We define an action of the Weyl group W of a simple Lie algebra \(\mathfrak {g}\) on a completion of the ring \({\mathcal Y}\) , which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra \(U_q(\widehat{\mathfrak {g}})\) . We prove that the subring of W-invariants of \({\mathcal Y}\) is precisely the ring of q-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of \(U_q(\widehat{\mathfrak {g}})\) . This resolves an old puzzle in the theory of q-characters. We also identify the screening operators, which were previously used to describe the ring of q-characters, as the subleading terms of simple reflections from W in a certain limit. Our results have already found applications to the study of the category \({\mathcal O}\) of representations of the Borel subalgebra of \(U_q(\widehat{\mathfrak {g}})\) in [Frenkel and Hernandez, Extended Baxter Relations and QQ-Systems for Quantum Affine Algebras, Comm. Math. Phys. 405:190, 2024. (arXiv:2312.13256)] and to the categorification of cluster algebras in [Geiss et al., Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case, Proc. Lond. Math. Soc. 3(129): e12630, 2024 (arXiv:2401.04616)].
