Generalized Demazure modules and fusion products

TL;DR

This paper studies the structure of Demazure modules over current algebras, proving that certain fusion products form generalized Demazure modules with explicit relations, and shows their parameter independence.

Contribution

It demonstrates that fusion products of Demazure modules are generalized Demazure modules with explicit defining relations and parameter independence.

Findings

Fusion product of Demazure modules forms a generalized Demazure module.

Fusion product is independent of the parameters chosen.

Provides generators and relations for specific generalized Demazure modules.

Abstract

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$ and let $\mathfrak{g}[t]$ be the corresponding current algebra. In this paper, we consider the $\mathfrak{g}[t]$-stable Demazure modules associated to integrable highest weight representations of the affine Lie algebra $\widehat{\mathfrak{g}}$. We prove that the fusion product of Demazure modules of a given level with a single Demazure module of a different level and with highest weight a multiple of $\theta$ is a generalized Demazure module, and also give defining relations. This also shows that the fusion product of such Demazure modules is independent of the chosen parameters. As a consequence we obtain generators and relations for certain types of generalized Demazure modules. We also establish a connection with the modules defined by Chari and Venkatesh.