Generalized Weyl modules and Demazure submodules of level-zero extremal weight modules

We study a relationship between the graded characters of generalized Weyl modules $W_{w\lambda }$, $w\in W$, over the positive part of the affine Lie algebra and those of specific quotients $V_{w}^- (\lambda) / X_{w}^- (\lambda)$,$w \in W$,oftheDemazuresubmodules$V_{w}^- (\lambda)$ of the extremal weight modules $V(\lambda )$ over the quantum affine algebra, where $W$ is the finite Weyl group and $\lambda$ is a dominant weight. More precisely, we prove that a specific quotient of the Demazure submodule is a quantum analog of a generalized Weyl module.