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

For a Dynkin quiver $Q$ (of type 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-Leclercs monoidal category $mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(Lmathfrak{g})$ via Nakajimas homomorphism. As an application, we show that Kang-Kashiwara-Kims 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-Leclercs category $mathcal{C}_Q$, assuming the simpleness of some poles of normalized R-matrices for type E.