Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra [Formula: see text] and any height function [Formula: see text] adapted to an orientation Q of the Dynkin diagram of [Formula: see text] , Hernandez–Leclerc introduced a certain category [Formula: see text] of representations of the quantum affine algebra [Formula: see text] , as well as a subcategory [Formula: see text] of [Formula: see text] whose complexified Grothendieck ring is isomorphic to the coordinate ring [Formula: see text] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism [Formula: see text] on a torus [Formula: see text] containing the image of [Formula: see text] under the truncated q-character morphism. We prove that the restriction of [Formula: see text] to [Formula: see text] coincides with the morphism [Formula: see text] recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in [Formula: see text] , as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of [Formula: see text] allows us to prove a conjecture by the first author on the distinguished values of [Formula: see text] on the flag minors of [Formula: see text] . We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh’s generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park’s recent constructions to define a cluster algebra [Formula: see text] as a subquotient of [Formula: see text] naturally containing [Formula: see text] , and suggest the existence of an analogue of the Mirković–Vilonen basis in [Formula: see text] on which the values of [Formula: see text] may be interpreted as certain equivariant multiplicities.