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

Abstract

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.

41 pages, introduction is revised, Lemma 3.2 is revised, the proofs of Lemma 3.8 and Corollary 4.8 are revised, added Example 3.9, Example 4.4 and Example 4.19, comments added on Remark 3.11, Remark 4.9, Remark 4.21 and Remark 4.30, typos corrected, minor corrections

References in corpus (2)

• Affinization of category O for quantum groups
• Lusztig data of Kashiwara-Nakashima tableaux in type D

Cited by in corpus (1)

• Affinization of $q$-oscillator representations of $U_q({\mathfrak{gl}}_n)$