Irreducible representations for toroidal Lie-algebras

Let W be a simple finite-dimensional Lie-algebra over the complex numbers C. The universal central extension of G circle times C[t(1)(+/- 1) .....t(n)(+/- 1)] is denoted by tau(0). We add degree derivations d(1) ..... d(n) to tau(0) and denote the resulting Lie-algebra by tau which we call a toroidal Lie-algebra, for n >= 2 it is known that the center of tau(0) is infinite dimensional. This intinite center, which is only an abelian ideal in tau, does not act as scalars on any irreducible representation of tau. In this paper, we prove that the study of irreducible representation of tau with finite-dimensional weight spaces is reduced to the study of irreducible representation for to tau(0) circle plus Cd-n with finite-dimensional weight spaces on which the center acts as scalars. In the process we prove an interesting result for n >= 2. Let (tau) over bar be the quotient of tau by the non-zero degree central operators. Then (tau) over bar does not admit representations with finite dimensional weight spaces where the zero degree center acts non-trivially. (c) 2005 Elsevier B.V. All rights reserved.