The motivation of the paper is the conjectural correspondence between quantum \({}^{L}\hat{\mathfrak{g}}\)-KdV Hamiltonians and affine \(\hat{\mathfrak{g}}\)-opers through \(Q\widetilde{Q}\)-systems for twisted quantum affine algebras. The author aimed to establish the twisted \(Q\widetilde{Q}\)-systems in this correspondence. Different from the non-twisted case, the author proposed a new method to avoid using shifted quantum affine algebras which has not been defined for twisted types, and to avoid using \(\hat{\mathfrak{sl}}_2\)-reductions in the case \(A_{2n}^{(2)}\).
For this purpose, the author first defined the category \(\mathcal{O}\) of the Borel subalgebra \(\mathcal{U}_q(\mathfrak{b}^{\sigma})\) of twisted quantum affine algebras by mimicking the non-twisted case in the work of D. Hernandez and M. Jimbo. The author proved that the simple objects in the category \(\mathcal{O}\) are the \(l\)-highest weight modules by standard arguments. More precisely, the allowed highest \(l\)-weights are those tuples of rational functions \((\psi_i(u))_{i \in I}\) which are compatible with the Dynkin diagram automorphism in the sense that \(\psi_{\sigma(i)}(u) = \psi_{i}(\omega u)\), where \(\omega\) is an \(M\)th root of unity, where \(M\) is the order of \(\sigma\). To prove it, the author needed to show that the positive and negative prefundamental representations \(L_{i,a}^{\pm}\) are in the category \(\mathcal{O}\). These prefundamental representations were constructed by D. Hernandez with the same method in non-twisted case. The author verified that these constructions still work in the twisted case.
Then the author defined the Grothendieck ring \(K_0(\mathcal{O})\) of the category \(\mathcal{O}\) and also the \(q\)-character of modules in \(\mathcal{O}\). He proved that the \(q\)-character is a ring isomorphism \[ \chi_q^{\sigma} : K_0(\mathcal{O}) \to \mathcal{E}_{l}^{\sigma} \] to some commutative ring. This is also true in the non-twisted case by the same argument. In particular, the Grothendieck ring \(K_0(\mathcal{O})\) is commutative.
The relation between the non-twisted quantum affine algebra \(\mathcal{U}_q(\mathcal{L}\mathfrak{g})\) and the twisted quantum affine algebra \(\mathcal{U}_q(\mathcal{L}\mathfrak{g})^{\sigma}\) associated by the automorphism \(\sigma\) of Dynkin diagram is useful to generalize known results in non-twisted types to twisted types. As an application, the author derived the \(TQ\)-relations for twisted types and relate them to the \(TQ\)-relations for non-twisted types.
But it is not clear if this relation holds in general. Moreover, the author proved that this conjecture holds for another particular class of representations in \(\mathcal{O}\): the representation \(X_{\bar{i},a} = L(\widetilde{\mathbf{\Psi}}_{i,a})\). As a consequence, the \(Q\widetilde{Q}\)-systems for twisted quantum affine algebras can be proved by applying the conjecture on \(X_{\bar{i},a}\) and on \(L_{\bar{i},a}^+\).
As an application of the \(Q\widetilde{Q}\)-systems of twisted types, the author derived the Bethe Ansatz equations for twisted quantum affine algebras by applying the transfer matrices on \(Q\widetilde{Q}\)-systems.