Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

Abstract

For a Dynkin quiver Q Q (of type A D E \mathrm {ADE} ), we consider a central completion of the convolution algebra of the equivariant K K -group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category C Q \mathcal {C}_{Q} of modules over the quantum loop algebra U q ( L g ) U_{q}(L\mathfrak {g}) via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q Q and Hernandez-Leclerc’s category C Q \mathcal {C}_{Q} , assuming the simpleness of some poles of normalized R R -matrices for type E \mathrm {E} .