Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$

We give a new realization of the prefundamental representations $L_{r,a}^{\pm }$ introduced by Hernandez and Jimbo, when the quantum loop algebra $U_q(\mathfrak{g})$ is of types $A_n^{(1)}$ and $D_n^{(1)}$, and the $r$-th fundamental weight ${\varpi }_r$ for types $A_n$ and $D_n$ is minuscule. We define an action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ on the unipotent quantum coordinate ring associated to the translation by $-{\varpi }_r$, and show that it is isomorphic to $L_{r,a}^{\pm }$. We then give a combinatorial realization of $L_{r,a}^{+}$ in terms of the Lusztig data of the dual PBW vectors.