Classification of Irreducible integrable modules for toroidal Lie-algebras with finite dimensional weight spaces

The study of Maps (X,G), the group of polynomial maps of a complex algebraic variety X into a complex algebraic group G, and its representations is only well developed in the case that X is a complex torus C. In this case Maps (X,G) is a loop group and the corresponding Lie-algebra Maps (X, ◦ G) is the loop algebra C[t, t]⊗ ◦ G. Here the representation comes to life only after one replaces Maps (X, ◦ G) by its universal central extension, the corresponding affine Lie-algebra. One then obtains the well known theory of highest weight modules, vertex representations, modular forms and character theory and so on. The next easiest case is presumably the case of n dimensional torus (C). So we consider the universal central extension of ◦ G ⊗C[t1 , · · · , t ± n ] which is referred to as the toroidal Lie-algebra τ in [EM ] and [MEY ]. The most interesting modules are the integrable modules (where the real root space acts locally nilpotently (see section 2)), as they lift to the corresponding group. Unlike the affine case where the central extension is one dimensional, the toroidal case has infinite dimensional centre which makes the theory more complicated. For the first time a large number of integrable (reducible) modules for toroidal Lie-algebras (simply laced case) have been constructed through the use of vertex operators in [EM] and [MEY]. In this paper we construct two classes of (Examples (4.1) and (4.2)) of irreducible integrable modules for toroidal Lie-algebras with finite dimensional