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 \(V_{w}^- (\lambda) / X_{w}^- (\lambda)\), \(w \in W\), of the Demazure submodules \(V_{w}^- (\lambda)\) 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.