Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra $$ and any height function $$ adapted to an orientation Q of the Dynkin diagram of $$ , Hernandez-Leclerc introduced a certain category $$ of representations of the quantum affine algebra $$ , as well as a subcategory $$ of $$ whose complexified Grothendieck ring is isomorphic to the coordinate ring $$ of a maximal unipotent subgroup. In this paper, we define an algebraic morphism $$ on a torus $$ containing the image of $$ under the truncated q-character morphism. We prove that the restriction of $$ to $$ coincides with the morphism $$ 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 $$ , as well as certain results by Brundan-Kleshchev-McNamara on the representation theory of quiver Hecke algebras. This alternative description of $$ allows us to prove a conjecture by the first author on the distinguished values of $$ on the flag minors of $$ . 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 $$ as a subquotient of $$ naturally containing $$ , and suggest the existence of an analogue of the Mirković-Vilonen basis in $$ on which the values of $$ may be interpreted as certain equivariant multiplicities.