Let \(G\) be a semisimple  complex Lie group and \(N\) a maximal unipotent subgroup of \(G\).  In their study of the equivariant homology of the affine Grassmannian \(Gr_{G^{\vee}}\), Baumann-Kamnitzer-Knutson introduced an algebraic morphism \(\bar D\) on the coordinate ring \(\mathbb{C}[N]\) providing a powerful tool to compare distinguished bases of this algebra, such as the Mirković-Vilonen basis arising from the geometric Satake correspondence.
      In this talk we will focus on the simply-laced case and present an alternative description of \(\bar D\) proposed in a joint work with Jian-Rong Li, that relies on Hernandez-Leclerc's categorification of the cluster structure of  \(\mathbb{C}[N]\) via finite-dimensional representations of affine quantum groups. We will then present a work in progress (also joint with Jian-Rong Li) aiming to establish a large family of non-trivial rational identities obtained by applying our construction to Frenkel-Reshetikhin's \(q\)-characters. If time allows, we will discuss possible interpretations of these identities in terms of equivariant homology, raising the question of natural geometric models associated to representations of affine quantum groups.