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

Abstract:

For a Dynkin quiver $Q$ (of type $\mathrm {ADE}$), we consider a central completion of the convolution algebra of the equivariant $K$-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 $\mathcal {C}_{Q}$ of modules over the quantum loop algebra $U_{q}(L\mathfrak {g})$ 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 $Q$ and Hernandez-Leclerc’s category $\mathcal {C}_{Q}$, assuming the simpleness of some poles of normalized $R$-matrices for type $\mathrm {E}$.