Abstract

 Equivariant multiplicities via representations of quantum affine algebrasElie CasbiFor any simply-laced type simple Lie algebra g and any height function ξ adapted to anorientation Q of the Dynkin diagram of g, Hernandez-Leclerc introduced a certain categoryC⁄ξ of representations of the quantum affine algebra Uqppgq, as well as a subcategory CQ ofC⁄ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring CrNs of amaximal unipotent subgroup. In this talk, I will present our construction of an algebraic morphism Drξ on a torus Y⁄ξ containing the image of K0pC⁄ξq under the truncated q-charactermorphism. We prove that the restriction of Drξ to K0pCQq coincides with the morphism D recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicitiesof Mirković-Vilonen cycles. I will begin by showing how the cluster structure of CrNs playeda key role in the proof of this result. I will also explain how this alternative description of Dallowed us to prove a conjecture from an earlier work of mine on the distinguished values ofD on the flag minors of CrNs. Finally I will conclude with some applications of our resultsand some perspectives of further developments.This is a joint work with Jianrong LI.