It follows from the above proposition that the number

e |{0

where V = VN ··· V1 V0 = 0 is a pseudo-Jordan-H¨older series for V ,is well-deﬁned. We call this number the pseudo-length of V .

4.5. We can now prove Theorem 1.

∈ Ifin ∈ ∈ Proof. Let V Ob .Foreachs Z,let ms Z+ be the multiplicity of Csδ mr ⊂ L(g) as a direct summand of V and set U1 = r∈Z Crδ . Clearly U1 V .For ∈ e (s) ms s Zwith ms =0fixanL (g)-module complement V of Csδ in V and let U = V (r).SinceV (r) ∩ Cmr = 0 it follows that U ∩ U = {0}. 2 r∈Z : mr=0 rδ 1 2 e Let v ∈ Vµ, µ ∈ wt (V ). If µ = sδ for some s ∈ Z such that ms =0,then ∈ (s) ∈ ∈ (s) ⊕ ms v V for all s Z and so v U2. Otherwise, since V = V Csδ ,wecan ∈ ms ∈ (s) write v uniquely as v = u1 + u2 for some u1 Csδ and u2 V . It follows from weight considerations that u2 ∈ U2 andsowehaveprovedthatV = U1 ⊕ U2.Note e that if U2 = 0, then by Lemma 3.5 wt (U2) is not a subset of Zδ. e It remains to show that U2 is a finite direct sum of indecomposable L (g)- modules. If U2 is indecomposable we are done. Otherwise, we can write U2 = e M1 ⊕ M2.Notethatwt(Mj) is not a subset of Zδ for j =1, 2 since otherwise by + Lemma 3.5 we would have a contradiction to the definition of U1. Hence dim Mj > + + ⊕ + + + + 0forj =1, 2, and since U2 = M1 M2 ,dimU2 =dimM1 +dimM2 .The + ∞ statement follows by repeating the argument and using the fact that dim U2 < (cf. Proposition 3.3).

Since by Theorem 1 every object in the category Ifin is a direct sum of indecomposables, it makes sense to deﬁne the blocks in that category.

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4.6. Let C be an abelian category in which any object is a direct sum of indecomposables. We say that two indecomposable objects Ui, i =1, 2, in a C are linked and write U1 ∼ U2 if there do not exist full abelian subcategories Ci, i =1, 2, such that Ui ∈ Ob Ci and C = C1 ⊕C2.IftheUi are decomposable, then they are said to be linked if every indecomposable summand of U1 is linked to every indecomposable summand of U2. This deﬁnes an equivalence relation on C, and the equivalence classes are called blocks. Each block is a full abelian subcategory, and the category C is a direct sum of the blocks. It is not hard to see that the following is an equivalent deﬁnition of linking. Two indecomposable objects U, V in C are linked if and only if there exists a family of indecomposable objects U1 = U, U2,...,Ul = V in C such that either HomC(Uk,Uk+1) =0orHom C(Uk+1,Uk) =0forall k =1,...,l− 1.

4.7. Following [5], let Ξ be the set of functions χ : C× → Γ with finite support. The addition of functions defines on Ξ the structure of an abelian group. Given × i ∈ I, a ∈ C ,setχi,a(z)=δa,z¯ i,where¯i denotes the canonical image of i in × Γ. Clearly Ξ is the free abelian group generated by the χi,a, i ∈ I, a ∈ C . Define an action of C× on Ξ by (a · χ)(z):=χ(az),a,z∈ C×. Let Ξ be the set of orbits in Ξ for this action and letχ ¯ be the C×-orbit of χ ∈ Ξ. In the rest of the paper, we shall prove the following: Theorem 2. The blocks in the category Ifin are parametrized by the elements of Ξ.

5. The category F and the functor L + 5.1. Let P be the set of I-tuples of polynomials π =(πi)i∈I in u with con- stant term 1. We regard P+ as a commutative monoid with multiplication defined × + component-wise. Let 1 =(1,...,1) and, for i ∈ I and a ∈ C ,leti,a ∈P be the I-tuple of polynomials with the polynomial (1 − au)intheith place and ∈ ∈ × oneeverywhereelse.Theelements i,a, i I,anda C generate the monoid P+ ∈P+ ∈ + ∈ + .Forπ ,setλπ = i∈I (deg πi)i P . Conversely, given λ P and λ(α∨) ∈ × i a C ,setπλ,a = i∈I i,a . ∈P+ ∈ We say that π =(πi)i∈I , π =(πi)i∈I are co-prime if for all i, j I,the ∈P+ polynomials πi and πj are co-prime. Clearly any π can be written, uniquely, ∈ + ∈ × ≤ ≤ as a product πλj ,aj for some λj P and aj C ,1 j k,andar = aj if r = j. + Let m : P → Z+ be the map defined by setting m(π) to be the maximal r non-negative integer r such that πi ∈ C[u ] for all i ∈ I. Lemma. Suppose that λ, µ ∈ P + and λ>µ.Letπ ∈P+ and a ∈ C× be such that i,a is coprime to π for some i ∈ I. Then either m(ππλ,a)=1or m(ππµ,a)=1. ∈P+ Proof. Observe that for all π =(πi)i∈ I , m(π ) > 1 implies that r µr(h)br ∈ ∈ =0forallh h, provided that π = r πµr,br where the br are distinct and µr + ∈ ∨ P . Indeed, for i I, r µr(αi )br is obviously the coefficient of u in the polynomial πi(u). Since we assume that m(π ) > 1, it follows from the definition of m(π ) ∈ that the coefficient of u in all the πi, i I, must be zero.

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k ≤ Write π = r=1 πλr ,ar where the ar are distinct. Obviously, a = ar for all 1 ≤ − ∈ + \{ } r k.Letβ = λ µ Q 0 . Suppose that m(πλπ) > 1. By the above ∈ ∈ argument, λ(h)a + r λr(h)ar =0forall h h.Sinceβ =0,thereexists i I ∨ ∨ ∨ − ∨ ∨ such that β(αi ) =0.Thenµ(αi )a+ r λr(αi )ar =(λ β)(αi )a+ r λr(αi )ar = − ∨ β(αi )a = 0 which implies m(πµπ)=1. 5.2. Deﬁnition. Let V be an L(g)-module and let 0 = v ∈ V .Wesaythatv is an -highest vector if + L(n )v =0, dimC U(L(h))v =1. The module V is said to be -highest weight if it is generated, as an L(g)-module, by an -highest weight vector.

+ Given π =(πi)i∈I ∈ P ,letW(π)betheL(g)-module generated by a vector wπ satisfying the relations: + ± L(n )wπ =0,hwπ = λπ(h)wπ, Λi,±kwπ = πi,kwπ, ∈ + k where h h, πi(u)= k≥0 πi,ku and − − − k deg πi 1 deg πi 1 | πi,ku = u πi(u )/(u πi(u ) u=0). k≥0 The modules W(π) are clearly -highest weight modules for L(g). Let F be the category of ﬁnite-dimensional representations of L(g). The following was proved in [2, 7, 8]. Proposition. (i) For all π ∈P+, W(π) is an indecomposable object in F. (ii) Any -highest weight module in F is a quotient of W(π) for some π ∈P+. (iii) The modules W(π) have a unique irreducible quotient V(π) and any irreducible module in F is isomorphic to V(π) for some π ∈P+. ∼ (iv) Let π, π ∈P+ be coprime. Then W(π) ⊗W(π) = W(ππ ).Inparticular, any quotient of W(ππ) is isomorphic to a tensor product of quotients of W(π) and W(π). × r (v) Let a ∈ C and let τa be the automorphism of L(g) deﬁned by τa(x ⊗ t )= r ⊗ r ∈ ∈ ∗W a x t for all x g, r Z.Thenτa (π) is isomorphic, as an L(g)-module, to W(π(au)).

+ Corollary. Let π ∈P and suppose that m = m(π) > 1. Then there exists ηm ∈ W ⊗ r −r ⊗ r EndC (π) such that ηm(wπ)=wπ and ηm((x t )w)=ζm (x t )ηπ(w) for th all x ∈ g, r ∈ Z and w ∈W(π),whereζm is an m primitive complex root of unity. Moreover, ηm is of order m. ∈ × ∗W W Proof. Let a C .Sinceτa (π) is isomorphic to (π) as a vector space, it follows from (v) that there exists a map ηπ,a ∈ HomC(W(π), W(π(au))) such r r r that ηπ,a(wπ)=wπ and ηπ,a((x ⊗ t )w)=a (x ⊗ t )ηπ,a(w) for all x ∈ g, r ∈ Z and w ∈ W .Setη = η −1 .Sincem = m(π) > 1, π(ζ u)=π(u)and m π,ζm m so ηm ∈ EndC W(π). The ﬁrst two properties of ηm are immediate. For the last, observe that since W(π) is generated by wπ as an L(g)-module, it follows from the properties of ηm that W(π) is a direct sum of eigenspaces of ηm, all eigenvalues th of ηm are m complex roots of unity, and ζm is an eigenvalue of ηm.

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Remark. The module V(πλ,a) is isomorphic to V (λ)asag-module, the L(g)-module structure being deﬁned by the evaluation at a,thatis,(x⊗tn)v = anxv for all x ∈ g, n ∈ Z and v ∈ V (λ). More generally, V(π), π ∈P+, is isomorphic to a tensor product of modules of the form V(πλ,a) with distinct a.

+ 5.3. The assignment i,a → χi,a extends to a surjective map of monoids P → Ξ. + Denote by χπ the image of π ∈P under this map. Given χ ∈ Ξ, let Fχ be the full subcategory of F whose objects have the following property: V(π) is an irreducible constituent of V in Fχ only if χ = χπ. The following result was proved in [5]. F F F F Theorem 3. We have = χ∈Ξ χ. Moreover the χ are the blocks in . 5.4. Deﬁne a functor L : F→Ifin by L(V )=V ⊗ C[t, t−1]withtheLe(g)-module structure given by: (x ⊗ tk)(v ⊗ tn)=(x ⊗ tk)v ⊗ tk+n,d(v ⊗ tn)=nv ⊗ tn for all x ∈ g, v ∈ V and k, n ∈ Z. Clearly L preserves direct sums and short exact sequences. It should be noted that the functor L is not essentially surjective. For example, the indecomposable module L(g) ⊕ C with the Le(g)-module structure given by

k r r+k r r (x ⊗ t )(y ⊗ t ,a)=([x, y] ⊗ t ,rδr,−k(x, y)g),d(y ⊗ t ,a)=(ry ⊗ t , 0) is not isomorphic to L(V ) for any V in F. Given any -highest weight L(g)-module generated by an -highest weight vector v,letLs(V )betheLe(g)-submodule of L(V ) generated by v ⊗ ts. The following result was proved in [7] (see also [3]). Proposition. Let π ∈P+, π = 1. (i) For 0 ≤ s

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∈ F ∈ Ifin Proposition. (i) If V Ob χ,thenL(V ) Ob χ . (ii) Suppose that V is an -highest weight quotient of W(π) for some π ∈P+ and let v be the canonical image of wπ in V .For0 ≤ s

Proof. Since V is finite dimensional, it has a Jordan-Hölder series V = VN ··· V = 0. By applying the functor L we obtain a filtration L(V )=L(V ) ··· 0 ∼ N L(V0)=0.ObservethatL(Vi)/L(Vi−1) = L(Vi/Vi−1) and thus is either simple or completely reducible. In the second case, it is a finite direct sum of simple ∼ ∼ objects unless Vi/Vi−1 = C in which case L(Vi/Vi−1) = r∈Z Crδ. Therefore, refining and dropping terms if necessary we can construct a pseudo-Jordan-Hölder series for L(V ) out of Jordan-Hölder series for V . The statement follows since the pseudo-Jordan-Hölder series is unique up to equivalence defined in Section 4.4. It suffices to prove (ii) when V = W(π)andwhenm = m(π) > 1. Clearly, L(W(π)) = Ls(W(π)),Ls(W(π)) = Lr(W(π)) if s = r (mod m). s∈Z ∈ W ⊗ r r ⊗ r th Define ηm EndC L( (π)) by ηm(w t )=ζmηm(w) t ,whereζm is an m primitive root of unity and ηm is a map from Corollary 5.2 (cf. [3, 2.6]). It is easy to check that η m ∈ EndLe(g) L(W(π)) and since η m is clearly of order m, it defines arepresentationofZ/mZ on L(W(π)). It follows that L(W(π)) is a direct sum of Z/mZ-isotypical components corresponding to m distinct irreducible characters of the finite abelian group Z/mZ. It remains to observe that Z/mZ acts by its s s W s W irreducible character corresponding to ζm on L ( (π)) and hence L ( (π)) is contained in a Z/mZ-isotypical component of L(W(π)). 5.6. For any object V in Ifin,letV # = V ∗ ⊂ V ∗ be the graded dual of V . µ∈Pe µ Then V # is an Le(g)-submodule of V ∗ and is in Ifin, and the functor sending V ∼ ∗ to V # is exact and contravariant. It is easy to see that L(V(π))# = L(V(π )), ∗ + with π ∈P satisfying λπ∗ = −w◦λπ where w◦ is the longest element of W .In ∈P+ − ∈ + \{ } ∗ − ∗ ∈ + particular, if πj , j =1, 2, and λπ1 λπ2 Q 0 ,thenλπ2 λπ1 / Q . 5.7.

× + Proposition. Let a ∈ C and let λj ∈ P , j =1, 2, be such that λ1 >λ2. 1 V V ∈P+ Set πj = πλj ,a and suppose that ExtF ( (π1), (π2)) =0. Assume that π is coprime to i,a for i ∈ I.

(i) If m(π1π)=1,thenL(V(π1π)) and L(V(π2π)) are linked. s (ii) If m(π1π) > 1,thenL(V(π2π)) is simple and is linked to L (V(π1π)) for some 0 ≤ s

(5.1) 0 −→ L(V(π2) ⊗V(π)) −→ L(V ⊗V(π)) −→ L(V(π1) ⊗V(π)) −→ 0.

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It is not hard to see (cf. [5]) that the module V is an -highest weight module for L(g) and is a quotient of W(π1). Moreover, since π and πj, j =1, 2, are coprime, it follows from Proposition 5.2 that V ⊗V(π)isan-highest weight quotient ∼ of W(π1π) and also that V(πj) ⊗V(π) = V(πjπ), j =1, 2. If m(π1π) = 1, then by Proposition 5.5(ii) we see that L(V ⊗V(π)) is indecomposable and part (i) is immediate. Otherwise, by Lemma 5.1, m(π2π)=1and so L(V(π2π)) is simple. By Proposition 5.5(ii), − m(π1π) 1 L(V ⊗V(π)) = Ls(V ⊗V(π)) s=0 − V m(π1π) 1 s V and L( (π1π)) = s=0 L ( (π1π)). It follows from (5.1) that the sequence s s 0 −→ L(V(π2π)) −→ L (V ⊗V(π)) −→ L (V(π1π)) −→ 0 s is exact for some 0 ≤ s

+ × Corollary. Let π ∈P and assume that m(π) > 1.Leta ∈ C be such that i,a, i ∈ I, is coprime to π.ThenL(V(ππθ,a)) ∼ L(V(π)). In particular, the modules Ls(V(π)) and Lr(V(π)) are linked for any 0 ≤ s, r < m(π).

1 Proof. By Lemma 5.1, m(πθ,aπ) = 1. Since ExtF (V(πθ,a), C) = 0 (cf. [5]), the result is immediate from the Proposition. 5.8.

+ Proposition. Let π, π ∈P and suppose that χπ =0.ThenL(V(ππ )) ∼ L(V(π)). Proof. Clearly it is suﬃcient to prove the statement for π = πβ,a where β ∈ P + ∩ Q ⊂ Q+. + + Suppose ﬁrst that πβ,a is co-prime with π.Sinceλ ∈ P ∩ Q , by [5, Propo- + + sition 1.2] there exists a sequence γr ∈ P ∩ Q , r =0,...,N, such that γ0 = β, 1 V V γN =0andExtF ( (πγr,a), (πγr+1,a)) = 0. It then follows from Proposition 5.7 V V and its Corollary that the module L( (πγr,aπ)) is linked to L( (πγs,aπ)) for all 0 ≤ r, s ≤ N, which implies the assertion. If πβ,a is not co-prime with π,thenwecanwriteπ = πµ,aπ1 where π1 is coprime with i,a, i ∈ I.Thenππβ,a = πµ+β,aπ1. Again, by [5, Proposition 1.2], + there exists a sequence νr ∈ P , r =0,...,K, such that ν0 = µ + β, νK = µ 1 V V and ExtF ( (πνr,a), (πνr+1,a)) = 0. Then it follows from Proposition 5.7 and its V ∼ V ≤ ≤ Corollary that L( (πνr,aπ1)) L( (πνs,aπ1)) for all 0 r, s K. ∈P+ V ∼ Corollary. Let πj , j =1, 2, be such that χπ1 = χπ2 .ThenL( (π1)) L(V(π2)).

Proof. Suppose ﬁrst that χπj =0,j =1, 2. Then χπj = 0 and it follows from the Proposition that the L(V(πj)), j =1, 2, are linked to L(C) and hence are linked. Suppose that χ = 0. We may assume, without loss of generality, that χ = πj π1 k − χπ2 . By the Proposition, we may assume that πj = r=1 πλj,r ,ar such that λ1,r + − − ± ∈ + λ2,r = βr βr , βr Q . Applying the Proposition again we conclude that k k L(V(π )) ∼ L(V(π π + )) = L(V(π π − )) ∼ L(V(π )). 2 2 r=1 βr ,ar 1 r=1 βr ,ar 1

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6. Characters of indecomposable objects in Ifin 6.1. In order to complete the proof of Theorem 2, it remains to establish the following Proposition. Proposition. Let V be an indecomposable object in Ifin.ThenV is an object Ifin ∈ in χ¯ for some χ Ξ. In order to prove this Proposition we will need to establish some properties of -highest weight Le(g)-modules.

∈ + e e 6.2. Given λ Pe ,letW (λ)betheleftL (g)-module generated by an element wλ subject to the relations + − ⊗ λ(α∨)+1 L(n )wλ =0,hwλ = λ(h)wλ, (xα 1) wλ =0, + for all α ∈ R and h ∈ he. Clearly, right multiplication by elements of U(L(h)) deﬁnes a structure of a right U(L(h))-module on W e(λ). The following can be found in [8].

+ Proposition. (i) Let Jλ =AnnU(L(h)) wλ. Given π ∈P with λπ = λ,there exists a maximal ideal Jπ ⊃ Jλ in U(L(h)) such that one has an isomorphism of L(g)-modules

∼ e W(π) = W (λ) ⊗U(L(h)) U(L(h))/Jπ.

Conversely if J is any maximal ideal in U(L(h))/Jλ,thenJ = Jπ for some π ∈P+ and ∼ e W(π) = W (λ) ⊗U(L(h)) U(L(h))/J. 6.3. As explained in the Introduction, the next theorem follows from [1], [13, 12]. It was also proved, by diﬀerent methods, in [8] for g = sl2,in[4]forg = sln, n ≥ 3, and in [10] for g of all simply laced types.

+ Theorem 4. Let λ ∈ P . Then there exists Nλ > 0 such that dim W(π)=Nλ for + all π ∈P satisfying λπ = λ.

e Corollary. As a U(L(h))/Jλ-module, W (λ) is free of rank Nλ.

Proof. Let Sλ = U(L(h))/Jλ. Then any maximal ideal in Sλ is of the form Jπ ∈P+ W ∼ e ⊗ for some π with λπ = λ.Since (π) = W (λ) Sλ Sλ/Jπ, it follows from e ⊗ the Theorem that dim W (λ) Sλ Sλ/J = Nλ for any maximal ideal J in Sλ.The statement follows by Nakayama’s Lemma.

±1 6.4. Let Kπ be the kernel of the homomorphism ψπ : U(L(h)) → C[t ]ofZ- ± ±s ⊃ graded algebras deﬁned by ψπ(Λi,±s)=πi,st .ThenKπ Jλπ and we set e e W (π)=W (λπ) ⊗U(L(h)) U(L(h))/Kπ. e e It is clear that any -highest weight L (g)-module is a quotient of W (π) ⊗ Csδ for some π ∈P+, s ∈ Z. Proposition. The module W e(π) is a free module over U(L(h))/K of rank ∼ π dim W(π).InparticularW e(π) = L0(W(π)) as Le(g)-modules.

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Proof. The first statement is a standard consequence of the Corollary of Theo- rem 4. For the second, let m = m(π)andletηm ∈ EndC W(π)bethemap from Corollary 5.2. By the proof of Proposition 5.5(ii), L0(W(π)) is spanned by ⊗ s ∈ ∈W −s th elements w t , s Z,wherew (π)satisfiesηm(w)=ζm w, ζm being an m ± complex primitive root of unity. Since πi,s = 0 unless m divides s, it follows that the formula ⊗ r ± ⊗ r±s (w t )Λi,±s = πi,s(w t ) defines a right action of U(L(h)) on L0(W(π)) which commutes with the left L(g)- 0 module action. It follows that L (W(π)) is a free module for U(L(h))/Kπ of rank dim W(π). The Proposition follows by noticing that L0(W(π)) is an Le(g)-module quotient of W e(π), that the quotient map is also a map of right U(L(h))-modules and 0 e that L (W(π)) and W (π)arefreeU(L(h))/Kπ-modules of the same rank. Corollary. Let V be an -highest weight Le(g)-module. Then V is a quotient of Ls(W(π)) for some π ∈P+, 0 ≤ s

φ ξ 0 −−−−→ V2 −−−−→ M −−−−→ V1 −−−−→ 0. ∈ Ifin ∈ Since Crδ Ob 0¯ for all r Z, we may assume that at least one of V1 or V2 is not isomorphic to Crδ for r ∈ Z. By passing to graded duals if necessary ∼ s + (cf. Section 5.6), we can assume that V1 = L (V(π1)) for some π1 ∈P with ∼ r π1 = 1,0≤ sλπ1 ,wehave + e L(n )mπ1 =0,M= U(L (g))mπ1 . e In particular, if we denote by M0 the U(L (h))-submodule of M generated by mπ1 , then we have M = M ∼ (V ) ⊕ (V ) 0 λπ1 +rδ = 1 λπ1 +rδ 2 λπ1 +rδ r∈Z r∈Z

as he-modules. If dim M ≤ 1 (in particular, if λ = λ ), then M is an -highest λπ1 +rδ π1 π2 e s weight module for L (g). Then by Proposition 6.4 M is a quotient of L (W(π1)). s fin Since Proposition 5.4 implies that L (W(π1)) ∈ Ob I it follows that M ∈ χ¯π1 Ifin Ob χ¯π . That is impossible ifχ ¯π2 =¯χπ1 and we obtain a contradiction. 1 ∈ If λπ1 = λπ2 = λ and dim Mλ+p0δ =2forsomep0 Z,thendimMλ+pδ =2 ∈ for all p = p0 (mod k)wherek =lcm(m(π1),m(π2)). Fix mπ1,l Mλ+(p0+kl)δ ⊗ p0+kl ⊗ p0+kl ∈ such that ξ(mπ1,l)=vπ1 t and set mπ2,l = φ(vπ2 t ) Mλ+(p0+kl)δ. { } ¯ Then mπ1,l, mπ2,l is a basis of Mλ+(p0+kl)δ.Letψπ denote the composition

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±1 → → ∈ of ψπ with the map C[t ] C, t 1. Sinceχ ¯π1 =¯χπ2 there exists l, s Z ∈ ¯ ¯ and x U(L(h))ks such that ψπ1 (x) = ψπ2 (x)and ¯ ¯ xmπ1,l = ψπ1 (x)mπ1,l+s + cmπ2,l+s,xmπ2,l = ψπ2 (x)mπ2,l+s, ∈ → where c C.Letγ : Mλ+(p0+(l+s)k)δ Mλ+(p0+kl)δ be the isomorphism of vector ◦ spaces deﬁned by γ(mπj ,l+s)=mπj ,l, j =1, 2. Then the matrix of γ x in { } the basis mπ1,l,mπ2,l is upper triangular with distinct diagonal entries and so ◦ γ x has two one-dimensional eigenspaces. Obviously, mπ2,l is an eigenvector of ◦ ¯ ∈ γ x corresponding to the eigenvalue ψπ2 (x). On the other hand, since mπ2,l e ∈ e U(L (h))mπ1,l, it follows that mπ2,l = ymπ1,l for some y U(L (h))0.Now

yγ(xmπ1,l)=γ(xymπ1,l), ◦ and hence mπ2,l is also an eigenvector of γ x corresponding to the eigenvalue ¯ ψπ1 (x), which is a contradiction. ∈ Ifin 1 Corollary. Suppose that Vi Ob χ¯i , i=1, 2,withχ¯1 =¯χ2.ThenExtIfin(V1,V2) =0. Proof. An induction on the pseudo-length (similar to the induction on length used in [5, Lemma 5.2] for F) yields the Corollary.

6.6. We are now able to prove Proposition 6.1.

Proof. Let V be an indecomposable object in Ifin. We proceed by induction on the ∈Ifin pseudo-length of V .IfV is simple, then it is clear from the deﬁnition that V χ¯π for some π ∈P+. Otherwise, we have an extension

(6.1) 0 −−−−→ V1 −−−−→ V −−−−→ U −−−−→ 0,

L(g) L(g) where either V1 = V or V1 ⊂ V and is simple. In any case, V1 is an object Ifin ∈ in χ¯ for some χ Ξ. Using Theorem 1, we can write k U = Uj j=0 mr ≥ where U0 = r∈Z Crδ , mr 0, and for j =1,...,k the module Uj is indecom- e posable with wt (Uj) ⊂ Zδ. Notice that the pseudo-length of Uj, j ≥ 1, is strictly L(g) smaller than that of V unless U is indecomposable and V1 = V . L(g) Suppose ﬁrst that V1 = V . Then, by the induction hypothesis, we see that Ifin ∈ for j =1,...,k, Uj is an object in χ¯j for some χj Ξ. Suppose thatχ ¯ =¯χj0 for some 0 ≤ j0 ≤ k. Then by Corollary 6.5 1 ∼ 1 ExtIfin(U, V1) = ExtIfin(Uj,V1).

j= j0 In other words, the exact sequence (6.1) is equivalent to 0 −−−−→ V −−−−→ U ⊕ V −−−−→ U ⊕ U −−−−→ 0, 1 j0 j0 j= j0 j

where 0 −−−−→ V −−−−→ V −−−−→ U −−−−→ 0 1 j= j0 j

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GRADED LEVEL ZERO INTEGRABLE REPRESENTATIONS 2939 1 represents an element of Ext fin(U ,V ). This is a contradiction since V j= j0 I j 1 L(g) is indecomposable. Finally, suppose that U is indecomposable and V1 = V . Then U L(g) = 0 and so we have a short exact sequence

0 −−−−→ U1 −−−−→ U −−−−→ U2 −−−−→ 0, L(g) where U1 is simple and U1 ⊂ U . Hence the pseudo-length of U2 is strictly smaller than that of U, and so we can apply the above argument and show that U is an fin 1 object in I for some χ ∈ Ξ. Ifχ ¯ =0,thenExt Ifin(U, V ) = 0 by Corollary 6.5, ∼χ¯ 1 and so V = V1 ⊕ U which is a contradiction. Thus,χ ¯ =0andV is an object Ifin in Ob 0¯ . List of notation g, h 2.1, p. 2924 Ifin 3.2, p. 2926 + I, αi, i 2.1, p. 2924 Vλ 3.3, p. 2927 R, R+, Q, Q+, P , P + 2.1, p. 2924 V L(g) 3.5, p. 2928 θ, θs 2.1, p. 2924 (·, ·)g 4, p. 2928 W 2.1, p. 2924 ∼ 4.6, p. 2931 ± ∨ xα , α 2.1, p. 2924 Ξ, Ξ 4.7, p. 2931 Γ, γ 2.1, p. 2924 χi,a 4.7, p. 2931 + V (λ), vλ 2.2, p. 2925 P 5.1, p. 2931 U(a) 2.3, p. 2925 i,a, πλ,a 5.1, p. 2931 e L(a), L (a), d 2.3, p. 2925 λπ, m(π) 5.1, p. 2931 he 2.4, p. 2925 F 5.2, p. 2932 δ 2.4, p. 2925 W(π), wπ, V(π), vπ 5.2, p. 2932 + Pe, Pe 2.4, p. 2925 τa 5.2, p. 2932 W 2.4, p. 2925 χπ 5.3, p. 2933 F rλ 2.4, p. 2925 χ 5.3, p. 2933 ± Λi (u), Λi,±k 2.5, p. 2926 L 5.4, p. 2933 I 3.1, p. 2926 Ls(V(π)) 5.4, p. 2933 e Ifin wt (V ) 3.1, p. 2926 χ 5.5, p. 2933 # Caδ 3.2, p. 2926 5.6, p. 2934

Acknowledgments We are grateful to Hiraku Nakajima for discussions relating to the dimension conjecture.

References

1. J. Beck and H. Nakajima, Crystal bases and two-sided cells of quantum affine algebras,Duke Math. J. 123 (2004), no. 2, 335–402 MR2066942 (2005e:17020) 2. V. Chari, Integrable representations of affine Lie-algebras, Invent. Math. 85 (1986), no. 2, 317–335 MR846931 (88a:17034) 3. V. Chari and J. Greenstein, Quantum loop modules, Represent. Theory 7 (2003), 56–80 (electronic) MR1973367 (2004b:17044) 4. V. Chari and S. Loktev, Weyl, Fusion and Demazure modules for the current algebra of slr+1, Adv. Math. 207 (2006), no. 2, 928–960. MR2271991 5. V. Chari and A. A. Moura, Spectral characters of finite-dimensional representations of affine algebras,J.Algebra279 (2004), no. 2, 820–839 MR2078944 (2005f:17002) 6. , Characters and blocks for finite-dimensional representations of quantum affine algebras,Int.Math.Res.Not.2005, no. 5, 257–298 MR2130797 (2006a:17021)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2940 VYJAYANTHI CHARI AND JACOB GREENSTEIN

7. V. Chari and A. Pressley, New unitary representations of loop groups, Math. Ann. 275 (1986), no. 1, 87–104 MR849057 (88f:17029) 8. , Weyl modules for classical and quantum affine algebras. Represent. Theory 5 (2001), 191–223 MR1850556 (2002g:17027) 9. P. I. Etingof and A. A. Moura, Elliptic central characters and blocks of finite dimensional representations of quantum affine algebras, Represent. Theory 7 (2003), 346–373 (electronic) MR2017062 (2004j:17017) 10. G. Fourier and P. Littelmann, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), no. 2, 566–593 MR2323538 11. J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, New York, 1978 MR499562 (81b:17007) 12. M. Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383–413 MR1262212 (95c:17024) 13. , On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117–175 MR1890649 (2002m:17013) 14. H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras,J.Amer.Math.Soc.14 (2001), no. 1, 145–238 (electronic) MR1808477 (2002i:17023) 15. , Extremal weight modules of quantum affine algebras, Adv. Stud. Pure Math. 40 (2004), 343–369 MR2074599 (2005g:17036) 16. S. Eswara Rao, Complete reducibility of integrable modules for the affine Lie (super)algebras, J. Algebra 264 (2003), no. 1, 269–278 MR1980697 (2004b:17016)

Department of Mathematics, University of California, Riverside, California 92521 E-mail address: [email protected] Department of Mathematics, University of California, Riverside, California 92521 E-mail address: [email protected]

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