Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra $\mathfrak{g}$ and any height function $\xi$ adapted to an orientation $Q$ of the Dynkin diagram of $\mathfrak{g}$, Hernandez-Leclerc introduced a certain category $\mathcal{C}^{\leq \xi}$ of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$, as well as a subcategory $\mathcal{C}_Q$ of $\mathcal{C}^{\leq \xi}$ whose complexified Grothendieck ring is isomorphic to the coordinate ring $\mathbb{C}[\mathbf{N}]$ of a maximal unipotent subgroup. In this paper, we define an algebraic morphism $\widetilde{D}_{\xi}$ on a torus $\mathcal{Y}^{\leq \xi}$ containing the image of $K_0(\mathcal{C}^{\leq \xi})$ under the truncated $q$-character morphism. We prove that the restriction of $\widetilde{D}_{\xi}$ to $K_0(\mathcal{C}_Q)$ coincides with the morphism $\overline{D}$ recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities of Mirkovi\'c-Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov-Reshetikhin modules in $\mathcal{C}_Q$, as well as certain results by Brundan-Kleshchev-McNamara on the representation theory of quiver Hecke algebras. This alternative description of $\overline{D}$ allows us to prove a conjecture by the first author on the distinguished values of $\overline{D}$ on the flag minors of $\mathbb{C}[\mathbf{N}]$. We also provide applications of our results from the perspective of Kang-Kashiwara-Kim-Oh's generalized Schur-Weyl duality. Finally, we define a cluster algebra $\overline{\mathcal{A}}_Q$ as a subquotient of $K_0(\mathcal{C}^{\leq \xi})$ naturally containing $\mathbb{C}[\mathbf{N}]$, and suggest the existence of an analogue of the Mirkovi\'c-Vilonen basis in $\overline{\mathcal{A}}_Q$ on which the values of $\widetilde{D}_{\xi}$ may be interpreted as certain equivariant multiplicities.