Classification of Harish-Chandra Modules for Current Algebras

Abstract

For any reductive Lie algebra g \mathfrak {g} and commutative, associative, unital algebra S S , we give a complete classification of the simple weight modules of g ⊗ S \mathfrak {g}\otimes S with finite weight multiplicities. In particular, any such module is parabolically induced from a simple admissible module for a Levi subalgebra. Conversely, all modules obtained in this way have finite weight multiplicities. These modules are isomorphic to tensor products of evaluation modules at distinct maximal ideals of S S . Our results also classify simple Harish-Chandra modules up to isomorphism for all central extensions of current algebras.