Kirillov–reshetikhin Modules Associated to G2

We define and study the Kirillov– Reshetikhin modules for algebras of type G2. We compute the graded character of these modules and verify that they are in accordance with the conjectures in [7], [8]. These results give the first complete description of families of Kirillov–Reshetikhin modules whose isotypical components have multiplicity bigger than one. Introduction In [2] we defined and studied a family of restricted modules for the current and twisted current algebras associated to a finite–dimensional classical simple Lie algebra g and a diagram automorphism of g of order two. These modules, which we called the restricted KirillovReshetikhin modules, are given by generators and relations and were denoted by KR(mωi), where σ is the diagram automorphism, i is a node of the Dynkin diagram of the subalgebra g0 of g consisting of the fixed points of σ, and m is a non–negative integer. Here we understand σ to be the identity in the untwisted case. They admit a natural grading which is compatible with the grading on the current algebras. In particular, the graded pieces are finite–dimensional modules for g0. It was proved in [2] that, regarded as g0-modules, there were no non–zero maps between the distinct graded pieces and, moreover, the multiplicity of any irreducible representation in a particular graded piece was at most one. In fact, the graded character was computed in [2] and verified to be in accordance with the conjectures in [7, Appendix A] and [8, Section 6] for the usual Kirillov-Reshetikhin modules for the corresponding quantum affine algebras. When g0 is an exceptional Lie algebra, the conjectures in these papers make it clear that for some nodes of the Dynkin diagram one or both of the aforementioned properties of the graded pieces may fail. The modules KR(mωi) are known to be isomorphic to the Demazure modules, further details can be found in [1], [2], [5], [6]. In this paper we define and study the modules for the current algebra associated to G2 and to the twisted current algebra associated to D4 and a diagram automorphism of order three. In both cases the fixed point subalgebra g0 is of type G2. We prove that the conjectures of [7] and [8] are true in these cases. In particular, there are now maps of g0–modules between the distinct non–zero graded pieces for KR(mωi) for some i and the multiplicity of an irreducible module in a graded piece can be greater than one. Moreover, our result on the graded character of the module KR(mω1) for G2 is actually an improvement on the conjectural graded–character formula in [7] which has some multiplicity–zero terms. The overall scheme of the proof is very similar to the one in [2]: we prove that the conjectural character formula is an upper bound for the character and then we prove that it is also a lower VC was partially supported by the NSF grant DMS-0500751. 1 2 VYJAYANTHI CHARI AND ADRIANO MOURA bound. However, one runs into difficulty almost immediately as the underlying combinatorics is rather more complicated. In order to prove the upper bound we use an elementary but useful result on representations of the 3–dimensional Heisenberg algebra. For the lower bound, as in [2], we first study some “fundamental” Kirillov-Reshetikhin modules and then realize the other modules as a submodule of a tensor product of the fundamental ones. But this time the fundamental modules are too big to be constructed explicitly as in [2]. To solve this we use the notion of fusion product of modules of the current algebra, which was introduced and studied in [3], [4]. The second step, in which involves studying graded quotients of tensor products of the fundamental Kirillov–Reshetikhin modules, is really much more complicated, since one has to prove not only that a particular representation occurs in a given grade, but also one has to determine its multiplicity. Identifying these quotients and proving that the isotypical components occur is non–trivial, since the projection of the natural vectors do not generate the desired g0–submodule. To solve this part we use the explicit description of some highest–weight vectors in tensor products of representations of sl2 and in tensor products of fundamental representations for g0. The paper is organized as follows. In section 1 we fix the basic notation and collect the results we will need for the proof. In section 2 we define the Kirillov-Reshetikhin modules, state the main theorem, and make the connection with the conjectures in [7] and [8]. We prove the theorem in sections 3 and 4. 1. Preliminaries 1.1. The Lie algebra G2 and its representations. Throughout this paper g0 will denote the Lie algebra of type G2, h0 a Cartan subalgebra of g0 and αi, i = 1, 2, a set of simple roots where we assume that α1 is short and α2 is long. Let R + l and R + s be the set of positive long and positive short roots respectively, R l = {α2, α2 + 3α1, 2α2 + 3α1}, R + s = {α1, α1 + α2, 2α1 + α2}. Given α ∈ R+ we denote by xα any non-zero element of (g0)±α. The subalgebras n ± 0 are defined in the obvious way by n±0 = ⊕α∈R+Cx ± α . Let ωi, i = 1, 2, be the fundamental weights and note that ω1 = 2α1 + α2 and ω2 = 2α1 + 3α2. Let P (resp. Q) be the integer lattice spanned by the fundamental weights (resp. simple roots) and let P+ (resp. Q+) be the Z+ span of the fundamental weights (resp. simple roots). Fix elements hαi , i = 1, 2, such that ωj(hαi) = δij for i, j = 1, 2. Then it is easy to see that [x + αi , x − αi ] is a non–zero multiple of hαi . Given a finite–dimensional g0–module V , we have V = ⊕λ∈PVλ, Vλ = {v ∈ V : hv = λ(h)v ∀ h ∈ h}. Let wt(V ) = {μ ∈ P : Vμ 6= 0} and given 0 6= v ∈ Vμ set wt(v) = μ. Let Z[P ] be the integral group ring of P with basis e(μ), μ ∈ P , and set ch(V ) = ∑ μ∈P dim(Vμ)e(μ). For λ ∈ P+, let V (λ) be the irreducible g0–module with highest weight λ and highest weight vector vλ. Thus V = U(g0)v, where n0 vλ = 0, hv = λ(h)v, (x − αi) iv = 0.