Demazure Filtrations of Tensor Product Modules and Character Formula

We study the structure of the finite-dimensional representations of $\mathfrak{sl}_2[t]$, the current Lie algebra type of $A_1$, which are obtained by taking tensor products of special Demazure modules. We show that these representations admit 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 derive an explicit expression for graded character of the tensor product of a local Weyl module with an irreducible $\mathfrak{sl}_2[t]$ module. In conjunction with the results of \cite{MR3210603}, our findings provide evidence for the conjecture in \cite{9} that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.