On some families of modules for the current algebra

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.