Irreducible representations for toroidal Lie-algebras

Let image be a simple finite-dimensional Lie-algebra over the complex numbers image. The universal central extension of image is denoted by τ0. We add degree derivations d1,…,dn to τ0 and denote the resulting Lie-algebra by τ which we call a toroidal Lie-algebra. For ngreater-or-equal, slanted2 it is known that the center of τ0 is infinite dimensional. This infinite center, which is only an abelian ideal in τ, does not act as scalars on any irreducible representation of τ. In this paper, we prove that the study of irreducible representation of τ with finite-dimensional weight spaces is reduced to the study of irreducible representation for image with finite-dimensional weight spaces on which the center acts as scalars. In the process we prove an interesting result for ngreater-or-equal, slanted2. Let image be the quotient of τ by the non-zero degree central operators. Then image does not admit representations with finite dimensional weight spaces where the zero degree center acts non-trivially.