BGG reciprocity for current algebras

TL;DR

This paper proves a BGG-type reciprocity for categories of graded representations of current algebras associated with affine Lie algebras, confirming a conjecture for a broad class of these algebras.

Contribution

It establishes the validity of BGG reciprocity for current algebras linked to any indecomposable affine Lie algebra, extending previous conjectures.

Findings

BGG reciprocity holds for the category of graded representations of these current algebras

The result applies to all indecomposable affine Lie algebras

Supports the conjecture by Bennett, Chari, and Manning

Abstract

It was conjectured by Bennett, Chari, and Manning that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.