The theory of Lie groups and their representations lies at the very heart of mathematics and physics and, as befits such a central subject, it can be developed from many different points of view. The geometric approach puts the emphasis on some homogeneous spaces of the Lie groups. The Lie groups themselves then appear as symmetry groups of the homogeneous spaces while the representations arise essentially from functions on the homogeneous spaces. This approach can also be interpreted in physical terms where the homogeneous space is viewed as a 'phasespace' and the linear space (of the representation) is viewed as its 'quantization'.These ideas apply best when G is a compact Lie group and the homogeneous space is the 'flag manifold' G/T, where T is a maximal torus. This flag manifold is actually a complex algebraic variety and its different projective embeddings correspond essentially to the different irreducible representations of G.This classical theory can be generalized in several major directions. If we relax the compactness condition of G then irreducible representations tend to be infinitedimensional and we enter the large arena created by Harish-Chandra. This has important connections with number theory, through the Langlands programme, and origins in physics through the pioneering work of Bargmann and Wigner on representations of the Poincare group.A different, and more far-reaching, generalization arises when we allow G itself to be an infinite-dimensional Lie group. This direction has again been pioneered by physicists for the more recent requirements of quantum field theory. Two kinds of infinite-Himensional Lie groups are relevant here:(1) Diff + (M), the group of differentiable automorphisms of an oriented manifold M;(2) Map (M, G), the group of differentiable maps from a manifold M into a compact Lie group G.So far only the 1-dimensional case when M = S 1 (the circle) has been seriously developed. In particular the group LG = Map(S\G)called the loop group of G is the subject of this book, and Diff + (5' 1 ) is closely associated with it.Surprisingly it turns out that LG, although it is infinite-dimensional, behaves in many ways like G itself. In particular the homogeneous space LG/T (where T denotes here the maximal torus of G embedded in LG as constant loops) is an 'infinitedimensional algebraic variety' with infinite-dimensional projective embeddings related to irreducible representations of LG. Moreover these representations are labelled by discrete parameters, similar to those for G, and have character formulae analogous to the classical Hermann Weyl formula.In addition to this theory for general G there are more concrete results for the unitary and orthogonal groups. Just as in the finite-dimensional case the study of these classical groups involves some basic linear algebra (or analysis) of Hilbert space. One starts by fixing a decomposition of a Hilbert space into two (infinite-dimensional) sub-spaces H=H + ®H_
We introduce and study the notion of a Weyl module for the classical affine algebras, these modules are universal finite-dimensional highest weight modules. We conjecture that the modules are the classical limit of a family of irreducible modules of the quantum affine algebra, and prove the conjecture in the case of s l 2 sl_2 . The conjecture implies also that the Weyl modules are the classical limits of the standard modules introduced by Nakajima and further studied by Varagnolo and Vasserot.
Let t be an arbitrary symmetrizable Kac-Moody Lie algebra and U q (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 U q (t)-module L(µ). These modules have many nice properties and are well understood,More generally, given any integral weight λ, Kashiwara [K] defined an integrable U q (t)-module V max (λ) generated by an extremal vector v λ . If w is any element of the Weyl group W of t, then one has, where w 0 ∈ W is such that w 0 λ is dominant integral. In the case when λ is not in the Tits cone, the module V max (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 w 0 ∈ W so that w 0 λ is dominant with respect to the underlying finite-dimensional simple Lie algebra of t. In as yet unpublished work, Kashiwara proves that V max (λ) ∼ = W q (w 0 λ), where W q (w 0 λ) is an integrable U q (t)-module defined by generators and relations analogous to the definition of L(µ).In [CP4], we studied the modules W q (λ) further. In particular, we showed that they have a family W q (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 W q (π). In this paper, we show that, if t is the affine Lie algebra associated to sl 2 and λ = m ∈ Z + , the modules W q (π) all have the same dimension 2 m . This is done by showing that the modules W q (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 modules W q (π) have a unique irreducible quotient V q (π), and we show that these are all the irreducible finitedimensional U q (t)-modules. In [CP1], [CP2], a similar classification was obtained by regarding q as a complex number and U q (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 W q (m) as being the space of invariants of the action of the Hecke algebra H m on the tensor product (V ⊗ C(q)[t, t −1 ]) ⊗m , 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 W q (π) depends only on λ, and we give a formula for this dimension.
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