WEYL, DEMAZURE AND FUSION MODULES FOR THE CURRENT ALGEBRA OF slr+1 VYJAYANTHI CHARI AND SERGEI LOKTEV

We construct a Poincare-Birkhoff–Witt type basis for the Weyl modules [CP1] of the current algebra of slr+1. As a corollary we prove the conjecture made in [CP1], [CP2] on the dimension of the Weyl modules in this case. Further, we relate the Weyl modules to the fusion modules defined in [FL] of the current algebra and the Demazure modules in level one representations of the corresponding affine algebra. In particular, this allows us to establish substantial cases of the conjectures in [FL] on the structure and graded character of the fusion modules. Introduction The study of finite–dimensional representations of quantum affine algebras has been of some interest in recent years and there are several different approaches to the subject, [CP1], [FR],[FM],[N],[Ka1] for instance. An approach that was developed in [CP1] was to study these representations by specializing to the case of classical affine algebras. It was noticed in that paper, that an irreducible finite–dimensional representation of a quantum affine algebra, in general specialized to an indecomposable but reducible representation of the affine Lie algebra. This behavior is seen in the representation theory of simple Lie algebras, when passing from the characteristic zero situation to the case of non–zero characteristic. The modules in non–zero characteristic which are obtained from the irreducible finite–dimensional modules 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 [CP1], in terms of generators and relations for both the classical and quantum affine Lie algebras. It was also conjectured there, that the Weyl modules of the affine algebra are the classical (q → 1) limit of the standard modules of the quantum affine algebra defined in [N]. In particular, a proof of the conjecture would give generators and relations for these representations, or equivalently, would prove that the standard modules are isomorphic to the quantum Weyl modules. This latter result was proved recently in [CdM] using some deep geometric results of Nakajima. 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 of the classical Weyl modules could be proved. This latter conjecture was established in [CP1] for sl2. Moreover, it was shown that it was enough to study the analogous modules for the current 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 slr+1, by constructing an explicit basis for the Weyl modules over the current algebra of slr+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 theory of affine and current algebras. Thus, we are able to establish a substantial case of conjectures in [FL] on the fusion modules for a current Lie algebras. The definition of these modules was motivated by studying the space of conformal blocks for affine Lie algebras. The fusion modules are given by a set of representations of the simple Lie algebra and a set of complex points, one for each representation. It was 1 2 VYJAYANTHI CHARI AND SERGEI LOKTEV conjectured that in the case of a simple Lie algebra and where the representations were irreducible 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 slr+1 by showing that the fusion module is isomorphic to a Weyl modules. Another well–studied family of modules are the Demazure modules. These modules for the current algebra are obtained by taking the current algebra module generated by the extremal vectors in modules 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 slr+1. Further, we 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 algebra can be regarded as a pull–back of Weyl modules for the affine Lie algebra, it follows that the slr+1–structure of the Demazure 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 irreducible as a module for the simple Lie algebra. But we are convinced that the conjecture of [CP1] is true for these modules and can be established in a similar way. Note however, that since 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 slr+1 once the conjecture on dimension of the Weyl modules is proved. The paper is organized as 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 exist between the Weyl modules, the fusion modules and the Demazure modules. In Section 2 we start proving the conjecture by identifying a basis for the Weyl module. The proof that 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 slr. This latter 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-3419.2005.1 and 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 slr+1. In particular, this establishes the conjecture on the dimension of the 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 which would be an immediate 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 zero matrices. However, we prefer to use the more general notation of simple Lie algebras since we expect the results to hold in that generality. 1.1.1. Let Z denote the set of integers, N the set of non–negative integers and N+ the set of positive integers. Let g = slr+1 be the Lie algebra of (r + 1)× (r + 1)–matrices of trace zero, h be the Cartan subalgebra of g consisting of diagonal matrices and αi, 1 ≤ i ≤ r, a set of simple roots for g with respect to h. For 1 ≤ i ≤ j ≤ r, let αi,j = (αi+ · · ·+αj) and let x + i,j (resp. x − i,j) be the (r+1)× (r+1)–matrix with 1 in the (i, j + 1) (resp. (j + 1, i))–position and 0 elsewhere. Define subalgebras n of g by n = ⊕ 1≤i≤j≤r Cxi,j . For 1 ≤ i ≤ r+1 let Hi be the diagonal matrix with 1 in the i place and zero elsewhere. The elements hi = Hi−Hi+1, 1 ≤ i ≤ r are a basis of h. Let {ωi : 1 ≤ i ≤ r} be the the set of fundamental weights of g, namely the basis of h which is dual to {αi : 1 ≤ i ≤ r}. Let P = ∑r i=1 Zωi, (resp. Q = ∑r i=1 Zαi) be the weight lattice (resp. root lattice) of g and set P = ∑r i=1 Nωi, (resp. Q + = ∑r i=1 Nαi). Given a Lie algebra a, we let U(a) denote the universal enveloping algebra a. 1.1.2. Let Z[P ] be the integral group ring of P with basis e(μ), μ ∈ P . If V is any finite–dimensional g–module with V = ⊕ μ∈P Vμ, Vμ = {v ∈ V : hv = μ(h)v, h ∈ h}, let chg(V ) = ∑ μ∈P dim(Vμ)e(μ) ∈ Z[P ] be the character of V . Given λ = ∑r i=1 miωi ∈ P , let V (λ) denote the irreducible finite–dimensional g–module with highest weight λ and highest weight vector vλ. More precisely, V (λ) = U(g)vλ with defining relations: (1.1) x+i,jvλ = 0, (h− λ(h))vλ = 0, (x − i,i) vλ = 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] is a Lie algebra with the Lie bracket being given by [x⊗f, y⊗g] = [x, y]⊗fg for all x, y ∈ a and f, g ∈ C[t]. We regard a as a subalgebra of a[t] by mapping x → x⊗1 for x ∈ a. We denote the maximal ideal of a[t] generated by elements of the form a⊗ t, a ∈ a, n > 0 by at[t]. The Lie algebra a[t] is a N–graded Lie algebra, the grading being given by powers of t and hence U(a[t]) is a N–graded algebra. Given a N–graded g[t]–module M = ⊕ s∈N Ms. Observe that Ms, s ∈ N are g–submodules of M . In the case when (1.2) M = ⊕ (μ,s)∈P×N Mμ,s, Mμ,s = {v ∈ Ms : hv = μ(h)v, h ∈ h}, recall, that the graded character of M to be the element of Z[P ][t] is defined as,