Demazure Modules, Chari–Venkatesh Modules and Fusion Products

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$. Given two non-negative integers $m$, $n$, we prove that the fusion product of $m$ copies of the level one Demazure module $D(1,\theta)$ with $n$ copies of the adjoint representation $\mathrm{ev}_0 V(\theta)$ is independent of the parameters and we give explicit defining relations. As a consequence, for $\mathfrak{g}$ simply laced, we show that the fusion product of a special family of Chari–Venkatesh modules is again a Chari–Venkatesh module. We also get a description of the truncated Weyl module associated to a multiple of $\theta$.