Generalized Baxter’s TQ-relations and the QQ-system are systems of algebraic relations in the category \({\mathcal O}\) of representations of the Borel subalgebra of the quantum affine algebra \(U_q(\widehat{\mathfrak {g}})\) , which we established in our earlier works (Frenkel and Hernandez in Duke Math J 164, 2407–2460, 2015; Commun Math Phys 362, 361–414, 2018). In the present paper, we conjecture a family of analogous relations labeled by elements of the Weyl group W of \(\mathfrak {g}\) , so that the original relations correspond to the identity element. These relations are closely connected to the W-symmetry of q-characters established in our earlier paper. We prove these relations for all \(w \in W\) if \(\mathfrak {g}\) has rank two, and we prove the extended TQ-relations if w is a simple reflection. We also generalize our results and conjectures to the shifted quantum affine algebras.
