Unipotent quantum coordinate ring and cominuscule prefundamental representations

Abstract

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 $L_{r,a\hat{I}{\cdot }_r}^{\pm }$ for some $\hat{I}{\cdot }_r\in {\mathbb{C}}^{\times }$.

v3: Corollary 3.7 corrected, v2: 38 pages, introduction revised, reference added, some part of Remark 4.9 is incorrect, so it is revised (this remark moved to Section 5), proof of Proposition 4.15 improved, formulas in Corollary 2.7 and Lemma 4.10 corrected, several remarks added, typos and some notations corrected, to appear in Journal of Algebra; v1: 36 pages