UC Riverside

For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C},$ we study the representations of the associated current algebra $\mathfrak{g}[t].$ We focus our attention on $V(\boldsymbol \xi)$ modules, which are a large family of indecomposable representations that are quotients of local Weyl modules and include Demazure modules when $\mathfrak{g}$ is simply laced. We establish three new presentations of $V(\boldsymbol \xi)$ modules, which show that they are finitely presented as quotients of local Weyl modules. We establish each presentation for $\mathfrak{sl}_2$ and then for an arbitrary $\mathfrak{g}.$ With these presentations, we establish the existence of two short exact sequences of $V(\boldsymbol \xi)$ modules. These short exact sequences were used to establish a character formula for the tensor product of a level 2 Demazure module and a local Weyl module.