Equivariant multiplicities via representations of quantum affine algebras.

For any simply-laced type simple Lie algebra $g$ and any height function $\xi$ adapted to an orientation Q of the Dynkin diagram of $g$ , Hernandez-Leclerc introduced a certain category $C^{\le \xi }$ of representations of the quantum affine algebra $U_q\left(\hat{g}\right)$ , as well as a subcategory $C_Q$ of $C^{\le \xi }$ whose complexified Grothendieck ring is isomorphic to the coordinate ring $C\left[N\right]$ of a maximal unipotent subgroup. In this paper, we define an algebraic morphism ${\tilde{D}}_{\xi }$ on a torus $Y^{\le \xi }$ containing the image of $K_0\left(C^{\le \xi }\right)$ under the truncated q-character morphism. We prove that the restriction of ${\tilde{D}}_{\xi }$ to $K_0\left(C_Q\right)$ coincides with the morphism $\overline{D}$ 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 $C_Q$ , 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 $C\left[N\right]$ . 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 ${\overline{A}}_Q$ as a subquotient of $K_0\left(C^{\le \xi }\right)$ naturally containing $C\left[N\right]$ , and suggest the existence of an analogue of the Mirković-Vilonen basis in ${\overline{A}}_Q$ on which the values of ${\tilde{D}}_{\xi }$ may be interpreted as certain equivariant multiplicities.