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We study the structure of the finite-dimensional representations of$\mathfrak{sl}_2[t]$, the current Lie algebra type of $A_1$, which are obtainedby taking tensor products of special Demazure modules. We show that theserepresentations admit a Demazure flag and obtain a closed formula for thegraded multiplicities of the level 2 Demazure modules in the filtration of thetensor product of two local Weyl modules for $\mathfrak{sl}_2[t]$. Furthermore,we derive an explicit expression for graded character of the tensor product ofa local Weyl module with an irreducible $\mathfrak{sl}_2[t]$ module. Inconjunction with the results of \cite{MR3210603}, our findings provide evidencefor the conjecture in \cite{9} that the tensor product of Demazure modules oflevels m and n respectively has a filtration by Demazure modules of level m +n.

Demazure Filtrations of Tensor Product Modules and Character Formula