Equivariant multiplicities via representations of quantum affine algebras

Abstract

AbstractFor any simply-laced type simple Lie algebra $$\mathfrak {g}$$ g and any height function $$\xi $$ ξ adapted to an orientation Q of the Dynkin diagram of $$\mathfrak {g}$$ g , Hernandez–Leclerc introduced a certain category $$\mathcal {C}^{\le \xi }$$ C ≤ ξ of representations of the quantum affine algebra $$U_q(\widehat{\mathfrak {g}})$$ U q ( g ^ ) , as well as a subcategory $$\mathcal {C}_Q$$ C Q of $$\mathcal {C}^{\le \xi }$$ C ≤ ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring $$\mathbb {C}[\textbf{N}]$$ C [ N ] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism $${\widetilde{D}}_{\xi }$$ D ~ ξ on a torus $$\mathcal {Y}^{\le \xi }$$ Y ≤ ξ containing the image of $$K_0(\mathcal {C}^{\le \xi })$$ K 0 ( C ≤ ξ ) under the truncated q-character morphism. We prove that the restriction of $${\widetilde{D}}_{\xi }$$ D ~ ξ to $$K_0(\mathcal {C}_Q)$$ K 0 ( C Q ) coincides with the morphism $$\overline{D}$$ 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 $$\mathcal {C}_Q$$ C Q , as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of $$\overline{D}$$ D ¯ allows us to prove a conjecture by the first author on the distinguished values of $$\overline{D}$$ D ¯ on the flag minors of $$\mathbb {C}[\textbf{N}]$$ C [ N ] . 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 $$\overline{\mathcal {A}}_Q$$ A ¯ Q as a subquotient of $$K_0(\mathcal {C}^{\le \xi })$$ K 0 ( C ≤ ξ ) naturally containing $$\mathbb {C}[\textbf{N}]$$ C [ N ] , and suggest the existence of an analogue of the Mirković–Vilonen basis in $$\overline{\mathcal {A}}_Q$$ A ¯ Q on which the values of $${\widetilde{D}}_{\xi }$$ D ~ ξ may be interpreted as certain equivariant multiplicities.