Elie Casbi, p 1, Representations of affine quantum groups and the homology of affine Grassmannians

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 for the Lamglands dual group, Baumann-Kamnitzer-Knutson introduced an algebraic morphism D on the coordinate ring 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 D proposed in a joint work with Jian-Rong Li, that relies on Hernandez-Leclerc’s categorification of the cluster structure of 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.