Weyl group twists and representations of quantum affine Borel algebras

We define categories Oʷ of representations of Borel subalgebras Uqb of quantum affine algebras Uqĝ, which come from the category O twisted by Weyl group elements w. We construct inductive systems of finite-dimensional Uqb-modules twisted by w, which provide representations in the category Oʷ. We also establish a classification of simple modules in these categories Oʷ. We explore convergent phenomenon of q-characters of representations of quantum affine algebras, which conjecturally give the q-characters of representations in Oʷ. Furthermore, we propose a conjecture concerning the relationship between the category O and the twisted category Oʷ, and we propose a possible connection with shifted quantum affine algebras.