The goal of this paper is to understand the graded limit of a family of irreducible prime representations of the quantum affine algebra associated to a simply-laced simple Lie algebra g. This family was introduced in [19,20] in the context of monoidal categorification of cluster algebras. The graded limit of a member of this family is an indecomposable graded module for the current algebra g[t]; or equivalently a module for the maximal standard parabolic subalgebra in the affine Lie algebra g. In the case when g is of type An the problem was studied in [4], where it was shown that the graded limit is isomorphic to a level two Demazure module. In this paper we study the case when g is of type Dn. We show that in certain cases the limit is a generalized Demazure module, i.e., it is a submodule of a tensor product of level one Demazure modules. We give a presentation of these modules and compute their graded character (and hence also the character of the prime representations) in terms of Demazure modules of level two.

No abstract