Weyl group twists and representations of quantum affine Borel algebras

Abstract

We define categories ${\mathcal{O}}^w$ of representations of Borel subalgebras ${\mathcal{U}}_q\mathfrak{b}$ of quantum affine algebras ${\mathcal{U}}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements $w$. We construct inductive systems of finite-dimensional ${\mathcal{U}}_q\mathfrak{b}$-modules twisted by $w$, which provide representations in the category ${\mathcal{O}}^w$. We also establish a classification of simple modules in these categories ${\mathcal{O}}^w$. We explore convergent phenomenon of $q$-characters of representations of quantum affine algebras, which conjecturally give the $q$-characters of representations in ${\mathcal{O}}^w$. Furthermore, we propose a conjecture concerning the relationship between the category $\mathcal{O}$ and the twisted category ${\mathcal{O}}^w$, and we propose a possible connection with shifted quantum affine algebras.

30 pages