Let \(G\) be a semisimple  complex Lie group of simply-laced type, \(B\) a Borel in \(G\) and \(T\) a maximal torus in \(B\). 
   In our last joint work with Jian-Rong Li, we exhibited new families of non-trivial rational identities arising from the representation theory of \(U_q(\widehat{\mathfrak{g}})\), the quantum affine algebra associated to the Lie algebra of \(G\). This was achieved by constructing a ring homomorphism from the torus containing  Frenkel-Reshetikin's \(q\)-characters of finite-dimensional representations of \(U_q(\widehat{\mathfrak{g}})\) to the ring of regular functions on the regular locus of the Lie algebra of \(T\). 
   We will present a geometric proof of these identities using schemes recently introduced by Francone-Leclerc called spaces of \((G,c)\)-bands. This provides a unified picture for our former results with Jian-Rong Li relating the rational identities obtained from \(q\)-characters to the map \(\bar{D}\) introduced by Baumann-Kamnitzer-Knutson in their study of the equivariant homology of Mirkovic-Vilonen cycles. 
   This is joint work with Ryo Fujita and Jian-Rong Li.