Resumen de Equivariant multiplicities via representations of quantum affine algebras
Elie Casbi, Jian Rong Li
For any simply-laced type simple Lie algebra g and any height function ξ adapted to an orientation Q of the Dynkin diagram of g, Hernandez–Leclerc introduced a certain category C≤ξ of representations of the quantum affine algebra Uq(gˆ), as well as a subcategory CQ of C≤ξ whose complexified Grothendieck ring isomorphic to the coordinate ring C[N] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism D˜ξ on a torus Y≤ξ containing the image of K0(C≤ξ) under the truncated q-character morphism. We prove that the restriction of D˜ξ to K0(CQ) coincides with the morphism 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 CQ, as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of Dallows us to prove a conjecture by the first author on the distinguished values of D on the flag minors of 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 AQ as a subquotient of K0(C≤ξ) naturally containing C[N], and suggest the existence of an analogue of the Mirković–Vilonen basis in AQ on which the values of D˜ξ may be interpreted as certain equivariant multiplicities.
