ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES

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1 REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 15, Pages (December 20, 2011) S (2011) ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES MATTHEW BENNETT, VYJAYANTHI CHARI, JACOB GREENSTEIN, AND NATHAN MANNING Abstract. Let g be a simple finite-dimensional Lie algebra and let A be a commutative associative algebra with unity. Global Weyl modules for the generalized loop algebra g A were defined by Chari and Pressley (2001) and Feigin and Loktev (2004) for any dominant integral weight λ of g by generators and relations and further studied by Chari, Fourier, and Khandai (2010). They are expected to play a role similar to that of Verma modules in the study of categories of representations of g A. One of the fundamental properties of Verma modules is that the space of morphisms between two Verma modules is either zero or one-dimensional and also that any non-zero morphism is injective. The aim of this paper is to establish an analogue of this property for global Weyl modules. This is done under certain restrictions on g, λ and A. A crucial tool is the construction of fundamental global Weyl modules in terms of fundamental local Weyl modules. Introduction In this paper, we continue the study of representations of loop algebras and, more generally, of Lie algebras of the form g A, whereg is a complex finite-dimensional simple Lie algebra and A is a commutative associative algebra with unity over the complex numbers. To be precise, we are interested in the category I A of g A- modules which are integrable as g-modules. One motivation for studying this category is that it is closely related to the categories F and F q of finite-dimensional representations, respectively, of affine Lie and quantum affine algebras, which has been a subject of considerable interest in recent years. The categories I A, F and F q fail to be semi-simple, and it was proved in [5] that irreducible representations of the quantum affine algebra specialize at q = 1 to reducible indecomposable representations of the loop algebra (obtained by taking A = C[t, t 1 ]). This phenomenon is analogous to the one observed in modular representation theory, where an irreducible finite-dimensional representation in characteristic zero becomes reducible by passing to characteristic p and is called a Weyl module. This analogy motivated the definition of Weyl modules (global and local) for loop algebras in [5]. Their study was pursued for more general rings A in [3] and [6]. Thus, given any dominant integral weight of the semi-simple Lie algebra g one can define an infinite-dimensional object W A (λ)ofi A, called the global Weyl module,via generators and relations. It was shown (see [3] for the most general case) that if A is finitely generated, then W A (λ) is a right module for a certain commutative finitely generated associative algebra A λ, which is canonically associated with A and λ. Received by the editors December 2, 2010 and, in revised form, March 9, Mathematics Subject Classification. Primary 17B10, 17B37. The second and third authors were partially supported by DMS (V.C.) and DMS (J.G.). c 2011 American Mathematical Society Reverts to public domain 28 years from publication 733

2 734 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING The local Weyl modules are obtained by tensoring the global Weyl module W A (λ) over A λ with simple A λ -modules or, equivalently, can be given via generators and relations. The local Weyl modules have been useful in understanding the blocks of the category F A of finite-dimensional representations of g A, and the quantum analogs are used to understand the blocks in F q. One motivation for this paper is to explore the use of the global Weyl modules to further understand the homological properties of the categories F A and I A. The global Weyl modules have nice universal properties, and one expects them to play a role similar to that of the Verma modules M(λ) in the study of the Bernstein-Gelfand-Gelfand category O for g. One of the most basic results about Verma modules is that Hom g (M(λ),M(μ)) is of dimension at most one and also that any non-zero map is injective. In this paper we prove the following analogue of this result. Theorem 1. Suppose that g = sl r+1 or g = sp 2r and A = C[t] or C[t ±1 ].Then { Hom g A (W A (μ),w A (λ)) 0, μ λ, = Aλ A λ, μ = λ, and every non-zero element of Hom g A (W A (λ),w A (λ)) is injective. A more general version of this statement is provided in Theorem 3. Note that instead of having dimension one, the Hom-spaces become free modules of rank one over A λ. This of course reflects the fact that the top weight space of M(λ) is one-dimensional, while for W A (λ) itisisomorphictoa λ as a A λ -module. Also, Hom g A (W A (μ),w A (λ)) = 0 unless μ λ in the standard partial order (cf. Corollary 1.9). It should be noted that, although the vanishing of Hom for μ < λ definitely fails for g of other types, as shown in the remark in 6.1, we still expect that all non-zero elements of Hom g A (W A (μ),w A (λ)) are injective. As is natural, one is guided by the representation theory of quantum affine algebras and the phenomena occurring in F q. It is interesting to note that Theorem 3, for instance, fails exactly when the irreducible finite-dimensional module in the quantum case specializes to a reducible module for the loop algebra. We conclude by noting that global Weyl modules are also defined for quantum affine algebras and similar questions can be posed in the quantum case. Some of these have been studied in [1] using crystal bases. However, it should be noted that the results in the quantum case do not specialize to the classical case are sometimes rather different. One of the reasons is that the local fundamental Weyl module is always irreducible in the quantum case, while its classical analogue is often reducible. We now explain in more detail the organization and results of this paper. In Section 1 we fix the notation and recall the basic facts on global Weyl modules that will be necessary for the sequel. The main results are formulated and discussed in Section 2, together with their relation with the quantum case. Section 3 contains properties of the algebras A λ and local Weyl modules which are needed later. One of the principal difficulties in working with global Weyl modules is their abstract definition. However, when A is the ring of (generalized) Laurent polynomials, it admits a natural bialgebra structure which can be used to reconstruct a fundamental global Weyl module W A (ω i ) from a local Weyl module. This explicit realization of fundamental Weyl modules plays a crucial role in proving our main results and occupies Section 4 (Proposition 4.4). It is not hard to deduce from the definition

3 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES 735 of global Weyl modules that there exists a canonical map from W A (λ) to a suitable tensor product of global fundamental Weyl modules. Our next result (Theorem 2 and its Corollary 2.2) studies the more general problem of describing the space of g A-module homomorphisms from W A (λ) to an arbitrary tensor product of fundamental global Weyl modules. The structure of the fundamental local Weyl modules was given in [3] for all classical simple Lie algebras and for certain nodes of exceptional Lie algebras. This information and the realization of fundamental global Weyl modules provided by Proposition 4.4 allows one to establish Theorem 2. In Theorem 3, we use Theorem 2 to compute Hom g A (W A (λ),w A (μ)) when A is the ring of polynomials or Laurent polynomials in one variable and under suitable conditions on μ. We are able to do this because one has a lot of information (see [1, 4, 5, 7, 9, 11]) on global Weyl modules in this case. We are also able to use a result of [6] and Theorem 2 of this paper to compute Hom g A (W A (λ),w A (μ)) when g is of type sl r+1, μ is a multiple of the first fundamental weight and A is the (generalized) ring of Laurent polynomials in several variables. This is done in Section 6 of the paper. A crucial step is to show that every non-zero homomorphism from W A (λ) to a suitable tensor product of fundamental global Weyl module is injective (Proposition 6.2). 1. Global Weyl modules In this section we establish the notation to be used in the rest of the paper and then recall the definition and some elementary properties of the global Weyl modules Let C be the field of complex numbers and let Z (respectively Z + )betheset of integers (respectively non-negative integers). Given two complex vector spaces V, W let V W (respectively, Hom(V,W)) denote their tensor product over C and (respectively the space of C-linear maps from V to W ). Given a commutative and associative algebra A over C, letmaxa be the maximal spectrum of A and denote A Mod the category of left A-modules. Given a right A-module M andanelementm M, the annihilating (right) ideal of m is Ann A m = {a A : m.a =0} For a complex Lie algebra a let U(a) be the associated universal enveloping algebra. The assignment x x 1+1 x for x a extends to a homomorphism of algebras Δ : U(a) U(a) U(a) and defines a bialgebra structure on U(a). In particular, if V,W are two a-modules, then V W and Hom C (V,W) are naturally U(a)-modules and W V = V W as U(a)-modules. One can also define the trivial a-module structure on C and we have V a = {v V : av =0} = Hom a (C,V). Suppose that A is an associative commutative algebra over C with unity. Then a A is naturally a Lie algebra, with the Lie bracket given by [x a, y b] =[x, y] ab, x, y a, a,b A. We shall identify a with the Lie subalgebra a 1ofa A. Note that for any algebra homomorphism ϕ : A A the canonical map 1 ϕ : g A g A is a homomorphism of Lie algebras and hence induces an algebra homomorphism U(g A) U(g A ).

4 736 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING 1.3. Let g be a finite-dimensional complex simple Lie algebra with a fixed Cartan subalgebra h. Let Φ be the corresponding root system and fix a set {α i : i I} h (where I = {1,...,dim h}) of simple roots for Φ. The root lattice Q is the Z-span of the simple roots while Q + is the Z + -span of the simple roots, and Φ + =Φ Q + denotes the set of positive roots in Φ. Let ht : Q + Z + be the homomorphism of free semi-groups defined by setting ht(α i )=1,i I. The restriction of the Killing form κ : g g C to h h induces a non-degenerate bilinear form (, ) onh, and we let {ω i : i I} h be the fundamental weights defined by 2(ω j,α i ) = δ i,j (α i,α i ), i, j I. Let P (respectively P + )bethez (respectively Z + ) span of the {ω i : i I} and note that Q P. Given λ, μ P we say that μ λ if and only if λ μ Q +. Clearly, is a partial order on P.The set Φ + has a unique maximal element with respect to this order which is denoted by θ and is called the highest root of Φ +. From now on, we normalize the bilinear form on h so that (θ, θ) = Given α Φ, let g α be the corresponding root space and define subalgebras n ± of g by n ± = g ±α. α Φ + We have isomorphisms of the vector spaces (1.1) g = n h n +, U(g) = U(n ) U(h) U(n + ). For α Φ +, fix elements x ± α g ±α and h α h spanning a Lie subalgebra of g isomorphic to sl 2, i.e., we have [h α,x ± α ]=±2x ± α, [x + α,x α ]=h α, and more generally, assume that the set {x ± α : α Φ + } {h i := h αi : i I} is a Chevalley basis for g Given an h-module V,wesaythatV is a weight module if V = μ h V μ, V μ = {v V : hv = μ(h)v, h h}, and we set wt V = {μ h : V μ 0}. IfdimV μ < for all μ h,letchv be the character of V, namely the element of the group ring Z[h ] given by, ch V = μ h dim V μ e(μ), where e(μ) Z[h ] is the element corresponding to μ h. Observe that for two such modules V 1 and V 2,wehave ch(v 1 V 2 )=chv 1 +chv 2, ch(v 1 V 2 )=chv 1 ch V For λ P +,letv(λ) betheleftg-module generated by an element v λ with defining relations hv λ = λ(h)v λ, x + α i v λ =0, (x α i ) λ(h α )+1 i v λ =0, where h h and i I. ItiswellknownthatV (λ) is an irreducible weight module, and dim V (λ) <, dim V (λ) λ =1, wt V (λ) λ Q +.

5 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES 737 If V is a finite-dimensional irreducible g-module, then there exists a unique λ P + such that V isomorphic to V (λ). We shall say that V is a locally finite-dimensional g-module if dim U(g)v <, v V. It is well known that a locally finite-dimensional g-module is isomorphic to a direct sum of irreducible finite-dimensional modules; moreover, V μ V n+ = Homg (V (μ),v), μ P Assume from now on that A is an associative commutative algebra over C with unity. We recall the definition of the global Weyl modules. These were first introduced and studied in the case when A = C[t, t 1 ] in [5] and then later in [6] in the general case. We shall, however, follow the approach developed in [3]. Definition. For λ P +,theglobal Weyl module W A (λ)istheleftu(g A)-module generated by an element w λ with defining relations, (n + A)w λ =0, hw λ = λ(h)w λ, (x α i ) λ(h α )+1 i w λ =0, where h h and i I. It is clear that W A (λ) is not isomorphic to W A (μ) ifλ μ. Suppose that ϕ is an algebra automorphism of A. The defining ideal of W A (λ) is clearly preserved by the automorphism of U(g A) induced by the Lie algebra automorphism 1 ϕ of g A (cf. 1.2), and we have an isomorphism of g A-modules, (1.2) W A (λ) = (1 ϕ) W A (λ) The following construction immediately shows that W A (λ) is non-zero. Given any ideal I of A, define an action of g A on V (λ) A/I by (x a)(v b) =xv āb, x g, a A, b A/I, where ā is the canonical image of a in A/I. In particular, if a/ I and h h is such that λ(h) 0wehave (1.3) (h a)(v λ 1) = λ(h)v λ ā 0. Clearly, if I Max A, then V (λ) A/I = V (λ) as g-modules and hence v λ 1 generates V (λ) A/I as a g-module (and so also as a g A-module). Since v λ 1 satisfies the defining relations of W A (λ), we see that V (λ) A/I is a non-zero quotient of W A (λ) Given a weight module V of g A, and a Lie subalgebra a of g A, set Vμ a = V μ V a, μ h. The following lemma is elementary. Lemma. For λ P + the module W A (λ) is a locally finite-dimensional g-module and we have W A (λ) = η Q + W A (λ) λ η,

6 738 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING which, in particular, means that wt W A (λ) λ Q +. If V is a g A-module which is locally finite-dimensional as a g-module, then we have an isomorphism of vector spaces, Hom g A (W A (λ),v) = V n+ A λ. We note the following immediate corollary that further extends the analogy with Verma modules. Corollary. Hom g A (W A (λ),w A (μ)) = 0 unless λ μ The weight spaces W A (λ) λ η are not necessarily finite-dimensional, and to understand them we proceed as follows. It is easily checked that one can regard W A (λ) as a right module for U(h A) by setting (uw λ )(h a) =u(h a)w λ, u U(g A), h h, a A. Since U(h A) is commutative, the algebra A λ defined by A λ = U(h A)/ Ann U(h A) w λ is a commutative associative algebra. It follows that W A (λ) isa(g A, A λ )- bimodule and that for all η Q +, the weight space W A (λ) λ η is a right A λ -module. Clearly, W A (λ) λ =Aλ A λ, where we regard A λ as a right A λ -module through the right multiplication. The following is immediate. Lemma. For λ P +, η Q + the subspaces W A (λ) n+ λ η and W A(λ) n+ A λ η are A λ - submodules of W A (λ) and we have W A (λ) n+ λ = W A (λ) n+ A λ = W A (λ) λ. Further, if μ P +, the space Hom g A (W A (μ),w A (λ)) has the natural structure of arighta λ -module and Hom g A (W A (μ),w A (λ)) = Aλ W A (λ) n+ A μ. Lemma. For λ P +, a A λ the assignment w λ w λ a extends to a well-defined homomorphism W A (λ) W A (λ) of g A-modules and we have Hom g A (W A (λ),w A (λ)) = Aλ A λ =Aλ W A (λ) n+ A λ. Proof. Since W A (λ) isa(g A, A λ )-bimodule, it follows that w λ a satisfies the defining relations of W A (λ) which yield the first statement of the Lemma. For the second, let π : W A (λ) W A (λ) be a non-zero g A-module map. Since W A (λ) λ = U(h A)w λ,thereexistsu π U(h A) such that π(w λ )=u π w λ.since π is non-zero, the image ũ π of u π in A λ is non-zero. Thus, we obtain a well-defined map Hom g A (W A (λ),w A (λ)) A λ given by π ũ π. It is clear that the map is an isomorphism of right A λ -modules and the lemma is proved.

7 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES The main results 2.1. For s =(s i ) i I Z I +,set (2.1) A s = A s i ω i, W A (s) = W A (ω i ) s i, i I i I w s = i I w s i ω i, where all tensor products are taken in the same (fixed) order. Given k, l Z + let R k,l be the algebra C[t ±1 1,...,t±1 k,u 1,...,u l ] with the convention that if k = 0 (respectively l = 0), R 0,l (respectively R k,0 ) is just the ring of polynomials (respectively Laurent polynomials) in l (respectively k) variables The main result of this paper is the following. Theorem 2. Assume that A = R k,l for some k, l Z +. For all s Z n + and μ P +, we have (2.2) Hom g A (W A (μ),w A (s)) ( ) = As W A (s) n+ A μ = (W A (ω i ) n+ A ) s i Inthecasewhengis a classical simple Lie algebra, we can make (2.2) more precise. Let I 0 be the set of i I such that α i occurs in θ with the coefficient 2/(α i,α i ). In particular, I 0 = I for g of type A or C. Givens =(s i ) i I Z I + and λ P +,letc s (λ) Z + be the coefficient of e(λ) in e(ω i ) si ( ) j + k 1 e(ω i 2j ) si, j i I 0 i/ I 0 0 j i/2 where ω 0 =0. Corollary. Let λ P + and s Z I +. Assume either that g is not an exceptional Lie algebra or that s i =0if i/ I 0. We have Hom g A (W A (λ),w A (s)) = As A c s(λ) s, where we use the convention that A c s(λ) s =0if c s (λ) = Our next result is the following. Recall from Lemma 1.11 that for all λ P + we have Hom g A (W A (λ),w A (λ)) = Aλ A λ. Theorem 3. Let A be the ring R 0,1 or R 1,0. For all μ = i I s iω i P + with s i =0if i/ I 0,andallλ P +, we have Hom g A (W A (λ),w A (μ)) = 0, if λ μ. Further, any non-zero element of Hom g A (W A (μ),w A (μ)) is injective. An analogous result holds when A = R k,l, k, l Z +, g = sl n+1 and μ = sω We now make some comments on the various restrictions in the main results. The proof of Theorem 2 relies on an explicit construction of the fundamental global Weyl modules in terms of certain finite-dimensional modules called the fundamental local Weyl modules. A crucial ingredient of this construction is a natural bialgebra structure of R k,l. The proof of Corollary 2.2 depends on a deeper understanding of the g-module structure of the local fundamental Weyl modules. These results are unavailable for the exceptional Lie algebras when k + l>1. In the case when k + l = 1, the structure of these modules for the exceptional algebras is known as a consequence of the work of many authors on the Kirillov-Reshetikhin conjecture i I μ.

8 740 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING (see [2] for extensive references on the subject). Hence, a precise statement of Corollary 2.2 could be made when k + l = 1 in a case by case and in a not very compact fashion. The interested reader is referred to [8] and [10] Before discussing Theorem 3, we make the following conjecture. Conjecture. Let A = R k,l for some k, l Z +. Then for all λ P + and s Z n +, any non-zero element of Hom g A (W A (λ),w A (s)) is injective. The proof of Theorem 3 will rely on the fact that this conjecture is true (see Section 5 of this paper) when k + l =1ands i =0ifi/ I 0 as well as on the fact that the fundamental local Weyl module is irreducible as a g-module if i I 0.We shall prove in Section 5 using Corollary 2.2 and the work of [6] that the conjecture is also true when g = sl r+1 and λ = sω 1. Remark 6.1 of this paper shows that Hom g A (W A (λ),w A (μ)) can be non-zero if we remove the restriction on μ Finally, we make some remarks on quantum analogs of this result. In the case of the quantum loop algebra, one also has analogous notions of global and local Weyl modules which were defined in [5], and one can construct the global fundamental Weyl module from the local Weyl module in a way analogous to the one given in this paper for A = C[t, t 1 ]. It was shown in [5] for the quantum loop algebra of sl 2 that the canonical map from the global Weyl module into the tensor product of fundamental global Weyl modules is injective. For the general quantum loop algebra, this was established by Beck and Nakajima ([1]) using crystal and global bases. They also describe the space of extremal weight vectors in the tensor product of quantum fundamental global Weyl modules. 3. The algebra A λ and the local Weyl modules In this section we recall some necessary results from [3] and also the definition and elementary properties of local Weyl modules For r Z + the symmetric group S r acts naturally on A r and on (Max A) r and we let (A r ) S r be the corresponding ring of invariants and (Max A) r /S r the set of orbits. If r = r r n, then we regard S r1 S rn as a subgroup of S r in the canonical way, i.e., S r1 permutes the first r 1 letters, S r2 the next r 2 letters and so on. Given λ = i I r iω i P +,set (3.1) (3.2) r λ = r i, S λ = S r1 S rn, A λ =(A r λ ) S λ, i I Max A λ =(MaxA) r λ /S λ. The algebra A λ is generated by elements of the form (3.3) sym i λ(a) =1 (r 1+ +r i 1 ) ( r i 1 k=0 The following was proved in [3, Theorem 4]. 1 k a 1 (r i k 1) ) 1 (r i+1+ +r n ), a A, i I.

9 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES 741 Proposition. Let A be a finitely generated commutative associative algebra over C with trivial Jacobson radical. Then the homomorphism of associative algebras U(h A) A λ defined by h i a sym i λ(a), i I, a A induces an isomorphism of algebras sym λ : A λ Aλ. In particular, if A is a finitely generated integral domain, then A λ is isomorphic to an integral subdomain of A r For λ, μ P +,itisclearthatthetensorproductw A (λ) W A (μ) hasthe natural structure of a (g A, A λ A μ )-module. We recall from [3] that, in fact, there exists a (g A, A λ+μ )-bimodule structure on W A (λ) W A (μ). It is clear from Definition 1.7 that the assignment w λ+μ w λ w μ defines a homomorphism τ λ,μ : W A (λ + μ) W A (λ) W A (μ) ofg A-modules. The restriction of this map to W A (λ + μ) λ+μ induces a homomorphism of algebras A λ+μ A λ A μ as follows. Consider the restriction of the comultiplication Δ of U(g A) tou(h A). It is not hard to see that Ann U(h A) U(h A) (w λ w μ ) Ann U(h A) w λ U(h A)+U(h A) Ann U(h A) w μ, and hence we have Δ(Ann U(h A) (w λ+μ )) Ann U(h A) w λ U(h A)+U(h A) Ann U(h A) w μ. It is now immediate that the comultiplication Δ : U(h A) U(h A) U(h A) induces a homomorphism of algebras Δ λ,μ : A λ+μ A λ A μ. This endows any right A λ A μ -module (hence, in particular, W A (λ) W A (μ)) with the structure of a right A λ+μ -module. It was shown in [3] that τ λ,μ is then a homomorphism of (g A, A λ+μ )-bimodules. Summarizing, we have Lemma. Let λ s P +, 1 s k and let λ = k s=1 λ s. The natural map W A (λ) W A (λ 1 ) W A (λ k ) given by w λ w λ1 w λk is a homomorphism of (g A, A λ )-bimodules For λ P +,letwa λ be the right exact functor from A λ Mod to the category of g A-modules given on objects by WAM λ = W A (λ) Aλ M, M A λ Mod. Clearly, WA λ M is a weight module for g and we have isomorphisms of vector spaces (WAM) λ λ η = (WA (λ)) λ η Aλ M, η Q +, (WAM) λ λ = (WA (λ) λ ) Aλ M = w λ C M. Moreover, WA λ M is generated as a g A-module by the space w λ C M and WAM λ = W μ A N λ = μ, M = Aλ N. The isomorphism classes of simple objects of A λ Modaregivenbythemaximal ideals of A λ.giveni Max A λ,thequotienta λ /I is a simple object of A λ Mod and has dimension one. The g A-modules WA λ A λ/i are called the local Weyl modules and when λ = ω i, i I we call them the fundamental local Weyl modules. It follows that (WA λ A λ/i) λ is also a one-dimensional vector space spanned by w λ,aλ /I = w λ 1. We note the following corollary.

10 742 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING Corollary. Suppose that M A λ Mod is finite-dimensional. Then WA λ M is finite-dimensional. In particular, the local Weyl modules are finite-dimensional and have a unique irreducible quotient VA λ M We note the following consequence of Proposition 3.1. Lemma. For i I, we have A ωi = A and WA (ω i ) is a finitely generated right A-module. The fundamental local Weyl modules are given by W ω i A A/I = W A(ω i ) A A/I, I Max A. In particular, we have (h a)w ωi,a/i =0, (h b)w ωi,a/i = ω i (h)w ωi b, h h, a I, b A The following lemma is a special case of a result proved in [3] and we include the proof in this case for the reader s convenience. Lemma. Let A be a finitely generated, commutative associative algebra. For I Max A there exists N Z + such that for all i I, (3.4) (g I N )W ω i A A/I =0. Proof. Since g is a simple Lie algebra (3.4) follows if we show that there exists N 0 such that (x θ IN )W ω i A A/I =0. Since W ω i A A/I = U(n A)w ωi,a/i, [x θ, n ]=0, it is enough to show that there exists N 0 such that (x θ IN )w ωi,a/i =0. We claim that is a consequence of showing that for j I there exists N j Z +, with (3.5) (x α j I N j )w ωi,a/i =0. To prove the claim, write x θ =[x α i1 [ [x α ip 1,x α ] ]forsomei i 1,...,i p p I and take N = p j=1 N i j. To prove (3.5), observe first that for j i I, k I and for all a A, x + α k (x α j a)w ωi,a/i = δ k,j (h j a)w ωi,i =0, by Lemma 3.4 and the defining relations of W A (ω i ). Thus, (x α j a)w ωi,a/i (W ω i A A/I)n+ and since ω i α j / P + we conclude that (x α j A)w ωi,a/i =0, j i. If j = i, then 0=(x + α i a)(x α i ) 2 w ωi =2((x α i 1)(h i a) (x α i a))w ωi. By Lemma 3.4, we have (h i a)w ωi,a/i =0ifa I, andsoweget and (3.5) is established. (x α i a)w ωi,a/i =0, a I

11 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES Fundamental global Weyl modules In this section we establish the main tool for proving Theorem 2. It is not, in general, clear how (or even if it is possible) to reconstruct the global Weyl module from a local Weyl module. The main result of this section is that it is possible to do so when λ = ω i and A = R k,l for some k, l Z We begin with a general construction. The Lie algebra (g A) A acts naturally on V A for any g A-module V. Suppose that A is a bialgebra with the comultiplication h : A A A (it is useful to recall that A is a commutative associative algebra with identity). Then the comultiplication map h induces a homomorphism of Lie algebras 1 h : g A g A A (cf. 1.2) and thus a g A-module structure on V A. Explicitly, the (g A, A)-bimodule structure on V A is given by the following formulas: (x a)(v b) = s (x a s)v a s b, (v b)a = v ba, v V,a,b A, where h(a) = s a s a s. We denote this bimodule by (V A) h and observe that it is a free right A-module of rank equal to dim C V. Itistrivialtoseethat(V A) h is a weight module for g A if V is a weight module for g A and that ((V A) h ) μ = V μ A, V n+ A A (V A) n+ A h. Moreover, if V 1,V 2 are g A-modules, one has a natural inclusion: (4.1) Hom g A (V 1,V 2 ) Hom g A ((V 1 A) h, (V 2 A) h ), η η 1. In particular, if V is reducible, then (V A) h is also a reducible (g A)-module Let h k,l : R k,l R k,l R k,l be the comultiplication given by h k,l (t ±1 s )=t ±1 s t ±1 s, h k,l (u r )=u r 1+1 u r, where 1 s k and 1 r l. Any monomial m R k,l can be written uniquely as a product of monomials: m = m u m t, m t C[t ±1 1,...,t±1 k ], m u C[u 1,...,u l ]. Set deg t ±1 s = ±1 and deg u r = 1 for 1 s k, 1 r l and let deg t m (respectively, deg u m) be the total degree of m t (respectively, m u )andforany f A define deg t f and deg u f in the obvious way. The next lemma is elementary. Lemma. Let m = m t m u be a monomial in R k,l. Then m t R k,l, h k,l(m t )= m t m t and (4.2) h k,l (m) =m m t + q m u,qm t m u,qm t = m t m+ q m u,qm t m u,qm t, where m u,q, m u,q are (scalar multiples of) monomials in the u r, 1 r l, such that if m u,q 0, m u,q 0,thendeg u m u,q < deg u m and deg u m u,q +deg u m u,q = deg u m.

12 744 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING 4.3. For the rest of the section A denotes the algebra R k,l for some k, l Z + and I the ideal of A generated by the elements {t 1 1,...,t k 1,u 1,...,u l }. Suppose that J Max R k,l. It is clear that there exists an algebra automorphism ϕ : R k,l R k,l such that ϕ(i) = J. As a consequence, we have an induced isomorphism of g A-modules, (4.3) W ω i A A/I = (1 ϕ) W ω i A A/J. Moreover, if we set h ϕ k,l =(ϕ ϕ) h k,l ϕ 1 : A A A, then h ϕ k,l also defines a bialgebra structure on A andwehaveanisomorphismof g A-modules (4.4) (1 ϕ) (W ω i A A/I A) h k,l = (W ω i A A/J A) h ϕ k,l. This becomes an isomorphism of (g A, A)-bimodules if we twist the right A-module structure of (W ω i A A/J A) h ϕ by ϕ. k,l 4.4. We now reconstruct the global fundamental Weyl module from a local one. Proposition. For all i I, the assignment w ωi w ωi,a/i 1 defines an isomorphism of (g A, A)-bimodules W A (ω i ) = (W ω i A A/I A) h k,l. Remark. It is clear from (4.4) and Section 1.7 that one can work with an arbitrary ideal J provided that h k,l is replaced by h ϕ k,l,whereϕis the unique automorphism of A such that ϕ(i) =J. Proof. The element w ωi,a/i 1 (W ω i A A/I A) h satisfies the relations in Definition 1.7 and hence the assignment w ωi w ωi,a/i 1 defines a homomorphism of g A-modules p : W A (ω i ) (W ω i A A/I A) h k,l. We begin by proving that p is a homomorphism of right A-modules. By Lemma 3.4 and the definition of the right module structure on W A (ω i )weseethat p((uw ωi )a) =p(u(h i a)w ωi )=u(h i a)(w ωi,a/i 1), u U(g A), a A. This shows that it is enough to prove that for any monomial m in A, wehave (4.5) (h i m)(w ωi,a/i 1) = w ωi,a/i m. Write m = m t m u and observe that m t 1 I while deg u m > 0 = m I. Using (4.2) we get h k,l (m) 1 m I A and since (h i I A)(w ωi,a/i 1) = 0 by Lemma 3.4, we have established (4.5). To prove that p is surjective we must show that U(g A)(w ωi,a/i 1) = (W ω i A A/I A) h k,l, and the remarks in Section 4.1 show that it is enough to prove (4.6) (W ω i A A/I) ω i η A U(g A)(w ωi,a/i 1), η Q +. The argument is by induction on ht η, the induction base with ht η = 0 being immediate from (4.5). For the inductive step assume that we have proved the

13 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES 745 result for all η Q + with ht η<k. To prove the result for htη = k, it suffices to prove that for all j I and all monomials m in A we have ((x α j m)w) g U(g A)(w ωi,a/i 1), where w (W ω i A A/I) ω i η with ht η = k 1andg A. For this, we argue by a further induction on deg u m.ifdeg u m =0,thenm = m t and we have ((x α j m)w) g =(x α j m)(w m 1 g) U(g A)(W ω i A A/I) ω i η. This proves that the induction on deg u m starts. If deg u m > 0weuse(4.2)toget ((x α j m)w) g =(x α j m)(w m 1 t g) q ((x α j m u,qm t )w) m u,qg. Both terms on the right-hand side are in U(g A)(W ω i A A/I) ω i η : the first by the induction hypothesis on ht η and the second by the induction hypothesis on deg u m. This completes the proof of the surjectivity of p. To prove that p is injective, recall from Section 4.1 that (W ω i A A/I A) h k,l is a free right A-module of rank equal to the dimension of W ω i A A/I. Hence if K is the kernel of p, we have an isomorphism of right A-modules, W A (ω i ) = K (W ω i A A/I A) h k,l. Using (4.3) we see that for any maximal ideal J in A, dim(w A (ω i ) A A/J) =dimw ω i A A/J =dimwω i A A/I =dim((w ω i A A/I A) h k,l A A/J). Therefore, K A A/J = 0 and so by Nakayama s Lemma, K J =0forallJ Max A. Thus, K =0. 5. Proof of Theorem 2 and Corollary 2.2 We continue to assume that A = R k,l, k, l Z + and that I is the maximal ideal of A generated by {t 1 1,...,t k 1,u 1,...,u l }. We also use the comultiplication h k,l and denote it by just h. LetM A A be the set of monomials in the generators u r, t ±1 s,1 r l, 1 s k The following proposition, together with Lemma 3.5 and Proposition 4.4, completes the proof of Theorem 2. Proposition. Suppose that V s, 1 s M are g A-modules such that there exists N Z + with (5.1) (n + I N )V s =0, 1 s M. Then ((V 1 A) h (V M A) h ) n+ A =(V n+ A 1 A) h (V n+ A M A) h.

14 746 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING 5.2. We need some notation. Let V be a g A-module and let K Z +. Define V K = {v V :(n + m)v =0, m M A, deg t m K}. Note that V 0 = V n+ A. The first step in the proof of Proposition 5.1 is the following. Lemma. Let V be a g A-module and K Z +.Then (5.2) ((V A) h ) K = V K A. In particular, (5.3) (V A) n+ A h = V n+ A A. Proof. Let v h (V A) h and write, v h = p v p g p,where{g p } p is a linearly independent subset of A. By (4.2) we have (x m)v h = p (x m)v p m t g p + p,q (x m u,qm t )v p m u,qm t g p, with deg u m u,q < deg u m. Since deg t m u,qm t =deg t m, it follows that V K A ((V A) h ) K. We prove the reverse inclusion by induction on deg u m. Let v h ((V A) h ) K and let deg t m K. Ifdeg u m =0,then 0=(x m)v h = p (x m)v p g p m. Since the set {g p m} p is also linearly independent, we see that (x m)v p =0for all p. Ifdeg u m > 0, we use (4.2) to get 0=(x m)v h = p (x m)v p m t g p + p,q (x m u,qm t )v p m u,qm t g p. Since deg u m u,q < deg u m all terms in the second sum are zero by the induction hypothesis, and the linear independence of the set {m t g p } p gives (x m)v p =0for all p Proposition. Let U, V be g A-modules and suppose that for some N Z +, Then (n + I N )V =0. (5.4) (U (V A) h ) n+ A = U n+ A (V n+ A A). Before proving this proposition, we establish Proposition 5.1. The argument is by induction on M, with (5.3) showing that induction begins at M =1. ForM>1, take U =(V 1 A) h (V M 1 A) h, V = V M. The induction hypothesis applies to U and together with Proposition 5.3 completes the inductive step.

15 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES Lemma. Let A = R k,l with k>0. Let V be a g A-module and suppose that (n + I N )V =0for some N Z +. Then for all K Z + we have V n+ A = V K. Proof. It suffices to prove that V K V K 1 for all K 1. Since (1 t ±1 1 )N I N we have N ( ) N (5.5) 0=(x m(1 t ±1 1 )N )v =(x m)v + ( 1) s (x mt ±s 1 s )v, s=1 for all x n +, m M A and v V. Suppose that v V K and take m M A with deg t m = K 1. If deg t m 0 (respectively, deg t m < 0), then deg t mt s 1 K (respectively, deg t mt s 1 K) for all s>0. Thus we conclude that all terms in the sum in (5.5) with the appropriate sign choice equal zero hence (x m)v =0 and so v V K Lemma. Let A = R 0,l.LetV be a g A-module and suppose that (n + I N )V =0 for some N Z +.LetK N Z +.Then (5.6) V n+ A A = {v h (V A) h :(n + m)v h =0, m M A, deg u m K}. Proof. Since (V A) h is a (g A, A)-bimodule the sets on both sides of (5.6) are right A-modules. Hence if v h is an element of the set on the right-hand side of (5.6), then v h u s j is also in the right-hand side of (5.6) for all s Z +. Write v h = p v p g p,where{g p } p is a linearly independent subset of A. Since the u j, 1 j l are primitive and u s j IN if s N, wehaveforall0 r N, N ( ) 0=(x u (K+N r) j )(v h )u (r) j = ((x u (s) j )v p ) u (K+N r s) j g p u (r) j s=0 p N ( ) K + N s ( = ((x u (s) j )v p ) u (K+N s) j g p ). r s=0 p We claim that the matrix C(N,K) =( ( ) K+N s r )0 s,r N is invertible. Assuming the claim, we get p ((x u (s) j )v p ) u (K+N s) j g p =0, 0 s N, and since the g p are linearly independent this implies that (x u (s) j )v p =0, 0 s N and so (x u s j )v h =0forallx n +, s Z +. Now, let m M A and let α Φ +. Then (h α m)v h is also an element of the right-hand side of (5.6) and hence by the preceding argument, we get 0=(x α 1)(h α m)v h =(h α m)(x α 1)v h 2(x α m)v h = (2x α m)v h, thus proving that v h (V A) n+ A h = V n+ A A by (5.3).

16 748 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING To prove the claim, let u be an indeterminate and let {p r C[u] :0 r N} be a collection of polynomials such that deg p r = r (in particular, we assume that p 0 is a non-zero constant polynomial). Then for any tuple (a 0,...,a N ) C N+1, we have det(p r (a s )) 0 r,s N = c det(a r s) 0 r,s N = c 0 r<s N (a s a r ), where c is the product of highest coeffcients of the p r,0 r N. Since ( u r) is a polynomial in u of degree r with highest coefficient 1/r!, we obtain with a s = N + K s, ( N det C(N,K) = r=1 ) 1 r! 0 r<s N (r s) =( 1) N(N+1)/ Now we have all the necessary ingredients to prove Proposition 5.3. Proof of Proposition 5.3. Let v h (U (V A) h ) n+ A and write v h = p,s w s v s,p g p, where {w s } s and {g p } p are linearly independent subsets of U and A, respectively. We have 0=(x m)v h = ( ) (5.7) ((x m)w s ) v s,p g p + w s (x m)(v s,p g p ) s,p (5.8) = ((x m)w s ) v s,p g p s,p + ( w s (x m)v s,p g p m t + (x m u,qm t )v s,p m u,qm t g p ). s,p q Suppose first that A = R k,l with k>0 and let K =max p deg t g p +1. If m is such that deg t m K, then the set {g p } p is linearly independent from the set {m u,qm t g p } p,q and hence we must have that (5.9) ((x m)w s ) v s,p g p =0, s,p w s (x m)(v s,p g p )=0 Using the linear independence of the elements {w s } s we conclude that for all s, (v s,p g p ) ((V A) h ) K = V K A = V n+ A A, p by (5.2) and Lemma 5.4. This proves Proposition 5.3 in the case when k>0. Suppose now that A = R 0,l and let N Z + be such that (n + I N )V =0. Let K = N +1+max p {deg u g p } and let x n +. If deg u m K, thenm I N and so (x m)v s,p =0. Furthermore,(x m u,q)v s,p 0 implies that deg u m u,q <N. By Lemma 4.2 it follows that deg u m u,q > max p {deg g p }. Therefore, the non-zero terms, if any, in the second sum in (5.8) are linearly independent from those in the first sum and we obtain (5.9). Furthermore, we have 0= (x m u,q)v s,p m u,q g p p {q :deg u m <N} u,q and as before we conclude that (x m u,q)v s,p =0whendeg u m u,q <N. Thus, (x m) ( p v s,p g p ) =0foralls and it remains to apply Lemma 5.5. s,p

17 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES We conclude this section with a proof of Corollary 2.2. The following is a special case of Theorem 7.1 and Proposition 7.7 of [3]. Theorem 4. Let I Max A, wherea = R k,l.then dim(w ω i A A A/I)n+ = dim(w ω i A A/I) ω i =1, i I 0, If g is of type B n or D n and i/ I 0,then where ω 0 =0. dim(w ω i A A/I)n+ A μ = 0, ( μ ω i 2j, j + k 1 j ), μ = ω i 2j,i 2j 0, Another way to formulate this result is: The subspace (W ω i A A is an A/I)n+ h-module with character given by ch(w ω i A A/I)n+ A = Hence, using Theorem 2, we get and Corollary 2.2 follows. j:i 2j 0 ( j + k 1 j ) e(ω i 2j ). chw A (s) n+ A = i I(ch(W ω i A A/I)n+ A ) s i, 6. Proof of Theorem 3 Given s Z I +, define μ s P + by μ s = i I s iω i and let τ s : W A (μ s ) W A (s) be the natural map of (g A, A)-bimodules defined in the above lemma and satisfying τ s (w μs )=w s. Since W A (s) μs = w s A s, we see that any nonzero element τ of Hom g A (W A (μ s ),W A (s)) is given by composing τ s with right multiplication by an element of A s, i.e. τ = τ s a with a A s Lemma. Assume that λ P + and s Z I + satisfy (i) any non-zero element of Hom g A (W A (λ),w A (s)) is injective, (ii) the map τ s : W A (μ s ) W A (s) is injective. Then any non-zero element of Hom g A (W A (λ),w A (μ s )) is injective. Moreover, if s i =0for all i/ I 0,then Hom g A (W A (λ),w A (μ s )) = 0, λ μ s. Proof. Let η Hom g A (W A (λ),w A (μ s )). If η 0,thenτ s η Hom g A (W A (λ), W A (s)) is non-zero since τ s is injective. Hence τ s η is injective which forces η to be injective. If we now assume that λ μ s and that s i =0ifi / I 0,thenit follows from Corollary 2.2 that Hom g A (W A (λ),w A (s)) = 0 and hence it follows that η = 0 in this case. Remark. Using Theorem 4, we see that if g is of type B n or D n with n 6and i =4,then(W ω 4 A A A/I)n+ ω 2 0 or equivalently, Hom g A (W ω 2 A A/I, Wω 4 A A/I) 0.

18 750 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING Using Proposition 4.4 and (4.1) we get Hom g A (W A (ω 2 ),W A (ω 4 )) 0, which in particular proves that the last assertion of the above Lemma and hence Theorem 3 fail in this case From now on, we shall assume that s Z I + is such that s i =0ifi/ I 0. By Corollary 2.2, we see that Hom g A (W A (λ),w A (μ s ))=0ifλ μ s and the first condition of Lemma 6.1 is trivially satisfied. Hence Theorem 3 will follow if we show that μ s satisfies both conditions in Lemma 6.1. By the discussion at the start of Section 5, we see that proving that μ s satisfies the first condition is equivalent to proving that τ s a is injective for all a A s. In other words, Theorem 3 follows if we establish the following. Proposition. Let s Z I + be such that s i =0if i / I 0. For all a A s, the canonical map τ μ a : W A (μ s ) W A (s) given by extending w μs w s a is injective, in the following cases: (i) A = R 0,1 or R 1,0, (ii) A = R k,l, g = sl n+1 and s =(s, 0,...,0) Z I +, s>0. The rest of the section is devoted to proving the proposition We begin by proving the following Lemma. Lemma. Let A be a finitely generated integral domain. Let s Z I + be such that s i =0if i/ I 0.Thenτ μs a : W A (μ s ) W A (s) is injective for a A s \{0} if and only if τ μs is injective. Proof. Consider the map ρ a : W A (s) W A (s) givenby ρ a (w) =wa, w W A (s). This is clearly a map of (g A, A s )-bimodules. Since W A (s) μs = w s A s, and A s is an integral domain, it follows that that the restriction of ρ a to W A (s) μs is injective, and so ker ρ a W A (s) μs = {0}. Since wt W A (s) μ s Q +, it follows that if ker ρ a is non-zero, there must exist w ker ρ a with (n + A)w =0. But this is impossible by Corollary 2.2 and hence ker ρ a =0. Sinceτ μs a = ρ a τ μs, the Lemma follows We now prove that τ μs is injective. This was proved in [5] for g = sl 2 and A = R 1,0 and in [6] for g = sl n+1, s =(s, 0,...,0) Z n +, s>0 and for any finitely generated integral domain A. Since τ s W A (μ s ) μs (U(g A)w =Aμ s s) μs A =Aμ s μ s, the following proposition completes the proof of Proposition 6.2. Proposition. Let μ P + and let π : W A (μ) W be a surjective map of (g A, A μ )-bi-modules such that the restriction of π to W A (μ) μ is an isomorphism of right A μ -modules. If A = R 0,1 or R 1,0 and μ = i I 0 s i ω i, then π is an isomorphism.

19 ON HOMOMORPHISMS BETWEEN GLOBAL WEYL MODULES Assume from now on that A is either R 0,1 or R 1,0. The following is well-known. Proposition. For all r Z +,thering(r r 0,1 )S r is isomorphic to R 0,r and (R r 1,0 )S r is isomorphic to C[t 1,t 2,...,t r,t 1 r ]. The proposition implies that Max A λ is an irreducible variety. Given λ P +, define D λ Max A λ by: I D λ if and only if the S rλ -orbit of sym λ I is of maximal size, i.e, sym λ I is the S rλ -orbit of ((t a 1,1 ),...,(t a 1,r1 ),...,(t a n,1 ),...,(t a n,rn )) (Max A) r λ for some a i,r C (respectively a i,r C ) with a i,r a j,s if (i, r) (j, s). The set of such orbits is clearly Zariski open in Max A λ. Since sym λ induces an isomorphism of algebraic varieties Max A λ Max A λ,we conclude that D λ is Zariski open, hence is dense in Max A λ. Therefore, given any non-zero a A λ there exists I D λ with a / I We shall need the following theorem. Theorem 5. Let A = R 0,1 or R 1,0 and let λ = i I r iω i P +. (i) The right A λ -module W A (λ) is free of rank d λ,where d λ = i I(dim W ω i A (A/I))r i, for any I Max A. (ii) Let I D λ.then WA(A λ λ /I) r i = W ω i A (A/I i,r), i I r=1 where I i,r Max A is the ideal generated by (t a i,r ). If, in addition, we have r i =0for i/ I 0,thenW λ A (A λ/i) is an irreducible g A- module. Part (i) of the Theorem was proved in [5] for sl 2,in[4]forsl r+1 and in [7] for algebras of type A, D, E. The general case can be deduced from the quantum case, using results of [1, 9, 11]. Part (ii) of the Theorem was proved in [5] in a different language and in [3] in the language of this paper Proof of Proposition 6.4. Let {w s } 1 s dμ be an A μ -basis of W A (μ) (cf. Theorem 5(i)). Then for all I max A μ, {w s 1} 1 s dμ is a C-basis of W μ A A μ/i. Suppose that w ker π and write w = d μ s=1 w s a s, a s A μ. If w 0,leta be the product of the non-zero elements of the set {a s :1 s d μ }. Since A μ is an integral domain we see that a 0. By the discussion in Section 6.5 we can choose I D μ with a / I. Then a s 0 implies that a s / I and hence w := w 1= d μ s=1 w s ā s 0,whereā s is the canonical image of a s in A μ /I. Notice that Theorem 5(ii) implies that W μ A (A μ/i) isasimpleg A-module. Since π is surjective, W is generated by π(w μ ). Setting W = π(w A (μ)i), we see that W μ = π((w A (μ)i) μ )=π(w μ )I.

20 752 M. BENNETT, V. CHARI, J. GREENSTEIN, AND N. MANNING In particular, this proves that π(w μ ) / W, hence W is a proper submodule of W and (W/W )I =0. This implies that π induces a well-defined non-zero surjective homomorphism of g A-modules π : W μ A (A μ/i) W/W 0. In fact, since W μ A (A μ/i) issimple, we see that π is an isomorphism. But now we have 0=π(w) = π( w), forcing w = 0 which is a contradiction caused by our assumption that w 0. The proof of Proposition 6.4 is complete. References [1] J. Beck and H. Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, MR (2005e:17020) [2] V. Chari and D. Hernandez, Beyond Kirillov-Reshetikhin modules, Quantum affine algebras, extended affine Lie algebras, and their applications, Contemp. Math., vol. 506, Amer. Math. Soc., Providence, RI, 2010, pp MR [3] V. Chari, G. Fourier, and T. Khandai, A categorical approach to Weyl modules, Transform. Groups 15 (2010), no. 3, MR [4] V. Chari and S. Loktev, Weyl, Demazure and fusion modules for the current algebra of sl r+1, Adv. Math. 207 (2006), no. 2, MR (2008a:17029) [5] V.ChariandA.Pressley,Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), (electronic). MR (2002g:17027) [6] B. Feigin and S. Loktev, Multi-dimensional Weyl modules and symmetric functions, Comm. Math. Phys. 251 (2004), no. 3, MR (2005m:17005) [7] G. Fourier and P. Littelmann, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), no. 2, MR (2008k:17005) [8] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Remarks on fermionic formula, (Raleigh, NC, 1998), Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp MR (2001m:81129) [9] M. Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, MR (2002m:17013) [10] M. Kleber, Combinatorial structure of finite-dimensional representations of Yangians: the simply-laced case, Internat. Math. Res. Notices 4 (1997), MR (98a:17018) [11] H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J.Amer.Math.Soc.14 (2001), no. 1, MR (2002i:17023) Department of Mathematics, University of California, Riverside, California address: mbenn002@math.ucr.edu Department of Mathematics, University of California, Riverside, California address: vyjayanthi.chari@ucr.edu Department of Mathematics, University of California, Riverside, California address: jacob.greenstein@ucr.edu Department of Mathematics, University of California, Riverside, California address: nmanning@math.ucr.edu