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