Generalized Demazure Modules and Prime Representations in Type $D_n$

Abstract

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 $\mathfrak{g}$. This family was introduced by David Hernandez and Bernard Leclerc 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 $\mathfrak{g}[t]$; or equivalently a module for the maximal standard parabolic subalgebra in the affine Lie algebra $\hat{\mathfrak{g}}$. In this paper we study the case when $\mathfrak{g}$ is of type $D_n$. 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.

To Kolya Reshetikhin, on his 60th birthday