Unipotent quantum coordinate ring and cominuscule prefundamental representations

We continue the study of realization of the prefundamental modules $L_{r,a}^{\pm }$, introduced by Hernandez and Jimbo, in terms of unipotent quantum coordinate rings as in [J-Kwon-Park, Int. Math. Res. Not., 2023]. We show that the ordinary character of $L_{r,a}^{\pm }$ is equal to that of the unipotent quantum coordinate ring $U_q^{-}(w_r)$ associated to fundamental $r$-th coweight. When $r$ is cominuscule, we prove that there exists a $U_q(\mathfrak{b})$-module structure on $U_q^{-}(w_r)$, which is isomorphic to $L_{r,a{\eta }_r}^{\pm }$ for some ${\eta }_r\in {\mathbb{C}}^{\times }$.