# Classification of Harish-Chandra modules for current algebras

## TL;DR

This paper classifies all simple weight modules with finite multiplicities for current algebras formed from reductive Lie algebras and commutative algebras, showing they are tensor products of evaluation modules and extending to central extensions.

## Contribution

It provides a complete classification of simple weight modules with finite multiplicities for current algebras, including a description via parabolic induction and evaluation modules.

## Key findings

- All such modules are tensor products of evaluation modules at distinct maximal ideals.
- Modules are parabolically induced from simple admissible modules of Levi subalgebras.
- Classification extends to simple Harish-Chandra modules for central extensions.

## Abstract

For any reductive Lie algebra $\mathfrak{g}$ and commutative, associative, unital algebra $S$, we give a complete classification of the simple weight modules of $\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 $S$. Our results also classify simple Harish-Chandra modules up to isomorphism for all central extensions of current algebras.