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(widehat{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}[textbf{N}] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism {widetilde{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 {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 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}[textbf{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 as a subquotient of K_0(mathcal {C}^{le xi }) naturally containing mathbb {C}[textbf{N}], and suggest the existence of an analogue of the Mirković–Vilonen basis in overline{mathcal {A}}_Q on which the values of {widetilde{D}}_{xi } may be interpreted as certain equivariant multiplicities.