  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.
