On Truncated Weyl Modules

We study structural properties of truncated Weyl modules. A truncated Weyl module $W_N(\lambda)$ is a local Weyl module for $\mathfrak g[t]_N = \mathfrak g \otimes \frac{\mathbb C[t]}{t^N\mathbb C[t]}$, where $\mathfrak g$ is a finite-dimensional simple Lie algebra. It has been conjectured that, if $N$ is sufficiently small with respect to $\lambda$, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when $\lambda$ is a multiple of certain fundamental weights, including all minuscule ones for simply laced $\mathfrak g$. We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that $\mathfrak g=\mathfrak{sl}_2$ related to Demazure flags and chains of inclusions of truncated Weyl modules.