Integrable and Weyl modules for quantum affine sl_2

Let t be an arbitrary symmetrizable Kac-Moody Lie algebra and Uq(t) the corresponding quantized enveloping algebra of t defined over C(q). If μ is a dominant integral weight of t then one can associate to it in a natural way an irreducible integrable Uq(t)-module L(μ). These modules have many nice properties and are well understood, [K], [L1]. More generally, given any integral weight λ, Kashiwara [K] defined an integrable Uq(t)-module V (λ) generated by an extremal vector vλ. If w is any element of the Weyl group W of t, then one has V (λ) ∼= V (wλ). Further, if λ is in the Tits cone, then V(λ) ∼= L(w0λ), where w0 ∈ W is such that w0λ is dominant integral. In the case when λ is not in the Tits cone, the module V (λ) is not irreducible and very little is known about it, although it is known that it admits a crystal basis, [K]. In the case when t is an affine Lie algebra, an integral weight λ is not in the Tits cone if and only if λ has level zero. Choose w0 ∈ W so that w0λ is dominant with respect to the underlying finite-dimensional simple Lie algebra of t. In as yet unpublished work, Kashiwara proves that V (λ) ∼=Wq(w0λ), where Wq(w0λ) is an integrable Uq(t)-module defined by generators and relations analogous to the definition of L(μ). In [CP4], we studied the modules Wq(λ) further. In particular, we showed that they have a family Wq(π) of non–isomorphic finite-dimensional quotients which are maximal, in the sense that any another finite-dimensional quotient is a proper quotient of some Wq(π). In this paper, we show that, if t is the affine Lie algebra associated to sl2 and λ = m ∈ Z , the modulesWq(π) all have the same dimension 2. This is done by showing that the modules Wq(π), under suitable conditions, have a q = 1 limit, which allows us to reduce to the study of the corresponding problem in the classical case carried out in [CP4]. The modulesWq(π) have a unique irreducible quotient Vq(π), and we show that these are all the irreducible finitedimensional Uq(t)-modules. In [CP1], [CP2], a similar classification was obtained by regarding q as a complex number and Uq(t) as an algebra over C; in the present situation, we have to allow modules defined over finite extensions of C(q). We are then able to realize the modules Wq(m) as being the space of invariants of the action of the Hecke algebra Hm on the tensor product (V ⊗C(q)[t, t ]), where V is a two-dimensional vector space over C(q). Again, this is done by reducing to the case of q = 1. In the last section, we indicate how to extend some of the results of this paper to the general case. We conjecture that the dimension of the modules Wq(π) depends only on λ, and we give a formula for this dimension.