Weyl group twists and representations of quantum affine Borel algebras

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.