Let G be a simple simply-connected algebraic group of type A, D, E. In the 60's Steinberg has described cross-sections A(c) of the set of regular conjugacy classes in G, depending on the choice of a Coxeter element c in the Weyl group of G. We define a (G,c)-band as a sequence (g(k)) of elements of G such that for every k, g(k+1) is equal to the product of g(k) by an element of A(c). We show that (G,c)-bands are the rational points of an infinite-dimensional affine integral scheme B(G,c), whose coordinate ring R(G,c) has the structure of a cluster algebra. The ring R(G,c) is closely related with a q-difference W-algebra associated with the loop group LG of G, introduced in the 90's by Frenkel and Reshetikhin. In their foundational paper on q-characters of finite-dimensional modules over quantum affine algebras, Frenkel and Reshetikhin conjectured that the q-character homomorphism can be described geometrically using a certain Miura transformation attached to this deformed W-algebra. I will explain how the above cluster structure on R(G,c) yields a proof of this conjecture. Joint work with Luca Francone.