On some families of modules for the current algebra

TL;DR

This paper constructs and analyzes a new family of graded modules for the current algebra associated with a semi-simple Lie algebra, revealing duality properties in type A cases.

Contribution

It introduces a novel family of modules for the current algebra, with explicit graded characters and duality phenomena in type A.

Findings

Constructed graded modules indexed by symmetric group representations.

Determined explicit graded characters of these modules.

Discovered a duality in the graded characters for type A Lie algebras.

Abstract

Given a finite-dimensional module, $V$, for a finite-dimensional, complex, semi-simple Lie algebra $\lie g$ and a positive integer $m$, we construct a family of graded modules for the current algebra $\lie g[t]$ indexed by simple $\CC\lie S_m$-modules. These modules have the additional structure of being free modules of finite rank for the ring of symmetric polynomials and so can be localized to give finite-dimensional graded $\lie g[t]$-modules. We determine the graded characters of these modules and show that if $\lie g$ is of type $A$ and $V$ the natural representation, these graded characters admit a curious duality.