Weyl Group Twists and Representations of Quantum Affine Borel Algebras

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.