Equivariant multiplicities via representations of quantum affine algebras

Abstract

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 ${\mathcal{C}}^{\le \xi }$ of representations of the quantum affine algebra $U_q(\hat{\mathfrak{g}})$, as well as a subcategory ${\mathcal{C}}_Q$ of ${\mathcal{C}}^{\le \xi }$ 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 ${\tilde{D}}_{\xi }$ on a torus ${\mathcal{Y}}^{\le \xi }$ containing the image of $K_0({\mathcal{C}}^{\le \xi })$ under the truncated $q$-character morphism. We prove that the restriction of ${\tilde{D}}_{\xi }$ 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 Mirković-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 $K_0({\mathcal{C}}^{\le \xi })$ naturally containing $\mathbb{C}[\mathbf{N}]$, and suggest the existence of an analogue of the Mirković-Vilonen basis in ${\overline{\mathcal{A}}}_Q$ on which the values of ${\tilde{D}}_{\xi }$ may be interpreted as certain equivariant multiplicities.

arXiv commentMinor changes following referee's suggestions. Accepted for publication in Selecta Mathematica (NS)

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