Irreducible representations for toroidal Lie-algebras

Abstract

Let G be a simple finite-dimensional Lie-algebra over the complex numbers C . The universal central extension of G ⊗ C [ t 1 ± 1 , … , t n ± 1 ] is denoted by τ 0 . We add degree derivations d 1 , … , d n to τ 0 and denote the resulting Lie-algebra by τ which we call a toroidal Lie-algebra. For n ⩾ 2 it is known that the center of τ 0 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 τ 0 ⊕ C d 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 τ ¯ be the quotient of τ by the non-zero degree central operators. Then τ ¯ does not admit representations with finite dimensional weight spaces where the zero degree center acts non-trivially.