Demazure Filtration of Tensor Product Modules of Current Lie Algebra of Type \(A_1\)

Abstract

In this paper we study the structure of finite-dimensional representations of the current Lie algebra of type \(A_1\), \(\mathfrak {sl}_2[t]\), which are obtained by taking tensor products of local Weyl modules with Demazure modules. We show that such a representation admits a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for \(\mathfrak {sl}_2[t]\). Furthermore, we show that the tensor product of a local Weyl module with an irreducible \(\mathfrak {sl}_2[t]\) module admits a Demazure filtration and derive the graded character of such tensor product modules. In conjunction with the results of Chari et al. (SIGMA Symmetry Integrability Geom. Methods Appl. 10(032), 2014), our findings provide evidence for the conjecture in Blanton (2017) that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.