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We give a new realization of the prefundamental representations $L^\pm_{r,a}$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 $r$-thfundamental weight $\varpi_r$ for types $A_n$ and $D_n$ is minuscule. We definean action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ onthe unipotent quantum coordinate ring associated to the translation by$-\varpi_r$, and show that it is isomorphic to $L^\pm_{r,a}$. We then give acombinatorial realization of $L^+_{r,a}$ in terms of the Lusztig data of thedual PBW vectors.

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